Cubic critical

Percentage Accurate: 52.1% → 84.5%

Time: 3.8s

Alternatives: 21

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: 52.1% 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: 84.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -7.9 \cdot 10^{+49}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.5, \frac{c}{b} \cdot a, -0.6666666666666666 \cdot b\right)}{a}\\ \mathbf{elif}\;b \leq 7.2 \cdot 10^{-112}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -7.9e+49)
   (/ (fma 0.5 (* (/ c b) a) (* -0.6666666666666666 b)) a)
   (if (<= b 7.2e-112)
     (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a))
     (/ (* -0.5 c) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -7.9e+49) {
		tmp = fma(0.5, ((c / b) * a), (-0.6666666666666666 * b)) / a;
	} else if (b <= 7.2e-112) {
		tmp = (-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a);
	} else {
		tmp = (-0.5 * c) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -7.9e+49)
		tmp = Float64(fma(0.5, Float64(Float64(c / b) * a), Float64(-0.6666666666666666 * b)) / a);
	elseif (b <= 7.2e-112)
		tmp = Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c)))) / Float64(3.0 * a));
	else
		tmp = Float64(Float64(-0.5 * c) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -7.9e+49], N[(N[(0.5 * N[(N[(c / b), $MachinePrecision] * a), $MachinePrecision] + N[(-0.6666666666666666 * b), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b, 7.2e-112], 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[(-0.5 * c), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -7.9000000000000004e49

   1. Initial program 59.7%

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

   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 93.7

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

   4. Applied rewrites 93.7%

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

   5. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift-*.f64 93.7

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

   7. Applied rewrites 93.7%

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

   if -7.9000000000000004e49 < b < 7.2000000000000002e-112

   1. Initial program 80.8%

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

   if 7.2000000000000002e-112 < b

   1. Initial program 21.3%

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

   2. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 81.8

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

   4. Applied rewrites 81.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r/ N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 81.8

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

   6. Applied rewrites 81.8%

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

Alternative 2: 84.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -7.9 \cdot 10^{+49}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.5, \frac{c}{b} \cdot a, -0.6666666666666666 \cdot b\right)}{a}\\ \mathbf{elif}\;b \leq 7.2 \cdot 10^{-112}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)} + \left(-b\right)}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -7.9e+49)
   (/ (fma 0.5 (* (/ c b) a) (* -0.6666666666666666 b)) a)
   (if (<= b 7.2e-112)
     (/ (+ (sqrt (fma (* -3.0 a) c (* b b))) (- b)) (* a 3.0))
     (/ (* -0.5 c) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -7.9e+49) {
		tmp = fma(0.5, ((c / b) * a), (-0.6666666666666666 * b)) / a;
	} else if (b <= 7.2e-112) {
		tmp = (sqrt(fma((-3.0 * a), c, (b * b))) + -b) / (a * 3.0);
	} else {
		tmp = (-0.5 * c) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -7.9e+49)
		tmp = Float64(fma(0.5, Float64(Float64(c / b) * a), Float64(-0.6666666666666666 * b)) / a);
	elseif (b <= 7.2e-112)
		tmp = Float64(Float64(sqrt(fma(Float64(-3.0 * a), c, Float64(b * b))) + Float64(-b)) / Float64(a * 3.0));
	else
		tmp = Float64(Float64(-0.5 * c) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -7.9e+49], N[(N[(0.5 * N[(N[(c / b), $MachinePrecision] * a), $MachinePrecision] + N[(-0.6666666666666666 * b), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b, 7.2e-112], 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[(-0.5 * c), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -7.9000000000000004e49

   1. Initial program 59.7%

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

   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 93.7

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

   4. Applied rewrites 93.7%

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

   5. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift-*.f64 93.7

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

   7. Applied rewrites 93.7%

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

   if -7.9000000000000004e49 < b < 7.2000000000000002e-112

   1. Initial program 80.8%

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

   3. Applied rewrites 80.8%

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

   if 7.2000000000000002e-112 < b

   1. Initial program 21.3%

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

   2. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 81.8

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

   4. Applied rewrites 81.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r/ N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 81.8

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

   6. Applied rewrites 81.8%

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

Alternative 3: 84.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -7.9 \cdot 10^{+49}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.5, \frac{c}{b} \cdot a, -0.6666666666666666 \cdot b\right)}{a}\\ \mathbf{elif}\;b \leq 7.2 \cdot 10^{-112}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(c \cdot a, -3, b \cdot b\right)}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -7.9e+49)
   (/ (fma 0.5 (* (/ c b) a) (* -0.6666666666666666 b)) a)
   (if (<= b 7.2e-112)
     (/ (+ (- b) (sqrt (fma (* c a) -3.0 (* b b)))) (* 3.0 a))
     (/ (* -0.5 c) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -7.9e+49) {
		tmp = fma(0.5, ((c / b) * a), (-0.6666666666666666 * b)) / a;
	} else if (b <= 7.2e-112) {
		tmp = (-b + sqrt(fma((c * a), -3.0, (b * b)))) / (3.0 * a);
	} else {
		tmp = (-0.5 * c) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -7.9e+49)
		tmp = Float64(fma(0.5, Float64(Float64(c / b) * a), Float64(-0.6666666666666666 * b)) / a);
	elseif (b <= 7.2e-112)
		tmp = Float64(Float64(Float64(-b) + sqrt(fma(Float64(c * a), -3.0, Float64(b * b)))) / Float64(3.0 * a));
	else
		tmp = Float64(Float64(-0.5 * c) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -7.9e+49], N[(N[(0.5 * N[(N[(c / b), $MachinePrecision] * a), $MachinePrecision] + N[(-0.6666666666666666 * b), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b, 7.2e-112], 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[(-0.5 * c), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -7.9000000000000004e49

   1. Initial program 59.7%

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

   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 93.7

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

   4. Applied rewrites 93.7%

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

   5. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift-*.f64 93.7

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

   7. Applied rewrites 93.7%

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

   if -7.9000000000000004e49 < b < 7.2000000000000002e-112

   1. Initial program 80.8%

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

      1. lift-*.f64 N/A

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

      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-*l* 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 80.7

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

   3. Applied rewrites 80.7%

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

   if 7.2000000000000002e-112 < b

   1. Initial program 21.3%

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

   2. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 81.8

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

   4. Applied rewrites 81.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r/ N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 81.8

