Cubic critical, medium range

Percentage Accurate: 31.8% → 95.4%

Time: 4.7s

Alternatives: 8

Speedup: 3.3×

Specification

\[\left(\left(1.1102230246251565 \cdot 10^{-16} < a \land a < 9007199254740992\right) \land \left(1.1102230246251565 \cdot 10^{-16} < b \land b < 9007199254740992\right)\right) \land \left(1.1102230246251565 \cdot 10^{-16} < c \land c < 9007199254740992\right)\]

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: 31.8% 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: 95.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(\left(\left(a \cdot a\right) \cdot a\right) \cdot \frac{c}{b \cdot b}, -1.0546875, \left(a \cdot a\right) \cdot -0.5625\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}, c, \frac{a}{b \cdot b} \cdot -0.375\right) \cdot c, c, -0.5 \cdot c\right)}{b} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/
  (fma
   (*
    (fma
     (/
      (fma (* (* (* a a) a) (/ c (* b b))) -1.0546875 (* (* a a) -0.5625))
      (* (* b b) (* b b)))
     c
     (* (/ a (* b b)) -0.375))
    c)
   c
   (* -0.5 c))
  b))

C

double code(double a, double b, double c) {
	return fma((fma((fma((((a * a) * a) * (c / (b * b))), -1.0546875, ((a * a) * -0.5625)) / ((b * b) * (b * b))), c, ((a / (b * b)) * -0.375)) * c), c, (-0.5 * c)) / b;
}

Julia

function code(a, b, c)
	return Float64(fma(Float64(fma(Float64(fma(Float64(Float64(Float64(a * a) * a) * Float64(c / Float64(b * b))), -1.0546875, Float64(Float64(a * a) * -0.5625)) / Float64(Float64(b * b) * Float64(b * b))), c, Float64(Float64(a / Float64(b * b)) * -0.375)) * c), c, Float64(-0.5 * c)) / b)
end

Wolfram

code[a_, b_, c_] := N[(N[(N[(N[(N[(N[(N[(N[(N[(a * a), $MachinePrecision] * a), $MachinePrecision] * N[(c / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -1.0546875 + N[(N[(a * a), $MachinePrecision] * -0.5625), $MachinePrecision]), $MachinePrecision] / N[(N[(b * b), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * c + N[(N[(a / N[(b * b), $MachinePrecision]), $MachinePrecision] * -0.375), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision] * c + N[(-0.5 * c), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]

Derivation

1. Initial program 31.8%

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

2. Taylor expanded in b around inf

   \[\leadsto \color{blue}{\frac{\frac{-9}{16} \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(\frac{-1}{2} \cdot c + \left(\frac{-3}{8} \cdot \frac{a \cdot {c}^{2}}{{b}^{2}} + \frac{-1}{6} \cdot \frac{\frac{81}{64} \cdot \left({a}^{4} \cdot {c}^{4}\right) + \frac{81}{16} \cdot \left({a}^{4} \cdot {c}^{4}\right)}{a \cdot {b}^{6}}\right)\right)}{b}} \]

4. Applied rewrites 95.4%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}, -0.5625, \mathsf{fma}\left(-0.5, c, \mathsf{fma}\left(\frac{\left(\left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right) \cdot \left(\left(c \cdot c\right) \cdot \left(c \cdot c\right)\right)\right) \cdot 6.328125}{\left(\left(\left(b \cdot b\right) \cdot b\right) \cdot \left(\left(b \cdot b\right) \cdot b\right)\right) \cdot a}, -0.16666666666666666, \frac{-0.375 \cdot \left(\left(c \cdot c\right) \cdot a\right)}{b \cdot b}\right)\right)\right)}{b}} \]

5. Taylor expanded in c around 0

   \[\leadsto \frac{c \cdot \left(c \cdot \left(\frac{-3}{8} \cdot \frac{a}{{b}^{2}} + c \cdot \left(\frac{-135}{128} \cdot \frac{{a}^{3} \cdot c}{{b}^{6}} + \frac{-9}{16} \cdot \frac{{a}^{2}}{{b}^{4}}\right)\right) - \frac{1}{2}\right)}{b} \]

7. Applied rewrites 95.3%

   \[\leadsto \frac{c \cdot \mathsf{fma}\left(c, \mathsf{fma}\left(-0.375, \frac{a}{b \cdot b}, c \cdot \mathsf{fma}\left(-1.0546875, \frac{\left(\left(a \cdot a\right) \cdot a\right) \cdot c}{\left(\left(b \cdot b\right) \cdot b\right) \cdot \left(\left(b \cdot b\right) \cdot b\right)}, -0.5625 \cdot \frac{a \cdot a}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}\right)\right), -0.5\right)}{b} \]

8. Taylor expanded in b around inf

   \[\leadsto \frac{c \cdot \mathsf{fma}\left(c, \mathsf{fma}\left(\frac{-3}{8}, \frac{a}{b \cdot b}, c \cdot \frac{\frac{-135}{128} \cdot \frac{{a}^{3} \cdot c}{{b}^{2}} + \frac{-9}{16} \cdot {a}^{2}}{{b}^{4}}\right), \frac{-1}{2}\right)}{b} \]
9. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{c \cdot \mathsf{fma}\left(c, \mathsf{fma}\left(\frac{-3}{8}, \frac{a}{b \cdot b}, c \cdot \frac{\frac{-135}{128} \cdot \frac{{a}^{3} \cdot c}{{b}^{2}} + \frac{-9}{16} \cdot {a}^{2}}{{b}^{4}}\right), \frac{-1}{2}\right)}{b} \]

   2. lower-fma.f64 N/A

      \[\leadsto \frac{c \cdot \mathsf{fma}\left(c, \mathsf{fma}\left(\frac{-3}{8}, \frac{a}{b \cdot b}, c \cdot \frac{\mathsf{fma}\left(\frac{-135}{128}, \frac{{a}^{3} \cdot c}{{b}^{2}}, \frac{-9}{16} \cdot {a}^{2}\right)}{{b}^{4}}\right), \frac{-1}{2}\right)}{b} \]

   3. lower-/.f64 N/A

      \[\leadsto \frac{c \cdot \mathsf{fma}\left(c, \mathsf{fma}\left(\frac{-3}{8}, \frac{a}{b \cdot b}, c \cdot \frac{\mathsf{fma}\left(\frac{-135}{128}, \frac{{a}^{3} \cdot c}{{b}^{2}}, \frac{-9}{16} \cdot {a}^{2}\right)}{{b}^{4}}\right), \frac{-1}{2}\right)}{b} \]

   4. pow3 N/A

      \[\leadsto \frac{c \cdot \mathsf{fma}\left(c, \mathsf{fma}\left(\frac{-3}{8}, \frac{a}{b \cdot b}, c \cdot \frac{\mathsf{fma}\left(\frac{-135}{128}, \frac{\left(\left(a \cdot a\right) \cdot a\right) \cdot c}{{b}^{2}}, \frac{-9}{16} \cdot {a}^{2}\right)}{{b}^{4}}\right), \frac{-1}{2}\right)}{b} \]

   5. lift-*.f64 N/A

      \[\leadsto \frac{c \cdot \mathsf{fma}\left(c, \mathsf{fma}\left(\frac{-3}{8}, \frac{a}{b \cdot b}, c \cdot \frac{\mathsf{fma}\left(\frac{-135}{128}, \frac{\left(\left(a \cdot a\right) \cdot a\right) \cdot c}{{b}^{2}}, \frac{-9}{16} \cdot {a}^{2}\right)}{{b}^{4}}\right), \frac{-1}{2}\right)}{b} \]

