Cubic critical, wide range

Percentage Accurate: 18.1% → 97.7%

Time: 6.2s

Alternatives: 9

Speedup: 3.3×

Specification

\[\left(\left(4.930380657631324 \cdot 10^{-32} < a \land a < 2.028240960365167 \cdot 10^{+31}\right) \land \left(4.930380657631324 \cdot 10^{-32} < b \land b < 2.028240960365167 \cdot 10^{+31}\right)\right) \land \left(4.930380657631324 \cdot 10^{-32} < c \land c < 2.028240960365167 \cdot 10^{+31}\right)\]

Math

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

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: 18.1% accurate, 1.0× speedup

Math

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

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: 97.7% accurate, 0.2× speedup

Math

\[\frac{\mathsf{fma}\left(\left(\left(\left(\left({b}^{-4} \cdot c\right) \cdot a\right) \cdot a\right) \cdot c\right) \cdot c, -0.5625, \mathsf{fma}\left(-0.5, c, \mathsf{fma}\left(\left(\frac{c}{b \cdot b} \cdot c\right) \cdot -0.375, a, \frac{-1.0546875}{{b}^{6} \cdot a} \cdot {\left(c \cdot a\right)}^{4}\right)\right)\right)}{b} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/
  (fma
   (* (* (* (* (* (pow b -4.0) c) a) a) c) c)
   -0.5625
   (fma
    -0.5
    c
    (fma
     (* (* (/ c (* b b)) c) -0.375)
     a
     (* (/ -1.0546875 (* (pow b 6.0) a)) (pow (* c a) 4.0)))))
  b))

C

double code(double a, double b, double c) {
	return fma((((((pow(b, -4.0) * c) * a) * a) * c) * c), -0.5625, fma(-0.5, c, fma((((c / (b * b)) * c) * -0.375), a, ((-1.0546875 / (pow(b, 6.0) * a)) * pow((c * a), 4.0))))) / b;
}

Julia

function code(a, b, c)
	return Float64(fma(Float64(Float64(Float64(Float64(Float64((b ^ -4.0) * c) * a) * a) * c) * c), -0.5625, fma(-0.5, c, fma(Float64(Float64(Float64(c / Float64(b * b)) * c) * -0.375), a, Float64(Float64(-1.0546875 / Float64((b ^ 6.0) * a)) * (Float64(c * a) ^ 4.0))))) / b)
end

Wolfram

code[a_, b_, c_] := N[(N[(N[(N[(N[(N[(N[(N[Power[b, -4.0], $MachinePrecision] * c), $MachinePrecision] * a), $MachinePrecision] * a), $MachinePrecision] * c), $MachinePrecision] * c), $MachinePrecision] * -0.5625 + N[(-0.5 * c + N[(N[(N[(N[(c / N[(b * b), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision] * -0.375), $MachinePrecision] * a + N[(N[(-1.0546875 / N[(N[Power[b, 6.0], $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] * N[Power[N[(c * a), $MachinePrecision], 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]

Derivation

1. Initial program 18.1%

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

2. Taylor expanded in b around inf

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

   1. lower-/.f64 N/A

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

4. Applied rewrites 97.7%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.5625, \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}}, \mathsf{fma}\left(-0.5, c, \mathsf{fma}\left(-0.375, \frac{a \cdot {c}^{2}}{{b}^{2}}, -0.16666666666666666 \cdot \frac{\mathsf{fma}\left(1.265625, {a}^{4} \cdot {c}^{4}, 5.0625 \cdot \left({a}^{4} \cdot {c}^{4}\right)\right)}{a \cdot {b}^{6}}\right)\right)\right)}{b}} \]