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

   6. Applied rewrites 81.8%

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

Alternative 4: 79.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.3 \cdot 10^{-80}:\\ \;\;\;\;\frac{\left(-b\right) \cdot \mathsf{fma}\left(\frac{c}{b \cdot b}, -1.5, \frac{2}{a}\right)}{3}\\ \mathbf{elif}\;b \leq 2.05 \cdot 10^{-129}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\left(a \cdot -3\right) \cdot c}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2.3e-80)
   (/ (* (- b) (fma (/ c (* b b)) -1.5 (/ 2.0 a))) 3.0)
   (if (<= b 2.05e-129)
     (/ (+ (- b) (sqrt (* (* a -3.0) c))) (* 3.0 a))
     (/ (* -0.5 c) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.3e-80) {
		tmp = (-b * fma((c / (b * b)), -1.5, (2.0 / a))) / 3.0;
	} else if (b <= 2.05e-129) {
		tmp = (-b + sqrt(((a * -3.0) * c))) / (3.0 * a);
	} else {
		tmp = (-0.5 * c) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2.3e-80)
		tmp = Float64(Float64(Float64(-b) * fma(Float64(c / Float64(b * b)), -1.5, Float64(2.0 / a))) / 3.0);
	elseif (b <= 2.05e-129)
		tmp = Float64(Float64(Float64(-b) + sqrt(Float64(Float64(a * -3.0) * c))) / Float64(3.0 * a));
	else
		tmp = Float64(Float64(-0.5 * c) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2.3e-80], N[(N[((-b) * N[(N[(c / N[(b * b), $MachinePrecision]), $MachinePrecision] * -1.5 + N[(2.0 / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 3.0), $MachinePrecision], If[LessEqual[b, 2.05e-129], N[(N[((-b) + N[Sqrt[N[(N[(a * -3.0), $MachinePrecision] * c), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(-0.5 * c), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -2.2999999999999998e-80

   1. Initial program 68.8%

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

   3. Applied rewrites 68.9%

      \[\leadsto \color{blue}{\frac{\sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)} + \left(-b\right)}{a \cdot 3}} \]
   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 68.8%

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

   6. Taylor expanded in b around -inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lift-neg.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. lower-/.f64 84.4

          \[\leadsto \frac{\left(-b\right) \cdot \mathsf{fma}\left(\frac{c}{b \cdot b}, -1.5, \frac{2}{a}\right)}{3} \]

   8. Applied rewrites 84.4%

      \[\leadsto \frac{\color{blue}{\left(-b\right) \cdot \mathsf{fma}\left(\frac{c}{b \cdot b}, -1.5, \frac{2}{a}\right)}}{3} \]

   if -2.2999999999999998e-80 < b < 2.05e-129

   1. Initial program 77.1%

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

   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 70.4

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

   4. Applied rewrites 70.4%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{-3 \cdot \left(c \cdot a\right)}}}{3 \cdot a} \]
   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. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. associate-*l* 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 70.5

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

   6. Applied rewrites 70.5%

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

   if 2.05e-129 < b

   1. Initial program 22.8%

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

   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 80.1

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

   4. Applied rewrites 80.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r/ N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 80.1

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

   6. Applied rewrites 80.1%

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

Alternative 5: 79.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.3 \cdot 10^{-80}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.5, \frac{c}{b} \cdot a, -0.6666666666666666 \cdot b\right)}{a}\\ \mathbf{elif}\;b \leq 2.05 \cdot 10^{-129}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\left(a \cdot -3\right) \cdot c}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2.3e-80)
   (/ (fma 0.5 (* (/ c b) a) (* -0.6666666666666666 b)) a)
   (if (<= b 2.05e-129)
     (/ (+ (- b) (sqrt (* (* a -3.0) c))) (* 3.0 a))
     (/ (* -0.5 c) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.3e-80) {
		tmp = fma(0.5, ((c / b) * a), (-0.6666666666666666 * b)) / a;
	} else if (b <= 2.05e-129) {
		tmp = (-b + sqrt(((a * -3.0) * c))) / (3.0 * a);
	} else {
		tmp = (-0.5 * c) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2.3e-80)
		tmp = Float64(fma(0.5, Float64(Float64(c / b) * a), Float64(-0.6666666666666666 * b)) / a);
	elseif (b <= 2.05e-129)
		tmp = Float64(Float64(Float64(-b) + sqrt(Float64(Float64(a * -3.0) * c))) / Float64(3.0 * a));
	else
		tmp = Float64(Float64(-0.5 * c) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2.3e-80], N[(N[(0.5 * N[(N[(c / b), $MachinePrecision] * a), $MachinePrecision] + N[(-0.6666666666666666 * b), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b, 2.05e-129], N[(N[((-b) + N[Sqrt[N[(N[(a * -3.0), $MachinePrecision] * c), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(-0.5 * c), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -2.2999999999999998e-80

   1. Initial program 68.8%

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

   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 84.5

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

   4. Applied rewrites 84.5%

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

   5. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift-*.f64 84.5

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

   7. Applied rewrites 84.5%

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

   if -2.2999999999999998e-80 < b < 2.05e-129

   1. Initial program 77.1%

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

   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 70.4

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

   4. Applied rewrites 70.4%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{-3 \cdot \left(c \cdot a\right)}}}{3 \cdot a} \]
   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. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. associate-*l* 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 70.5

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

   6. Applied rewrites 70.5%

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

   if 2.05e-129 < b

   1. Initial program 22.8%

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

   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 80.1

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

   4. Applied rewrites 80.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r/ N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 80.1

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

   6. Applied rewrites 80.1%

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

Alternative 6: 79.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.3 \cdot 10^{-80}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.5, \frac{c}{b} \cdot a, -0.6666666666666666 \cdot b\right)}{a}\\ \mathbf{elif}\;b \leq 2.05 \cdot 10^{-129}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{-3 \cdot \left(c \cdot a\right)}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2.3e-80)
   (/ (fma 0.5 (* (/ c b) a) (* -0.6666666666666666 b)) a)
   (if (<= b 2.05e-129)
     (/ (+ (- b) (sqrt (* -3.0 (* c a)))) (* 3.0 a))
     (/ (* -0.5 c) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.3e-80) {
		tmp = fma(0.5, ((c / b) * a), (-0.6666666666666666 * b)) / a;
	} else if (b <= 2.05e-129) {
		tmp = (-b + sqrt((-3.0 * (c * a)))) / (3.0 * a);
	} else {
		tmp = (-0.5 * c) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2.3e-80)
		tmp = Float64(fma(0.5, Float64(Float64(c / b) * a), Float64(-0.6666666666666666 * b)) / a);
	elseif (b <= 2.05e-129)
		tmp = Float64(Float64(Float64(-b) + sqrt(Float64(-3.0 * Float64(c * a)))) / Float64(3.0 * a));
	else
		tmp = Float64(Float64(-0.5 * c) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2.3e-80], N[(N[(0.5 * N[(N[(c / b), $MachinePrecision] * a), $MachinePrecision] + N[(-0.6666666666666666 * b), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b, 2.05e-129], N[(N[((-b) + N[Sqrt[N[(-3.0 * N[(c * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(-0.5 * c), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -2.2999999999999998e-80

