Cubic critical

Percentage Accurate: 51.7% → 90.2%

Time: 5.6s

Alternatives: 7

Speedup: N/A×

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: 51.7% 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: 90.2% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-0.5, \frac{c}{\left|b\right|}, 0.3333333333333333 \cdot \frac{\mathsf{fma}\left(-1, b, \left|b\right|\right)}{a}\right)\\ t_1 := \frac{-1 \cdot b + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq -2 \cdot 10^{-222}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+296}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0
         (fma
          -0.5
          (/ c (fabs b))
          (* 0.3333333333333333 (/ (fma -1.0 b (fabs b)) a))))
        (t_1 (/ (+ (* -1.0 b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a))))
   (if (<= t_1 (- INFINITY))
     t_0
     (if (<= t_1 -2e-222)
       t_1
       (if (<= t_1 0.0) (* (/ c b) -0.5) (if (<= t_1 5e+296) t_1 t_0))))))

C

double code(double a, double b, double c) {
	double t_0 = fma(-0.5, (c / fabs(b)), (0.3333333333333333 * (fma(-1.0, b, fabs(b)) / a)));
	double t_1 = ((-1.0 * b) + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = t_0;
	} else if (t_1 <= -2e-222) {
		tmp = t_1;
	} else if (t_1 <= 0.0) {
		tmp = (c / b) * -0.5;
	} else if (t_1 <= 5e+296) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(-0.5, Float64(c / abs(b)), Float64(0.3333333333333333 * Float64(fma(-1.0, b, abs(b)) / a)))
	t_1 = Float64(Float64(Float64(-1.0 * b) + sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c)))) / Float64(3.0 * a))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = t_0;
	elseif (t_1 <= -2e-222)
		tmp = t_1;
	elseif (t_1 <= 0.0)
		tmp = Float64(Float64(c / b) * -0.5);
	elseif (t_1 <= 5e+296)
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(-0.5 * N[(c / N[Abs[b], $MachinePrecision]), $MachinePrecision] + N[(0.3333333333333333 * N[(N[(-1.0 * b + N[Abs[b], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(-1.0 * b), $MachinePrecision] + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], t$95$0, If[LessEqual[t$95$1, -2e-222], t$95$1, If[LessEqual[t$95$1, 0.0], N[(N[(c / b), $MachinePrecision] * -0.5), $MachinePrecision], If[LessEqual[t$95$1, 5e+296], t$95$1, t$95$0]]]]]]

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)) < -inf.0 or 5.0000000000000001e296 < (/.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 17.7%

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

      1. lift-*.f64 N/A

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

      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-neg.f64 N/A

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

      4. lift-+.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

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

   4. Applied rewrites 17.9%

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(-1, b, {\left(\mathsf{fma}\left({b}^{1}, {b}^{1}, -3 \cdot \left(c \cdot a\right)\right)\right)}^{0.5}\right)}{3}}{a}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-fma.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/l/ N/A

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

      11. lower-/.f64 N/A

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

   6. Applied rewrites 17.9%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left({\left(\mathsf{fma}\left(\left|b\right|, \left|b\right|, \left(a \cdot -3\right) \cdot c\right)\right)}^{0.25}, {\left(\mathsf{fma}\left(\left|b\right|, \left|b\right|, \left(a \cdot -3\right) \cdot c\right)\right)}^{0.25}, b \cdot -1\right)}{3 \cdot a}} \]

   7. Taylor expanded in c around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lift-fabs.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. div-add N/A

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

      6. lower-+.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift-fabs.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 66.7

          \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{\left|b\right|}, 0.3333333333333333 \cdot \left(\frac{\left|b\right|}{a} + \frac{-1 \cdot b}{a}\right)\right) \]

   9. Applied rewrites 66.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5, \frac{c}{\left|b\right|}, 0.3333333333333333 \cdot \left(\frac{\left|b\right|}{a} + \frac{-1 \cdot b}{a}\right)\right)} \]

   10. Taylor expanded in a around 0

       \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{c}{\left|b\right|}, \frac{1}{3} \cdot \frac{\left|b\right| + -1 \cdot b}{a}\right) \]
   11. Step-by-step derivation