6. Applied rewrites 97.7%

   \[\leadsto \frac{\mathsf{fma}\left({b}^{-4} \cdot \left(\left(\left(a \cdot a\right) \cdot c\right) \cdot \left(c \cdot c\right)\right), -0.5625, \mathsf{fma}\left(-0.375 \cdot a, c \cdot \frac{c}{b \cdot b}, \mathsf{fma}\left({\left(c \cdot a\right)}^{4} \cdot \frac{6.328125}{{b}^{6} \cdot a}, -0.16666666666666666, -0.5 \cdot c\right)\right)\right)}{b} \]
7. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left({b}^{-4} \cdot \left(\left(\left(a \cdot a\right) \cdot c\right) \cdot \left(c \cdot c\right)\right), \frac{-9}{16}, \mathsf{fma}\left(\frac{-3}{8} \cdot a, c \cdot \frac{c}{b \cdot b}, \mathsf{fma}\left({\left(c \cdot a\right)}^{4} \cdot \frac{\frac{405}{64}}{{b}^{6} \cdot a}, \frac{-1}{6}, \frac{-1}{2} \cdot c\right)\right)\right)}{b} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left({b}^{-4} \cdot \left(\left(\left(a \cdot a\right) \cdot c\right) \cdot \left(c \cdot c\right)\right), \frac{-9}{16}, \mathsf{fma}\left(\frac{-3}{8} \cdot a, c \cdot \frac{c}{b \cdot b}, \mathsf{fma}\left({\left(c \cdot a\right)}^{4} \cdot \frac{\frac{405}{64}}{{b}^{6} \cdot a}, \frac{-1}{6}, \frac{-1}{2} \cdot c\right)\right)\right)}{b} \]

   3. associate-*r* N/A

      \[\leadsto \frac{\mathsf{fma}\left(\left({b}^{-4} \cdot \left(\left(a \cdot a\right) \cdot c\right)\right) \cdot \left(c \cdot c\right), \frac{-9}{16}, \mathsf{fma}\left(\frac{-3}{8} \cdot a, c \cdot \frac{c}{b \cdot b}, \mathsf{fma}\left({\left(c \cdot a\right)}^{4} \cdot \frac{\frac{405}{64}}{{b}^{6} \cdot a}, \frac{-1}{6}, \frac{-1}{2} \cdot c\right)\right)\right)}{b} \]

   4. lift-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\left({b}^{-4} \cdot \left(\left(a \cdot a\right) \cdot c\right)\right) \cdot \left(c \cdot c\right), \frac{-9}{16}, \mathsf{fma}\left(\frac{-3}{8} \cdot a, c \cdot \frac{c}{b \cdot b}, \mathsf{fma}\left({\left(c \cdot a\right)}^{4} \cdot \frac{\frac{405}{64}}{{b}^{6} \cdot a}, \frac{-1}{6}, \frac{-1}{2} \cdot c\right)\right)\right)}{b} \]

   5. associate-*r* N/A

      \[\leadsto \frac{\mathsf{fma}\left(\left(\left({b}^{-4} \cdot \left(\left(a \cdot a\right) \cdot c\right)\right) \cdot c\right) \cdot c, \frac{-9}{16}, \mathsf{fma}\left(\frac{-3}{8} \cdot a, c \cdot \frac{c}{b \cdot b}, \mathsf{fma}\left({\left(c \cdot a\right)}^{4} \cdot \frac{\frac{405}{64}}{{b}^{6} \cdot a}, \frac{-1}{6}, \frac{-1}{2} \cdot c\right)\right)\right)}{b} \]

   6. lower-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\left(\left({b}^{-4} \cdot \left(\left(a \cdot a\right) \cdot c\right)\right) \cdot c\right) \cdot c, \frac{-9}{16}, \mathsf{fma}\left(\frac{-3}{8} \cdot a, c \cdot \frac{c}{b \cdot b}, \mathsf{fma}\left({\left(c \cdot a\right)}^{4} \cdot \frac{\frac{405}{64}}{{b}^{6} \cdot a}, \frac{-1}{6}, \frac{-1}{2} \cdot c\right)\right)\right)}{b} \]

   7. lower-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\left(\left({b}^{-4} \cdot \left(\left(a \cdot a\right) \cdot c\right)\right) \cdot c\right) \cdot c, \frac{-9}{16}, \mathsf{fma}\left(\frac{-3}{8} \cdot a, c \cdot \frac{c}{b \cdot b}, \mathsf{fma}\left({\left(c \cdot a\right)}^{4} \cdot \frac{\frac{405}{64}}{{b}^{6} \cdot a}, \frac{-1}{6}, \frac{-1}{2} \cdot c\right)\right)\right)}{b} \]

   8. lift-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\left(\left({b}^{-4} \cdot \left(\left(a \cdot a\right) \cdot c\right)\right) \cdot c\right) \cdot c, \frac{-9}{16}, \mathsf{fma}\left(\frac{-3}{8} \cdot a, c \cdot \frac{c}{b \cdot b}, \mathsf{fma}\left({\left(c \cdot a\right)}^{4} \cdot \frac{\frac{405}{64}}{{b}^{6} \cdot a}, \frac{-1}{6}, \frac{-1}{2} \cdot c\right)\right)\right)}{b} \]