   1. Initial program 68.8%

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

   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 84.5

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

   4. Applied rewrites 84.5%

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

   5. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift-*.f64 84.5

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

   7. Applied rewrites 84.5%

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

   if -2.2999999999999998e-80 < b < 2.05e-129

   1. Initial program 77.1%

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

   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 70.4

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

   4. Applied rewrites 70.4%

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

   if 2.05e-129 < b

   1. Initial program 22.8%

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

   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 80.1

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

   4. Applied rewrites 80.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r/ N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 80.1

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

   6. Applied rewrites 80.1%

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

Alternative 7: 79.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.3 \cdot 10^{-80}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.5, \frac{c}{b} \cdot a, -0.6666666666666666 \cdot b\right)}{a}\\ \mathbf{elif}\;b \leq 7.2 \cdot 10^{-112}:\\ \;\;\;\;\frac{\sqrt{\left(-3 \cdot a\right) \cdot c}}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2.3e-80)
   (/ (fma 0.5 (* (/ c b) a) (* -0.6666666666666666 b)) a)
   (if (<= b 7.2e-112)
     (/ (sqrt (* (* -3.0 a) c)) (* a 3.0))
     (/ (* -0.5 c) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.3e-80) {
		tmp = fma(0.5, ((c / b) * a), (-0.6666666666666666 * b)) / a;
	} else if (b <= 7.2e-112) {
		tmp = sqrt(((-3.0 * a) * c)) / (a * 3.0);
	} else {
		tmp = (-0.5 * c) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2.3e-80)
		tmp = Float64(fma(0.5, Float64(Float64(c / b) * a), Float64(-0.6666666666666666 * b)) / a);
	elseif (b <= 7.2e-112)
		tmp = Float64(sqrt(Float64(Float64(-3.0 * a) * c)) / Float64(a * 3.0));
	else
		tmp = Float64(Float64(-0.5 * c) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2.3e-80], N[(N[(0.5 * N[(N[(c / b), $MachinePrecision] * a), $MachinePrecision] + N[(-0.6666666666666666 * b), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b, 7.2e-112], N[(N[Sqrt[N[(N[(-3.0 * a), $MachinePrecision] * c), $MachinePrecision]], $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(-0.5 * c), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -2.2999999999999998e-80

   1. Initial program 68.8%

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

   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 84.5

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

   4. Applied rewrites 84.5%

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

   5. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift-*.f64 84.5

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

   7. Applied rewrites 84.5%

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

   if -2.2999999999999998e-80 < b < 7.2000000000000002e-112

   1. Initial program 76.4%

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

   3. Applied rewrites 76.4%

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

   4. Taylor expanded in a around inf

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. *-commutative N/A

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

      4. *-commutative N/A

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

      5. pow2 N/A

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

      6. *-commutative N/A

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

      7. +-commutative N/A

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

      8. metadata-eval N/A

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

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

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

      10. associate-*l* N/A

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

      11. pow2 N/A

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

      12. lift-neg.f64 N/A

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

      13. +-commutative N/A

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

      14. lift-neg.f64 N/A

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

   6. Applied rewrites 68.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 68.8

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

   8. Applied rewrites 68.8%

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

   if 7.2000000000000002e-112 < b

   1. Initial program 21.3%

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

   2. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 81.8

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

   4. Applied rewrites 81.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r/ N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 81.8

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

   6. Applied rewrites 81.8%

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

Alternative 8: 79.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.3 \cdot 10^{-80}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{c}{b}, \frac{b}{a} \cdot -0.6666666666666666\right)\\ \mathbf{elif}\;b \leq 7.2 \cdot 10^{-112}:\\ \;\;\;\;\frac{\sqrt{\left(-3 \cdot a\right) \cdot c}}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2.3e-80)
   (fma 0.5 (/ c b) (* (/ b a) -0.6666666666666666))
   (if (<= b 7.2e-112)
     (/ (sqrt (* (* -3.0 a) c)) (* a 3.0))
     (/ (* -0.5 c) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.3e-80) {
		tmp = fma(0.5, (c / b), ((b / a) * -0.6666666666666666));
	} else if (b <= 7.2e-112) {
		tmp = sqrt(((-3.0 * a) * c)) / (a * 3.0);
	} else {
		tmp = (-0.5 * c) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2.3e-80)
		tmp = fma(0.5, Float64(c / b), Float64(Float64(b / a) * -0.6666666666666666));
	elseif (b <= 7.2e-112)
		tmp = Float64(sqrt(Float64(Float64(-3.0 * a) * c)) / Float64(a * 3.0));
	else
		tmp = Float64(Float64(-0.5 * c) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2.3e-80], N[(0.5 * N[(c / b), $MachinePrecision] + N[(N[(b / a), $MachinePrecision] * -0.6666666666666666), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 7.2e-112], N[(N[Sqrt[N[(N[(-3.0 * a), $MachinePrecision] * c), $MachinePrecision]], $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(-0.5 * c), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -2.2999999999999998e-80

   1. Initial program 68.8%

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

   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 84.5

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

   4. Applied rewrites 84.5%

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

   5. Taylor expanded in a around inf

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 84.6

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

   7. Applied rewrites 84.6%

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

   if -2.2999999999999998e-80 < b < 7.2000000000000002e-112

   1. Initial program 76.4%

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

   3. Applied rewrites 76.4%

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

   4. Taylor expanded in a around inf

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. *-commutative N/A

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

      4. *-commutative N/A

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

      5. pow2 N/A

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

      6. *-commutative N/A

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

      7. +-commutative N/A

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

      8. metadata-eval N/A

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

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

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

      10. associate-*l* N/A

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

      11. pow2 N/A

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

      12. lift-neg.f64 N/A

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

      13. +-commutative N/A

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

      14. lift-neg.f64 N/A

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

   6. Applied rewrites 68.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 68.8

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

   8. Applied rewrites 68.8%

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

   if 7.2000000000000002e-112 < b

   1. Initial program 21.3%

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

   2. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 81.8

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

   4. Applied rewrites 81.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r/ N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 81.8