       1. lower-/.f64 N/A

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

       2. mul-1-neg N/A

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

       3. +-commutative N/A

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

       4. mul-1-neg N/A

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

       5. lower-fma.f64 N/A

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

       6. lift-fabs.f64 82.0

          \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{\left|b\right|}, 0.3333333333333333 \cdot \frac{\mathsf{fma}\left(-1, b, \left|b\right|\right)}{a}\right) \]

   12. Applied rewrites 82.0%

       \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{\left|b\right|}, 0.3333333333333333 \cdot \frac{\mathsf{fma}\left(-1, b, \left|b\right|\right)}{a}\right) \]

   if -inf.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)) < -2.0000000000000001e-222 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)) < 5.0000000000000001e296

   1. Initial program 90.0%

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

   if -2.0000000000000001e-222 < (/.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 13.8%

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

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b}} \]
   4. 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 100.0

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

   5. Applied rewrites 100.0%

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

4. Final simplification 89.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{-1 \cdot b + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \frac{c}{\left|b\right|}, 0.3333333333333333 \cdot \frac{\mathsf{fma}\left(-1, b, \left|b\right|\right)}{a}\right)\\ \mathbf{elif}\;\frac{-1 \cdot b + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \leq -2 \cdot 10^{-222}:\\ \;\;\;\;\frac{-1 \cdot b + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\\ \mathbf{elif}\;\frac{-1 \cdot b + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \leq 0:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \mathbf{elif}\;\frac{-1 \cdot b + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \leq 5 \cdot 10^{+296}:\\ \;\;\;\;\frac{-1 \cdot b + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \frac{c}{\left|b\right|}, 0.3333333333333333 \cdot \frac{\mathsf{fma}\left(-1, b, \left|b\right|\right)}{a}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 90.1% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-0.5, \frac{c}{\left|b\right|}, 0.3333333333333333 \cdot \frac{\mathsf{fma}\left(-1, b, \left|b\right|\right)}{a}\right)\\ t_1 := \frac{-1 \cdot b + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\\ t_2 := \mathsf{fma}\left(-0.3333333333333333, \frac{b}{a}, 0.3333333333333333 \cdot \left({a}^{-1} \cdot {\left(\mathsf{fma}\left(-3, a \cdot c, b \cdot b\right)\right)}^{0.5}\right)\right)\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq -2 \cdot 10^{-222}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+296}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0
         (fma
          -0.5
          (/ c (fabs b))
          (* 0.3333333333333333 (/ (fma -1.0 b (fabs b)) a))))
        (t_1 (/ (+ (* -1.0 b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))
        (t_2
         (fma
          -0.3333333333333333
          (/ b a)
          (*
           0.3333333333333333
           (* (pow a -1.0) (pow (fma -3.0 (* a c) (* b b)) 0.5))))))
   (if (<= t_1 (- INFINITY))
     t_0
     (if (<= t_1 -2e-222)
       t_2
       (if (<= t_1 0.0) (* (/ c b) -0.5) (if (<= t_1 5e+296) t_2 t_0))))))

C

double code(double a, double b, double c) {
	double t_0 = fma(-0.5, (c / fabs(b)), (0.3333333333333333 * (fma(-1.0, b, fabs(b)) / a)));
	double t_1 = ((-1.0 * b) + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a);
	double t_2 = fma(-0.3333333333333333, (b / a), (0.3333333333333333 * (pow(a, -1.0) * pow(fma(-3.0, (a * c), (b * b)), 0.5))));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = t_0;
	} else if (t_1 <= -2e-222) {
		tmp = t_2;
	} else if (t_1 <= 0.0) {
		tmp = (c / b) * -0.5;
	} else if (t_1 <= 5e+296) {
		tmp = t_2;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(-0.5, Float64(c / abs(b)), Float64(0.3333333333333333 * Float64(fma(-1.0, b, abs(b)) / a)))
	t_1 = Float64(Float64(Float64(-1.0 * b) + sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c)))) / Float64(3.0 * a))
	t_2 = fma(-0.3333333333333333, Float64(b / a), Float64(0.3333333333333333 * Float64((a ^ -1.0) * (fma(-3.0, Float64(a * c), Float64(b * b)) ^ 0.5))))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = t_0;
	elseif (t_1 <= -2e-222)
		tmp = t_2;
	elseif (t_1 <= 0.0)
		tmp = Float64(Float64(c / b) * -0.5);
	elseif (t_1 <= 5e+296)
		tmp = t_2;
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(-0.5 * N[(c / N[Abs[b], $MachinePrecision]), $MachinePrecision] + N[(0.3333333333333333 * N[(N[(-1.0 * b + N[Abs[b], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(-1.0 * b), $MachinePrecision] + 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[(-0.3333333333333333 * N[(b / a), $MachinePrecision] + N[(0.3333333333333333 * N[(N[Power[a, -1.0], $MachinePrecision] * N[Power[N[(-3.0 * N[(a * c), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], t$95$0, If[LessEqual[t$95$1, -2e-222], t$95$2, If[LessEqual[t$95$1, 0.0], N[(N[(c / b), $MachinePrecision] * -0.5), $MachinePrecision], If[LessEqual[t$95$1, 5e+296], t$95$2, t$95$0]]]]]]]