   9. *-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left(\left(\left({b}^{-4} \cdot \left(c \cdot \left(a \cdot a\right)\right)\right) \cdot c\right) \cdot c, \frac{-9}{16}, \mathsf{fma}\left(\frac{-3}{8} \cdot a, c \cdot \frac{c}{b \cdot b}, \mathsf{fma}\left({\left(c \cdot a\right)}^{4} \cdot \frac{\frac{405}{64}}{{b}^{6} \cdot a}, \frac{-1}{6}, \frac{-1}{2} \cdot c\right)\right)\right)}{b} \]

   10. associate-*r* N/A

       \[\leadsto \frac{\mathsf{fma}\left(\left(\left(\left({b}^{-4} \cdot c\right) \cdot \left(a \cdot a\right)\right) \cdot c\right) \cdot c, \frac{-9}{16}, \mathsf{fma}\left(\frac{-3}{8} \cdot a, c \cdot \frac{c}{b \cdot b}, \mathsf{fma}\left({\left(c \cdot a\right)}^{4} \cdot \frac{\frac{405}{64}}{{b}^{6} \cdot a}, \frac{-1}{6}, \frac{-1}{2} \cdot c\right)\right)\right)}{b} \]

   11. lower-*.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\left(\left(\left({b}^{-4} \cdot c\right) \cdot \left(a \cdot a\right)\right) \cdot c\right) \cdot c, \frac{-9}{16}, \mathsf{fma}\left(\frac{-3}{8} \cdot a, c \cdot \frac{c}{b \cdot b}, \mathsf{fma}\left({\left(c \cdot a\right)}^{4} \cdot \frac{\frac{405}{64}}{{b}^{6} \cdot a}, \frac{-1}{6}, \frac{-1}{2} \cdot c\right)\right)\right)}{b} \]

   12. lower-*.f64 97.7

       \[\leadsto \frac{\mathsf{fma}\left(\left(\left(\left({b}^{-4} \cdot c\right) \cdot \left(a \cdot a\right)\right) \cdot c\right) \cdot c, -0.5625, \mathsf{fma}\left(-0.375 \cdot a, c \cdot \frac{c}{b \cdot b}, \mathsf{fma}\left({\left(c \cdot a\right)}^{4} \cdot \frac{6.328125}{{b}^{6} \cdot a}, -0.16666666666666666, -0.5 \cdot c\right)\right)\right)}{b} \]

   13. lift-fma.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\left(\left(\left({b}^{-4} \cdot c\right) \cdot \left(a \cdot a\right)\right) \cdot c\right) \cdot c, \frac{-9}{16}, \left(\frac{-3}{8} \cdot a\right) \cdot \left(c \cdot \frac{c}{b \cdot b}\right) + \mathsf{fma}\left({\left(c \cdot a\right)}^{4} \cdot \frac{\frac{405}{64}}{{b}^{6} \cdot a}, \frac{-1}{6}, \frac{-1}{2} \cdot c\right)\right)}{b} \]

8. Applied rewrites 97.7%

   \[\leadsto \frac{\mathsf{fma}\left(\left(\left(\left({b}^{-4} \cdot c\right) \cdot \left(a \cdot a\right)\right) \cdot c\right) \cdot c, -0.5625, \mathsf{fma}\left(-0.5, c, \mathsf{fma}\left(\left(\frac{c}{b \cdot b} \cdot c\right) \cdot -0.375, a, \frac{-1.0546875}{{b}^{6} \cdot a} \cdot {\left(c \cdot a\right)}^{4}\right)\right)\right)}{b} \]
9. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*r* N/A

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

   4. lower-*.f64 N/A

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

   5. lower-*.f64 97.7

      \[\leadsto \frac{\mathsf{fma}\left(\left(\left(\left(\left({b}^{-4} \cdot c\right) \cdot a\right) \cdot a\right) \cdot c\right) \cdot c, -0.5625, \mathsf{fma}\left(-0.5, c, \mathsf{fma}\left(\left(\frac{c}{b \cdot b} \cdot c\right) \cdot -0.375, a, \frac{-1.0546875}{{b}^{6} \cdot a} \cdot {\left(c \cdot a\right)}^{4}\right)\right)\right)}{b} \]

