Cubic critical

Percentage Accurate: 51.9% → 89.8%

Time: 5.1s

Alternatives: 19

Speedup: 2.2×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 51.9% 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: 89.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\left|b\right| - b}{a}\\ t_1 := \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\\ t_2 := \sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)}\\ t_3 := \frac{c}{\left|b\right|}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+210}:\\ \;\;\;\;\mathsf{fma}\left(t\_0, 0.3333333333333333, \left(\frac{-0.375 \cdot \left(c \cdot a\right)}{\left(b \cdot b\right) \cdot b} - \frac{0.5}{\left|b\right|}\right) \cdot c\right)\\ \mathbf{elif}\;t\_1 \leq -5 \cdot 10^{-219}:\\ \;\;\;\;\frac{\frac{t\_2 + \left(-b\right)}{3}}{a}\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;\mathsf{fma}\left(t\_0, 0.3333333333333333, t\_3 \cdot -0.5\right)\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+263}:\\ \;\;\;\;\frac{-b}{a \cdot 3} + \frac{t\_2}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_3, -0.5, t\_0 \cdot 0.3333333333333333\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (- (fabs b) b) a))
        (t_1 (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))
        (t_2 (sqrt (fma (* -3.0 a) c (* b b))))
        (t_3 (/ c (fabs b))))
   (if (<= t_1 -1e+210)
     (fma
      t_0
      0.3333333333333333
      (* (- (/ (* -0.375 (* c a)) (* (* b b) b)) (/ 0.5 (fabs b))) c))
     (if (<= t_1 -5e-219)
       (/ (/ (+ t_2 (- b)) 3.0) a)
       (if (<= t_1 0.0)
         (fma t_0 0.3333333333333333 (* t_3 -0.5))
         (if (<= t_1 2e+263)
           (+ (/ (- b) (* a 3.0)) (/ t_2 (* a 3.0)))
           (fma t_3 -0.5 (* t_0 0.3333333333333333))))))))

C

double code(double a, double b, double c) {
	double t_0 = (fabs(b) - b) / a;
	double t_1 = (-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a);
	double t_2 = sqrt(fma((-3.0 * a), c, (b * b)));
	double t_3 = c / fabs(b);
	double tmp;
	if (t_1 <= -1e+210) {
		tmp = fma(t_0, 0.3333333333333333, ((((-0.375 * (c * a)) / ((b * b) * b)) - (0.5 / fabs(b))) * c));
	} else if (t_1 <= -5e-219) {
		tmp = ((t_2 + -b) / 3.0) / a;
	} else if (t_1 <= 0.0) {
		tmp = fma(t_0, 0.3333333333333333, (t_3 * -0.5));
	} else if (t_1 <= 2e+263) {
		tmp = (-b / (a * 3.0)) + (t_2 / (a * 3.0));
	} else {
		tmp = fma(t_3, -0.5, (t_0 * 0.3333333333333333));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = Float64(Float64(abs(b) - b) / a)
	t_1 = Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c)))) / Float64(3.0 * a))
	t_2 = sqrt(fma(Float64(-3.0 * a), c, Float64(b * b)))
	t_3 = Float64(c / abs(b))
	tmp = 0.0
	if (t_1 <= -1e+210)
		tmp = fma(t_0, 0.3333333333333333, Float64(Float64(Float64(Float64(-0.375 * Float64(c * a)) / Float64(Float64(b * b) * b)) - Float64(0.5 / abs(b))) * c));
	elseif (t_1 <= -5e-219)
		tmp = Float64(Float64(Float64(t_2 + Float64(-b)) / 3.0) / a);
	elseif (t_1 <= 0.0)
		tmp = fma(t_0, 0.3333333333333333, Float64(t_3 * -0.5));
	elseif (t_1 <= 2e+263)
		tmp = Float64(Float64(Float64(-b) / Float64(a * 3.0)) + Float64(t_2 / Float64(a * 3.0)));
	else
		tmp = fma(t_3, -0.5, Float64(t_0 * 0.3333333333333333));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(N[Abs[b], $MachinePrecision] - b), $MachinePrecision] / a), $MachinePrecision]}, Block[{t$95$1 = 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]}, Block[{t$95$2 = N[Sqrt[N[(N[(-3.0 * a), $MachinePrecision] * c + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(c / N[Abs[b], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e+210], N[(t$95$0 * 0.3333333333333333 + N[(N[(N[(N[(-0.375 * N[(c * a), $MachinePrecision]), $MachinePrecision] / N[(N[(b * b), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision] - N[(0.5 / N[Abs[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -5e-219], N[(N[(N[(t$95$2 + (-b)), $MachinePrecision] / 3.0), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[t$95$1, 0.0], N[(t$95$0 * 0.3333333333333333 + N[(t$95$3 * -0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e+263], N[(N[((-b) / N[(a * 3.0), $MachinePrecision]), $MachinePrecision] + N[(t$95$2 / N[(a * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$3 * -0.5 + N[(t$95$0 * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -9.99999999999999927e209

   1. Initial program 51.9%

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

   2. Taylor expanded in c around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. pow2 N/A

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

      5. rem-sqrt-square-rev N/A

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

      6. lower-fabs.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower--.f64 N/A

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

      11. pow2 N/A

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

      12. rem-sqrt-square-rev N/A

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

      13. lower-fabs.f64 68.2

          \[\leadsto \mathsf{fma}\left(\frac{c}{\left|b\right|}, -0.5, \frac{\left|b\right| - b}{a} \cdot 0.3333333333333333\right) \]

   4. Applied rewrites 68.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{c}{\left|b\right|}, -0.5, \frac{\left|b\right| - b}{a} \cdot 0.3333333333333333\right)} \]

   5. Taylor expanded in c around 0

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{\sqrt{{b}^{2}} - b}{a} + c \cdot \left(\frac{-3}{8} \cdot \frac{a \cdot c}{{\left(\sqrt{{b}^{2}}\right)}^{3}} - \frac{1}{2} \cdot \frac{1}{\sqrt{{b}^{2}}}\right)} \]

   7. Applied rewrites 60.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{c \cdot a}{\left|b\right| \cdot \left(b \cdot b\right)} \cdot -0.375 - \frac{0.5}{\left|b\right|}, c, \frac{\left|b\right| - b}{a} \cdot 0.3333333333333333\right)} \]

   9. Applied rewrites 61.3%

      \[\leadsto \mathsf{fma}\left(\frac{\left|b\right| - b}{a}, \color{blue}{0.3333333333333333}, \left(\frac{-0.375 \cdot \left(c \cdot a\right)}{\left(b \cdot b\right) \cdot b} - \frac{0.5}{\left|b\right|}\right) \cdot c\right) \]

   if -9.99999999999999927e209 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -5.0000000000000002e-219

   1. Initial program 51.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-neg.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/r* N/A

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

      11. lower-/.f64 N/A

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

   3. Applied rewrites 51.9%

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

   if -5.0000000000000002e-219 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -0.0

   1. Initial program 51.9%

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

   2. Taylor expanded in c around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. pow2 N/A

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

      5. rem-sqrt-square-rev N/A

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

      6. lower-fabs.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower--.f64 N/A

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

      11. pow2 N/A

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

      12. rem-sqrt-square-rev N/A

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

      13. lower-fabs.f64 68.2

          \[\leadsto \mathsf{fma}\left(\frac{c}{\left|b\right|}, -0.5, \frac{\left|b\right| - b}{a} \cdot 0.3333333333333333\right) \]

   4. Applied rewrites 68.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{c}{\left|b\right|}, -0.5, \frac{\left|b\right| - b}{a} \cdot 0.3333333333333333\right)} \]
   5. Step-by-step derivation

      1. lift-fma.f64 N/A

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

      2. lift-fabs.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-fabs.f64 N/A

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

      8. lift--.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. lift--.f64 N/A

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

      12. lift-fabs.f64 N/A

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

      13. lift-/.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lift-/.f64 N/A

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

      17. lift-fabs.f64 68.2

          \[\leadsto \mathsf{fma}\left(\frac{\left|b\right| - b}{a}, 0.3333333333333333, \frac{c}{\left|b\right|} \cdot -0.5\right) \]

   6. Applied rewrites 68.2%

      \[\leadsto \mathsf{fma}\left(\frac{\left|b\right| - b}{a}, \color{blue}{0.3333333333333333}, \frac{c}{\left|b\right|} \cdot -0.5\right) \]

   if -0.0 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < 2.00000000000000003e263

   1. Initial program 51.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. div-add N/A

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

      10. lower-+.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

   3. Applied rewrites 51.4%

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

   if 2.00000000000000003e263 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a))

   1. Initial program 51.9%

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

   2. Taylor expanded in c around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. pow2 N/A

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

      5. rem-sqrt-square-rev N/A

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

      6. lower-fabs.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower--.f64 N/A

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

      11. pow2 N/A

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

      12. rem-sqrt-square-rev N/A

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

      13. lower-fabs.f64 68.2

          \[\leadsto \mathsf{fma}\left(\frac{c}{\left|b\right|}, -0.5, \frac{\left|b\right| - b}{a} \cdot 0.3333333333333333\right) \]

   4. Applied rewrites 68.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{c}{\left|b\right|}, -0.5, \frac{\left|b\right| - b}{a} \cdot 0.3333333333333333\right)} \]
3. Recombined 5 regimes into one program.
4. Add Preprocessing