   6. lift-*.f64 N/A

      \[\leadsto \frac{c \cdot \mathsf{fma}\left(c, \mathsf{fma}\left(\frac{-3}{8}, \frac{a}{b \cdot b}, c \cdot \frac{\mathsf{fma}\left(\frac{-135}{128}, \frac{\left(\left(a \cdot a\right) \cdot a\right) \cdot c}{{b}^{2}}, \frac{-9}{16} \cdot {a}^{2}\right)}{{b}^{4}}\right), \frac{-1}{2}\right)}{b} \]

   7. lift-*.f64 N/A

      \[\leadsto \frac{c \cdot \mathsf{fma}\left(c, \mathsf{fma}\left(\frac{-3}{8}, \frac{a}{b \cdot b}, c \cdot \frac{\mathsf{fma}\left(\frac{-135}{128}, \frac{\left(\left(a \cdot a\right) \cdot a\right) \cdot c}{{b}^{2}}, \frac{-9}{16} \cdot {a}^{2}\right)}{{b}^{4}}\right), \frac{-1}{2}\right)}{b} \]

   8. pow2 N/A

      \[\leadsto \frac{c \cdot \mathsf{fma}\left(c, \mathsf{fma}\left(\frac{-3}{8}, \frac{a}{b \cdot b}, c \cdot \frac{\mathsf{fma}\left(\frac{-135}{128}, \frac{\left(\left(a \cdot a\right) \cdot a\right) \cdot c}{b \cdot b}, \frac{-9}{16} \cdot {a}^{2}\right)}{{b}^{4}}\right), \frac{-1}{2}\right)}{b} \]

   9. lift-*.f64 N/A

      \[\leadsto \frac{c \cdot \mathsf{fma}\left(c, \mathsf{fma}\left(\frac{-3}{8}, \frac{a}{b \cdot b}, c \cdot \frac{\mathsf{fma}\left(\frac{-135}{128}, \frac{\left(\left(a \cdot a\right) \cdot a\right) \cdot c}{b \cdot b}, \frac{-9}{16} \cdot {a}^{2}\right)}{{b}^{4}}\right), \frac{-1}{2}\right)}{b} \]

   10. lower-*.f64 N/A

       \[\leadsto \frac{c \cdot \mathsf{fma}\left(c, \mathsf{fma}\left(\frac{-3}{8}, \frac{a}{b \cdot b}, c \cdot \frac{\mathsf{fma}\left(\frac{-135}{128}, \frac{\left(\left(a \cdot a\right) \cdot a\right) \cdot c}{b \cdot b}, \frac{-9}{16} \cdot {a}^{2}\right)}{{b}^{4}}\right), \frac{-1}{2}\right)}{b} \]

   11. pow2 N/A

       \[\leadsto \frac{c \cdot \mathsf{fma}\left(c, \mathsf{fma}\left(\frac{-3}{8}, \frac{a}{b \cdot b}, c \cdot \frac{\mathsf{fma}\left(\frac{-135}{128}, \frac{\left(\left(a \cdot a\right) \cdot a\right) \cdot c}{b \cdot b}, \frac{-9}{16} \cdot \left(a \cdot a\right)\right)}{{b}^{4}}\right), \frac{-1}{2}\right)}{b} \]

   12. lift-*.f64 N/A

       \[\leadsto \frac{c \cdot \mathsf{fma}\left(c, \mathsf{fma}\left(\frac{-3}{8}, \frac{a}{b \cdot b}, c \cdot \frac{\mathsf{fma}\left(\frac{-135}{128}, \frac{\left(\left(a \cdot a\right) \cdot a\right) \cdot c}{b \cdot b}, \frac{-9}{16} \cdot \left(a \cdot a\right)\right)}{{b}^{4}}\right), \frac{-1}{2}\right)}{b} \]

   13. sqr-pow N/A

       \[\leadsto \frac{c \cdot \mathsf{fma}\left(c, \mathsf{fma}\left(\frac{-3}{8}, \frac{a}{b \cdot b}, c \cdot \frac{\mathsf{fma}\left(\frac{-135}{128}, \frac{\left(\left(a \cdot a\right) \cdot a\right) \cdot c}{b \cdot b}, \frac{-9}{16} \cdot \left(a \cdot a\right)\right)}{{b}^{\left(\frac{4}{2}\right)} \cdot {b}^{\left(\frac{4}{2}\right)}}\right), \frac{-1}{2}\right)}{b} \]

   14. metadata-eval N/A

       \[\leadsto \frac{c \cdot \mathsf{fma}\left(c, \mathsf{fma}\left(\frac{-3}{8}, \frac{a}{b \cdot b}, c \cdot \frac{\mathsf{fma}\left(\frac{-135}{128}, \frac{\left(\left(a \cdot a\right) \cdot a\right) \cdot c}{b \cdot b}, \frac{-9}{16} \cdot \left(a \cdot a\right)\right)}{{b}^{2} \cdot {b}^{\left(\frac{4}{2}\right)}}\right), \frac{-1}{2}\right)}{b} \]

   15. metadata-eval N/A

       \[\leadsto \frac{c \cdot \mathsf{fma}\left(c, \mathsf{fma}\left(\frac{-3}{8}, \frac{a}{b \cdot b}, c \cdot \frac{\mathsf{fma}\left(\frac{-135}{128}, \frac{\left(\left(a \cdot a\right) \cdot a\right) \cdot c}{b \cdot b}, \frac{-9}{16} \cdot \left(a \cdot a\right)\right)}{{b}^{2} \cdot {b}^{2}}\right), \frac{-1}{2}\right)}{b} \]

10. Applied rewrites 95.3%

    \[\leadsto \frac{c \cdot \mathsf{fma}\left(c, \mathsf{fma}\left(-0.375, \frac{a}{b \cdot b}, c \cdot \frac{\mathsf{fma}\left(-1.0546875, \frac{\left(\left(a \cdot a\right) \cdot a\right) \cdot c}{b \cdot b}, -0.5625 \cdot \left(a \cdot a\right)\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}\right), -0.5\right)}{b} \]

12. Applied rewrites 95.4%

    \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(\left(\left(a \cdot a\right) \cdot a\right) \cdot \frac{c}{b \cdot b}, -1.0546875, \left(a \cdot a\right) \cdot -0.5625\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}, c, \frac{a}{b \cdot b} \cdot -0.375\right) \cdot c, c, -0.5 \cdot c\right)}{b} \]
13. Add Preprocessing

Alternative 2: 95.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \frac{c \cdot \mathsf{fma}\left(c, \mathsf{fma}\left(-0.375, \frac{a}{b \cdot b}, c \cdot \frac{\mathsf{fma}\left(a \cdot \frac{c}{b \cdot b}, -1.0546875, -0.5625\right) \cdot \left(a \cdot a\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}\right), -0.5\right)}{b} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/
  (*
   c
   (fma
    c
    (fma
     -0.375
     (/ a (* b b))
     (*
      c
      (/
       (* (fma (* a (/ c (* b b))) -1.0546875 -0.5625) (* a a))
       (* (* b b) (* b b)))))
    -0.5))
  b))

C

double code(double a, double b, double c) {
	return (c * fma(c, fma(-0.375, (a / (b * b)), (c * ((fma((a * (c / (b * b))), -1.0546875, -0.5625) * (a * a)) / ((b * b) * (b * b))))), -0.5)) / b;
}

Julia

function code(a, b, c)
	return Float64(Float64(c * fma(c, fma(-0.375, Float64(a / Float64(b * b)), Float64(c * Float64(Float64(fma(Float64(a * Float64(c / Float64(b * b))), -1.0546875, -0.5625) * Float64(a * a)) / Float64(Float64(b * b) * Float64(b * b))))), -0.5)) / b)
end