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

   6. Applied rewrites 81.8%

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

Alternative 9: 79.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.3 \cdot 10^{-80}:\\ \;\;\;\;\frac{-2 \cdot b}{3 \cdot a}\\ \mathbf{elif}\;b \leq 7.2 \cdot 10^{-112}:\\ \;\;\;\;\frac{\sqrt{\left(-3 \cdot a\right) \cdot c}}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2.3e-80)
   (/ (* -2.0 b) (* 3.0 a))
   (if (<= b 7.2e-112)
     (/ (sqrt (* (* -3.0 a) c)) (* a 3.0))
     (/ (* -0.5 c) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.3e-80) {
		tmp = (-2.0 * b) / (3.0 * a);
	} else if (b <= 7.2e-112) {
		tmp = sqrt(((-3.0 * a) * c)) / (a * 3.0);
	} else {
		tmp = (-0.5 * c) / b;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-2.3d-80)) then
        tmp = ((-2.0d0) * b) / (3.0d0 * a)
    else if (b <= 7.2d-112) then
        tmp = sqrt((((-3.0d0) * a) * c)) / (a * 3.0d0)
    else
        tmp = ((-0.5d0) * c) / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.3e-80) {
		tmp = (-2.0 * b) / (3.0 * a);
	} else if (b <= 7.2e-112) {
		tmp = Math.sqrt(((-3.0 * a) * c)) / (a * 3.0);
	} else {
		tmp = (-0.5 * c) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -2.3e-80:
		tmp = (-2.0 * b) / (3.0 * a)
	elif b <= 7.2e-112:
		tmp = math.sqrt(((-3.0 * a) * c)) / (a * 3.0)
	else:
		tmp = (-0.5 * c) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2.3e-80)
		tmp = Float64(Float64(-2.0 * b) / Float64(3.0 * a));
	elseif (b <= 7.2e-112)
		tmp = Float64(sqrt(Float64(Float64(-3.0 * a) * c)) / Float64(a * 3.0));
	else
		tmp = Float64(Float64(-0.5 * c) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -2.3e-80)
		tmp = (-2.0 * b) / (3.0 * a);
	elseif (b <= 7.2e-112)
		tmp = sqrt(((-3.0 * a) * c)) / (a * 3.0);
	else
		tmp = (-0.5 * c) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2.3e-80], N[(N[(-2.0 * b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 7.2e-112], N[(N[Sqrt[N[(N[(-3.0 * a), $MachinePrecision] * c), $MachinePrecision]], $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(-0.5 * c), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -2.2999999999999998e-80

   1. Initial program 68.8%

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

   2. Taylor expanded in b around -inf

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

      1. lower-*.f64 84.2

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

   4. Applied rewrites 84.2%

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

   if -2.2999999999999998e-80 < b < 7.2000000000000002e-112

   1. Initial program 76.4%

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

   3. Applied rewrites 76.4%

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

   4. Taylor expanded in a around inf

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. *-commutative N/A

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

      4. *-commutative N/A

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

      5. pow2 N/A

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

      6. *-commutative N/A

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

      7. +-commutative N/A

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

      8. metadata-eval N/A

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

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

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

      10. associate-*l* N/A

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

      11. pow2 N/A

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

      12. lift-neg.f64 N/A

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

      13. +-commutative N/A

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

      14. lift-neg.f64 N/A

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

   6. Applied rewrites 68.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 68.8

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

   8. Applied rewrites 68.8%

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

   if 7.2000000000000002e-112 < b

   1. Initial program 21.3%

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

   2. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 81.8

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

   4. Applied rewrites 81.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r/ N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 81.8

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

   6. Applied rewrites 81.8%

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

Alternative 10: 79.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.3 \cdot 10^{-80}:\\ \;\;\;\;\frac{-2 \cdot b}{3 \cdot a}\\ \mathbf{elif}\;b \leq 7.2 \cdot 10^{-112}:\\ \;\;\;\;\frac{\sqrt{-3 \cdot \left(c \cdot a\right)}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2.3e-80)
   (/ (* -2.0 b) (* 3.0 a))
   (if (<= b 7.2e-112)
     (/ (sqrt (* -3.0 (* c a))) (* 3.0 a))
     (/ (* -0.5 c) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.3e-80) {
		tmp = (-2.0 * b) / (3.0 * a);
	} else if (b <= 7.2e-112) {
		tmp = sqrt((-3.0 * (c * a))) / (3.0 * a);
	} else {
		tmp = (-0.5 * c) / b;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-2.3d-80)) then
        tmp = ((-2.0d0) * b) / (3.0d0 * a)
    else if (b <= 7.2d-112) then
        tmp = sqrt(((-3.0d0) * (c * a))) / (3.0d0 * a)
    else
        tmp = ((-0.5d0) * c) / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.3e-80) {
		tmp = (-2.0 * b) / (3.0 * a);
	} else if (b <= 7.2e-112) {
		tmp = Math.sqrt((-3.0 * (c * a))) / (3.0 * a);
	} else {
		tmp = (-0.5 * c) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -2.3e-80:
		tmp = (-2.0 * b) / (3.0 * a)
	elif b <= 7.2e-112:
		tmp = math.sqrt((-3.0 * (c * a))) / (3.0 * a)
	else:
		tmp = (-0.5 * c) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2.3e-80)
		tmp = Float64(Float64(-2.0 * b) / Float64(3.0 * a));
	elseif (b <= 7.2e-112)
		tmp = Float64(sqrt(Float64(-3.0 * Float64(c * a))) / Float64(3.0 * a));
	else
		tmp = Float64(Float64(-0.5 * c) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -2.3e-80)
		tmp = (-2.0 * b) / (3.0 * a);
	elseif (b <= 7.2e-112)
		tmp = sqrt((-3.0 * (c * a))) / (3.0 * a);
	else
		tmp = (-0.5 * c) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2.3e-80], N[(N[(-2.0 * b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 7.2e-112], N[(N[Sqrt[N[(-3.0 * N[(c * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(-0.5 * c), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -2.2999999999999998e-80

   1. Initial program 68.8%

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

   2. Taylor expanded in b around -inf

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

      1. lower-*.f64 84.2

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

   4. Applied rewrites 84.2%

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

   if -2.2999999999999998e-80 < b < 7.2000000000000002e-112

   1. Initial program 76.4%

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

   2. Taylor expanded in a around 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 68.7

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

   4. Applied rewrites 68.7%

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

   if 7.2000000000000002e-112 < b

   1. Initial program 21.3%

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

   2. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 81.8

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

   4. Applied rewrites 81.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r/ N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 81.8