Alternative 2: 89.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)}\\ t_1 := \frac{c}{\left|b\right|}\\ t_2 := \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\\ t_3 := \frac{\left|b\right| - b}{a}\\ t_4 := \mathsf{fma}\left(t\_1, -0.5, t\_3 \cdot 0.3333333333333333\right)\\ \mathbf{if}\;t\_2 \leq -1 \cdot 10^{+210}:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;t\_2 \leq -5 \cdot 10^{-219}:\\ \;\;\;\;\frac{\frac{t\_0 + \left(-b\right)}{3}}{a}\\ \mathbf{elif}\;t\_2 \leq 0:\\ \;\;\;\;\mathsf{fma}\left(t\_3, 0.3333333333333333, t\_1 \cdot -0.5\right)\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+263}:\\ \;\;\;\;\frac{-b}{a \cdot 3} + \frac{t\_0}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;t\_4\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (fma (* -3.0 a) c (* b b))))
        (t_1 (/ c (fabs b)))
        (t_2 (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))
        (t_3 (/ (- (fabs b) b) a))
        (t_4 (fma t_1 -0.5 (* t_3 0.3333333333333333))))
   (if (<= t_2 -1e+210)
     t_4
     (if (<= t_2 -5e-219)
       (/ (/ (+ t_0 (- b)) 3.0) a)
       (if (<= t_2 0.0)
         (fma t_3 0.3333333333333333 (* t_1 -0.5))
         (if (<= t_2 2e+263)
           (+ (/ (- b) (* a 3.0)) (/ t_0 (* a 3.0)))
           t_4))))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt(fma((-3.0 * a), c, (b * b)));
	double t_1 = c / fabs(b);
	double t_2 = (-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a);
	double t_3 = (fabs(b) - b) / a;
	double t_4 = fma(t_1, -0.5, (t_3 * 0.3333333333333333));
	double tmp;
	if (t_2 <= -1e+210) {
		tmp = t_4;
	} else if (t_2 <= -5e-219) {
		tmp = ((t_0 + -b) / 3.0) / a;
	} else if (t_2 <= 0.0) {
		tmp = fma(t_3, 0.3333333333333333, (t_1 * -0.5));
	} else if (t_2 <= 2e+263) {
		tmp = (-b / (a * 3.0)) + (t_0 / (a * 3.0));
	} else {
		tmp = t_4;
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = sqrt(fma(Float64(-3.0 * a), c, Float64(b * b)))
	t_1 = Float64(c / abs(b))
	t_2 = Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c)))) / Float64(3.0 * a))
	t_3 = Float64(Float64(abs(b) - b) / a)
	t_4 = fma(t_1, -0.5, Float64(t_3 * 0.3333333333333333))
	tmp = 0.0
	if (t_2 <= -1e+210)
		tmp = t_4;
	elseif (t_2 <= -5e-219)
		tmp = Float64(Float64(Float64(t_0 + Float64(-b)) / 3.0) / a);
	elseif (t_2 <= 0.0)
		tmp = fma(t_3, 0.3333333333333333, Float64(t_1 * -0.5));
	elseif (t_2 <= 2e+263)
		tmp = Float64(Float64(Float64(-b) / Float64(a * 3.0)) + Float64(t_0 / Float64(a * 3.0)));
	else
		tmp = t_4;
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Sqrt[N[(N[(-3.0 * a), $MachinePrecision] * c + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(c / N[Abs[b], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = 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]}, Block[{t$95$3 = N[(N[(N[Abs[b], $MachinePrecision] - b), $MachinePrecision] / a), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$1 * -0.5 + N[(t$95$3 * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -1e+210], t$95$4, If[LessEqual[t$95$2, -5e-219], N[(N[(N[(t$95$0 + (-b)), $MachinePrecision] / 3.0), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[t$95$2, 0.0], N[(t$95$3 * 0.3333333333333333 + N[(t$95$1 * -0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 2e+263], N[(N[((-b) / N[(a * 3.0), $MachinePrecision]), $MachinePrecision] + N[(t$95$0 / N[(a * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$4]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -9.99999999999999927e209 or 2.00000000000000003e263 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a))

   1. Initial program 51.9%

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

   2. Taylor expanded in c around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. pow2 N/A

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

      5. rem-sqrt-square-rev N/A

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

      6. lower-fabs.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower--.f64 N/A

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

      11. pow2 N/A

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

      12. rem-sqrt-square-rev N/A

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

      13. lower-fabs.f64 68.2

          \[\leadsto \mathsf{fma}\left(\frac{c}{\left|b\right|}, -0.5, \frac{\left|b\right| - b}{a} \cdot 0.3333333333333333\right) \]

   4. Applied rewrites 68.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{c}{\left|b\right|}, -0.5, \frac{\left|b\right| - b}{a} \cdot 0.3333333333333333\right)} \]

   if -9.99999999999999927e209 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -5.0000000000000002e-219

   1. Initial program 51.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-neg.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/r* N/A

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

      11. lower-/.f64 N/A

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

   3. Applied rewrites 51.9%

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

   if -5.0000000000000002e-219 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -0.0

   1. Initial program 51.9%

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

   2. Taylor expanded in c around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. pow2 N/A

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

      5. rem-sqrt-square-rev N/A

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

      6. lower-fabs.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower--.f64 N/A

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

      11. pow2 N/A

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

      12. rem-sqrt-square-rev N/A

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

      13. lower-fabs.f64 68.2

          \[\leadsto \mathsf{fma}\left(\frac{c}{\left|b\right|}, -0.5, \frac{\left|b\right| - b}{a} \cdot 0.3333333333333333\right) \]

   4. Applied rewrites 68.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{c}{\left|b\right|}, -0.5, \frac{\left|b\right| - b}{a} \cdot 0.3333333333333333\right)} \]
   5. Step-by-step derivation

      1. lift-fma.f64 N/A

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

      2. lift-fabs.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-fabs.f64 N/A

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

      8. lift--.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. lift--.f64 N/A

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

      12. lift-fabs.f64 N/A

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

      13. lift-/.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lift-/.f64 N/A

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

      17. lift-fabs.f64 68.2

          \[\leadsto \mathsf{fma}\left(\frac{\left|b\right| - b}{a}, 0.3333333333333333, \frac{c}{\left|b\right|} \cdot -0.5\right) \]

   6. Applied rewrites 68.2%

      \[\leadsto \mathsf{fma}\left(\frac{\left|b\right| - b}{a}, \color{blue}{0.3333333333333333}, \frac{c}{\left|b\right|} \cdot -0.5\right) \]

   if -0.0 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < 2.00000000000000003e263

   1. Initial program 51.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. div-add N/A

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

      10. lower-+.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

   3. Applied rewrites 51.4%

      \[\leadsto \color{blue}{\frac{-b}{a \cdot 3} + \frac{\sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)}}{a \cdot 3}} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 3: 89.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c}{\left|b\right|}\\ t_1 := \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\\ t_2 := \frac{\left|b\right| - b}{a}\\ t_3 := \mathsf{fma}\left(t\_0, -0.5, t\_2 \cdot 0.3333333333333333\right)\\ t_4 := \sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)} + \left(-b\right)\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+210}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_1 \leq -5 \cdot 10^{-219}:\\ \;\;\;\;\frac{\frac{t\_4}{3}}{a}\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;\mathsf{fma}\left(t\_2, 0.3333333333333333, t\_0 \cdot -0.5\right)\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+263}:\\ \;\;\;\;\frac{t\_4}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ c (fabs b)))
        (t_1 (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))
        (t_2 (/ (- (fabs b) b) a))
        (t_3 (fma t_0 -0.5 (* t_2 0.3333333333333333)))
        (t_4 (+ (sqrt (fma (* -3.0 a) c (* b b))) (- b))))
   (if (<= t_1 -1e+210)
     t_3
     (if (<= t_1 -5e-219)
       (/ (/ t_4 3.0) a)
       (if (<= t_1 0.0)
         (fma t_2 0.3333333333333333 (* t_0 -0.5))
         (if (<= t_1 2e+263) (/ t_4 (* a 3.0)) t_3))))))

C

double code(double a, double b, double c) {
	double t_0 = c / fabs(b);
	double t_1 = (-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a);
	double t_2 = (fabs(b) - b) / a;
	double t_3 = fma(t_0, -0.5, (t_2 * 0.3333333333333333));
	double t_4 = sqrt(fma((-3.0 * a), c, (b * b))) + -b;
	double tmp;
	if (t_1 <= -1e+210) {
		tmp = t_3;
	} else if (t_1 <= -5e-219) {
		tmp = (t_4 / 3.0) / a;
	} else if (t_1 <= 0.0) {
		tmp = fma(t_2, 0.3333333333333333, (t_0 * -0.5));
	} else if (t_1 <= 2e+263) {
		tmp = t_4 / (a * 3.0);
	} else {
		tmp = t_3;
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = Float64(c / abs(b))
	t_1 = Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c)))) / Float64(3.0 * a))
	t_2 = Float64(Float64(abs(b) - b) / a)
	t_3 = fma(t_0, -0.5, Float64(t_2 * 0.3333333333333333))
	t_4 = Float64(sqrt(fma(Float64(-3.0 * a), c, Float64(b * b))) + Float64(-b))
	tmp = 0.0
	if (t_1 <= -1e+210)
		tmp = t_3;
	elseif (t_1 <= -5e-219)
		tmp = Float64(Float64(t_4 / 3.0) / a);
	elseif (t_1 <= 0.0)
		tmp = fma(t_2, 0.3333333333333333, Float64(t_0 * -0.5));
	elseif (t_1 <= 2e+263)
		tmp = Float64(t_4 / Float64(a * 3.0));
	else
		tmp = t_3;
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(c / N[Abs[b], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = 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]}, Block[{t$95$2 = N[(N[(N[Abs[b], $MachinePrecision] - b), $MachinePrecision] / a), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$0 * -0.5 + N[(t$95$2 * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[Sqrt[N[(N[(-3.0 * a), $MachinePrecision] * c + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + (-b)), $MachinePrecision]}, If[LessEqual[t$95$1, -1e+210], t$95$3, If[LessEqual[t$95$1, -5e-219], N[(N[(t$95$4 / 3.0), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[t$95$1, 0.0], N[(t$95$2 * 0.3333333333333333 + N[(t$95$0 * -0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e+263], N[(t$95$4 / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], t$95$3]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -9.99999999999999927e209 or 2.00000000000000003e263 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a))

   1. Initial program 51.9%

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

   2. Taylor expanded in c around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. pow2 N/A

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

      5. rem-sqrt-square-rev N/A

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

      6. lower-fabs.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower--.f64 N/A

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

      11. pow2 N/A

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

      12. rem-sqrt-square-rev N/A

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

      13. lower-fabs.f64 68.2

          \[\leadsto \mathsf{fma}\left(\frac{c}{\left|b\right|}, -0.5, \frac{\left|b\right| - b}{a} \cdot 0.3333333333333333\right) \]

   4. Applied rewrites 68.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{c}{\left|b\right|}, -0.5, \frac{\left|b\right| - b}{a} \cdot 0.3333333333333333\right)} \]

   if -9.99999999999999927e209 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -5.0000000000000002e-219

   1. Initial program 51.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-neg.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/r* N/A

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

      11. lower-/.f64 N/A

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

   3. Applied rewrites 51.9%

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

   if -5.0000000000000002e-219 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -0.0

   1. Initial program 51.9%

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

   2. Taylor expanded in c around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. pow2 N/A

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

      5. rem-sqrt-square-rev N/A

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

      6. lower-fabs.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower--.f64 N/A

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

      11. pow2 N/A

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

      12. rem-sqrt-square-rev N/A

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

      13. lower-fabs.f64 68.2

          \[\leadsto \mathsf{fma}\left(\frac{c}{\left|b\right|}, -0.5, \frac{\left|b\right| - b}{a} \cdot 0.3333333333333333\right) \]

   4. Applied rewrites 68.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{c}{\left|b\right|}, -0.5, \frac{\left|b\right| - b}{a} \cdot 0.3333333333333333\right)} \]
   5. Step-by-step derivation

      1. lift-fma.f64 N/A

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

      2. lift-fabs.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-fabs.f64 N/A

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

      8. lift--.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. lift--.f64 N/A

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

      12. lift-fabs.f64 N/A

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

      13. lift-/.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lift-/.f64 N/A

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

      17. lift-fabs.f64 68.2

          \[\leadsto \mathsf{fma}\left(\frac{\left|b\right| - b}{a}, 0.3333333333333333, \frac{c}{\left|b\right|} \cdot -0.5\right) \]