Wolfram

code[a_, b_, c_] := N[(N[(c * N[(c * N[(-0.375 * N[(a / N[(b * b), $MachinePrecision]), $MachinePrecision] + N[(c * N[(N[(N[(N[(a * N[(c / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -1.0546875 + -0.5625), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision] / N[(N[(b * b), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -0.5), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]

Derivation

1. Initial program 31.8%

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

2. Taylor expanded in b around inf

   \[\leadsto \color{blue}{\frac{\frac{-9}{16} \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(\frac{-1}{2} \cdot c + \left(\frac{-3}{8} \cdot \frac{a \cdot {c}^{2}}{{b}^{2}} + \frac{-1}{6} \cdot \frac{\frac{81}{64} \cdot \left({a}^{4} \cdot {c}^{4}\right) + \frac{81}{16} \cdot \left({a}^{4} \cdot {c}^{4}\right)}{a \cdot {b}^{6}}\right)\right)}{b}} \]

4. Applied rewrites 95.4%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}, -0.5625, \mathsf{fma}\left(-0.5, c, \mathsf{fma}\left(\frac{\left(\left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right) \cdot \left(\left(c \cdot c\right) \cdot \left(c \cdot c\right)\right)\right) \cdot 6.328125}{\left(\left(\left(b \cdot b\right) \cdot b\right) \cdot \left(\left(b \cdot b\right) \cdot b\right)\right) \cdot a}, -0.16666666666666666, \frac{-0.375 \cdot \left(\left(c \cdot c\right) \cdot a\right)}{b \cdot b}\right)\right)\right)}{b}} \]

5. Taylor expanded in c around 0

   \[\leadsto \frac{c \cdot \left(c \cdot \left(\frac{-3}{8} \cdot \frac{a}{{b}^{2}} + c \cdot \left(\frac{-135}{128} \cdot \frac{{a}^{3} \cdot c}{{b}^{6}} + \frac{-9}{16} \cdot \frac{{a}^{2}}{{b}^{4}}\right)\right) - \frac{1}{2}\right)}{b} \]

7. Applied rewrites 95.3%

   \[\leadsto \frac{c \cdot \mathsf{fma}\left(c, \mathsf{fma}\left(-0.375, \frac{a}{b \cdot b}, c \cdot \mathsf{fma}\left(-1.0546875, \frac{\left(\left(a \cdot a\right) \cdot a\right) \cdot c}{\left(\left(b \cdot b\right) \cdot b\right) \cdot \left(\left(b \cdot b\right) \cdot b\right)}, -0.5625 \cdot \frac{a \cdot a}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}\right)\right), -0.5\right)}{b} \]

8. Taylor expanded in b around inf

   \[\leadsto \frac{c \cdot \mathsf{fma}\left(c, \mathsf{fma}\left(\frac{-3}{8}, \frac{a}{b \cdot b}, c \cdot \frac{\frac{-135}{128} \cdot \frac{{a}^{3} \cdot c}{{b}^{2}} + \frac{-9}{16} \cdot {a}^{2}}{{b}^{4}}\right), \frac{-1}{2}\right)}{b} \]
9. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{c \cdot \mathsf{fma}\left(c, \mathsf{fma}\left(\frac{-3}{8}, \frac{a}{b \cdot b}, c \cdot \frac{\frac{-135}{128} \cdot \frac{{a}^{3} \cdot c}{{b}^{2}} + \frac{-9}{16} \cdot {a}^{2}}{{b}^{4}}\right), \frac{-1}{2}\right)}{b} \]

   2. lower-fma.f64 N/A

      \[\leadsto \frac{c \cdot \mathsf{fma}\left(c, \mathsf{fma}\left(\frac{-3}{8}, \frac{a}{b \cdot b}, c \cdot \frac{\mathsf{fma}\left(\frac{-135}{128}, \frac{{a}^{3} \cdot c}{{b}^{2}}, \frac{-9}{16} \cdot {a}^{2}\right)}{{b}^{4}}\right), \frac{-1}{2}\right)}{b} \]

   3. lower-/.f64 N/A

      \[\leadsto \frac{c \cdot \mathsf{fma}\left(c, \mathsf{fma}\left(\frac{-3}{8}, \frac{a}{b \cdot b}, c \cdot \frac{\mathsf{fma}\left(\frac{-135}{128}, \frac{{a}^{3} \cdot c}{{b}^{2}}, \frac{-9}{16} \cdot {a}^{2}\right)}{{b}^{4}}\right), \frac{-1}{2}\right)}{b} \]

   4. pow3 N/A

      \[\leadsto \frac{c \cdot \mathsf{fma}\left(c, \mathsf{fma}\left(\frac{-3}{8}, \frac{a}{b \cdot b}, c \cdot \frac{\mathsf{fma}\left(\frac{-135}{128}, \frac{\left(\left(a \cdot a\right) \cdot a\right) \cdot c}{{b}^{2}}, \frac{-9}{16} \cdot {a}^{2}\right)}{{b}^{4}}\right), \frac{-1}{2}\right)}{b} \]

   5. lift-*.f64 N/A

      \[\leadsto \frac{c \cdot \mathsf{fma}\left(c, \mathsf{fma}\left(\frac{-3}{8}, \frac{a}{b \cdot b}, c \cdot \frac{\mathsf{fma}\left(\frac{-135}{128}, \frac{\left(\left(a \cdot a\right) \cdot a\right) \cdot c}{{b}^{2}}, \frac{-9}{16} \cdot {a}^{2}\right)}{{b}^{4}}\right), \frac{-1}{2}\right)}{b} \]

   6. lift-*.f64 N/A

      \[\leadsto \frac{c \cdot \mathsf{fma}\left(c, \mathsf{fma}\left(\frac{-3}{8}, \frac{a}{b \cdot b}, c \cdot \frac{\mathsf{fma}\left(\frac{-135}{128}, \frac{\left(\left(a \cdot a\right) \cdot a\right) \cdot c}{{b}^{2}}, \frac{-9}{16} \cdot {a}^{2}\right)}{{b}^{4}}\right), \frac{-1}{2}\right)}{b} \]

   7. lift-*.f64 N/A

      \[\leadsto \frac{c \cdot \mathsf{fma}\left(c, \mathsf{fma}\left(\frac{-3}{8}, \frac{a}{b \cdot b}, c \cdot \frac{\mathsf{fma}\left(\frac{-135}{128}, \frac{\left(\left(a \cdot a\right) \cdot a\right) \cdot c}{{b}^{2}}, \frac{-9}{16} \cdot {a}^{2}\right)}{{b}^{4}}\right), \frac{-1}{2}\right)}{b} \]

   8. pow2 N/A

      \[\leadsto \frac{c \cdot \mathsf{fma}\left(c, \mathsf{fma}\left(\frac{-3}{8}, \frac{a}{b \cdot b}, c \cdot \frac{\mathsf{fma}\left(\frac{-135}{128}, \frac{\left(\left(a \cdot a\right) \cdot a\right) \cdot c}{b \cdot b}, \frac{-9}{16} \cdot {a}^{2}\right)}{{b}^{4}}\right), \frac{-1}{2}\right)}{b} \]

   9. lift-*.f64 N/A

      \[\leadsto \frac{c \cdot \mathsf{fma}\left(c, \mathsf{fma}\left(\frac{-3}{8}, \frac{a}{b \cdot b}, c \cdot \frac{\mathsf{fma}\left(\frac{-135}{128}, \frac{\left(\left(a \cdot a\right) \cdot a\right) \cdot c}{b \cdot b}, \frac{-9}{16} \cdot {a}^{2}\right)}{{b}^{4}}\right), \frac{-1}{2}\right)}{b} \]