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)) < -inf.0 or 5.0000000000000001e296 < (/.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 17.7%

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

      1. lift-*.f64 N/A

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

      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-neg.f64 N/A

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

      4. lift-+.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

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

   4. Applied rewrites 17.9%

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(-1, b, {\left(\mathsf{fma}\left({b}^{1}, {b}^{1}, -3 \cdot \left(c \cdot a\right)\right)\right)}^{0.5}\right)}{3}}{a}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-fma.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/l/ N/A

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

      11. lower-/.f64 N/A

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

   6. Applied rewrites 17.9%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left({\left(\mathsf{fma}\left(\left|b\right|, \left|b\right|, \left(a \cdot -3\right) \cdot c\right)\right)}^{0.25}, {\left(\mathsf{fma}\left(\left|b\right|, \left|b\right|, \left(a \cdot -3\right) \cdot c\right)\right)}^{0.25}, b \cdot -1\right)}{3 \cdot a}} \]

   7. Taylor expanded in c around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lift-fabs.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. div-add N/A

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

      6. lower-+.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift-fabs.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 66.7

          \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{\left|b\right|}, 0.3333333333333333 \cdot \left(\frac{\left|b\right|}{a} + \frac{-1 \cdot b}{a}\right)\right) \]

   9. Applied rewrites 66.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5, \frac{c}{\left|b\right|}, 0.3333333333333333 \cdot \left(\frac{\left|b\right|}{a} + \frac{-1 \cdot b}{a}\right)\right)} \]

   10. Taylor expanded in a around 0

       \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{c}{\left|b\right|}, \frac{1}{3} \cdot \frac{\left|b\right| + -1 \cdot b}{a}\right) \]
   11. Step-by-step derivation

       1. lower-/.f64 N/A

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

       2. mul-1-neg N/A

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

       3. +-commutative N/A

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

       4. mul-1-neg N/A

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

       5. lower-fma.f64 N/A

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

       6. lift-fabs.f64 82.0

          \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{\left|b\right|}, 0.3333333333333333 \cdot \frac{\mathsf{fma}\left(-1, b, \left|b\right|\right)}{a}\right) \]

   12. Applied rewrites 82.0%

       \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{\left|b\right|}, 0.3333333333333333 \cdot \frac{\mathsf{fma}\left(-1, b, \left|b\right|\right)}{a}\right) \]

   if -inf.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)) < -2.0000000000000001e-222 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)) < 5.0000000000000001e296

   1. Initial program 90.0%

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

      1. lift-*.f64 N/A

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

      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-neg.f64 N/A

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

      4. lift-+.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

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

   4. Applied rewrites 90.0%

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(-1, b, {\left(\mathsf{fma}\left({b}^{1}, {b}^{1}, -3 \cdot \left(c \cdot a\right)\right)\right)}^{0.5}\right)}{3}}{a}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-fma.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/l/ N/A