   6. Applied rewrites 68.2%

      \[\leadsto \mathsf{fma}\left(\frac{\left|b\right| - b}{a}, \color{blue}{0.3333333333333333}, \frac{c}{\left|b\right|} \cdot -0.5\right) \]

   if -0.0 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < 2.00000000000000003e263

   1. Initial program 51.9%

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

   3. Applied rewrites 51.9%

      \[\leadsto \color{blue}{\frac{\sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)} + \left(-b\right)}{a \cdot 3}} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 4: 88.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c}{\left|b\right|}\\ t_1 := \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\\ t_2 := \frac{\left|b\right| - b}{a}\\ t_3 := \mathsf{fma}\left(t\_0, -0.5, t\_2 \cdot 0.3333333333333333\right)\\ t_4 := \frac{\sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)} + \left(-b\right)}{a \cdot 3}\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+256}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_1 \leq -5 \cdot 10^{-219}:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;\mathsf{fma}\left(t\_2, 0.3333333333333333, t\_0 \cdot -0.5\right)\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+263}:\\ \;\;\;\;t\_4\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ c (fabs b)))
        (t_1 (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))
        (t_2 (/ (- (fabs b) b) a))
        (t_3 (fma t_0 -0.5 (* t_2 0.3333333333333333)))
        (t_4 (/ (+ (sqrt (fma (* -3.0 a) c (* b b))) (- b)) (* a 3.0))))
   (if (<= t_1 -5e+256)
     t_3
     (if (<= t_1 -5e-219)
       t_4
       (if (<= t_1 0.0)
         (fma t_2 0.3333333333333333 (* t_0 -0.5))
         (if (<= t_1 2e+263) t_4 t_3))))))

C

double code(double a, double b, double c) {
	double t_0 = c / fabs(b);
	double t_1 = (-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a);
	double t_2 = (fabs(b) - b) / a;
	double t_3 = fma(t_0, -0.5, (t_2 * 0.3333333333333333));
	double t_4 = (sqrt(fma((-3.0 * a), c, (b * b))) + -b) / (a * 3.0);
	double tmp;
	if (t_1 <= -5e+256) {
		tmp = t_3;
	} else if (t_1 <= -5e-219) {
		tmp = t_4;
	} else if (t_1 <= 0.0) {
		tmp = fma(t_2, 0.3333333333333333, (t_0 * -0.5));
	} else if (t_1 <= 2e+263) {
		tmp = t_4;
	} else {
		tmp = t_3;
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = Float64(c / abs(b))
	t_1 = Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c)))) / Float64(3.0 * a))
	t_2 = Float64(Float64(abs(b) - b) / a)
	t_3 = fma(t_0, -0.5, Float64(t_2 * 0.3333333333333333))
	t_4 = Float64(Float64(sqrt(fma(Float64(-3.0 * a), c, Float64(b * b))) + Float64(-b)) / Float64(a * 3.0))
	tmp = 0.0
	if (t_1 <= -5e+256)
		tmp = t_3;
	elseif (t_1 <= -5e-219)
		tmp = t_4;
	elseif (t_1 <= 0.0)
		tmp = fma(t_2, 0.3333333333333333, Float64(t_0 * -0.5));
	elseif (t_1 <= 2e+263)
		tmp = t_4;
	else
		tmp = t_3;
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(c / N[Abs[b], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = 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]}, Block[{t$95$2 = N[(N[(N[Abs[b], $MachinePrecision] - b), $MachinePrecision] / a), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$0 * -0.5 + N[(t$95$2 * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = 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]}, If[LessEqual[t$95$1, -5e+256], t$95$3, If[LessEqual[t$95$1, -5e-219], t$95$4, If[LessEqual[t$95$1, 0.0], N[(t$95$2 * 0.3333333333333333 + N[(t$95$0 * -0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e+263], t$95$4, t$95$3]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -5.00000000000000015e256 or 2.00000000000000003e263 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a))

   1. Initial program 51.9%

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

   2. Taylor expanded in c around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. pow2 N/A

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

      5. rem-sqrt-square-rev N/A

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

      6. lower-fabs.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower--.f64 N/A

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

      11. pow2 N/A

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

      12. rem-sqrt-square-rev N/A

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

      13. lower-fabs.f64 68.2

          \[\leadsto \mathsf{fma}\left(\frac{c}{\left|b\right|}, -0.5, \frac{\left|b\right| - b}{a} \cdot 0.3333333333333333\right) \]

   4. Applied rewrites 68.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{c}{\left|b\right|}, -0.5, \frac{\left|b\right| - b}{a} \cdot 0.3333333333333333\right)} \]

   if -5.00000000000000015e256 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -5.0000000000000002e-219 or -0.0 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < 2.00000000000000003e263

   1. Initial program 51.9%

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

   3. Applied rewrites 51.9%

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

   if -5.0000000000000002e-219 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -0.0

   1. Initial program 51.9%

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

   2. Taylor expanded in c around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. pow2 N/A

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

      5. rem-sqrt-square-rev N/A

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

      6. lower-fabs.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower--.f64 N/A

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

      11. pow2 N/A

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

      12. rem-sqrt-square-rev N/A

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

      13. lower-fabs.f64 68.2

          \[\leadsto \mathsf{fma}\left(\frac{c}{\left|b\right|}, -0.5, \frac{\left|b\right| - b}{a} \cdot 0.3333333333333333\right) \]

   4. Applied rewrites 68.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{c}{\left|b\right|}, -0.5, \frac{\left|b\right| - b}{a} \cdot 0.3333333333333333\right)} \]
   5. Step-by-step derivation

      1. lift-fma.f64 N/A

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

      2. lift-fabs.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-fabs.f64 N/A

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

      8. lift--.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. lift--.f64 N/A

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

      12. lift-fabs.f64 N/A

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

      13. lift-/.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lift-/.f64 N/A

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

      17. lift-fabs.f64 68.2

          \[\leadsto \mathsf{fma}\left(\frac{\left|b\right| - b}{a}, 0.3333333333333333, \frac{c}{\left|b\right|} \cdot -0.5\right) \]

   6. Applied rewrites 68.2%

      \[\leadsto \mathsf{fma}\left(\frac{\left|b\right| - b}{a}, \color{blue}{0.3333333333333333}, \frac{c}{\left|b\right|} \cdot -0.5\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 5: 81.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -5.7 \cdot 10^{-21}:\\ \;\;\;\;\frac{\left(-b\right) \cdot \mathsf{fma}\left(a \cdot \frac{c}{b \cdot b}, -0.5, 0.6666666666666666\right)}{a}\\ \mathbf{elif}\;b \leq 4.5 \cdot 10^{-117}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\left(-3 \cdot a\right) \cdot c}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -5.7e-21)
   (/ (* (- b) (fma (* a (/ c (* b b))) -0.5 0.6666666666666666)) a)
   (if (<= b 4.5e-117)
     (/ (+ (- b) (sqrt (* (* -3.0 a) c))) (* 3.0 a))
     (/ (* -0.5 c) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -5.7e-21) {
		tmp = (-b * fma((a * (c / (b * b))), -0.5, 0.6666666666666666)) / a;
	} else if (b <= 4.5e-117) {
		tmp = (-b + sqrt(((-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 <= -5.7e-21)
		tmp = Float64(Float64(Float64(-b) * fma(Float64(a * Float64(c / Float64(b * b))), -0.5, 0.6666666666666666)) / a);
	elseif (b <= 4.5e-117)
		tmp = Float64(Float64(Float64(-b) + sqrt(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, -5.7e-21], N[(N[((-b) * N[(N[(a * N[(c / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.5 + 0.6666666666666666), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b, 4.5e-117], N[(N[((-b) + N[Sqrt[N[(N[(-3.0 * a), $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 < -5.6999999999999996e-21

   1. Initial program 51.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-neg.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/r* N/A

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

      11. lower-/.f64 N/A

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

   3. Applied rewrites 51.9%

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

   4. Taylor expanded in b around -inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lift-neg.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 34.2

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

   6. Applied rewrites 34.2%

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

   if -5.6999999999999996e-21 < b < 4.49999999999999969e-117

   1. Initial program 51.9%

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

   2. Taylor expanded in a around inf

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{-3 \cdot \left(a \cdot c\right)}}}{3 \cdot a} \]
   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 33.6

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

   4. Applied rewrites 33.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*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. lift-*.f64 33.7

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

   6. Applied rewrites 33.7%

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

   if 4.49999999999999969e-117 < b

   1. Initial program 51.9%

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

   2. Taylor expanded in c around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. pow2 N/A

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

      5. rem-sqrt-square-rev N/A

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

      6. lower-fabs.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower--.f64 N/A

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

      11. pow2 N/A

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

      12. rem-sqrt-square-rev N/A

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

      13. lower-fabs.f64 68.2

          \[\leadsto \mathsf{fma}\left(\frac{c}{\left|b\right|}, -0.5, \frac{\left|b\right| - b}{a} \cdot 0.3333333333333333\right) \]

   4. Applied rewrites 68.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{c}{\left|b\right|}, -0.5, \frac{\left|b\right| - b}{a} \cdot 0.3333333333333333\right)} \]

   5. Taylor expanded in b around inf

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

      1. lift-neg.f64 N/A

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

      2. +-commutative N/A

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

      3. pow2 N/A

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

      4. associate-*l* N/A

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

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

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

      6. metadata-eval N/A

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

      7. +-commutative N/A

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

      8. associate-*l* N/A

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

      9. pow2 N/A

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

      10. lift-neg.f64 N/A

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

      11. associate-/l/ N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 35.6

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

   7. Applied rewrites 35.6%

      \[\leadsto \color{blue}{\frac{c}{b} \cdot -0.5} \]
   8. 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 35.6

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

   9. Applied rewrites 35.6%

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

Alternative 6: 80.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -5.7 \cdot 10^{-21}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.5, \frac{c}{b} \cdot a, -0.6666666666666666 \cdot b\right)}{a}\\ \mathbf{elif}\;b \leq 4.5 \cdot 10^{-117}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\left(-3 \cdot a\right) \cdot c}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -5.7e-21)
   (/ (fma 0.5 (* (/ c b) a) (* -0.6666666666666666 b)) a)
   (if (<= b 4.5e-117)
     (/ (+ (- b) (sqrt (* (* -3.0 a) c))) (* 3.0 a))
     (/ (* -0.5 c) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -5.7e-21) {
		tmp = fma(0.5, ((c / b) * a), (-0.6666666666666666 * b)) / a;
	} else if (b <= 4.5e-117) {
		tmp = (-b + sqrt(((-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 <= -5.7e-21)
		tmp = Float64(fma(0.5, Float64(Float64(c / b) * a), Float64(-0.6666666666666666 * b)) / a);
	elseif (b <= 4.5e-117)
		tmp = Float64(Float64(Float64(-b) + sqrt(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, -5.7e-21], N[(N[(0.5 * N[(N[(c / b), $MachinePrecision] * a), $MachinePrecision] + N[(-0.6666666666666666 * b), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b, 4.5e-117], N[(N[((-b) + N[Sqrt[N[(N[(-3.0 * a), $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 < -5.6999999999999996e-21