   10. lower-*.f64 N/A

       \[\leadsto \frac{c \cdot \mathsf{fma}\left(c, \mathsf{fma}\left(\frac{-3}{8}, \frac{a}{b \cdot b}, c \cdot \frac{\mathsf{fma}\left(\frac{-135}{128}, \frac{\left(\left(a \cdot a\right) \cdot a\right) \cdot c}{b \cdot b}, \frac{-9}{16} \cdot {a}^{2}\right)}{{b}^{4}}\right), \frac{-1}{2}\right)}{b} \]

   11. pow2 N/A

       \[\leadsto \frac{c \cdot \mathsf{fma}\left(c, \mathsf{fma}\left(\frac{-3}{8}, \frac{a}{b \cdot b}, c \cdot \frac{\mathsf{fma}\left(\frac{-135}{128}, \frac{\left(\left(a \cdot a\right) \cdot a\right) \cdot c}{b \cdot b}, \frac{-9}{16} \cdot \left(a \cdot a\right)\right)}{{b}^{4}}\right), \frac{-1}{2}\right)}{b} \]

   12. lift-*.f64 N/A

       \[\leadsto \frac{c \cdot \mathsf{fma}\left(c, \mathsf{fma}\left(\frac{-3}{8}, \frac{a}{b \cdot b}, c \cdot \frac{\mathsf{fma}\left(\frac{-135}{128}, \frac{\left(\left(a \cdot a\right) \cdot a\right) \cdot c}{b \cdot b}, \frac{-9}{16} \cdot \left(a \cdot a\right)\right)}{{b}^{4}}\right), \frac{-1}{2}\right)}{b} \]

   13. sqr-pow N/A

       \[\leadsto \frac{c \cdot \mathsf{fma}\left(c, \mathsf{fma}\left(\frac{-3}{8}, \frac{a}{b \cdot b}, c \cdot \frac{\mathsf{fma}\left(\frac{-135}{128}, \frac{\left(\left(a \cdot a\right) \cdot a\right) \cdot c}{b \cdot b}, \frac{-9}{16} \cdot \left(a \cdot a\right)\right)}{{b}^{\left(\frac{4}{2}\right)} \cdot {b}^{\left(\frac{4}{2}\right)}}\right), \frac{-1}{2}\right)}{b} \]

   14. metadata-eval N/A

       \[\leadsto \frac{c \cdot \mathsf{fma}\left(c, \mathsf{fma}\left(\frac{-3}{8}, \frac{a}{b \cdot b}, c \cdot \frac{\mathsf{fma}\left(\frac{-135}{128}, \frac{\left(\left(a \cdot a\right) \cdot a\right) \cdot c}{b \cdot b}, \frac{-9}{16} \cdot \left(a \cdot a\right)\right)}{{b}^{2} \cdot {b}^{\left(\frac{4}{2}\right)}}\right), \frac{-1}{2}\right)}{b} \]

   15. metadata-eval N/A

       \[\leadsto \frac{c \cdot \mathsf{fma}\left(c, \mathsf{fma}\left(\frac{-3}{8}, \frac{a}{b \cdot b}, c \cdot \frac{\mathsf{fma}\left(\frac{-135}{128}, \frac{\left(\left(a \cdot a\right) \cdot a\right) \cdot c}{b \cdot b}, \frac{-9}{16} \cdot \left(a \cdot a\right)\right)}{{b}^{2} \cdot {b}^{2}}\right), \frac{-1}{2}\right)}{b} \]

10. Applied rewrites 95.3%

    \[\leadsto \frac{c \cdot \mathsf{fma}\left(c, \mathsf{fma}\left(-0.375, \frac{a}{b \cdot b}, c \cdot \frac{\mathsf{fma}\left(-1.0546875, \frac{\left(\left(a \cdot a\right) \cdot a\right) \cdot c}{b \cdot b}, -0.5625 \cdot \left(a \cdot a\right)\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}\right), -0.5\right)}{b} \]

11. Taylor expanded in a around 0

    \[\leadsto \frac{c \cdot \mathsf{fma}\left(c, \mathsf{fma}\left(\frac{-3}{8}, \frac{a}{b \cdot b}, c \cdot \frac{{a}^{2} \cdot \left(\frac{-135}{128} \cdot \frac{a \cdot c}{{b}^{2}} - \frac{9}{16}\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}\right), \frac{-1}{2}\right)}{b} \]
12. Step-by-step derivation

    1. *-commutative N/A

       \[\leadsto \frac{c \cdot \mathsf{fma}\left(c, \mathsf{fma}\left(\frac{-3}{8}, \frac{a}{b \cdot b}, c \cdot \frac{\left(\frac{-135}{128} \cdot \frac{a \cdot c}{{b}^{2}} - \frac{9}{16}\right) \cdot {a}^{2}}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}\right), \frac{-1}{2}\right)}{b} \]

    2. lower-*.f64 N/A

       \[\leadsto \frac{c \cdot \mathsf{fma}\left(c, \mathsf{fma}\left(\frac{-3}{8}, \frac{a}{b \cdot b}, c \cdot \frac{\left(\frac{-135}{128} \cdot \frac{a \cdot c}{{b}^{2}} - \frac{9}{16}\right) \cdot {a}^{2}}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}\right), \frac{-1}{2}\right)}{b} \]

    3. negate-sub N/A

       \[\leadsto \frac{c \cdot \mathsf{fma}\left(c, \mathsf{fma}\left(\frac{-3}{8}, \frac{a}{b \cdot b}, c \cdot \frac{\left(\frac{-135}{128} \cdot \frac{a \cdot c}{{b}^{2}} + \left(\mathsf{neg}\left(\frac{9}{16}\right)\right)\right) \cdot {a}^{2}}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}\right), \frac{-1}{2}\right)}{b} \]

    4. *-commutative N/A

       \[\leadsto \frac{c \cdot \mathsf{fma}\left(c, \mathsf{fma}\left(\frac{-3}{8}, \frac{a}{b \cdot b}, c \cdot \frac{\left(\frac{a \cdot c}{{b}^{2}} \cdot \frac{-135}{128} + \left(\mathsf{neg}\left(\frac{9}{16}\right)\right)\right) \cdot {a}^{2}}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}\right), \frac{-1}{2}\right)}{b} \]

    5. metadata-eval N/A

       \[\leadsto \frac{c \cdot \mathsf{fma}\left(c, \mathsf{fma}\left(\frac{-3}{8}, \frac{a}{b \cdot b}, c \cdot \frac{\left(\frac{a \cdot c}{{b}^{2}} \cdot \frac{-135}{128} + \frac{-9}{16}\right) \cdot {a}^{2}}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}\right), \frac{-1}{2}\right)}{b} \]

    6. lower-fma.f64 N/A

       \[\leadsto \frac{c \cdot \mathsf{fma}\left(c, \mathsf{fma}\left(\frac{-3}{8}, \frac{a}{b \cdot b}, c \cdot \frac{\mathsf{fma}\left(\frac{a \cdot c}{{b}^{2}}, \frac{-135}{128}, \frac{-9}{16}\right) \cdot {a}^{2}}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}\right), \frac{-1}{2}\right)}{b} \]

    7. associate-/l* N/A

       \[\leadsto \frac{c \cdot \mathsf{fma}\left(c, \mathsf{fma}\left(\frac{-3}{8}, \frac{a}{b \cdot b}, c \cdot \frac{\mathsf{fma}\left(a \cdot \frac{c}{{b}^{2}}, \frac{-135}{128}, \frac{-9}{16}\right) \cdot {a}^{2}}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}\right), \frac{-1}{2}\right)}{b} \]