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

      11. lower-/.f64 N/A

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

   6. Applied rewrites 89.7%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left({\left(\mathsf{fma}\left(\left|b\right|, \left|b\right|, \left(a \cdot -3\right) \cdot c\right)\right)}^{0.25}, {\left(\mathsf{fma}\left(\left|b\right|, \left|b\right|, \left(a \cdot -3\right) \cdot c\right)\right)}^{0.25}, b \cdot -1\right)}{3 \cdot a}} \]

   7. Taylor expanded in b around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. inv-pow N/A

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

      6. lift-pow.f64 N/A

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

      7. pow1/2 N/A

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

      8. lower-pow.f64 N/A

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

      9. pow2 N/A

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

      10. sqr-abs-rev N/A

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

      11. pow2 N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. pow2 N/A

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

      15. lift-*.f64 89.7

          \[\leadsto \mathsf{fma}\left(-0.3333333333333333, \frac{b}{a}, 0.3333333333333333 \cdot \left({a}^{-1} \cdot {\left(\mathsf{fma}\left(-3, a \cdot c, b \cdot b\right)\right)}^{0.5}\right)\right) \]

   9. Applied rewrites 89.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.3333333333333333, \frac{b}{a}, 0.3333333333333333 \cdot \left({a}^{-1} \cdot {\left(\mathsf{fma}\left(-3, a \cdot c, b \cdot b\right)\right)}^{0.5}\right)\right)} \]

   if -2.0000000000000001e-222 < (/.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 13.8%

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

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b}} \]
   4. 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 100.0

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

   5. Applied rewrites 100.0%

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

4. Final simplification 88.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{-1 \cdot b + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \frac{c}{\left|b\right|}, 0.3333333333333333 \cdot \frac{\mathsf{fma}\left(-1, b, \left|b\right|\right)}{a}\right)\\ \mathbf{elif}\;\frac{-1 \cdot b + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \leq -2 \cdot 10^{-222}:\\ \;\;\;\;\mathsf{fma}\left(-0.3333333333333333, \frac{b}{a}, 0.3333333333333333 \cdot \left({a}^{-1} \cdot {\left(\mathsf{fma}\left(-3, a \cdot c, b \cdot b\right)\right)}^{0.5}\right)\right)\\ \mathbf{elif}\;\frac{-1 \cdot b + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \leq 0:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \mathbf{elif}\;\frac{-1 \cdot b + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \leq 5 \cdot 10^{+296}:\\ \;\;\;\;\mathsf{fma}\left(-0.3333333333333333, \frac{b}{a}, 0.3333333333333333 \cdot \left({a}^{-1} \cdot {\left(\mathsf{fma}\left(-3, a \cdot c, b \cdot b\right)\right)}^{0.5}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \frac{c}{\left|b\right|}, 0.3333333333333333 \cdot \frac{\mathsf{fma}\left(-1, b, \left|b\right|\right)}{a}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 68.1% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(-0.5, \frac{c}{\left|b\right|}, 0.3333333333333333 \cdot \frac{\mathsf{fma}\left(-1, b, \left|b\right|\right)}{a}\right) \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (fma -0.5 (/ c (fabs b)) (* 0.3333333333333333 (/ (fma -1.0 b (fabs b)) a))))

C

double code(double a, double b, double c) {
	return fma(-0.5, (c / fabs(b)), (0.3333333333333333 * (fma(-1.0, b, fabs(b)) / a)));
}

Julia

function code(a, b, c)
	return fma(-0.5, Float64(c / abs(b)), Float64(0.3333333333333333 * Float64(fma(-1.0, b, abs(b)) / a)))
end

Wolfram

code[a_, b_, c_] := N[(-0.5 * N[(c / N[Abs[b], $MachinePrecision]), $MachinePrecision] + N[(0.3333333333333333 * N[(N[(-1.0 * b + N[Abs[b], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 46.9%

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

   1. lift-*.f64 N/A

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

   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-neg.f64 N/A

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

   4. lift-+.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

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

4. Applied rewrites 46.9%

   \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(-1, b, {\left(\mathsf{fma}\left({b}^{1}, {b}^{1}, -3 \cdot \left(c \cdot a\right)\right)\right)}^{0.5}\right)}{3}}{a}} \]
5. Step-by-step derivation