   1. Initial program 51.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-neg.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/r* N/A

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

      11. lower-/.f64 N/A

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

   3. Applied rewrites 51.9%

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

   4. Taylor expanded in b around -inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lift-neg.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 34.2

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

   6. Applied rewrites 34.2%

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

   7. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift-/.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 34.9

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

   9. Applied rewrites 34.9%

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

   if -5.6999999999999996e-21 < b < 4.49999999999999969e-117

   1. Initial program 51.9%

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

   2. Taylor expanded in a around inf

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{-3 \cdot \left(a \cdot c\right)}}}{3 \cdot a} \]
   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 33.6

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

   4. Applied rewrites 33.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*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. lift-*.f64 33.7

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

   6. Applied rewrites 33.7%

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

   if 4.49999999999999969e-117 < b

   1. Initial program 51.9%

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

   2. Taylor expanded in c around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. pow2 N/A

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

      5. rem-sqrt-square-rev N/A

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

      6. lower-fabs.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower--.f64 N/A

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

      11. pow2 N/A

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

      12. rem-sqrt-square-rev N/A

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

      13. lower-fabs.f64 68.2

          \[\leadsto \mathsf{fma}\left(\frac{c}{\left|b\right|}, -0.5, \frac{\left|b\right| - b}{a} \cdot 0.3333333333333333\right) \]

   4. Applied rewrites 68.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{c}{\left|b\right|}, -0.5, \frac{\left|b\right| - b}{a} \cdot 0.3333333333333333\right)} \]

   5. Taylor expanded in b around inf

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

      1. lift-neg.f64 N/A

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

      2. +-commutative N/A

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

      3. pow2 N/A

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

      4. associate-*l* N/A

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

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

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

      6. metadata-eval N/A

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

      7. +-commutative N/A

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

      8. associate-*l* N/A

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

      9. pow2 N/A

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

      10. lift-neg.f64 N/A

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

      11. associate-/l/ N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 35.6

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

   7. Applied rewrites 35.6%

      \[\leadsto \color{blue}{\frac{c}{b} \cdot -0.5} \]
   8. 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 35.6

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

   9. Applied rewrites 35.6%

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

Alternative 7: 80.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -5.7 \cdot 10^{-21}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.5, \frac{c}{b} \cdot a, -0.6666666666666666 \cdot b\right)}{a}\\ \mathbf{elif}\;b \leq 4.5 \cdot 10^{-117}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{-3 \cdot \left(c \cdot a\right)}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -5.7e-21)
   (/ (fma 0.5 (* (/ c b) a) (* -0.6666666666666666 b)) a)
   (if (<= b 4.5e-117)
     (/ (+ (- 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 <= -5.7e-21) {
		tmp = fma(0.5, ((c / b) * a), (-0.6666666666666666 * b)) / a;
	} else if (b <= 4.5e-117) {
		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 <= -5.7e-21)
		tmp = Float64(fma(0.5, Float64(Float64(c / b) * a), Float64(-0.6666666666666666 * b)) / a);
	elseif (b <= 4.5e-117)
		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, -5.7e-21], N[(N[(0.5 * N[(N[(c / b), $MachinePrecision] * a), $MachinePrecision] + N[(-0.6666666666666666 * b), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b, 4.5e-117], 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 < -5.6999999999999996e-21

   1. Initial program 51.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-neg.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/r* N/A

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

      11. lower-/.f64 N/A

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

   3. Applied rewrites 51.9%

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

   4. Taylor expanded in b around -inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lift-neg.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 34.2

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

   6. Applied rewrites 34.2%

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

   7. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift-/.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 34.9

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

   9. Applied rewrites 34.9%

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

   if -5.6999999999999996e-21 < b < 4.49999999999999969e-117

   1. Initial program 51.9%

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

   2. Taylor expanded in a around inf

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{-3 \cdot \left(a \cdot c\right)}}}{3 \cdot a} \]
   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 33.6

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

   4. Applied rewrites 33.6%

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

   if 4.49999999999999969e-117 < b

   1. Initial program 51.9%

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

   2. Taylor expanded in c around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. pow2 N/A

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

      5. rem-sqrt-square-rev N/A

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

      6. lower-fabs.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower--.f64 N/A

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

      11. pow2 N/A

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

      12. rem-sqrt-square-rev N/A

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

      13. lower-fabs.f64 68.2

          \[\leadsto \mathsf{fma}\left(\frac{c}{\left|b\right|}, -0.5, \frac{\left|b\right| - b}{a} \cdot 0.3333333333333333\right) \]

   4. Applied rewrites 68.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{c}{\left|b\right|}, -0.5, \frac{\left|b\right| - b}{a} \cdot 0.3333333333333333\right)} \]

   5. Taylor expanded in b around inf

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

      1. lift-neg.f64 N/A

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

      2. +-commutative N/A

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

      3. pow2 N/A

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

      4. associate-*l* N/A

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

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

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

      6. metadata-eval N/A

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

      7. +-commutative N/A

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

      8. associate-*l* N/A

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

      9. pow2 N/A

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

      10. lift-neg.f64 N/A

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

      11. associate-/l/ N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 35.6

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

   7. Applied rewrites 35.6%

      \[\leadsto \color{blue}{\frac{c}{b} \cdot -0.5} \]
   8. 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 35.6

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

   9. Applied rewrites 35.6%

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

Alternative 8: 80.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -5.7 \cdot 10^{-21}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.5, \frac{c}{b} \cdot a, -0.6666666666666666 \cdot b\right)}{a}\\ \mathbf{elif}\;b \leq 4.5 \cdot 10^{-117}:\\ \;\;\;\;\frac{\sqrt{\left(-3 \cdot a\right) \cdot c}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -5.7e-21)
   (/ (fma 0.5 (* (/ c b) a) (* -0.6666666666666666 b)) a)
   (if (<= b 4.5e-117)
     (/ (sqrt (* (* -3.0 a) c)) (* 3.0 a))
     (/ (* -0.5 c) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -5.7e-21) {
		tmp = fma(0.5, ((c / b) * a), (-0.6666666666666666 * b)) / a;
	} else if (b <= 4.5e-117) {
		tmp = sqrt(((-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 <= -5.7e-21)
		tmp = Float64(fma(0.5, Float64(Float64(c / b) * a), Float64(-0.6666666666666666 * b)) / a);
	elseif (b <= 4.5e-117)
		tmp = Float64(sqrt(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, -5.7e-21], N[(N[(0.5 * N[(N[(c / b), $MachinePrecision] * a), $MachinePrecision] + N[(-0.6666666666666666 * b), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b, 4.5e-117], N[(N[Sqrt[N[(N[(-3.0 * a), $MachinePrecision] * c), $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 < -5.6999999999999996e-21

   1. Initial program 51.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-neg.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/r* N/A

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

      11. lower-/.f64 N/A

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

   3. Applied rewrites 51.9%

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

   4. Taylor expanded in b around -inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lift-neg.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 34.2

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

   6. Applied rewrites 34.2%

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

   7. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift-/.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 34.9

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

   9. Applied rewrites 34.9%

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

   if -5.6999999999999996e-21 < b < 4.49999999999999969e-117

   1. Initial program 51.9%

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

   2. Taylor expanded in b around 0

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

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

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

      2. metadata-eval N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 28.8

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

   4. Applied rewrites 28.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

      6. lift-*.f64 28.8

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

   6. Applied rewrites 28.8%

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

   if 4.49999999999999969e-117 < b

   1. Initial program 51.9%

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

   2. Taylor expanded in c around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. pow2 N/A

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

      5. rem-sqrt-square-rev N/A

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

      6. lower-fabs.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower--.f64 N/A

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

      11. pow2 N/A

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

      12. rem-sqrt-square-rev N/A

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

      13. lower-fabs.f64 68.2

          \[\leadsto \mathsf{fma}\left(\frac{c}{\left|b\right|}, -0.5, \frac{\left|b\right| - b}{a} \cdot 0.3333333333333333\right) \]

   4. Applied rewrites 68.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{c}{\left|b\right|}, -0.5, \frac{\left|b\right| - b}{a} \cdot 0.3333333333333333\right)} \]

   5. Taylor expanded in b around inf

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

      1. lift-neg.f64 N/A

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

      2. +-commutative N/A

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

      3. pow2 N/A

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

      4. associate-*l* N/A

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

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

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

      6. metadata-eval N/A

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

      7. +-commutative N/A

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

      8. associate-*l* N/A

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

      9. pow2 N/A

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

      10. lift-neg.f64 N/A

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

      11. associate-/l/ N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 35.6

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

   7. Applied rewrites 35.6%

      \[\leadsto \color{blue}{\frac{c}{b} \cdot -0.5} \]
   8. 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 35.6

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

   9. Applied rewrites 35.6%

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

Alternative 9: 80.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -5.7 \cdot 10^{-21}:\\ \;\;\;\;\frac{-2 \cdot b}{3 \cdot a}\\ \mathbf{elif}\;b \leq 4.5 \cdot 10^{-117}:\\ \;\;\;\;\frac{\sqrt{\left(-3 \cdot a\right) \cdot c}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -5.7e-21)
   (/ (* -2.0 b) (* 3.0 a))
   (if (<= b 4.5e-117)
     (/ (sqrt (* (* -3.0 a) c)) (* 3.0 a))
     (/ (* -0.5 c) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -5.7e-21) {
		tmp = (-2.0 * b) / (3.0 * a);
	} else if (b <= 4.5e-117) {
		tmp = sqrt(((-3.0 * a) * c)) / (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 <= (-5.7d-21)) then
        tmp = ((-2.0d0) * b) / (3.0d0 * a)
    else if (b <= 4.5d-117) then
        tmp = sqrt((((-3.0d0) * a) * c)) / (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 <= -5.7e-21) {
		tmp = (-2.0 * b) / (3.0 * a);
	} else if (b <= 4.5e-117) {
		tmp = Math.sqrt(((-3.0 * a) * c)) / (3.0 * a);
	} else {
		tmp = (-0.5 * c) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -5.7e-21:
		tmp = (-2.0 * b) / (3.0 * a)
	elif b <= 4.5e-117:
		tmp = math.sqrt(((-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 <= -5.7e-21)
		tmp = Float64(Float64(-2.0 * b) / Float64(3.0 * a));
	elseif (b <= 4.5e-117)
		tmp = Float64(sqrt(Float64(Float64(-3.0 * a) * c)) / 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 <= -5.7e-21)
		tmp = (-2.0 * b) / (3.0 * a);
	elseif (b <= 4.5e-117)
		tmp = sqrt(((-3.0 * a) * c)) / (3.0 * a);
	else
		tmp = (-0.5 * c) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -5.7e-21], N[(N[(-2.0 * b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 4.5e-117], N[(N[Sqrt[N[(N[(-3.0 * a), $MachinePrecision] * c), $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 < -5.6999999999999996e-21