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

   6. Applied rewrites 81.8%

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

Alternative 11: 71.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{c}{a} \cdot -3}\\ \mathbf{if}\;b \leq -2.15 \cdot 10^{-146}:\\ \;\;\;\;\frac{-2 \cdot b}{3 \cdot a}\\ \mathbf{elif}\;b \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{t\_0}{3}\\ \mathbf{elif}\;b \leq 7.2 \cdot 10^{-131}:\\ \;\;\;\;t\_0 \cdot -0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (* (/ c a) -3.0))))
   (if (<= b -2.15e-146)
     (/ (* -2.0 b) (* 3.0 a))
     (if (<= b -5e-310)
       (/ t_0 3.0)
       (if (<= b 7.2e-131) (* t_0 -0.3333333333333333) (/ (* -0.5 c) b))))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt(((c / a) * -3.0));
	double tmp;
	if (b <= -2.15e-146) {
		tmp = (-2.0 * b) / (3.0 * a);
	} else if (b <= -5e-310) {
		tmp = t_0 / 3.0;
	} else if (b <= 7.2e-131) {
		tmp = t_0 * -0.3333333333333333;
	} 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) :: t_0
    real(8) :: tmp
    t_0 = sqrt(((c / a) * (-3.0d0)))
    if (b <= (-2.15d-146)) then
        tmp = ((-2.0d0) * b) / (3.0d0 * a)
    else if (b <= (-5d-310)) then
        tmp = t_0 / 3.0d0
    else if (b <= 7.2d-131) then
        tmp = t_0 * (-0.3333333333333333d0)
    else
        tmp = ((-0.5d0) * c) / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = Math.sqrt(((c / a) * -3.0));
	double tmp;
	if (b <= -2.15e-146) {
		tmp = (-2.0 * b) / (3.0 * a);
	} else if (b <= -5e-310) {
		tmp = t_0 / 3.0;
	} else if (b <= 7.2e-131) {
		tmp = t_0 * -0.3333333333333333;
	} else {
		tmp = (-0.5 * c) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = math.sqrt(((c / a) * -3.0))
	tmp = 0
	if b <= -2.15e-146:
		tmp = (-2.0 * b) / (3.0 * a)
	elif b <= -5e-310:
		tmp = t_0 / 3.0
	elif b <= 7.2e-131:
		tmp = t_0 * -0.3333333333333333
	else:
		tmp = (-0.5 * c) / b
	return tmp

Julia

function code(a, b, c)
	t_0 = sqrt(Float64(Float64(c / a) * -3.0))
	tmp = 0.0
	if (b <= -2.15e-146)
		tmp = Float64(Float64(-2.0 * b) / Float64(3.0 * a));
	elseif (b <= -5e-310)
		tmp = Float64(t_0 / 3.0);
	elseif (b <= 7.2e-131)
		tmp = Float64(t_0 * -0.3333333333333333);
	else
		tmp = Float64(Float64(-0.5 * c) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = sqrt(((c / a) * -3.0));
	tmp = 0.0;
	if (b <= -2.15e-146)
		tmp = (-2.0 * b) / (3.0 * a);
	elseif (b <= -5e-310)
		tmp = t_0 / 3.0;
	elseif (b <= 7.2e-131)
		tmp = t_0 * -0.3333333333333333;
	else
		tmp = (-0.5 * c) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Sqrt[N[(N[(c / a), $MachinePrecision] * -3.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[b, -2.15e-146], N[(N[(-2.0 * b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, -5e-310], N[(t$95$0 / 3.0), $MachinePrecision], If[LessEqual[b, 7.2e-131], N[(t$95$0 * -0.3333333333333333), $MachinePrecision], N[(N[(-0.5 * c), $MachinePrecision] / b), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if b < -2.15e-146

   1. Initial program 70.9%

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

   2. Taylor expanded in b around -inf

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

      1. lower-*.f64 79.3

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

   4. Applied rewrites 79.3%

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

   if -2.15e-146 < b < -4.999999999999985e-310

   1. Initial program 76.9%

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

   3. Applied rewrites 76.9%

      \[\leadsto \color{blue}{\frac{\sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)} + \left(-b\right)}{a \cdot 3}} \]
   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 76.8%

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

   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-prod N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-sqrt.f64 36.9

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

   8. Applied rewrites 36.9%

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

   if -4.999999999999985e-310 < b < 7.1999999999999999e-131

   1. Initial program 72.6%

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

   2. Taylor expanded in a around -inf

      \[\leadsto \color{blue}{\frac{-1}{3} \cdot \left(\sqrt{\frac{c}{a}} \cdot \left(\sqrt{-1} \cdot \sqrt{3}\right)\right)} \]
   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 33.8

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

   4. Applied rewrites 33.8%

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

   if 7.1999999999999999e-131 < b

   1. Initial program 22.8%

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

   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 80.0

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

   4. Applied rewrites 80.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r/ N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 80.0