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-fma.f64 N/A

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

   4. lift-pow.f64 N/A

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

   5. lift-pow.f64 N/A

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

   6. lift-pow.f64 N/A

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

   7. lift-fma.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. associate-/l/ N/A

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

   11. lower-/.f64 N/A

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

6. Applied rewrites 44.7%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left({\left(\mathsf{fma}\left(\left|b\right|, \left|b\right|, \left(a \cdot -3\right) \cdot c\right)\right)}^{0.25}, {\left(\mathsf{fma}\left(\left|b\right|, \left|b\right|, \left(a \cdot -3\right) \cdot c\right)\right)}^{0.25}, b \cdot -1\right)}{3 \cdot a}} \]

7. Taylor expanded in c around 0

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

   1. lower-fma.f64 N/A

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

   2. lower-/.f64 N/A

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

   3. lift-fabs.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. div-add N/A

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

   6. lower-+.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. lift-fabs.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lower-*.f64 60.1

       \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{\left|b\right|}, 0.3333333333333333 \cdot \left(\frac{\left|b\right|}{a} + \frac{-1 \cdot b}{a}\right)\right) \]

9. Applied rewrites 60.1%

   \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5, \frac{c}{\left|b\right|}, 0.3333333333333333 \cdot \left(\frac{\left|b\right|}{a} + \frac{-1 \cdot b}{a}\right)\right)} \]

10. Taylor expanded in a around 0

    \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{c}{\left|b\right|}, \frac{1}{3} \cdot \frac{\left|b\right| + -1 \cdot b}{a}\right) \]
11. Step-by-step derivation

    1. lower-/.f64 N/A

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

    2. mul-1-neg N/A

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

    3. +-commutative N/A

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

    4. mul-1-neg N/A

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

    5. lower-fma.f64 N/A

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

    6. lift-fabs.f64 68.7

       \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{\left|b\right|}, 0.3333333333333333 \cdot \frac{\mathsf{fma}\left(-1, b, \left|b\right|\right)}{a}\right) \]

12. Applied rewrites 68.7%

    \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{\left|b\right|}, 0.3333333333333333 \cdot \frac{\mathsf{fma}\left(-1, b, \left|b\right|\right)}{a}\right) \]
13. Add Preprocessing

Alternative 4: 68.1% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 4.9 \cdot 10^{-307}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a \cdot \frac{c}{\left|b\right|}, -0.5, \mathsf{fma}\left(-1, b, \left|b\right|\right) \cdot 0.3333333333333333\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 4.9e-307)
   (/
    (fma
     (* a (/ c (fabs b)))
     -0.5
     (* (fma -1.0 b (fabs b)) 0.3333333333333333))
    a)
   (* (/ c b) -0.5)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 4.9e-307) {
		tmp = fma((a * (c / fabs(b))), -0.5, (fma(-1.0, b, fabs(b)) * 0.3333333333333333)) / a;
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 4.9e-307)
		tmp = Float64(fma(Float64(a * Float64(c / abs(b))), -0.5, Float64(fma(-1.0, b, abs(b)) * 0.3333333333333333)) / a);
	else
		tmp = Float64(Float64(c / b) * -0.5);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 4.9e-307], N[(N[(N[(a * N[(c / N[Abs[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.5 + N[(N[(-1.0 * b + N[Abs[b], $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[(N[(c / b), $MachinePrecision] * -0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 4.9000000000000002e-307

   1. Initial program 67.1%

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

      1. lift-*.f64 N/A

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

      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-neg.f64 N/A

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

      4. lift-+.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

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

   4. Applied rewrites 67.3%

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(-1, b, {\left(\mathsf{fma}\left({b}^{1}, {b}^{1}, -3 \cdot \left(c \cdot a\right)\right)\right)}^{0.5}\right)}{3}}{a}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-fma.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/l/ N/A

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

      11. lower-/.f64 N/A

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

   6. Applied rewrites 67.2%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left({\left(\mathsf{fma}\left(\left|b\right|, \left|b\right|, \left(a \cdot -3\right) \cdot c\right)\right)}^{0.25}, {\left(\mathsf{fma}\left(\left|b\right|, \left|b\right|, \left(a \cdot -3\right) \cdot c\right)\right)}^{0.25}, b \cdot -1\right)}{3 \cdot a}} \]