   1. Initial program 51.9%

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

   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 35.1

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

   4. Applied rewrites 35.1%

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

   if -5.6999999999999996e-21 < b < 4.49999999999999969e-117

   1. Initial program 51.9%

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

   2. Taylor expanded in b around 0

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

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

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

      2. metadata-eval N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 28.8

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

   4. Applied rewrites 28.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

      6. lift-*.f64 28.8

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

   6. Applied rewrites 28.8%

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

   if 4.49999999999999969e-117 < b

   1. Initial program 51.9%

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

   2. Taylor expanded in c around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. pow2 N/A

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

      5. rem-sqrt-square-rev N/A

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

      6. lower-fabs.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower--.f64 N/A

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

      11. pow2 N/A

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

      12. rem-sqrt-square-rev N/A

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

      13. lower-fabs.f64 68.2

          \[\leadsto \mathsf{fma}\left(\frac{c}{\left|b\right|}, -0.5, \frac{\left|b\right| - b}{a} \cdot 0.3333333333333333\right) \]

   4. Applied rewrites 68.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{c}{\left|b\right|}, -0.5, \frac{\left|b\right| - b}{a} \cdot 0.3333333333333333\right)} \]

   5. Taylor expanded in b around inf

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

      1. lift-neg.f64 N/A

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

      2. +-commutative N/A

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

      3. pow2 N/A

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

      4. associate-*l* N/A

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

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

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

      6. metadata-eval N/A

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

      7. +-commutative N/A

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

      8. associate-*l* N/A

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

      9. pow2 N/A

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

      10. lift-neg.f64 N/A

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

      11. associate-/l/ N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 35.6

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

   7. Applied rewrites 35.6%

      \[\leadsto \color{blue}{\frac{c}{b} \cdot -0.5} \]
   8. 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 35.6

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

   9. Applied rewrites 35.6%

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

Alternative 10: 80.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -5.7 \cdot 10^{-21}:\\ \;\;\;\;\frac{-2 \cdot b}{3 \cdot a}\\ \mathbf{elif}\;b \leq 4.5 \cdot 10^{-117}:\\ \;\;\;\;\frac{\sqrt{-3 \cdot \left(c \cdot a\right)}}{a} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -5.7e-21)
   (/ (* -2.0 b) (* 3.0 a))
   (if (<= b 4.5e-117)
     (* (/ (sqrt (* -3.0 (* c a))) a) 0.3333333333333333)
     (/ (* -0.5 c) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -5.7e-21) {
		tmp = (-2.0 * b) / (3.0 * a);
	} else if (b <= 4.5e-117) {
		tmp = (sqrt((-3.0 * (c * a))) / a) * 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 <= (-5.7d-21)) then
        tmp = ((-2.0d0) * b) / (3.0d0 * a)
    else if (b <= 4.5d-117) then
        tmp = (sqrt(((-3.0d0) * (c * a))) / a) * 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 <= -5.7e-21) {
		tmp = (-2.0 * b) / (3.0 * a);
	} else if (b <= 4.5e-117) {
		tmp = (Math.sqrt((-3.0 * (c * a))) / a) * 0.3333333333333333;
	} else {
		tmp = (-0.5 * c) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -5.7e-21:
		tmp = (-2.0 * b) / (3.0 * a)
	elif b <= 4.5e-117:
		tmp = (math.sqrt((-3.0 * (c * a))) / a) * 0.3333333333333333
	else:
		tmp = (-0.5 * c) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -5.7e-21)
		tmp = Float64(Float64(-2.0 * b) / Float64(3.0 * a));
	elseif (b <= 4.5e-117)
		tmp = Float64(Float64(sqrt(Float64(-3.0 * Float64(c * a))) / a) * 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 <= -5.7e-21)
		tmp = (-2.0 * b) / (3.0 * a);
	elseif (b <= 4.5e-117)
		tmp = (sqrt((-3.0 * (c * a))) / a) * 0.3333333333333333;
	else
		tmp = (-0.5 * c) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -5.7e-21], N[(N[(-2.0 * b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 4.5e-117], N[(N[(N[Sqrt[N[(-3.0 * N[(c * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / a), $MachinePrecision] * 0.3333333333333333), $MachinePrecision], N[(N[(-0.5 * c), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -5.6999999999999996e-21

   1. Initial program 51.9%

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

   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 35.1

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

   4. Applied rewrites 35.1%

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

   if -5.6999999999999996e-21 < b < 4.49999999999999969e-117

   1. Initial program 51.9%

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

   2. Taylor expanded in b around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

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

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

      5. metadata-eval N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 28.8

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

   4. Applied rewrites 28.8%

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

   if 4.49999999999999969e-117 < b

   1. Initial program 51.9%

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

   2. Taylor expanded in c around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. pow2 N/A

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

      5. rem-sqrt-square-rev N/A

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

      6. lower-fabs.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower--.f64 N/A

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

      11. pow2 N/A

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

      12. rem-sqrt-square-rev N/A

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

      13. lower-fabs.f64 68.2

          \[\leadsto \mathsf{fma}\left(\frac{c}{\left|b\right|}, -0.5, \frac{\left|b\right| - b}{a} \cdot 0.3333333333333333\right) \]

   4. Applied rewrites 68.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{c}{\left|b\right|}, -0.5, \frac{\left|b\right| - b}{a} \cdot 0.3333333333333333\right)} \]

   5. Taylor expanded in b around inf

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

      1. lift-neg.f64 N/A

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

      2. +-commutative N/A

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

      3. pow2 N/A

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

      4. associate-*l* N/A

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

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

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

      6. metadata-eval N/A

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

      7. +-commutative N/A

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

      8. associate-*l* N/A

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

      9. pow2 N/A

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

      10. lift-neg.f64 N/A

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

      11. associate-/l/ N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 35.6

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

   7. Applied rewrites 35.6%

      \[\leadsto \color{blue}{\frac{c}{b} \cdot -0.5} \]
   8. 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 35.6

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

   9. Applied rewrites 35.6%

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

Alternative 11: 72.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -5.7 \cdot 10^{-21}:\\ \;\;\;\;\frac{-2 \cdot b}{3 \cdot a}\\ \mathbf{elif}\;b \leq 6 \cdot 10^{-102}:\\ \;\;\;\;\sqrt{\frac{c}{a} \cdot -3} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -5.7e-21)
   (/ (* -2.0 b) (* 3.0 a))
   (if (<= b 6e-102)
     (* (sqrt (* (/ c a) -3.0)) 0.3333333333333333)
     (/ (* -0.5 c) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -5.7e-21) {
		tmp = (-2.0 * b) / (3.0 * a);
	} else if (b <= 6e-102) {
		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 <= (-5.7d-21)) then
        tmp = ((-2.0d0) * b) / (3.0d0 * a)
    else if (b <= 6d-102) 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 <= -5.7e-21) {
		tmp = (-2.0 * b) / (3.0 * a);
	} else if (b <= 6e-102) {
		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 <= -5.7e-21:
		tmp = (-2.0 * b) / (3.0 * a)
	elif b <= 6e-102:
		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 <= -5.7e-21)
		tmp = Float64(Float64(-2.0 * b) / Float64(3.0 * a));
	elseif (b <= 6e-102)
		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 <= -5.7e-21)
		tmp = (-2.0 * b) / (3.0 * a);
	elseif (b <= 6e-102)
		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, -5.7e-21], N[(N[(-2.0 * b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 6e-102], 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 < -5.6999999999999996e-21

   1. Initial program 51.9%

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

   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 35.1

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

   4. Applied rewrites 35.1%

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

   if -5.6999999999999996e-21 < b < 6e-102

   1. Initial program 51.9%

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

   2. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. rem-square-sqrt N/A

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

      4. lower-sqrt.f64 N/A

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

      5. rem-square-sqrt N/A

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

      6. *-commutative 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 17.3

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

   4. Applied rewrites 17.3%

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

   if 6e-102 < b

   1. Initial program 51.9%

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

   2. Taylor expanded in c around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. pow2 N/A

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

      5. rem-sqrt-square-rev N/A

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

      6. lower-fabs.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower--.f64 N/A

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

      11. pow2 N/A

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

      12. rem-sqrt-square-rev N/A

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

      13. lower-fabs.f64 68.2

          \[\leadsto \mathsf{fma}\left(\frac{c}{\left|b\right|}, -0.5, \frac{\left|b\right| - b}{a} \cdot 0.3333333333333333\right) \]

   4. Applied rewrites 68.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{c}{\left|b\right|}, -0.5, \frac{\left|b\right| - b}{a} \cdot 0.3333333333333333\right)} \]

   5. Taylor expanded in b around inf

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

      1. lift-neg.f64 N/A

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

      2. +-commutative N/A

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

      3. pow2 N/A

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

      4. associate-*l* N/A

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

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

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

      6. metadata-eval N/A

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

      7. +-commutative N/A

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

      8. associate-*l* N/A

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

      9. pow2 N/A

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

      10. lift-neg.f64 N/A

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

      11. associate-/l/ N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 35.6

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

   7. Applied rewrites 35.6%

      \[\leadsto \color{blue}{\frac{c}{b} \cdot -0.5} \]
   8. 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 35.6

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

   9. Applied rewrites 35.6%

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

Alternative 12: 70.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -4.4 \cdot 10^{-163}:\\ \;\;\;\;\frac{-2 \cdot b}{3 \cdot a}\\ \mathbf{elif}\;b \leq 2.4 \cdot 10^{-116}:\\ \;\;\;\;-0.3333333333333333 \cdot \sqrt{\frac{c}{a} \cdot -3}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -4.4e-163)
   (/ (* -2.0 b) (* 3.0 a))
   (if (<= b 2.4e-116)
     (* -0.3333333333333333 (sqrt (* (/ c a) -3.0)))
     (/ (* -0.5 c) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -4.4e-163) {
		tmp = (-2.0 * b) / (3.0 * a);
	} else if (b <= 2.4e-116) {
		tmp = -0.3333333333333333 * sqrt(((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 <= (-4.4d-163)) then
        tmp = ((-2.0d0) * b) / (3.0d0 * a)
    else if (b <= 2.4d-116) then
        tmp = (-0.3333333333333333d0) * sqrt(((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 <= -4.4e-163) {
		tmp = (-2.0 * b) / (3.0 * a);
	} else if (b <= 2.4e-116) {
		tmp = -0.3333333333333333 * Math.sqrt(((c / a) * -3.0));
	} else {
		tmp = (-0.5 * c) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -4.4e-163:
		tmp = (-2.0 * b) / (3.0 * a)
	elif b <= 2.4e-116:
		tmp = -0.3333333333333333 * math.sqrt(((c / a) * -3.0))
	else:
		tmp = (-0.5 * c) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -4.4e-163)
		tmp = Float64(Float64(-2.0 * b) / Float64(3.0 * a));
	elseif (b <= 2.4e-116)
		tmp = Float64(-0.3333333333333333 * sqrt(Float64(Float64(c / 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 <= -4.4e-163)
		tmp = (-2.0 * b) / (3.0 * a);
	elseif (b <= 2.4e-116)
		tmp = -0.3333333333333333 * sqrt(((c / a) * -3.0));
	else
		tmp = (-0.5 * c) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -4.4e-163], N[(N[(-2.0 * b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 2.4e-116], N[(-0.3333333333333333 * N[Sqrt[N[(N[(c / a), $MachinePrecision] * -3.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(-0.5 * c), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -4.40000000000000022e-163