    8. lower-*.f64 N/A

       \[\leadsto \frac{c \cdot \mathsf{fma}\left(c, \mathsf{fma}\left(\frac{-3}{8}, \frac{a}{b \cdot b}, c \cdot \frac{\mathsf{fma}\left(a \cdot \frac{c}{{b}^{2}}, \frac{-135}{128}, \frac{-9}{16}\right) \cdot {a}^{2}}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}\right), \frac{-1}{2}\right)}{b} \]

    9. lower-/.f64 N/A

       \[\leadsto \frac{c \cdot \mathsf{fma}\left(c, \mathsf{fma}\left(\frac{-3}{8}, \frac{a}{b \cdot b}, c \cdot \frac{\mathsf{fma}\left(a \cdot \frac{c}{{b}^{2}}, \frac{-135}{128}, \frac{-9}{16}\right) \cdot {a}^{2}}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}\right), \frac{-1}{2}\right)}{b} \]

    10. pow2 N/A

        \[\leadsto \frac{c \cdot \mathsf{fma}\left(c, \mathsf{fma}\left(\frac{-3}{8}, \frac{a}{b \cdot b}, c \cdot \frac{\mathsf{fma}\left(a \cdot \frac{c}{b \cdot b}, \frac{-135}{128}, \frac{-9}{16}\right) \cdot {a}^{2}}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}\right), \frac{-1}{2}\right)}{b} \]

    11. lift-*.f64 N/A

        \[\leadsto \frac{c \cdot \mathsf{fma}\left(c, \mathsf{fma}\left(\frac{-3}{8}, \frac{a}{b \cdot b}, c \cdot \frac{\mathsf{fma}\left(a \cdot \frac{c}{b \cdot b}, \frac{-135}{128}, \frac{-9}{16}\right) \cdot {a}^{2}}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}\right), \frac{-1}{2}\right)}{b} \]

    12. pow2 N/A

        \[\leadsto \frac{c \cdot \mathsf{fma}\left(c, \mathsf{fma}\left(\frac{-3}{8}, \frac{a}{b \cdot b}, c \cdot \frac{\mathsf{fma}\left(a \cdot \frac{c}{b \cdot b}, \frac{-135}{128}, \frac{-9}{16}\right) \cdot \left(a \cdot a\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}\right), \frac{-1}{2}\right)}{b} \]

    13. lift-*.f64 95.3

        \[\leadsto \frac{c \cdot \mathsf{fma}\left(c, \mathsf{fma}\left(-0.375, \frac{a}{b \cdot b}, c \cdot \frac{\mathsf{fma}\left(a \cdot \frac{c}{b \cdot b}, -1.0546875, -0.5625\right) \cdot \left(a \cdot a\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}\right), -0.5\right)}{b} \]

13. Applied rewrites 95.3%

    \[\leadsto \frac{c \cdot \mathsf{fma}\left(c, \mathsf{fma}\left(-0.375, \frac{a}{b \cdot b}, c \cdot \frac{\mathsf{fma}\left(a \cdot \frac{c}{b \cdot b}, -1.0546875, -0.5625\right) \cdot \left(a \cdot a\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}\right), -0.5\right)}{b} \]
14. Add Preprocessing

Alternative 3: 93.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}, -0.5625, \mathsf{fma}\left(\frac{\left(c \cdot c\right) \cdot a}{b \cdot b}, -0.375, -0.5 \cdot c\right)\right)}{b} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/
  (fma
   (/ (* (* a a) (* (* c c) c)) (* (* b b) (* b b)))
   -0.5625
   (fma (/ (* (* c c) a) (* b b)) -0.375 (* -0.5 c)))
  b))

C

double code(double a, double b, double c) {
	return fma((((a * a) * ((c * c) * c)) / ((b * b) * (b * b))), -0.5625, fma((((c * c) * a) / (b * b)), -0.375, (-0.5 * c))) / b;
}

Julia

function code(a, b, c)
	return Float64(fma(Float64(Float64(Float64(a * a) * Float64(Float64(c * c) * c)) / Float64(Float64(b * b) * Float64(b * b))), -0.5625, fma(Float64(Float64(Float64(c * c) * a) / Float64(b * b)), -0.375, Float64(-0.5 * c))) / b)
end

Wolfram

code[a_, b_, c_] := N[(N[(N[(N[(N[(a * a), $MachinePrecision] * N[(N[(c * c), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision] / N[(N[(b * b), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.5625 + N[(N[(N[(N[(c * c), $MachinePrecision] * a), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision] * -0.375 + N[(-0.5 * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]

Derivation

1. Initial program 31.8%

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

2. Taylor expanded in b around inf

   \[\leadsto \color{blue}{\frac{\frac{-9}{16} \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(\frac{-1}{2} \cdot c + \frac{-3}{8} \cdot \frac{a \cdot {c}^{2}}{{b}^{2}}\right)}{b}} \]
3. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{\frac{-9}{16} \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(\frac{-1}{2} \cdot c + \frac{-3}{8} \cdot \frac{a \cdot {c}^{2}}{{b}^{2}}\right)}{\color{blue}{b}} \]

4. Applied rewrites 93.7%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}, -0.5625, \mathsf{fma}\left(\frac{\left(c \cdot c\right) \cdot a}{b \cdot b}, -0.375, -0.5 \cdot c\right)\right)}{b}} \]
5. Add Preprocessing

Alternative 4: 93.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(c, c \cdot \frac{\mathsf{fma}\left(\left(a \cdot a\right) \cdot \frac{c}{b \cdot b}, -0.5625, -0.375 \cdot a\right)}{b \cdot b}, c \cdot -0.5\right)}{b} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/
  (fma
   c
   (* c (/ (fma (* (* a a) (/ c (* b b))) -0.5625 (* -0.375 a)) (* b b)))
   (* c -0.5))
  b))

C

double code(double a, double b, double c) {
	return fma(c, (c * (fma(((a * a) * (c / (b * b))), -0.5625, (-0.375 * a)) / (b * b))), (c * -0.5)) / b;
}

Julia

function code(a, b, c)
	return Float64(fma(c, Float64(c * Float64(fma(Float64(Float64(a * a) * Float64(c / Float64(b * b))), -0.5625, Float64(-0.375 * a)) / Float64(b * b))), Float64(c * -0.5)) / b)
end

Wolfram

code[a_, b_, c_] := N[(N[(c * N[(c * N[(N[(N[(N[(a * a), $MachinePrecision] * N[(c / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.5625 + N[(-0.375 * a), $MachinePrecision]), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(c * -0.5), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]

Derivation

1. Initial program 31.8%

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

2. Taylor expanded in b around inf

   \[\leadsto \color{blue}{\frac{\frac{-9}{16} \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(\frac{-1}{2} \cdot c + \left(\frac{-3}{8} \cdot \frac{a \cdot {c}^{2}}{{b}^{2}} + \frac{-1}{6} \cdot \frac{\frac{81}{64} \cdot \left({a}^{4} \cdot {c}^{4}\right) + \frac{81}{16} \cdot \left({a}^{4} \cdot {c}^{4}\right)}{a \cdot {b}^{6}}\right)\right)}{b}} \]

4. Applied rewrites 95.4%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}, -0.5625, \mathsf{fma}\left(-0.5, c, \mathsf{fma}\left(\frac{\left(\left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right) \cdot \left(\left(c \cdot c\right) \cdot \left(c \cdot c\right)\right)\right) \cdot 6.328125}{\left(\left(\left(b \cdot b\right) \cdot b\right) \cdot \left(\left(b \cdot b\right) \cdot b\right)\right) \cdot a}, -0.16666666666666666, \frac{-0.375 \cdot \left(\left(c \cdot c\right) \cdot a\right)}{b \cdot b}\right)\right)\right)}{b}} \]