   7. Taylor expanded in c around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lift-fabs.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. div-add N/A

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

      6. lower-+.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift-fabs.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 67.6

          \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{\left|b\right|}, 0.3333333333333333 \cdot \left(\frac{\left|b\right|}{a} + \frac{-1 \cdot b}{a}\right)\right) \]

   9. Applied rewrites 67.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5, \frac{c}{\left|b\right|}, 0.3333333333333333 \cdot \left(\frac{\left|b\right|}{a} + \frac{-1 \cdot b}{a}\right)\right)} \]

   10. Taylor expanded in a around 0

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

       1. lower-/.f64 N/A

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

   12. Applied rewrites 67.6%

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(a \cdot \frac{c}{\left|b\right|}, -0.5, \mathsf{fma}\left(-1, b, \left|b\right|\right) \cdot 0.3333333333333333\right)}{a}} \]

   if 4.9000000000000002e-307 < b

   1. Initial program 29.2%

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

   3. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 69.7

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

   5. Applied rewrites 69.7%

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

Alternative 5: 68.0% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -4 \cdot 10^{-311}:\\ \;\;\;\;\left(\frac{0.6666666666666666}{a} \cdot \left(-1 \cdot b\right)\right) \cdot \left(-1 \cdot -1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -4e-311)
   (* (* (/ 0.6666666666666666 a) (* -1.0 b)) (* -1.0 -1.0))
   (* (/ c b) -0.5)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -4e-311) {
		tmp = ((0.6666666666666666 / a) * (-1.0 * b)) * (-1.0 * -1.0);
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-4d-311)) then
        tmp = ((0.6666666666666666d0 / a) * ((-1.0d0) * b)) * ((-1.0d0) * (-1.0d0))
    else
        tmp = (c / b) * (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -4e-311) {
		tmp = ((0.6666666666666666 / a) * (-1.0 * b)) * (-1.0 * -1.0);
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -4e-311:
		tmp = ((0.6666666666666666 / a) * (-1.0 * b)) * (-1.0 * -1.0)
	else:
		tmp = (c / b) * -0.5
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -4e-311)
		tmp = Float64(Float64(Float64(0.6666666666666666 / a) * Float64(-1.0 * b)) * Float64(-1.0 * -1.0));
	else
		tmp = Float64(Float64(c / b) * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -4e-311)
		tmp = ((0.6666666666666666 / a) * (-1.0 * b)) * (-1.0 * -1.0);
	else
		tmp = (c / b) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -4e-311], N[(N[(N[(0.6666666666666666 / a), $MachinePrecision] * N[(-1.0 * b), $MachinePrecision]), $MachinePrecision] * N[(-1.0 * -1.0), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] * -0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -3.99999999999979e-311

   1. Initial program 66.9%

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

      1. lift-*.f64 N/A

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

      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-neg.f64 N/A

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

      4. lift-+.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

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

   4. Applied rewrites 67.1%

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(-1, b, {\left(\mathsf{fma}\left({b}^{1}, {b}^{1}, -3 \cdot \left(c \cdot a\right)\right)\right)}^{0.5}\right)}{3}}{a}} \]

   5. Taylor expanded in b around -inf

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

      1. associate-/l/ N/A

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

      2. mul-1-neg N/A

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

      3. unpow1/2 N/A

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

      4. pow-prod-up N/A

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

      5. metadata-eval N/A

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

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

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

      7. pow2 N/A

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

      8. metadata-eval N/A

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

      9. *-commutative N/A

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

      10. associate-*l* N/A

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

      11. *-commutative N/A

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

   7. Applied rewrites 65.1%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(0.6666666666666666, {a}^{-1}, \frac{c}{b \cdot b} \cdot -0.5\right) \cdot b\right) \cdot -1} \]

   8. Taylor expanded in a around 0

      \[\leadsto \left(\frac{\frac{2}{3}}{a} \cdot b\right) \cdot -1 \]
   9. Step-by-step derivation

      1. lower-/.f64 67.9

         \[\leadsto \left(\frac{0.6666666666666666}{a} \cdot b\right) \cdot -1 \]