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

   6. Applied rewrites 80.0%

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

Alternative 12: 71.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.15 \cdot 10^{-146}:\\ \;\;\;\;\frac{-2 \cdot b}{3 \cdot a}\\ \mathbf{elif}\;b \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{\frac{-3 \cdot c}{a}} \cdot 0.3333333333333333\\ \mathbf{elif}\;b \leq 7.2 \cdot 10^{-131}:\\ \;\;\;\;\sqrt{\frac{c}{a} \cdot -3} \cdot -0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2.15e-146)
   (/ (* -2.0 b) (* 3.0 a))
   (if (<= b -5e-310)
     (* (sqrt (/ (* -3.0 c) a)) 0.3333333333333333)
     (if (<= b 7.2e-131)
       (* (sqrt (* (/ c a) -3.0)) -0.3333333333333333)
       (/ (* -0.5 c) b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.15e-146) {
		tmp = (-2.0 * b) / (3.0 * a);
	} else if (b <= -5e-310) {
		tmp = sqrt(((-3.0 * c) / a)) * 0.3333333333333333;
	} else if (b <= 7.2e-131) {
		tmp = sqrt(((c / a) * -3.0)) * -0.3333333333333333;
	} else {
		tmp = (-0.5 * c) / b;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-2.15d-146)) then
        tmp = ((-2.0d0) * b) / (3.0d0 * a)
    else if (b <= (-5d-310)) then
        tmp = sqrt((((-3.0d0) * c) / a)) * 0.3333333333333333d0
    else if (b <= 7.2d-131) then
        tmp = sqrt(((c / a) * (-3.0d0))) * (-0.3333333333333333d0)
    else
        tmp = ((-0.5d0) * c) / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.15e-146) {
		tmp = (-2.0 * b) / (3.0 * a);
	} else if (b <= -5e-310) {
		tmp = Math.sqrt(((-3.0 * c) / a)) * 0.3333333333333333;
	} else if (b <= 7.2e-131) {
		tmp = Math.sqrt(((c / a) * -3.0)) * -0.3333333333333333;
	} else {
		tmp = (-0.5 * c) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -2.15e-146:
		tmp = (-2.0 * b) / (3.0 * a)
	elif b <= -5e-310:
		tmp = math.sqrt(((-3.0 * c) / a)) * 0.3333333333333333
	elif b <= 7.2e-131:
		tmp = math.sqrt(((c / a) * -3.0)) * -0.3333333333333333
	else:
		tmp = (-0.5 * c) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2.15e-146)
		tmp = Float64(Float64(-2.0 * b) / Float64(3.0 * a));
	elseif (b <= -5e-310)
		tmp = Float64(sqrt(Float64(Float64(-3.0 * c) / a)) * 0.3333333333333333);
	elseif (b <= 7.2e-131)
		tmp = Float64(sqrt(Float64(Float64(c / a) * -3.0)) * -0.3333333333333333);
	else
		tmp = Float64(Float64(-0.5 * c) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -2.15e-146)
		tmp = (-2.0 * b) / (3.0 * a);
	elseif (b <= -5e-310)
		tmp = sqrt(((-3.0 * c) / a)) * 0.3333333333333333;
	elseif (b <= 7.2e-131)
		tmp = sqrt(((c / a) * -3.0)) * -0.3333333333333333;
	else
		tmp = (-0.5 * c) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2.15e-146], N[(N[(-2.0 * b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, -5e-310], N[(N[Sqrt[N[(N[(-3.0 * c), $MachinePrecision] / a), $MachinePrecision]], $MachinePrecision] * 0.3333333333333333), $MachinePrecision], If[LessEqual[b, 7.2e-131], N[(N[Sqrt[N[(N[(c / a), $MachinePrecision] * -3.0), $MachinePrecision]], $MachinePrecision] * -0.3333333333333333), $MachinePrecision], N[(N[(-0.5 * c), $MachinePrecision] / b), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if b < -2.15e-146

   1. Initial program 70.9%

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

   2. Taylor expanded in b around -inf

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

      1. lower-*.f64 79.3

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

   4. Applied rewrites 79.3%

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

   if -2.15e-146 < b < -4.999999999999985e-310

   1. Initial program 76.9%

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

   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 36.8

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

   4. Applied rewrites 36.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r/ N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 36.8

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

   6. Applied rewrites 36.8%

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

   if -4.999999999999985e-310 < b < 7.1999999999999999e-131

   1. Initial program 72.6%

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

   2. Taylor expanded in a around -inf

      \[\leadsto \color{blue}{\frac{-1}{3} \cdot \left(\sqrt{\frac{c}{a}} \cdot \left(\sqrt{-1} \cdot \sqrt{3}\right)\right)} \]
   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 33.8

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

   4. Applied rewrites 33.8%

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

   if 7.1999999999999999e-131 < b

   1. Initial program 22.8%

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

   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 80.0

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

   4. Applied rewrites 80.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r/ N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 80.0

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

   6. Applied rewrites 80.0%

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

Alternative 13: 71.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{c}{a} \cdot -3}\\ \mathbf{if}\;b \leq -2.15 \cdot 10^{-146}:\\ \;\;\;\;\frac{-2 \cdot b}{3 \cdot a}\\ \mathbf{elif}\;b \leq -5 \cdot 10^{-310}:\\ \;\;\;\;t\_0 \cdot 0.3333333333333333\\ \mathbf{elif}\;b \leq 7.2 \cdot 10^{-131}:\\ \;\;\;\;t\_0 \cdot -0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (* (/ c a) -3.0))))
   (if (<= b -2.15e-146)
     (/ (* -2.0 b) (* 3.0 a))
     (if (<= b -5e-310)
       (* t_0 0.3333333333333333)
       (if (<= b 7.2e-131) (* t_0 -0.3333333333333333) (/ (* -0.5 c) b))))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt(((c / a) * -3.0));
	double tmp;
	if (b <= -2.15e-146) {
		tmp = (-2.0 * b) / (3.0 * a);
	} else if (b <= -5e-310) {
		tmp = t_0 * 0.3333333333333333;
	} else if (b <= 7.2e-131) {
		tmp = t_0 * -0.3333333333333333;
	} 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) :: t_0
    real(8) :: tmp
    t_0 = sqrt(((c / a) * (-3.0d0)))
    if (b <= (-2.15d-146)) then
        tmp = ((-2.0d0) * b) / (3.0d0 * a)
    else if (b <= (-5d-310)) then
        tmp = t_0 * 0.3333333333333333d0
    else if (b <= 7.2d-131) then
        tmp = t_0 * (-0.3333333333333333d0)
    else
        tmp = ((-0.5d0) * c) / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = Math.sqrt(((c / a) * -3.0));
	double tmp;
	if (b <= -2.15e-146) {
		tmp = (-2.0 * b) / (3.0 * a);
	} else if (b <= -5e-310) {
		tmp = t_0 * 0.3333333333333333;
	} else if (b <= 7.2e-131) {
		tmp = t_0 * -0.3333333333333333;
	} else {
		tmp = (-0.5 * c) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = math.sqrt(((c / a) * -3.0))
	tmp = 0
	if b <= -2.15e-146:
		tmp = (-2.0 * b) / (3.0 * a)
	elif b <= -5e-310:
		tmp = t_0 * 0.3333333333333333
	elif b <= 7.2e-131:
		tmp = t_0 * -0.3333333333333333
	else:
		tmp = (-0.5 * c) / b
	return tmp

Julia

function code(a, b, c)
	t_0 = sqrt(Float64(Float64(c / a) * -3.0))
	tmp = 0.0
	if (b <= -2.15e-146)
		tmp = Float64(Float64(-2.0 * b) / Float64(3.0 * a));
	elseif (b <= -5e-310)
		tmp = Float64(t_0 * 0.3333333333333333);
	elseif (b <= 7.2e-131)
		tmp = Float64(t_0 * -0.3333333333333333);
	else
		tmp = Float64(Float64(-0.5 * c) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = sqrt(((c / a) * -3.0));
	tmp = 0.0;
	if (b <= -2.15e-146)
		tmp = (-2.0 * b) / (3.0 * a);
	elseif (b <= -5e-310)
		tmp = t_0 * 0.3333333333333333;
	elseif (b <= 7.2e-131)
		tmp = t_0 * -0.3333333333333333;
	else
		tmp = (-0.5 * c) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Sqrt[N[(N[(c / a), $MachinePrecision] * -3.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[b, -2.15e-146], N[(N[(-2.0 * b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, -5e-310], N[(t$95$0 * 0.3333333333333333), $MachinePrecision], If[LessEqual[b, 7.2e-131], N[(t$95$0 * -0.3333333333333333), $MachinePrecision], N[(N[(-0.5 * c), $MachinePrecision] / b), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if b < -2.15e-146