10. Applied rewrites 97.7%

    \[\leadsto \frac{\mathsf{fma}\left(\left(\left(\left(\left({b}^{-4} \cdot c\right) \cdot a\right) \cdot a\right) \cdot c\right) \cdot c, -0.5625, \mathsf{fma}\left(-0.5, c, \mathsf{fma}\left(\left(\frac{c}{b \cdot b} \cdot c\right) \cdot -0.375, a, \frac{-1.0546875}{{b}^{6} \cdot a} \cdot {\left(c \cdot a\right)}^{4}\right)\right)\right)}{b} \]
11. Add Preprocessing

Alternative 2: 97.6% accurate, 0.2× speedup

Math

\[\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{c}{b \cdot b} \cdot a, -0.375, -0.5\right), c, \mathsf{fma}\left({\left(c \cdot a\right)}^{4} \cdot -1.0546875, \frac{{b}^{-6}}{a}, -0.5625 \cdot \left(\left(\left(\left(\left({b}^{-4} \cdot c\right) \cdot a\right) \cdot a\right) \cdot c\right) \cdot c\right)\right)\right)}{b} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/
  (fma
   (fma (* (/ c (* b b)) a) -0.375 -0.5)
   c
   (fma
    (* (pow (* c a) 4.0) -1.0546875)
    (/ (pow b -6.0) a)
    (* -0.5625 (* (* (* (* (* (pow b -4.0) c) a) a) c) c))))
  b))

C

double code(double a, double b, double c) {
	return fma(fma(((c / (b * b)) * a), -0.375, -0.5), c, fma((pow((c * a), 4.0) * -1.0546875), (pow(b, -6.0) / a), (-0.5625 * (((((pow(b, -4.0) * c) * a) * a) * c) * c)))) / b;
}

Julia

function code(a, b, c)
	return Float64(fma(fma(Float64(Float64(c / Float64(b * b)) * a), -0.375, -0.5), c, fma(Float64((Float64(c * a) ^ 4.0) * -1.0546875), Float64((b ^ -6.0) / a), Float64(-0.5625 * Float64(Float64(Float64(Float64(Float64((b ^ -4.0) * c) * a) * a) * c) * c)))) / b)
end

Wolfram

code[a_, b_, c_] := N[(N[(N[(N[(N[(c / N[(b * b), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision] * -0.375 + -0.5), $MachinePrecision] * c + N[(N[(N[Power[N[(c * a), $MachinePrecision], 4.0], $MachinePrecision] * -1.0546875), $MachinePrecision] * N[(N[Power[b, -6.0], $MachinePrecision] / a), $MachinePrecision] + N[(-0.5625 * N[(N[(N[(N[(N[(N[Power[b, -4.0], $MachinePrecision] * c), $MachinePrecision] * a), $MachinePrecision] * a), $MachinePrecision] * c), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]

Derivation

1. Initial program 18.1%

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

2. Taylor expanded in b around inf

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

   1. lower-/.f64 N/A

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

4. Applied rewrites 97.7%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.5625, \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}}, \mathsf{fma}\left(-0.5, c, \mathsf{fma}\left(-0.375, \frac{a \cdot {c}^{2}}{{b}^{2}}, -0.16666666666666666 \cdot \frac{\mathsf{fma}\left(1.265625, {a}^{4} \cdot {c}^{4}, 5.0625 \cdot \left({a}^{4} \cdot {c}^{4}\right)\right)}{a \cdot {b}^{6}}\right)\right)\right)}{b}} \]

6. Applied rewrites 97.7%

   \[\leadsto \frac{\mathsf{fma}\left({b}^{-4} \cdot \left(\left(\left(a \cdot a\right) \cdot c\right) \cdot \left(c \cdot c\right)\right), -0.5625, \mathsf{fma}\left(-0.375 \cdot a, c \cdot \frac{c}{b \cdot b}, \mathsf{fma}\left({\left(c \cdot a\right)}^{4} \cdot \frac{6.328125}{{b}^{6} \cdot a}, -0.16666666666666666, -0.5 \cdot c\right)\right)\right)}{b} \]
7. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left({b}^{-4} \cdot \left(\left(\left(a \cdot a\right) \cdot c\right) \cdot \left(c \cdot c\right)\right), \frac{-9}{16}, \mathsf{fma}\left(\frac{-3}{8} \cdot a, c \cdot \frac{c}{b \cdot b}, \mathsf{fma}\left({\left(c \cdot a\right)}^{4} \cdot \frac{\frac{405}{64}}{{b}^{6} \cdot a}, \frac{-1}{6}, \frac{-1}{2} \cdot c\right)\right)\right)}{b} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left({b}^{-4} \cdot \left(\left(\left(a \cdot a\right) \cdot c\right) \cdot \left(c \cdot c\right)\right), \frac{-9}{16}, \mathsf{fma}\left(\frac{-3}{8} \cdot a, c \cdot \frac{c}{b \cdot b}, \mathsf{fma}\left({\left(c \cdot a\right)}^{4} \cdot \frac{\frac{405}{64}}{{b}^{6} \cdot a}, \frac{-1}{6}, \frac{-1}{2} \cdot c\right)\right)\right)}{b} \]