   10. Applied rewrites 67.9%

       \[\leadsto \left(\frac{0.6666666666666666}{a} \cdot b\right) \cdot -1 \]

   if -3.99999999999979e-311 < b

   1. Initial program 29.8%

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

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b}} \]
   4. 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 69.2

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

   5. Applied rewrites 69.2%

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

4. Final simplification 68.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4 \cdot 10^{-311}:\\ \;\;\;\;\left(\frac{0.6666666666666666}{a} \cdot \left(-1 \cdot b\right)\right) \cdot \left(-1 \cdot -1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 35.3% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \left(\frac{0.6666666666666666}{a} \cdot \left(-1 \cdot b\right)\right) \cdot \left(-1 \cdot -1\right) \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (* (* (/ 0.6666666666666666 a) (* -1.0 b)) (* -1.0 -1.0)))

C

double code(double a, double b, double c) {
	return ((0.6666666666666666 / a) * (-1.0 * b)) * (-1.0 * -1.0);
}

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 / a) * ((-1.0d0) * b)) * ((-1.0d0) * (-1.0d0))
end function

Java

public static double code(double a, double b, double c) {
	return ((0.6666666666666666 / a) * (-1.0 * b)) * (-1.0 * -1.0);
}

Python

def code(a, b, c):
	return ((0.6666666666666666 / a) * (-1.0 * b)) * (-1.0 * -1.0)

Julia

function code(a, b, c)
	return Float64(Float64(Float64(0.6666666666666666 / a) * Float64(-1.0 * b)) * Float64(-1.0 * -1.0))
end

MATLAB

function tmp = code(a, b, c)
	tmp = ((0.6666666666666666 / a) * (-1.0 * b)) * (-1.0 * -1.0);
end

Wolfram

code[a_, b_, c_] := N[(N[(N[(0.6666666666666666 / a), $MachinePrecision] * N[(-1.0 * b), $MachinePrecision]), $MachinePrecision] * N[(-1.0 * -1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 46.9%

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

   1. lift-*.f64 N/A

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

   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-neg.f64 N/A

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

   4. lift-+.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

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

4. Applied rewrites 46.9%

   \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(-1, b, {\left(\mathsf{fma}\left({b}^{1}, {b}^{1}, -3 \cdot \left(c \cdot a\right)\right)\right)}^{0.5}\right)}{3}}{a}} \]

5. Taylor expanded in b around -inf

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

   1. associate-/l/ N/A

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

   2. mul-1-neg N/A

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

   3. unpow1/2 N/A

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

   4. pow-prod-up N/A

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

   5. metadata-eval N/A

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

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

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

   7. pow2 N/A

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

   8. metadata-eval N/A

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

   9. *-commutative N/A

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

   10. associate-*l* N/A

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

   11. *-commutative N/A

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

7. Applied rewrites 31.1%

   \[\leadsto \color{blue}{\left(\mathsf{fma}\left(0.6666666666666666, {a}^{-1}, \frac{c}{b \cdot b} \cdot -0.5\right) \cdot b\right) \cdot -1} \]

8. Taylor expanded in a around 0

   \[\leadsto \left(\frac{\frac{2}{3}}{a} \cdot b\right) \cdot -1 \]
9. Step-by-step derivation

   1. lower-/.f64 32.7

      \[\leadsto \left(\frac{0.6666666666666666}{a} \cdot b\right) \cdot -1 \]

10. Applied rewrites 32.7%

    \[\leadsto \left(\frac{0.6666666666666666}{a} \cdot b\right) \cdot -1 \]

11. Final simplification 32.7%

    \[\leadsto \left(\frac{0.6666666666666666}{a} \cdot \left(-1 \cdot b\right)\right) \cdot \left(-1 \cdot -1\right) \]
12. Add Preprocessing

Alternative 7: 35.3% accurate, N/A× 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 46.9%

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

3. Taylor expanded in b around -inf

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

   1. lower-*.f64 N/A

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

   2. lower-/.f64 32.7

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

5. Applied rewrites 32.7%

   \[\leadsto \color{blue}{-0.6666666666666666 \cdot \frac{b}{a}} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2025066 
(FPCore (a b c)
  :name "Cubic critical"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))