   1. Initial program 70.9%

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

   2. Taylor expanded in b around -inf

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

      1. lower-*.f64 79.3

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

   4. Applied rewrites 79.3%

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

   if -2.15e-146 < b < -4.999999999999985e-310

   1. Initial program 76.9%

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

   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 36.8

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

   4. Applied rewrites 36.8%

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

   if -4.999999999999985e-310 < b < 7.1999999999999999e-131

   1. Initial program 72.6%

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

   2. Taylor expanded in a around -inf

      \[\leadsto \color{blue}{\frac{-1}{3} \cdot \left(\sqrt{\frac{c}{a}} \cdot \left(\sqrt{-1} \cdot \sqrt{3}\right)\right)} \]
   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 33.8

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

   4. Applied rewrites 33.8%

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

   if 7.1999999999999999e-131 < b

   1. Initial program 22.8%

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

   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 80.0

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

   4. Applied rewrites 80.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r/ N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 80.0

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

   6. Applied rewrites 80.0%

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

Alternative 14: 70.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.1 \cdot 10^{-225}:\\ \;\;\;\;\frac{-2 \cdot b}{3 \cdot a}\\ \mathbf{elif}\;b \leq 7.2 \cdot 10^{-131}:\\ \;\;\;\;\sqrt{\frac{c}{a} \cdot -3} \cdot -0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2.1e-225)
   (/ (* -2.0 b) (* 3.0 a))
   (if (<= b 7.2e-131)
     (* (sqrt (* (/ c a) -3.0)) -0.3333333333333333)
     (/ (* -0.5 c) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.1e-225) {
		tmp = (-2.0 * b) / (3.0 * a);
	} else if (b <= 7.2e-131) {
		tmp = sqrt(((c / a) * -3.0)) * -0.3333333333333333;
	} else {
		tmp = (-0.5 * c) / b;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-2.1d-225)) then
        tmp = ((-2.0d0) * b) / (3.0d0 * a)
    else if (b <= 7.2d-131) then
        tmp = sqrt(((c / a) * (-3.0d0))) * (-0.3333333333333333d0)
    else
        tmp = ((-0.5d0) * c) / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.1e-225) {
		tmp = (-2.0 * b) / (3.0 * a);
	} else if (b <= 7.2e-131) {
		tmp = Math.sqrt(((c / a) * -3.0)) * -0.3333333333333333;
	} else {
		tmp = (-0.5 * c) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -2.1e-225:
		tmp = (-2.0 * b) / (3.0 * a)
	elif b <= 7.2e-131:
		tmp = math.sqrt(((c / a) * -3.0)) * -0.3333333333333333
	else:
		tmp = (-0.5 * c) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2.1e-225)
		tmp = Float64(Float64(-2.0 * b) / Float64(3.0 * a));
	elseif (b <= 7.2e-131)
		tmp = Float64(sqrt(Float64(Float64(c / a) * -3.0)) * -0.3333333333333333);
	else
		tmp = Float64(Float64(-0.5 * c) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -2.1e-225)
		tmp = (-2.0 * b) / (3.0 * a);
	elseif (b <= 7.2e-131)
		tmp = sqrt(((c / a) * -3.0)) * -0.3333333333333333;
	else
		tmp = (-0.5 * c) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2.1e-225], N[(N[(-2.0 * b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 7.2e-131], N[(N[Sqrt[N[(N[(c / a), $MachinePrecision] * -3.0), $MachinePrecision]], $MachinePrecision] * -0.3333333333333333), $MachinePrecision], N[(N[(-0.5 * c), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -2.1e-225

   1. Initial program 71.3%

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

   2. Taylor expanded in b around -inf

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

      1. lower-*.f64 73.6

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

   4. Applied rewrites 73.6%

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

   if -2.1e-225 < b < 7.1999999999999999e-131

   1. Initial program 74.5%

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

   2. Taylor expanded in a around -inf

      \[\leadsto \color{blue}{\frac{-1}{3} \cdot \left(\sqrt{\frac{c}{a}} \cdot \left(\sqrt{-1} \cdot \sqrt{3}\right)\right)} \]
   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 34.2

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

   4. Applied rewrites 34.2%

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

   if 7.1999999999999999e-131 < b

   1. Initial program 22.8%

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

   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 80.0

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

   4. Applied rewrites 80.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r/ N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 80.0

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

   6. Applied rewrites 80.0%

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

Alternative 15: 67.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 4.2 \cdot 10^{-293}:\\ \;\;\;\;\frac{-2 \cdot b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 4.2e-293) (/ (* -2.0 b) (* 3.0 a)) (/ (* -0.5 c) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 4.2e-293) {
		tmp = (-2.0 * b) / (3.0 * 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 <= 4.2d-293) then
        tmp = ((-2.0d0) * b) / (3.0d0 * a)
    else
        tmp = ((-0.5d0) * c) / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 4.2e-293) {
		tmp = (-2.0 * b) / (3.0 * a);
	} else {
		tmp = (-0.5 * c) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 4.2e-293:
		tmp = (-2.0 * b) / (3.0 * a)
	else:
		tmp = (-0.5 * c) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 4.2e-293)
		tmp = Float64(Float64(-2.0 * b) / Float64(3.0 * a));
	else
		tmp = Float64(Float64(-0.5 * c) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 4.2e-293)
		tmp = (-2.0 * b) / (3.0 * a);
	else
		tmp = (-0.5 * c) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 4.2e-293], N[(N[(-2.0 * b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(-0.5 * c), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 4.2000000000000001e-293

   1. Initial program 72.0%

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

   2. Taylor expanded in b around -inf

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

      1. lower-*.f64 66.6

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

   4. Applied rewrites 66.6%

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

   if 4.2000000000000001e-293 < b

   1. Initial program 32.0%

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

   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 68.0

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

   4. Applied rewrites 68.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r/ N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 68.0