   3. associate-*r* N/A

      \[\leadsto \frac{\mathsf{fma}\left(\left({b}^{-4} \cdot \left(\left(a \cdot a\right) \cdot c\right)\right) \cdot \left(c \cdot c\right), \frac{-9}{16}, \mathsf{fma}\left(\frac{-3}{8} \cdot a, c \cdot \frac{c}{b \cdot b}, \mathsf{fma}\left({\left(c \cdot a\right)}^{4} \cdot \frac{\frac{405}{64}}{{b}^{6} \cdot a}, \frac{-1}{6}, \frac{-1}{2} \cdot c\right)\right)\right)}{b} \]

   4. lift-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\left({b}^{-4} \cdot \left(\left(a \cdot a\right) \cdot c\right)\right) \cdot \left(c \cdot c\right), \frac{-9}{16}, \mathsf{fma}\left(\frac{-3}{8} \cdot a, c \cdot \frac{c}{b \cdot b}, \mathsf{fma}\left({\left(c \cdot a\right)}^{4} \cdot \frac{\frac{405}{64}}{{b}^{6} \cdot a}, \frac{-1}{6}, \frac{-1}{2} \cdot c\right)\right)\right)}{b} \]

   5. associate-*r* N/A

      \[\leadsto \frac{\mathsf{fma}\left(\left(\left({b}^{-4} \cdot \left(\left(a \cdot a\right) \cdot c\right)\right) \cdot c\right) \cdot c, \frac{-9}{16}, \mathsf{fma}\left(\frac{-3}{8} \cdot a, c \cdot \frac{c}{b \cdot b}, \mathsf{fma}\left({\left(c \cdot a\right)}^{4} \cdot \frac{\frac{405}{64}}{{b}^{6} \cdot a}, \frac{-1}{6}, \frac{-1}{2} \cdot c\right)\right)\right)}{b} \]

   6. lower-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\left(\left({b}^{-4} \cdot \left(\left(a \cdot a\right) \cdot c\right)\right) \cdot c\right) \cdot c, \frac{-9}{16}, \mathsf{fma}\left(\frac{-3}{8} \cdot a, c \cdot \frac{c}{b \cdot b}, \mathsf{fma}\left({\left(c \cdot a\right)}^{4} \cdot \frac{\frac{405}{64}}{{b}^{6} \cdot a}, \frac{-1}{6}, \frac{-1}{2} \cdot c\right)\right)\right)}{b} \]

   7. lower-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\left(\left({b}^{-4} \cdot \left(\left(a \cdot a\right) \cdot c\right)\right) \cdot c\right) \cdot c, \frac{-9}{16}, \mathsf{fma}\left(\frac{-3}{8} \cdot a, c \cdot \frac{c}{b \cdot b}, \mathsf{fma}\left({\left(c \cdot a\right)}^{4} \cdot \frac{\frac{405}{64}}{{b}^{6} \cdot a}, \frac{-1}{6}, \frac{-1}{2} \cdot c\right)\right)\right)}{b} \]

   8. lift-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\left(\left({b}^{-4} \cdot \left(\left(a \cdot a\right) \cdot c\right)\right) \cdot c\right) \cdot c, \frac{-9}{16}, \mathsf{fma}\left(\frac{-3}{8} \cdot a, c \cdot \frac{c}{b \cdot b}, \mathsf{fma}\left({\left(c \cdot a\right)}^{4} \cdot \frac{\frac{405}{64}}{{b}^{6} \cdot a}, \frac{-1}{6}, \frac{-1}{2} \cdot c\right)\right)\right)}{b} \]

   9. *-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left(\left(\left({b}^{-4} \cdot \left(c \cdot \left(a \cdot a\right)\right)\right) \cdot c\right) \cdot c, \frac{-9}{16}, \mathsf{fma}\left(\frac{-3}{8} \cdot a, c \cdot \frac{c}{b \cdot b}, \mathsf{fma}\left({\left(c \cdot a\right)}^{4} \cdot \frac{\frac{405}{64}}{{b}^{6} \cdot a}, \frac{-1}{6}, \frac{-1}{2} \cdot c\right)\right)\right)}{b} \]