   1. Initial program 51.9%

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

   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 35.1

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

   4. Applied rewrites 35.1%

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

   if -4.40000000000000022e-163 < b < 2.39999999999999993e-116

   1. Initial program 51.9%

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

   2. Taylor expanded in a around -inf

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

      1. lower-*.f64 N/A

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

      2. rem-square-sqrt N/A

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

      3. lower-sqrt.f64 N/A

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

      4. rem-square-sqrt N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 17.5

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

   4. Applied rewrites 17.5%

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

   if 2.39999999999999993e-116 < b

   1. Initial program 51.9%

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

   2. Taylor expanded in c around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. pow2 N/A

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

      5. rem-sqrt-square-rev N/A

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

      6. lower-fabs.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower--.f64 N/A

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

      11. pow2 N/A

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

      12. rem-sqrt-square-rev N/A

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

      13. lower-fabs.f64 68.2

          \[\leadsto \mathsf{fma}\left(\frac{c}{\left|b\right|}, -0.5, \frac{\left|b\right| - b}{a} \cdot 0.3333333333333333\right) \]

   4. Applied rewrites 68.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{c}{\left|b\right|}, -0.5, \frac{\left|b\right| - b}{a} \cdot 0.3333333333333333\right)} \]

   5. Taylor expanded in b around inf

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

      1. lift-neg.f64 N/A

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

      2. +-commutative N/A

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

      3. pow2 N/A

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

      4. associate-*l* N/A

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

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

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

      6. metadata-eval N/A

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

      7. +-commutative N/A

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

      8. associate-*l* N/A

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

      9. pow2 N/A

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

      10. lift-neg.f64 N/A

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

      11. associate-/l/ N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 35.6

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

   7. Applied rewrites 35.6%

      \[\leadsto \color{blue}{\frac{c}{b} \cdot -0.5} \]
   8. 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 35.6

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

   9. Applied rewrites 35.6%

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

Alternative 13: 68.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 8 \cdot 10^{-253}:\\ \;\;\;\;\frac{\left|b\right| - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 8e-253) (/ (- (fabs b) b) (* 3.0 a)) (/ (* -0.5 c) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 8e-253) {
		tmp = (fabs(b) - 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 <= 8d-253) then
        tmp = (abs(b) - 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 <= 8e-253) {
		tmp = (Math.abs(b) - b) / (3.0 * a);
	} else {
		tmp = (-0.5 * c) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 8e-253:
		tmp = (math.fabs(b) - b) / (3.0 * a)
	else:
		tmp = (-0.5 * c) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 8e-253)
		tmp = Float64(Float64(abs(b) - 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 <= 8e-253)
		tmp = (abs(b) - b) / (3.0 * a);
	else
		tmp = (-0.5 * c) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 8e-253], N[(N[(N[Abs[b], $MachinePrecision] - 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 < 8.0000000000000005e-253

   1. Initial program 51.9%

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

   2. Taylor expanded in a around 0

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

      1. lower--.f64 N/A

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

      2. pow2 N/A

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

      3. rem-sqrt-square-rev N/A

         \[\leadsto \frac{\left|b\right| - b}{3 \cdot a} \]

      4. lower-fabs.f64 44.1

         \[\leadsto \frac{\left|b\right| - b}{3 \cdot a} \]

   4. Applied rewrites 44.1%

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

   if 8.0000000000000005e-253 < b

   1. Initial program 51.9%

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

   2. Taylor expanded in c around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. pow2 N/A

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

      5. rem-sqrt-square-rev N/A

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

      6. lower-fabs.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower--.f64 N/A

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

      11. pow2 N/A

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

      12. rem-sqrt-square-rev N/A

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

      13. lower-fabs.f64 68.2

          \[\leadsto \mathsf{fma}\left(\frac{c}{\left|b\right|}, -0.5, \frac{\left|b\right| - b}{a} \cdot 0.3333333333333333\right) \]

   4. Applied rewrites 68.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{c}{\left|b\right|}, -0.5, \frac{\left|b\right| - b}{a} \cdot 0.3333333333333333\right)} \]

   5. Taylor expanded in b around inf

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

      1. lift-neg.f64 N/A

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

      2. +-commutative N/A

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

      3. pow2 N/A

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

      4. associate-*l* N/A

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

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

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

      6. metadata-eval N/A

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

      7. +-commutative N/A

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

      8. associate-*l* N/A

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

      9. pow2 N/A

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

      10. lift-neg.f64 N/A

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

      11. associate-/l/ N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 35.6

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

   7. Applied rewrites 35.6%

      \[\leadsto \color{blue}{\frac{c}{b} \cdot -0.5} \]
   8. 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 35.6

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

   9. Applied rewrites 35.6%

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

Alternative 14: 68.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 8 \cdot 10^{-253}:\\ \;\;\;\;\frac{-2 \cdot b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 8e-253) (/ (* -2.0 b) (* 3.0 a)) (/ (* -0.5 c) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 8e-253) {
		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 <= 8d-253) 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 <= 8e-253) {
		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 <= 8e-253:
		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 <= 8e-253)
		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 <= 8e-253)
		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, 8e-253], 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 < 8.0000000000000005e-253

   1. Initial program 51.9%

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

   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 35.1

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

   4. Applied rewrites 35.1%

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

   if 8.0000000000000005e-253 < b

   1. Initial program 51.9%

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

   2. Taylor expanded in c around 0

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

      1. *-commutative N/A

         \[\leadsto \frac{c}{\sqrt{{b}^{2}}} \cdot \frac{-1}{2} + \color{blue}{\frac{1}{3}} \cdot \frac{\sqrt{{b}^{2}} - b}{a} \]

      2. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{c}{\sqrt{{b}^{2}}}, \color{blue}{\frac{-1}{2}}, \frac{1}{3} \cdot \frac{\sqrt{{b}^{2}} - b}{a}\right) \]

      3. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{c}{\sqrt{{b}^{2}}}, \frac{-1}{2}, \frac{1}{3} \cdot \frac{\sqrt{{b}^{2}} - b}{a}\right) \]

      4. pow2 N/A

         \[\leadsto \mathsf{fma}\left(\frac{c}{\sqrt{b \cdot b}}, \frac{-1}{2}, \frac{1}{3} \cdot \frac{\sqrt{{b}^{2}} - b}{a}\right) \]

      5. rem-sqrt-square-rev N/A

         \[\leadsto \mathsf{fma}\left(\frac{c}{\left|b\right|}, \frac{-1}{2}, \frac{1}{3} \cdot \frac{\sqrt{{b}^{2}} - b}{a}\right) \]

      6. lower-fabs.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{c}{\left|b\right|}, \frac{-1}{2}, \frac{1}{3} \cdot \frac{\sqrt{{b}^{2}} - b}{a}\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{c}{\left|b\right|}, \frac{-1}{2}, \frac{\sqrt{{b}^{2}} - b}{a} \cdot \frac{1}{3}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{c}{\left|b\right|}, \frac{-1}{2}, \frac{\sqrt{{b}^{2}} - b}{a} \cdot \frac{1}{3}\right) \]

      9. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{c}{\left|b\right|}, \frac{-1}{2}, \frac{\sqrt{{b}^{2}} - b}{a} \cdot \frac{1}{3}\right) \]

      10. lower--.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{c}{\left|b\right|}, \frac{-1}{2}, \frac{\sqrt{{b}^{2}} - b}{a} \cdot \frac{1}{3}\right) \]

      11. pow2 N/A

          \[\leadsto \mathsf{fma}\left(\frac{c}{\left|b\right|}, \frac{-1}{2}, \frac{\sqrt{b \cdot b} - b}{a} \cdot \frac{1}{3}\right) \]

      12. rem-sqrt-square-rev N/A

          \[\leadsto \mathsf{fma}\left(\frac{c}{\left|b\right|}, \frac{-1}{2}, \frac{\left|b\right| - b}{a} \cdot \frac{1}{3}\right) \]

      13. lower-fabs.f64 68.2

          \[\leadsto \mathsf{fma}\left(\frac{c}{\left|b\right|}, -0.5, \frac{\left|b\right| - b}{a} \cdot 0.3333333333333333\right) \]

   4. Applied rewrites 68.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{c}{\left|b\right|}, -0.5, \frac{\left|b\right| - b}{a} \cdot 0.3333333333333333\right)} \]

   5. Taylor expanded in b around inf

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b}} \]
   6. Step-by-step derivation

      1. lift-neg.f64 N/A

         \[\leadsto \frac{-1}{2} \cdot \frac{c}{b} \]

      2. +-commutative N/A

         \[\leadsto \frac{-1}{2} \cdot \frac{c}{b} \]

      3. pow2 N/A

         \[\leadsto \frac{-1}{2} \cdot \frac{c}{b} \]

      4. associate-*l* N/A

         \[\leadsto \frac{-1}{2} \cdot \frac{c}{b} \]

      5. fp-cancel-sub-sign-inv N/A

         \[\leadsto \frac{-1}{2} \cdot \frac{c}{b} \]

      6. metadata-eval N/A

         \[\leadsto \frac{-1}{2} \cdot \frac{c}{b} \]

      7. +-commutative N/A

         \[\leadsto \frac{-1}{2} \cdot \frac{c}{b} \]

      8. associate-*l* N/A

         \[\leadsto \frac{-1}{2} \cdot \frac{c}{b} \]

      9. pow2 N/A

         \[\leadsto \frac{-1}{2} \cdot \frac{c}{b} \]

      10. lift-neg.f64 N/A

          \[\leadsto \frac{-1}{2} \cdot \frac{c}{b} \]

      11. associate-/l/ N/A

          \[\leadsto \color{blue}{\frac{-1}{2}} \cdot \frac{c}{b} \]

      12. *-commutative N/A

          \[\leadsto \frac{c}{b} \cdot \color{blue}{\frac{-1}{2}} \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{c}{b} \cdot \color{blue}{\frac{-1}{2}} \]

      14. lower-/.f64 35.6

          \[\leadsto \frac{c}{b} \cdot -0.5 \]

   7. Applied rewrites 35.6%

      \[\leadsto \color{blue}{\frac{c}{b} \cdot -0.5} \]
   8. 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 35.6

         \[\leadsto \frac{-0.5 \cdot c}{b} \]