5. Taylor expanded in c around 0

   \[\leadsto \frac{c \cdot \left(c \cdot \left(\frac{-3}{8} \cdot \frac{a}{{b}^{2}} + c \cdot \left(\frac{-135}{128} \cdot \frac{{a}^{3} \cdot c}{{b}^{6}} + \frac{-9}{16} \cdot \frac{{a}^{2}}{{b}^{4}}\right)\right) - \frac{1}{2}\right)}{b} \]

7. Applied rewrites 95.3%

   \[\leadsto \frac{c \cdot \mathsf{fma}\left(c, \mathsf{fma}\left(-0.375, \frac{a}{b \cdot b}, c \cdot \mathsf{fma}\left(-1.0546875, \frac{\left(\left(a \cdot a\right) \cdot a\right) \cdot c}{\left(\left(b \cdot b\right) \cdot b\right) \cdot \left(\left(b \cdot b\right) \cdot b\right)}, -0.5625 \cdot \frac{a \cdot a}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}\right)\right), -0.5\right)}{b} \]

9. Applied rewrites 95.4%

   \[\leadsto \frac{\mathsf{fma}\left(c, c \cdot \mathsf{fma}\left(-0.375, \frac{a}{b \cdot b}, c \cdot \mathsf{fma}\left(-1.0546875, \frac{\left(\left(a \cdot a\right) \cdot a\right) \cdot c}{\left(\left(b \cdot b\right) \cdot b\right) \cdot \left(\left(b \cdot b\right) \cdot b\right)}, -0.5625 \cdot \frac{a \cdot a}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}\right)\right), c \cdot -0.5\right)}{b} \]

10. Taylor expanded in b around inf

    \[\leadsto \frac{\mathsf{fma}\left(c, c \cdot \frac{\frac{-9}{16} \cdot \frac{{a}^{2} \cdot c}{{b}^{2}} + \frac{-3}{8} \cdot a}{{b}^{2}}, c \cdot \frac{-1}{2}\right)}{b} \]
11. Step-by-step derivation

    1. lower-/.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(c, c \cdot \frac{\frac{-9}{16} \cdot \frac{{a}^{2} \cdot c}{{b}^{2}} + \frac{-3}{8} \cdot a}{{b}^{2}}, c \cdot \frac{-1}{2}\right)}{b} \]

    2. *-commutative N/A

       \[\leadsto \frac{\mathsf{fma}\left(c, c \cdot \frac{\frac{{a}^{2} \cdot c}{{b}^{2}} \cdot \frac{-9}{16} + \frac{-3}{8} \cdot a}{{b}^{2}}, c \cdot \frac{-1}{2}\right)}{b} \]

    3. lower-fma.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(c, c \cdot \frac{\mathsf{fma}\left(\frac{{a}^{2} \cdot c}{{b}^{2}}, \frac{-9}{16}, \frac{-3}{8} \cdot a\right)}{{b}^{2}}, c \cdot \frac{-1}{2}\right)}{b} \]

    4. associate-/l* N/A

       \[\leadsto \frac{\mathsf{fma}\left(c, c \cdot \frac{\mathsf{fma}\left({a}^{2} \cdot \frac{c}{{b}^{2}}, \frac{-9}{16}, \frac{-3}{8} \cdot a\right)}{{b}^{2}}, c \cdot \frac{-1}{2}\right)}{b} \]

    5. lower-*.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(c, c \cdot \frac{\mathsf{fma}\left({a}^{2} \cdot \frac{c}{{b}^{2}}, \frac{-9}{16}, \frac{-3}{8} \cdot a\right)}{{b}^{2}}, c \cdot \frac{-1}{2}\right)}{b} \]

    6. pow2 N/A

       \[\leadsto \frac{\mathsf{fma}\left(c, c \cdot \frac{\mathsf{fma}\left(\left(a \cdot a\right) \cdot \frac{c}{{b}^{2}}, \frac{-9}{16}, \frac{-3}{8} \cdot a\right)}{{b}^{2}}, c \cdot \frac{-1}{2}\right)}{b} \]

    7. lift-*.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(c, c \cdot \frac{\mathsf{fma}\left(\left(a \cdot a\right) \cdot \frac{c}{{b}^{2}}, \frac{-9}{16}, \frac{-3}{8} \cdot a\right)}{{b}^{2}}, c \cdot \frac{-1}{2}\right)}{b} \]

    8. lower-/.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(c, c \cdot \frac{\mathsf{fma}\left(\left(a \cdot a\right) \cdot \frac{c}{{b}^{2}}, \frac{-9}{16}, \frac{-3}{8} \cdot a\right)}{{b}^{2}}, c \cdot \frac{-1}{2}\right)}{b} \]

    9. pow2 N/A

       \[\leadsto \frac{\mathsf{fma}\left(c, c \cdot \frac{\mathsf{fma}\left(\left(a \cdot a\right) \cdot \frac{c}{b \cdot b}, \frac{-9}{16}, \frac{-3}{8} \cdot a\right)}{{b}^{2}}, c \cdot \frac{-1}{2}\right)}{b} \]

    10. lift-*.f64 N/A

        \[\leadsto \frac{\mathsf{fma}\left(c, c \cdot \frac{\mathsf{fma}\left(\left(a \cdot a\right) \cdot \frac{c}{b \cdot b}, \frac{-9}{16}, \frac{-3}{8} \cdot a\right)}{{b}^{2}}, c \cdot \frac{-1}{2}\right)}{b} \]

    11. lower-*.f64 N/A

        \[\leadsto \frac{\mathsf{fma}\left(c, c \cdot \frac{\mathsf{fma}\left(\left(a \cdot a\right) \cdot \frac{c}{b \cdot b}, \frac{-9}{16}, \frac{-3}{8} \cdot a\right)}{{b}^{2}}, c \cdot \frac{-1}{2}\right)}{b} \]

    12. pow2 N/A

        \[\leadsto \frac{\mathsf{fma}\left(c, c \cdot \frac{\mathsf{fma}\left(\left(a \cdot a\right) \cdot \frac{c}{b \cdot b}, \frac{-9}{16}, \frac{-3}{8} \cdot a\right)}{b \cdot b}, c \cdot \frac{-1}{2}\right)}{b} \]

    13. lift-*.f64 93.7

        \[\leadsto \frac{\mathsf{fma}\left(c, c \cdot \frac{\mathsf{fma}\left(\left(a \cdot a\right) \cdot \frac{c}{b \cdot b}, -0.5625, -0.375 \cdot a\right)}{b \cdot b}, c \cdot -0.5\right)}{b} \]

12. Applied rewrites 93.7%

    \[\leadsto \frac{\mathsf{fma}\left(c, c \cdot \frac{\mathsf{fma}\left(\left(a \cdot a\right) \cdot \frac{c}{b \cdot b}, -0.5625, -0.375 \cdot a\right)}{b \cdot b}, c \cdot -0.5\right)}{b} \]
13. Add Preprocessing

Alternative 5: 93.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \frac{c \cdot \mathsf{fma}\left(c, \frac{\mathsf{fma}\left(-0.5625, \frac{\left(a \cdot a\right) \cdot c}{b \cdot b}, -0.375 \cdot a\right)}{b \cdot b}, -0.5\right)}{b} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/
  (*
   c
   (fma
    c
    (/ (fma -0.5625 (/ (* (* a a) c) (* b b)) (* -0.375 a)) (* b b))
    -0.5))
  b))

C

double code(double a, double b, double c) {
	return (c * fma(c, (fma(-0.5625, (((a * a) * c) / (b * b)), (-0.375 * a)) / (b * b)), -0.5)) / b;
}