   10. associate-*r* N/A

       \[\leadsto \frac{\mathsf{fma}\left(\left(\left(\left({b}^{-4} \cdot c\right) \cdot \left(a \cdot a\right)\right) \cdot c\right) \cdot c, \frac{-9}{16}, \mathsf{fma}\left(\frac{-3}{8} \cdot a, c \cdot \frac{c}{b \cdot b}, \mathsf{fma}\left({\left(c \cdot a\right)}^{4} \cdot \frac{\frac{405}{64}}{{b}^{6} \cdot a}, \frac{-1}{6}, \frac{-1}{2} \cdot c\right)\right)\right)}{b} \]

   11. lower-*.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\left(\left(\left({b}^{-4} \cdot c\right) \cdot \left(a \cdot a\right)\right) \cdot c\right) \cdot c, \frac{-9}{16}, \mathsf{fma}\left(\frac{-3}{8} \cdot a, c \cdot \frac{c}{b \cdot b}, \mathsf{fma}\left({\left(c \cdot a\right)}^{4} \cdot \frac{\frac{405}{64}}{{b}^{6} \cdot a}, \frac{-1}{6}, \frac{-1}{2} \cdot c\right)\right)\right)}{b} \]

   12. lower-*.f64 97.7

       \[\leadsto \frac{\mathsf{fma}\left(\left(\left(\left({b}^{-4} \cdot c\right) \cdot \left(a \cdot a\right)\right) \cdot c\right) \cdot c, -0.5625, \mathsf{fma}\left(-0.375 \cdot a, c \cdot \frac{c}{b \cdot b}, \mathsf{fma}\left({\left(c \cdot a\right)}^{4} \cdot \frac{6.328125}{{b}^{6} \cdot a}, -0.16666666666666666, -0.5 \cdot c\right)\right)\right)}{b} \]

   13. lift-fma.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\left(\left(\left({b}^{-4} \cdot c\right) \cdot \left(a \cdot a\right)\right) \cdot c\right) \cdot c, \frac{-9}{16}, \left(\frac{-3}{8} \cdot a\right) \cdot \left(c \cdot \frac{c}{b \cdot b}\right) + \mathsf{fma}\left({\left(c \cdot a\right)}^{4} \cdot \frac{\frac{405}{64}}{{b}^{6} \cdot a}, \frac{-1}{6}, \frac{-1}{2} \cdot c\right)\right)}{b} \]

8. Applied rewrites 97.7%

   \[\leadsto \frac{\mathsf{fma}\left(\left(\left(\left({b}^{-4} \cdot c\right) \cdot \left(a \cdot a\right)\right) \cdot c\right) \cdot c, -0.5625, \mathsf{fma}\left(-0.5, c, \mathsf{fma}\left(\left(\frac{c}{b \cdot b} \cdot c\right) \cdot -0.375, a, \frac{-1.0546875}{{b}^{6} \cdot a} \cdot {\left(c \cdot a\right)}^{4}\right)\right)\right)}{b} \]

10. Applied rewrites 97.6%

    \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{c}{b \cdot b} \cdot a, -0.375, -0.5\right), c, \mathsf{fma}\left({\left(c \cdot a\right)}^{4} \cdot -1.0546875, \frac{{b}^{-6}}{a}, -0.5625 \cdot \left(\left(\left(\left(\left({b}^{-4} \cdot c\right) \cdot a\right) \cdot a\right) \cdot c\right) \cdot c\right)\right)\right)}{b} \]
11. Add Preprocessing

Alternative 3: 96.9% accurate, 0.3× speedup

Math

\[\mathsf{fma}\left(-0.5, \frac{c}{b}, a \cdot \mathsf{fma}\left(-0.5625, \frac{a \cdot {c}^{3}}{{b}^{5}}, -0.375 \cdot \frac{{c}^{2}}{{b}^{3}}\right)\right) \]

FPCore

(FPCore (a b c)
 :precision binary64
 (fma
  -0.5
  (/ c b)
  (*
   a
   (fma
    -0.5625
    (/ (* a (pow c 3.0)) (pow b 5.0))
    (* -0.375 (/ (pow c 2.0) (pow b 3.0)))))))