   9. Applied rewrites 35.6%

      \[\leadsto \frac{-0.5 \cdot c}{\color{blue}{b}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 68.0% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 8 \cdot 10^{-253}:\\ \;\;\;\;\frac{-0.6666666666666666 \cdot b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 8e-253) (/ (* -0.6666666666666666 b) a) (/ (* -0.5 c) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 8e-253) {
		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 <= 8d-253) 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 <= 8e-253) {
		tmp = (-0.6666666666666666 * b) / a;
	} else {
		tmp = (-0.5 * c) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 8e-253:
		tmp = (-0.6666666666666666 * b) / a
	else:
		tmp = (-0.5 * c) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 8e-253)
		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 <= 8e-253)
		tmp = (-0.6666666666666666 * b) / a;
	else
		tmp = (-0.5 * c) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 8e-253], 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 < 8.0000000000000005e-253

   1. Initial program 51.9%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{\color{blue}{3 \cdot a}} \]

      2. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}} \]

      3. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}{3 \cdot a} \]

      4. lift-neg.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left(b\right)\right)} + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

      5. lift-sqrt.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \color{blue}{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}{3 \cdot a} \]

      6. lift--.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\color{blue}{b \cdot b - \left(3 \cdot a\right) \cdot c}}}{3 \cdot a} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\color{blue}{b \cdot b} - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{b \cdot b - \color{blue}{\left(3 \cdot a\right) \cdot c}}}{3 \cdot a} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{b \cdot b - \color{blue}{\left(3 \cdot a\right)} \cdot c}}{3 \cdot a} \]

      10. associate-/r* N/A

          \[\leadsto \color{blue}{\frac{\frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3}}{a}} \]

      11. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3}}{a}} \]

   3. Applied rewrites 51.9%

      \[\leadsto \color{blue}{\frac{\frac{\sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)} + \left(-b\right)}{3}}{a}} \]

   4. Taylor expanded in b around -inf

      \[\leadsto \frac{\color{blue}{\frac{-2}{3} \cdot b}}{a} \]
   5. Step-by-step derivation

      1. lower-*.f64 35.0

         \[\leadsto \frac{-0.6666666666666666 \cdot \color{blue}{b}}{a} \]

   6. Applied rewrites 35.0%

      \[\leadsto \frac{\color{blue}{-0.6666666666666666 \cdot b}}{a} \]

   if 8.0000000000000005e-253 < b

   1. Initial program 51.9%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

   2. Taylor expanded in c around 0

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{\sqrt{{b}^{2}}} + \frac{1}{3} \cdot \frac{\sqrt{{b}^{2}} - b}{a}} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{c}{\sqrt{{b}^{2}}} \cdot \frac{-1}{2} + \color{blue}{\frac{1}{3}} \cdot \frac{\sqrt{{b}^{2}} - b}{a} \]

      2. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{c}{\sqrt{{b}^{2}}}, \color{blue}{\frac{-1}{2}}, \frac{1}{3} \cdot \frac{\sqrt{{b}^{2}} - b}{a}\right) \]

      3. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{c}{\sqrt{{b}^{2}}}, \frac{-1}{2}, \frac{1}{3} \cdot \frac{\sqrt{{b}^{2}} - b}{a}\right) \]

      4. pow2 N/A

         \[\leadsto \mathsf{fma}\left(\frac{c}{\sqrt{b \cdot b}}, \frac{-1}{2}, \frac{1}{3} \cdot \frac{\sqrt{{b}^{2}} - b}{a}\right) \]

      5. rem-sqrt-square-rev N/A

         \[\leadsto \mathsf{fma}\left(\frac{c}{\left|b\right|}, \frac{-1}{2}, \frac{1}{3} \cdot \frac{\sqrt{{b}^{2}} - b}{a}\right) \]

      6. lower-fabs.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{c}{\left|b\right|}, \frac{-1}{2}, \frac{1}{3} \cdot \frac{\sqrt{{b}^{2}} - b}{a}\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{c}{\left|b\right|}, \frac{-1}{2}, \frac{\sqrt{{b}^{2}} - b}{a} \cdot \frac{1}{3}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{c}{\left|b\right|}, \frac{-1}{2}, \frac{\sqrt{{b}^{2}} - b}{a} \cdot \frac{1}{3}\right) \]

      9. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{c}{\left|b\right|}, \frac{-1}{2}, \frac{\sqrt{{b}^{2}} - b}{a} \cdot \frac{1}{3}\right) \]

      10. lower--.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{c}{\left|b\right|}, \frac{-1}{2}, \frac{\sqrt{{b}^{2}} - b}{a} \cdot \frac{1}{3}\right) \]

      11. pow2 N/A

          \[\leadsto \mathsf{fma}\left(\frac{c}{\left|b\right|}, \frac{-1}{2}, \frac{\sqrt{b \cdot b} - b}{a} \cdot \frac{1}{3}\right) \]

      12. rem-sqrt-square-rev N/A

          \[\leadsto \mathsf{fma}\left(\frac{c}{\left|b\right|}, \frac{-1}{2}, \frac{\left|b\right| - b}{a} \cdot \frac{1}{3}\right) \]

      13. lower-fabs.f64 68.2

          \[\leadsto \mathsf{fma}\left(\frac{c}{\left|b\right|}, -0.5, \frac{\left|b\right| - b}{a} \cdot 0.3333333333333333\right) \]

   4. Applied rewrites 68.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{c}{\left|b\right|}, -0.5, \frac{\left|b\right| - b}{a} \cdot 0.3333333333333333\right)} \]

   5. Taylor expanded in b around inf

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b}} \]
   6. Step-by-step derivation

      1. lift-neg.f64 N/A

         \[\leadsto \frac{-1}{2} \cdot \frac{c}{b} \]

      2. +-commutative N/A

         \[\leadsto \frac{-1}{2} \cdot \frac{c}{b} \]

      3. pow2 N/A

         \[\leadsto \frac{-1}{2} \cdot \frac{c}{b} \]

      4. associate-*l* N/A

         \[\leadsto \frac{-1}{2} \cdot \frac{c}{b} \]

      5. fp-cancel-sub-sign-inv N/A

         \[\leadsto \frac{-1}{2} \cdot \frac{c}{b} \]

      6. metadata-eval N/A

         \[\leadsto \frac{-1}{2} \cdot \frac{c}{b} \]

      7. +-commutative N/A

         \[\leadsto \frac{-1}{2} \cdot \frac{c}{b} \]

      8. associate-*l* N/A

         \[\leadsto \frac{-1}{2} \cdot \frac{c}{b} \]

      9. pow2 N/A

         \[\leadsto \frac{-1}{2} \cdot \frac{c}{b} \]

      10. lift-neg.f64 N/A

          \[\leadsto \frac{-1}{2} \cdot \frac{c}{b} \]

      11. associate-/l/ N/A

          \[\leadsto \color{blue}{\frac{-1}{2}} \cdot \frac{c}{b} \]

      12. *-commutative N/A

          \[\leadsto \frac{c}{b} \cdot \color{blue}{\frac{-1}{2}} \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{c}{b} \cdot \color{blue}{\frac{-1}{2}} \]

      14. lower-/.f64 35.6

          \[\leadsto \frac{c}{b} \cdot -0.5 \]

   7. Applied rewrites 35.6%

      \[\leadsto \color{blue}{\frac{c}{b} \cdot -0.5} \]
   8. 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 35.6

         \[\leadsto \frac{-0.5 \cdot c}{b} \]

   9. Applied rewrites 35.6%

      \[\leadsto \frac{-0.5 \cdot c}{\color{blue}{b}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 16: 68.0% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 8 \cdot 10^{-253}:\\ \;\;\;\;-0.6666666666666666 \cdot \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 8e-253) (* -0.6666666666666666 (/ b a)) (/ (* -0.5 c) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 8e-253) {
		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 <= 8d-253) 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 <= 8e-253) {
		tmp = -0.6666666666666666 * (b / a);
	} else {
		tmp = (-0.5 * c) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 8e-253:
		tmp = -0.6666666666666666 * (b / a)
	else:
		tmp = (-0.5 * c) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 8e-253)
		tmp = Float64(-0.6666666666666666 * Float64(b / a));
	else
		tmp = Float64(Float64(-0.5 * c) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 8e-253)
		tmp = -0.6666666666666666 * (b / a);
	else
		tmp = (-0.5 * c) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 8e-253], N[(-0.6666666666666666 * N[(b / a), $MachinePrecision]), $MachinePrecision], N[(N[(-0.5 * c), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 8.0000000000000005e-253

   1. Initial program 51.9%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

   2. Taylor expanded in b around -inf

      \[\leadsto \color{blue}{\frac{-2}{3} \cdot \frac{b}{a}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{-2}{3} \cdot \color{blue}{\frac{b}{a}} \]

      2. lower-/.f64 35.0

         \[\leadsto -0.6666666666666666 \cdot \frac{b}{\color{blue}{a}} \]

   4. Applied rewrites 35.0%

      \[\leadsto \color{blue}{-0.6666666666666666 \cdot \frac{b}{a}} \]

   if 8.0000000000000005e-253 < b

   1. Initial program 51.9%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

   2. Taylor expanded in c around 0

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{\sqrt{{b}^{2}}} + \frac{1}{3} \cdot \frac{\sqrt{{b}^{2}} - b}{a}} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{c}{\sqrt{{b}^{2}}} \cdot \frac{-1}{2} + \color{blue}{\frac{1}{3}} \cdot \frac{\sqrt{{b}^{2}} - b}{a} \]

      2. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{c}{\sqrt{{b}^{2}}}, \color{blue}{\frac{-1}{2}}, \frac{1}{3} \cdot \frac{\sqrt{{b}^{2}} - b}{a}\right) \]

      3. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{c}{\sqrt{{b}^{2}}}, \frac{-1}{2}, \frac{1}{3} \cdot \frac{\sqrt{{b}^{2}} - b}{a}\right) \]

      4. pow2 N/A

         \[\leadsto \mathsf{fma}\left(\frac{c}{\sqrt{b \cdot b}}, \frac{-1}{2}, \frac{1}{3} \cdot \frac{\sqrt{{b}^{2}} - b}{a}\right) \]

      5. rem-sqrt-square-rev N/A

         \[\leadsto \mathsf{fma}\left(\frac{c}{\left|b\right|}, \frac{-1}{2}, \frac{1}{3} \cdot \frac{\sqrt{{b}^{2}} - b}{a}\right) \]

      6. lower-fabs.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{c}{\left|b\right|}, \frac{-1}{2}, \frac{1}{3} \cdot \frac{\sqrt{{b}^{2}} - b}{a}\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{c}{\left|b\right|}, \frac{-1}{2}, \frac{\sqrt{{b}^{2}} - b}{a} \cdot \frac{1}{3}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{c}{\left|b\right|}, \frac{-1}{2}, \frac{\sqrt{{b}^{2}} - b}{a} \cdot \frac{1}{3}\right) \]

      9. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{c}{\left|b\right|}, \frac{-1}{2}, \frac{\sqrt{{b}^{2}} - b}{a} \cdot \frac{1}{3}\right) \]

      10. lower--.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{c}{\left|b\right|}, \frac{-1}{2}, \frac{\sqrt{{b}^{2}} - b}{a} \cdot \frac{1}{3}\right) \]