Julia

function code(a, b, c)
	return Float64(Float64(c * fma(c, Float64(fma(-0.5625, Float64(Float64(Float64(a * a) * c) / Float64(b * b)), Float64(-0.375 * a)) / Float64(b * b)), -0.5)) / b)
end

Wolfram

code[a_, b_, c_] := N[(N[(c * N[(c * N[(N[(-0.5625 * N[(N[(N[(a * a), $MachinePrecision] * c), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision] + N[(-0.375 * a), $MachinePrecision]), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision] + -0.5), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]

Derivation

1. Initial program 31.8%

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

2. Taylor expanded in b around inf

   \[\leadsto \color{blue}{\frac{\frac{-9}{16} \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(\frac{-1}{2} \cdot c + \left(\frac{-3}{8} \cdot \frac{a \cdot {c}^{2}}{{b}^{2}} + \frac{-1}{6} \cdot \frac{\frac{81}{64} \cdot \left({a}^{4} \cdot {c}^{4}\right) + \frac{81}{16} \cdot \left({a}^{4} \cdot {c}^{4}\right)}{a \cdot {b}^{6}}\right)\right)}{b}} \]

4. Applied rewrites 95.4%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}, -0.5625, \mathsf{fma}\left(-0.5, c, \mathsf{fma}\left(\frac{\left(\left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right) \cdot \left(\left(c \cdot c\right) \cdot \left(c \cdot c\right)\right)\right) \cdot 6.328125}{\left(\left(\left(b \cdot b\right) \cdot b\right) \cdot \left(\left(b \cdot b\right) \cdot b\right)\right) \cdot a}, -0.16666666666666666, \frac{-0.375 \cdot \left(\left(c \cdot c\right) \cdot a\right)}{b \cdot b}\right)\right)\right)}{b}} \]

5. Taylor expanded in c around 0

   \[\leadsto \frac{c \cdot \left(c \cdot \left(\frac{-3}{8} \cdot \frac{a}{{b}^{2}} + c \cdot \left(\frac{-135}{128} \cdot \frac{{a}^{3} \cdot c}{{b}^{6}} + \frac{-9}{16} \cdot \frac{{a}^{2}}{{b}^{4}}\right)\right) - \frac{1}{2}\right)}{b} \]

7. Applied rewrites 95.3%

   \[\leadsto \frac{c \cdot \mathsf{fma}\left(c, \mathsf{fma}\left(-0.375, \frac{a}{b \cdot b}, c \cdot \mathsf{fma}\left(-1.0546875, \frac{\left(\left(a \cdot a\right) \cdot a\right) \cdot c}{\left(\left(b \cdot b\right) \cdot b\right) \cdot \left(\left(b \cdot b\right) \cdot b\right)}, -0.5625 \cdot \frac{a \cdot a}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}\right)\right), -0.5\right)}{b} \]

8. Taylor expanded in b around inf

   \[\leadsto \frac{c \cdot \mathsf{fma}\left(c, \frac{\frac{-9}{16} \cdot \frac{{a}^{2} \cdot c}{{b}^{2}} + \frac{-3}{8} \cdot a}{{b}^{2}}, \frac{-1}{2}\right)}{b} \]
9. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{c \cdot \mathsf{fma}\left(c, \frac{\frac{-9}{16} \cdot \frac{{a}^{2} \cdot c}{{b}^{2}} + \frac{-3}{8} \cdot a}{{b}^{2}}, \frac{-1}{2}\right)}{b} \]

   2. lower-fma.f64 N/A

      \[\leadsto \frac{c \cdot \mathsf{fma}\left(c, \frac{\mathsf{fma}\left(\frac{-9}{16}, \frac{{a}^{2} \cdot c}{{b}^{2}}, \frac{-3}{8} \cdot a\right)}{{b}^{2}}, \frac{-1}{2}\right)}{b} \]

   3. lower-/.f64 N/A

      \[\leadsto \frac{c \cdot \mathsf{fma}\left(c, \frac{\mathsf{fma}\left(\frac{-9}{16}, \frac{{a}^{2} \cdot c}{{b}^{2}}, \frac{-3}{8} \cdot a\right)}{{b}^{2}}, \frac{-1}{2}\right)}{b} \]

   4. lower-*.f64 N/A

      \[\leadsto \frac{c \cdot \mathsf{fma}\left(c, \frac{\mathsf{fma}\left(\frac{-9}{16}, \frac{{a}^{2} \cdot c}{{b}^{2}}, \frac{-3}{8} \cdot a\right)}{{b}^{2}}, \frac{-1}{2}\right)}{b} \]

   5. pow2 N/A

      \[\leadsto \frac{c \cdot \mathsf{fma}\left(c, \frac{\mathsf{fma}\left(\frac{-9}{16}, \frac{\left(a \cdot a\right) \cdot c}{{b}^{2}}, \frac{-3}{8} \cdot a\right)}{{b}^{2}}, \frac{-1}{2}\right)}{b} \]

   6. lift-*.f64 N/A

      \[\leadsto \frac{c \cdot \mathsf{fma}\left(c, \frac{\mathsf{fma}\left(\frac{-9}{16}, \frac{\left(a \cdot a\right) \cdot c}{{b}^{2}}, \frac{-3}{8} \cdot a\right)}{{b}^{2}}, \frac{-1}{2}\right)}{b} \]

   7. pow2 N/A

      \[\leadsto \frac{c \cdot \mathsf{fma}\left(c, \frac{\mathsf{fma}\left(\frac{-9}{16}, \frac{\left(a \cdot a\right) \cdot c}{b \cdot b}, \frac{-3}{8} \cdot a\right)}{{b}^{2}}, \frac{-1}{2}\right)}{b} \]

   8. lift-*.f64 N/A

      \[\leadsto \frac{c \cdot \mathsf{fma}\left(c, \frac{\mathsf{fma}\left(\frac{-9}{16}, \frac{\left(a \cdot a\right) \cdot c}{b \cdot b}, \frac{-3}{8} \cdot a\right)}{{b}^{2}}, \frac{-1}{2}\right)}{b} \]

   9. lower-*.f64 N/A

      \[\leadsto \frac{c \cdot \mathsf{fma}\left(c, \frac{\mathsf{fma}\left(\frac{-9}{16}, \frac{\left(a \cdot a\right) \cdot c}{b \cdot b}, \frac{-3}{8} \cdot a\right)}{{b}^{2}}, \frac{-1}{2}\right)}{b} \]

   10. pow2 N/A

       \[\leadsto \frac{c \cdot \mathsf{fma}\left(c, \frac{\mathsf{fma}\left(\frac{-9}{16}, \frac{\left(a \cdot a\right) \cdot c}{b \cdot b}, \frac{-3}{8} \cdot a\right)}{b \cdot b}, \frac{-1}{2}\right)}{b} \]

   11. lift-*.f64 93.6

       \[\leadsto \frac{c \cdot \mathsf{fma}\left(c, \frac{\mathsf{fma}\left(-0.5625, \frac{\left(a \cdot a\right) \cdot c}{b \cdot b}, -0.375 \cdot a\right)}{b \cdot b}, -0.5\right)}{b} \]

10. Applied rewrites 93.6%

    \[\leadsto \frac{c \cdot \mathsf{fma}\left(c, \frac{\mathsf{fma}\left(-0.5625, \frac{\left(a \cdot a\right) \cdot c}{b \cdot b}, -0.375 \cdot a\right)}{b \cdot b}, -0.5\right)}{b} \]
11. Add Preprocessing