C

double code(double a, double b, double c) {
	return fma(-0.5, (c / b), (a * fma(-0.5625, ((a * pow(c, 3.0)) / pow(b, 5.0)), (-0.375 * (pow(c, 2.0) / pow(b, 3.0))))));
}

Julia

function code(a, b, c)
	return fma(-0.5, Float64(c / b), Float64(a * fma(-0.5625, Float64(Float64(a * (c ^ 3.0)) / (b ^ 5.0)), Float64(-0.375 * Float64((c ^ 2.0) / (b ^ 3.0))))))
end

Wolfram

code[a_, b_, c_] := N[(-0.5 * N[(c / b), $MachinePrecision] + N[(a * N[(-0.5625 * N[(N[(a * N[Power[c, 3.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision] + N[(-0.375 * N[(N[Power[c, 2.0], $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 18.1%

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

2. Taylor expanded in b around inf

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

   1. lower-/.f64 N/A

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

4. Applied rewrites 97.7%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.5625, \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}}, \mathsf{fma}\left(-0.5, c, \mathsf{fma}\left(-0.375, \frac{a \cdot {c}^{2}}{{b}^{2}}, -0.16666666666666666 \cdot \frac{\mathsf{fma}\left(1.265625, {a}^{4} \cdot {c}^{4}, 5.0625 \cdot \left({a}^{4} \cdot {c}^{4}\right)\right)}{a \cdot {b}^{6}}\right)\right)\right)}{b}} \]

5. Taylor expanded in a around 0

   \[\leadsto \frac{-1}{2} \cdot \frac{c}{b} + \color{blue}{a \cdot \left(\frac{-9}{16} \cdot \frac{a \cdot {c}^{3}}{{b}^{5}} + \frac{-3}{8} \cdot \frac{{c}^{2}}{{b}^{3}}\right)} \]
6. Step-by-step derivation

   1. lower-fma.f64 N/A

      \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{c}{\color{blue}{b}}, a \cdot \left(\frac{-9}{16} \cdot \frac{a \cdot {c}^{3}}{{b}^{5}} + \frac{-3}{8} \cdot \frac{{c}^{2}}{{b}^{3}}\right)\right) \]

   2. lower-/.f64 N/A

      \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{c}{b}, a \cdot \left(\frac{-9}{16} \cdot \frac{a \cdot {c}^{3}}{{b}^{5}} + \frac{-3}{8} \cdot \frac{{c}^{2}}{{b}^{3}}\right)\right) \]

   3. lower-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{c}{b}, a \cdot \left(\frac{-9}{16} \cdot \frac{a \cdot {c}^{3}}{{b}^{5}} + \frac{-3}{8} \cdot \frac{{c}^{2}}{{b}^{3}}\right)\right) \]

   4. lower-fma.f64 N/A

      \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{c}{b}, a \cdot \mathsf{fma}\left(\frac{-9}{16}, \frac{a \cdot {c}^{3}}{{b}^{5}}, \frac{-3}{8} \cdot \frac{{c}^{2}}{{b}^{3}}\right)\right) \]

   5. lower-/.f64 N/A

      \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{c}{b}, a \cdot \mathsf{fma}\left(\frac{-9}{16}, \frac{a \cdot {c}^{3}}{{b}^{5}}, \frac{-3}{8} \cdot \frac{{c}^{2}}{{b}^{3}}\right)\right) \]

   6. lower-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{c}{b}, a \cdot \mathsf{fma}\left(\frac{-9}{16}, \frac{a \cdot {c}^{3}}{{b}^{5}}, \frac{-3}{8} \cdot \frac{{c}^{2}}{{b}^{3}}\right)\right) \]

   7. lower-pow.f64 N/A

      \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{c}{b}, a \cdot \mathsf{fma}\left(\frac{-9}{16}, \frac{a \cdot {c}^{3}}{{b}^{5}}, \frac{-3}{8} \cdot \frac{{c}^{2}}{{b}^{3}}\right)\right) \]

   8. lower-pow.f64 N/A

      \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{c}{b}, a \cdot \mathsf{fma}\left(\frac{-9}{16}, \frac{a \cdot {c}^{3}}{{b}^{5}}, \frac{-3}{8} \cdot \frac{{c}^{2}}{{b}^{3}}\right)\right) \]

   9. lower-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{c}{b}, a \cdot \mathsf{fma}\left(\frac{-9}{16}, \frac{a \cdot {c}^{3}}{{b}^{5}}, \frac{-3}{8} \cdot \frac{{c}^{2}}{{b}^{3}}\right)\right) \]