      11. pow2 N/A

          \[\leadsto \mathsf{fma}\left(\frac{c}{\left|b\right|}, \frac{-1}{2}, \frac{\sqrt{b \cdot b} - b}{a} \cdot \frac{1}{3}\right) \]

      12. rem-sqrt-square-rev N/A

          \[\leadsto \mathsf{fma}\left(\frac{c}{\left|b\right|}, \frac{-1}{2}, \frac{\left|b\right| - b}{a} \cdot \frac{1}{3}\right) \]

      13. lower-fabs.f64 68.2

          \[\leadsto \mathsf{fma}\left(\frac{c}{\left|b\right|}, -0.5, \frac{\left|b\right| - b}{a} \cdot 0.3333333333333333\right) \]

   4. Applied rewrites 68.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{c}{\left|b\right|}, -0.5, \frac{\left|b\right| - b}{a} \cdot 0.3333333333333333\right)} \]

   5. Taylor expanded in b around inf

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b}} \]
   6. Step-by-step derivation

      1. lift-neg.f64 N/A

         \[\leadsto \frac{-1}{2} \cdot \frac{c}{b} \]

      2. +-commutative N/A

         \[\leadsto \frac{-1}{2} \cdot \frac{c}{b} \]

      3. pow2 N/A

         \[\leadsto \frac{-1}{2} \cdot \frac{c}{b} \]

      4. associate-*l* N/A

         \[\leadsto \frac{-1}{2} \cdot \frac{c}{b} \]

      5. fp-cancel-sub-sign-inv N/A

         \[\leadsto \frac{-1}{2} \cdot \frac{c}{b} \]

      6. metadata-eval N/A

         \[\leadsto \frac{-1}{2} \cdot \frac{c}{b} \]

      7. +-commutative N/A

         \[\leadsto \frac{-1}{2} \cdot \frac{c}{b} \]

      8. associate-*l* N/A

         \[\leadsto \frac{-1}{2} \cdot \frac{c}{b} \]

      9. pow2 N/A

         \[\leadsto \frac{-1}{2} \cdot \frac{c}{b} \]

      10. lift-neg.f64 N/A

          \[\leadsto \frac{-1}{2} \cdot \frac{c}{b} \]

      11. associate-/l/ N/A

          \[\leadsto \color{blue}{\frac{-1}{2}} \cdot \frac{c}{b} \]

      12. *-commutative N/A

          \[\leadsto \frac{c}{b} \cdot \color{blue}{\frac{-1}{2}} \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{c}{b} \cdot \color{blue}{\frac{-1}{2}} \]

      14. lower-/.f64 35.6

          \[\leadsto \frac{c}{b} \cdot -0.5 \]

   7. Applied rewrites 35.6%

      \[\leadsto \color{blue}{\frac{c}{b} \cdot -0.5} \]
   8. 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 35.6

         \[\leadsto \frac{-0.5 \cdot c}{b} \]

   9. Applied rewrites 35.6%

      \[\leadsto \frac{-0.5 \cdot c}{\color{blue}{b}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 17: 68.0% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 8 \cdot 10^{-253}:\\ \;\;\;\;-0.6666666666666666 \cdot \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 8e-253) (* -0.6666666666666666 (/ b a)) (* (/ c b) -0.5)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 8e-253) {
		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 <= 8d-253) 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 <= 8e-253) {
		tmp = -0.6666666666666666 * (b / a);
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 8e-253:
		tmp = -0.6666666666666666 * (b / a)
	else:
		tmp = (c / b) * -0.5
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 8e-253)
		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 <= 8e-253)
		tmp = -0.6666666666666666 * (b / a);
	else
		tmp = (c / b) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 8e-253], 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 < 8.0000000000000005e-253

   1. Initial program 51.9%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

   2. Taylor expanded in b around -inf

      \[\leadsto \color{blue}{\frac{-2}{3} \cdot \frac{b}{a}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{-2}{3} \cdot \color{blue}{\frac{b}{a}} \]

      2. lower-/.f64 35.0

         \[\leadsto -0.6666666666666666 \cdot \frac{b}{\color{blue}{a}} \]

   4. Applied rewrites 35.0%

      \[\leadsto \color{blue}{-0.6666666666666666 \cdot \frac{b}{a}} \]

   if 8.0000000000000005e-253 < b

   1. Initial program 51.9%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

   2. Taylor expanded in b around inf

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b}} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{c}{b} \cdot \color{blue}{\frac{-1}{2}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{c}{b} \cdot \color{blue}{\frac{-1}{2}} \]

      3. lower-/.f64 35.6

         \[\leadsto \frac{c}{b} \cdot -0.5 \]

   4. Applied rewrites 35.6%

      \[\leadsto \color{blue}{\frac{c}{b} \cdot -0.5} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 18: 35.0% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ -0.6666666666666666 \cdot \frac{b}{a} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (* -0.6666666666666666 (/ b a)))

C

double code(double a, double b, double c) {
	return -0.6666666666666666 * (b / a);
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(a, b, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = (-0.6666666666666666d0) * (b / a)
end function

Java

public static double code(double a, double b, double c) {
	return -0.6666666666666666 * (b / a);
}

Python

def code(a, b, c):
	return -0.6666666666666666 * (b / a)

Julia

function code(a, b, c)
	return Float64(-0.6666666666666666 * Float64(b / a))
end

MATLAB

function tmp = code(a, b, c)
	tmp = -0.6666666666666666 * (b / a);
end

Wolfram

code[a_, b_, c_] := N[(-0.6666666666666666 * N[(b / a), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 51.9%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

2. Taylor expanded in b around -inf

   \[\leadsto \color{blue}{\frac{-2}{3} \cdot \frac{b}{a}} \]
3. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \frac{-2}{3} \cdot \color{blue}{\frac{b}{a}} \]

   2. lower-/.f64 35.0

      \[\leadsto -0.6666666666666666 \cdot \frac{b}{\color{blue}{a}} \]

4. Applied rewrites 35.0%

   \[\leadsto \color{blue}{-0.6666666666666666 \cdot \frac{b}{a}} \]
5. Add Preprocessing

Alternative 19: 15.4% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ -0.3333333333333333 \cdot \frac{b}{a} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (* -0.3333333333333333 (/ b a)))

C

double code(double a, double b, double c) {
	return -0.3333333333333333 * (b / a);
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(a, b, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = (-0.3333333333333333d0) * (b / a)
end function

Java

public static double code(double a, double b, double c) {
	return -0.3333333333333333 * (b / a);
}

Python

def code(a, b, c):
	return -0.3333333333333333 * (b / a)

Julia

function code(a, b, c)
	return Float64(-0.3333333333333333 * Float64(b / a))
end

MATLAB

function tmp = code(a, b, c)
	tmp = -0.3333333333333333 * (b / a);
end

Wolfram

code[a_, b_, c_] := N[(-0.3333333333333333 * N[(b / a), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 51.9%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

2. Taylor expanded in c around 0

   \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{\sqrt{{b}^{2}}} + \frac{1}{3} \cdot \frac{\sqrt{{b}^{2}} - b}{a}} \]
3. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \frac{c}{\sqrt{{b}^{2}}} \cdot \frac{-1}{2} + \color{blue}{\frac{1}{3}} \cdot \frac{\sqrt{{b}^{2}} - b}{a} \]

   2. lower-fma.f64 N/A

      \[\leadsto \mathsf{fma}\left(\frac{c}{\sqrt{{b}^{2}}}, \color{blue}{\frac{-1}{2}}, \frac{1}{3} \cdot \frac{\sqrt{{b}^{2}} - b}{a}\right) \]

   3. lower-/.f64 N/A

      \[\leadsto \mathsf{fma}\left(\frac{c}{\sqrt{{b}^{2}}}, \frac{-1}{2}, \frac{1}{3} \cdot \frac{\sqrt{{b}^{2}} - b}{a}\right) \]

   4. pow2 N/A

      \[\leadsto \mathsf{fma}\left(\frac{c}{\sqrt{b \cdot b}}, \frac{-1}{2}, \frac{1}{3} \cdot \frac{\sqrt{{b}^{2}} - b}{a}\right) \]

   5. rem-sqrt-square-rev N/A

      \[\leadsto \mathsf{fma}\left(\frac{c}{\left|b\right|}, \frac{-1}{2}, \frac{1}{3} \cdot \frac{\sqrt{{b}^{2}} - b}{a}\right) \]

   6. lower-fabs.f64 N/A

      \[\leadsto \mathsf{fma}\left(\frac{c}{\left|b\right|}, \frac{-1}{2}, \frac{1}{3} \cdot \frac{\sqrt{{b}^{2}} - b}{a}\right) \]

   7. *-commutative N/A

      \[\leadsto \mathsf{fma}\left(\frac{c}{\left|b\right|}, \frac{-1}{2}, \frac{\sqrt{{b}^{2}} - b}{a} \cdot \frac{1}{3}\right) \]

   8. lower-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(\frac{c}{\left|b\right|}, \frac{-1}{2}, \frac{\sqrt{{b}^{2}} - b}{a} \cdot \frac{1}{3}\right) \]

   9. lower-/.f64 N/A

      \[\leadsto \mathsf{fma}\left(\frac{c}{\left|b\right|}, \frac{-1}{2}, \frac{\sqrt{{b}^{2}} - b}{a} \cdot \frac{1}{3}\right) \]

   10. lower--.f64 N/A

       \[\leadsto \mathsf{fma}\left(\frac{c}{\left|b\right|}, \frac{-1}{2}, \frac{\sqrt{{b}^{2}} - b}{a} \cdot \frac{1}{3}\right) \]

   11. pow2 N/A

       \[\leadsto \mathsf{fma}\left(\frac{c}{\left|b\right|}, \frac{-1}{2}, \frac{\sqrt{b \cdot b} - b}{a} \cdot \frac{1}{3}\right) \]

   12. rem-sqrt-square-rev N/A

       \[\leadsto \mathsf{fma}\left(\frac{c}{\left|b\right|}, \frac{-1}{2}, \frac{\left|b\right| - b}{a} \cdot \frac{1}{3}\right) \]

   13. lower-fabs.f64 68.2

       \[\leadsto \mathsf{fma}\left(\frac{c}{\left|b\right|}, -0.5, \frac{\left|b\right| - b}{a} \cdot 0.3333333333333333\right) \]

4. Applied rewrites 68.2%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{c}{\left|b\right|}, -0.5, \frac{\left|b\right| - b}{a} \cdot 0.3333333333333333\right)} \]

5. Taylor expanded in b around inf

   \[\leadsto \frac{-1}{3} \cdot \color{blue}{\frac{b}{a}} \]
6. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \frac{-1}{3} \cdot \frac{b}{\color{blue}{a}} \]

   2. lift-/.f64 15.4

      \[\leadsto -0.3333333333333333 \cdot \frac{b}{a} \]

7. Applied rewrites 15.4%

   \[\leadsto -0.3333333333333333 \cdot \color{blue}{\frac{b}{a}} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2025143 
(FPCore (a b c)
  :name "Cubic critical"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))