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

   6. Applied rewrites 68.0%

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

Alternative 16: 67.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 4.2 \cdot 10^{-293}:\\ \;\;\;\;\left(-b\right) \cdot \frac{0.6666666666666666}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 4.2e-293) (* (- b) (/ 0.6666666666666666 a)) (/ (* -0.5 c) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 4.2e-293) {
		tmp = -b * (0.6666666666666666 / 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 <= 4.2d-293) then
        tmp = -b * (0.6666666666666666d0 / a)
    else
        tmp = ((-0.5d0) * c) / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 4.2e-293) {
		tmp = -b * (0.6666666666666666 / a);
	} else {
		tmp = (-0.5 * c) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 4.2e-293:
		tmp = -b * (0.6666666666666666 / a)
	else:
		tmp = (-0.5 * c) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 4.2e-293)
		tmp = Float64(Float64(-b) * Float64(0.6666666666666666 / a));
	else
		tmp = Float64(Float64(-0.5 * c) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 4.2e-293)
		tmp = -b * (0.6666666666666666 / a);
	else
		tmp = (-0.5 * c) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 4.2e-293], N[((-b) * N[(0.6666666666666666 / a), $MachinePrecision]), $MachinePrecision], N[(N[(-0.5 * c), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 4.2000000000000001e-293

   1. Initial program 72.0%

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

   2. Taylor expanded in b around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(b \cdot \left(\frac{-1}{2} \cdot \frac{c}{{b}^{2}} + \frac{2}{3} \cdot \frac{1}{a}\right)\right)} \]
   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 65.7

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

   4. Applied rewrites 65.7%

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

   5. Taylor expanded in a around 0

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

      1. lift-/.f64 66.6

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

   7. Applied rewrites 66.6%

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

   if 4.2000000000000001e-293 < b

   1. Initial program 32.0%

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

   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 68.0

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

   4. Applied rewrites 68.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r/ N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 68.0

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

   6. Applied rewrites 68.0%

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

Alternative 17: 67.3% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 4.2 \cdot 10^{-293}:\\ \;\;\;\;\frac{-0.6666666666666666 \cdot b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 4.2e-293) (/ (* -0.6666666666666666 b) a) (/ (* -0.5 c) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 4.2e-293) {
		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 <= 4.2d-293) 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 <= 4.2e-293) {
		tmp = (-0.6666666666666666 * b) / a;
	} else {
		tmp = (-0.5 * c) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 4.2e-293:
		tmp = (-0.6666666666666666 * b) / a
	else:
		tmp = (-0.5 * c) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 4.2e-293)
		tmp = Float64(Float64(-0.6666666666666666 * b) / a);
	else
		tmp = Float64(Float64(-0.5 * c) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 4.2e-293)
		tmp = (-0.6666666666666666 * b) / a;
	else
		tmp = (-0.5 * c) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 4.2e-293], N[(N[(-0.6666666666666666 * b), $MachinePrecision] / a), $MachinePrecision], N[(N[(-0.5 * c), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 4.2000000000000001e-293

   1. Initial program 72.0%

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

   2. Taylor expanded in b around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 66.6

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

   4. Applied rewrites 66.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 66.6

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

   6. Applied rewrites 66.6%

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

   if 4.2000000000000001e-293 < b

   1. Initial program 32.0%

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

   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 68.0

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

   4. Applied rewrites 68.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r/ N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 68.0

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

   6. Applied rewrites 68.0%

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

Alternative 18: 67.3% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 4.2 \cdot 10^{-293}:\\ \;\;\;\;\frac{-0.6666666666666666 \cdot b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 4.2e-293) (/ (* -0.6666666666666666 b) a) (* (/ c b) -0.5)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 4.2e-293) {
		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 <= 4.2d-293) 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 <= 4.2e-293) {
		tmp = (-0.6666666666666666 * b) / a;
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 4.2e-293:
		tmp = (-0.6666666666666666 * b) / a
	else:
		tmp = (c / b) * -0.5
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 4.2e-293)
		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 <= 4.2e-293)
		tmp = (-0.6666666666666666 * b) / a;
	else
		tmp = (c / b) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 4.2e-293], 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 < 4.2000000000000001e-293

   1. Initial program 72.0%

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

   2. Taylor expanded in b around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 66.6

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

   4. Applied rewrites 66.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 66.6

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

   6. Applied rewrites 66.6%

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

   if 4.2000000000000001e-293 < b

   1. Initial program 32.0%

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

   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 68.0

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

   4. Applied rewrites 68.0%

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

Alternative 19: 67.3% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 4.2 \cdot 10^{-293}:\\ \;\;\;\;-0.6666666666666666 \cdot \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 4.2e-293) (* -0.6666666666666666 (/ b a)) (* (/ c b) -0.5)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 4.2e-293) {
		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 <= 4.2d-293) 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 <= 4.2e-293) {
		tmp = -0.6666666666666666 * (b / a);
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 4.2e-293:
		tmp = -0.6666666666666666 * (b / a)
	else:
		tmp = (c / b) * -0.5
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 4.2e-293)
		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 <= 4.2e-293)
		tmp = -0.6666666666666666 * (b / a);
	else
		tmp = (c / b) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 4.2e-293], N[(-0.6666666666666666 * N[(b / a), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] * -0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 4.2000000000000001e-293

   1. Initial program 72.0%

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

   2. Taylor expanded in b around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 66.6

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

   4. Applied rewrites 66.6%

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

   if 4.2000000000000001e-293 < b

   1. Initial program 32.0%

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

   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 68.0

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

   4. Applied rewrites 68.0%

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

Alternative 20: 42.4% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 8.5 \cdot 10^{+84}:\\ \;\;\;\;-0.6666666666666666 \cdot \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 8.5e+84) (* -0.6666666666666666 (/ b a)) (* 0.5 (/ c b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 8.5e+84) {
		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 <= 8.5d+84) 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 <= 8.5e+84) {
		tmp = -0.6666666666666666 * (b / a);
	} else {
		tmp = 0.5 * (c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 8.5e+84:
		tmp = -0.6666666666666666 * (b / a)
	else:
		tmp = 0.5 * (c / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 8.5e+84)
		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 <= 8.5e+84)
		tmp = -0.6666666666666666 * (b / a);
	else
		tmp = 0.5 * (c / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 8.5e+84], 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 < 8.5000000000000008e84

   1. Initial program 65.1%

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

   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 44.7

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

   4. Applied rewrites 44.7%

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

   if 8.5000000000000008e84 < b

   1. Initial program 9.5%

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

   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. lower-/.f64 34.9

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

   7. Applied rewrites 34.9%

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

Alternative 21: 11.0% accurate, 3.3× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = 0.5d0 * (c / b)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 52.1%

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

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 34.1

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

4. Applied rewrites 34.1%

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

5. Taylor expanded in a around inf

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

   1. lower-*.f64 N/A

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

   2. lower-/.f64 11.0

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

7. Applied rewrites 11.0%

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

Reproduce

herbie shell --seed 2025123 
(FPCore (a b c)
  :name "Cubic critical"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))