Alternative 6: 90.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(\frac{\left(c \cdot c\right) \cdot a}{b \cdot b}, -0.375, -0.5 \cdot c\right)}{b} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (fma (/ (* (* c c) a) (* b b)) -0.375 (* -0.5 c)) b))

C

double code(double a, double b, double c) {
	return fma((((c * c) * a) / (b * b)), -0.375, (-0.5 * c)) / b;
}

Julia

function code(a, b, c)
	return Float64(fma(Float64(Float64(Float64(c * c) * a) / Float64(b * b)), -0.375, Float64(-0.5 * c)) / b)
end

Wolfram

code[a_, b_, c_] := N[(N[(N[(N[(N[(c * c), $MachinePrecision] * a), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision] * -0.375 + N[(-0.5 * c), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]

Derivation

1. Initial program 31.8%

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

2. Taylor expanded in b around inf

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

   1. lower-/.f64 N/A

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

   2. +-commutative N/A

      \[\leadsto \frac{\frac{-3}{8} \cdot \frac{a \cdot {c}^{2}}{{b}^{2}} + \frac{-1}{2} \cdot c}{b} \]

   3. *-commutative N/A

      \[\leadsto \frac{\frac{a \cdot {c}^{2}}{{b}^{2}} \cdot \frac{-3}{8} + \frac{-1}{2} \cdot c}{b} \]

   4. lower-fma.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{a \cdot {c}^{2}}{{b}^{2}}, \frac{-3}{8}, \frac{-1}{2} \cdot c\right)}{b} \]

   5. lower-/.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{a \cdot {c}^{2}}{{b}^{2}}, \frac{-3}{8}, \frac{-1}{2} \cdot c\right)}{b} \]

   6. *-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{{c}^{2} \cdot a}{{b}^{2}}, \frac{-3}{8}, \frac{-1}{2} \cdot c\right)}{b} \]

   7. lower-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{{c}^{2} \cdot a}{{b}^{2}}, \frac{-3}{8}, \frac{-1}{2} \cdot c\right)}{b} \]

   8. unpow2 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(c \cdot c\right) \cdot a}{{b}^{2}}, \frac{-3}{8}, \frac{-1}{2} \cdot c\right)}{b} \]

   9. lower-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(c \cdot c\right) \cdot a}{{b}^{2}}, \frac{-3}{8}, \frac{-1}{2} \cdot c\right)}{b} \]

   10. pow2 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(c \cdot c\right) \cdot a}{b \cdot b}, \frac{-3}{8}, \frac{-1}{2} \cdot c\right)}{b} \]

   11. lift-*.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(c \cdot c\right) \cdot a}{b \cdot b}, \frac{-3}{8}, \frac{-1}{2} \cdot c\right)}{b} \]

   12. lower-*.f64 90.4

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

4. Applied rewrites 90.4%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{\left(c \cdot c\right) \cdot a}{b \cdot b}, -0.375, -0.5 \cdot c\right)}{b}} \]
5. Add Preprocessing

Alternative 7: 90.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \frac{c \cdot \mathsf{fma}\left(-0.375, \frac{a \cdot c}{b \cdot b}, -0.5\right)}{b} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (* c (fma -0.375 (/ (* a c) (* b b)) -0.5)) b))

C

double code(double a, double b, double c) {
	return (c * fma(-0.375, ((a * c) / (b * b)), -0.5)) / b;
}

Julia

function code(a, b, c)
	return Float64(Float64(c * fma(-0.375, Float64(Float64(a * c) / Float64(b * b)), -0.5)) / b)
end

Wolfram

code[a_, b_, c_] := N[(N[(c * N[(-0.375 * N[(N[(a * c), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision] + -0.5), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]

Derivation

1. Initial program 31.8%

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

2. Taylor expanded in b around inf

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

   1. lower-/.f64 N/A

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

   2. +-commutative N/A

      \[\leadsto \frac{\frac{-3}{8} \cdot \frac{a \cdot {c}^{2}}{{b}^{2}} + \frac{-1}{2} \cdot c}{b} \]

   3. *-commutative N/A

      \[\leadsto \frac{\frac{a \cdot {c}^{2}}{{b}^{2}} \cdot \frac{-3}{8} + \frac{-1}{2} \cdot c}{b} \]

   4. lower-fma.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{a \cdot {c}^{2}}{{b}^{2}}, \frac{-3}{8}, \frac{-1}{2} \cdot c\right)}{b} \]

   5. lower-/.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{a \cdot {c}^{2}}{{b}^{2}}, \frac{-3}{8}, \frac{-1}{2} \cdot c\right)}{b} \]

   6. *-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{{c}^{2} \cdot a}{{b}^{2}}, \frac{-3}{8}, \frac{-1}{2} \cdot c\right)}{b} \]

   7. lower-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{{c}^{2} \cdot a}{{b}^{2}}, \frac{-3}{8}, \frac{-1}{2} \cdot c\right)}{b} \]

   8. unpow2 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(c \cdot c\right) \cdot a}{{b}^{2}}, \frac{-3}{8}, \frac{-1}{2} \cdot c\right)}{b} \]

   9. lower-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(c \cdot c\right) \cdot a}{{b}^{2}}, \frac{-3}{8}, \frac{-1}{2} \cdot c\right)}{b} \]

   10. pow2 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(c \cdot c\right) \cdot a}{b \cdot b}, \frac{-3}{8}, \frac{-1}{2} \cdot c\right)}{b} \]

   11. lift-*.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(c \cdot c\right) \cdot a}{b \cdot b}, \frac{-3}{8}, \frac{-1}{2} \cdot c\right)}{b} \]

   12. lower-*.f64 90.4

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

4. Applied rewrites 90.4%

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

5. Taylor expanded in c around 0

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

   1. lower-*.f64 N/A

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

   2. negate-sub N/A

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

   3. metadata-eval N/A

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

   4. lower-fma.f64 N/A

      \[\leadsto \frac{c \cdot \mathsf{fma}\left(\frac{-3}{8}, \frac{a \cdot c}{{b}^{2}}, \frac{-1}{2}\right)}{b} \]

   5. lower-/.f64 N/A

      \[\leadsto \frac{c \cdot \mathsf{fma}\left(\frac{-3}{8}, \frac{a \cdot c}{{b}^{2}}, \frac{-1}{2}\right)}{b} \]

   6. lower-*.f64 N/A

      \[\leadsto \frac{c \cdot \mathsf{fma}\left(\frac{-3}{8}, \frac{a \cdot c}{{b}^{2}}, \frac{-1}{2}\right)}{b} \]

   7. pow2 N/A

      \[\leadsto \frac{c \cdot \mathsf{fma}\left(\frac{-3}{8}, \frac{a \cdot c}{b \cdot b}, \frac{-1}{2}\right)}{b} \]

   8. lift-*.f64 90.3

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

7. Applied rewrites 90.3%

   \[\leadsto \frac{c \cdot \mathsf{fma}\left(-0.375, \frac{a \cdot c}{b \cdot b}, -0.5\right)}{b} \]
8. Add Preprocessing

Alternative 8: 80.9% accurate, 3.3× speedup

Math

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

FPCore

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

C

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

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 = (c / b) * (-0.5d0)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 31.8%

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

2. Taylor expanded in a around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-/.f64 80.9

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

4. Applied rewrites 80.9%

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

Reproduce

herbie shell --seed 2025119 
(FPCore (a b c)
  :name "Cubic critical, medium range"
  :precision binary64
  :pre (and (and (and (< 1.1102230246251565e-16 a) (< a 9007199254740992.0)) (and (< 1.1102230246251565e-16 b) (< b 9007199254740992.0))) (and (< 1.1102230246251565e-16 c) (< c 9007199254740992.0)))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))