   10. lower-/.f64 N/A

       \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{c}{b}, a \cdot \mathsf{fma}\left(\frac{-9}{16}, \frac{a \cdot {c}^{3}}{{b}^{5}}, \frac{-3}{8} \cdot \frac{{c}^{2}}{{b}^{3}}\right)\right) \]

7. Applied rewrites 96.9%

   \[\leadsto \mathsf{fma}\left(-0.5, \color{blue}{\frac{c}{b}}, a \cdot \mathsf{fma}\left(-0.5625, \frac{a \cdot {c}^{3}}{{b}^{5}}, -0.375 \cdot \frac{{c}^{2}}{{b}^{3}}\right)\right) \]
8. Add Preprocessing

Alternative 4: 96.9% accurate, 0.3× speedup

Math

\[\frac{\mathsf{fma}\left(\left(\left(\left({b}^{-4} \cdot c\right) \cdot \left(a \cdot a\right)\right) \cdot c\right) \cdot c, -0.5625, \mathsf{fma}\left(-0.5, c, -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}}\right)\right)}{b} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/
  (fma
   (* (* (* (* (pow b -4.0) c) (* a a)) c) c)
   -0.5625
   (fma -0.5 c (* -0.375 (/ (* a (pow c 2.0)) (pow b 2.0)))))
  b))

C

double code(double a, double b, double c) {
	return fma(((((pow(b, -4.0) * c) * (a * a)) * c) * c), -0.5625, fma(-0.5, c, (-0.375 * ((a * pow(c, 2.0)) / pow(b, 2.0))))) / b;
}

Julia

function code(a, b, c)
	return Float64(fma(Float64(Float64(Float64(Float64((b ^ -4.0) * c) * Float64(a * a)) * c) * c), -0.5625, fma(-0.5, c, Float64(-0.375 * Float64(Float64(a * (c ^ 2.0)) / (b ^ 2.0))))) / b)
end

Wolfram

code[a_, b_, c_] := N[(N[(N[(N[(N[(N[(N[Power[b, -4.0], $MachinePrecision] * c), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision] * c), $MachinePrecision] * -0.5625 + N[(-0.5 * c + N[(-0.375 * N[(N[(a * N[Power[c, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]

Derivation

1. Initial program 18.1%

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

2. Taylor expanded in b around inf

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

   1. lower-/.f64 N/A

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

4. Applied rewrites 97.7%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.5625, \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}}, \mathsf{fma}\left(-0.5, c, \mathsf{fma}\left(-0.375, \frac{a \cdot {c}^{2}}{{b}^{2}}, -0.16666666666666666 \cdot \frac{\mathsf{fma}\left(1.265625, {a}^{4} \cdot {c}^{4}, 5.0625 \cdot \left({a}^{4} \cdot {c}^{4}\right)\right)}{a \cdot {b}^{6}}\right)\right)\right)}{b}} \]

6. Applied rewrites 97.7%

   \[\leadsto \frac{\mathsf{fma}\left({b}^{-4} \cdot \left(\left(\left(a \cdot a\right) \cdot c\right) \cdot \left(c \cdot c\right)\right), -0.5625, \mathsf{fma}\left(-0.375 \cdot a, c \cdot \frac{c}{b \cdot b}, \mathsf{fma}\left({\left(c \cdot a\right)}^{4} \cdot \frac{6.328125}{{b}^{6} \cdot a}, -0.16666666666666666, -0.5 \cdot c\right)\right)\right)}{b} \]
7. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left({b}^{-4} \cdot \left(\left(\left(a \cdot a\right) \cdot c\right) \cdot \left(c \cdot c\right)\right), \frac{-9}{16}, \mathsf{fma}\left(\frac{-3}{8} \cdot a, c \cdot \frac{c}{b \cdot b}, \mathsf{fma}\left({\left(c \cdot a\right)}^{4} \cdot \frac{\frac{405}{64}}{{b}^{6} \cdot a}, \frac{-1}{6}, \frac{-1}{2} \cdot c\right)\right)\right)}{b} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left({b}^{-4} \cdot \left(\left(\left(a \cdot a\right) \cdot c\right) \cdot \left(c \cdot c\right)\right), \frac{-9}{16}, \mathsf{fma}\left(\frac{-3}{8} \cdot a, c \cdot \frac{c}{b \cdot b}, \mathsf{fma}\left({\left(c \cdot a\right)}^{4} \cdot \frac{\frac{405}{64}}{{b}^{6} \cdot a}, \frac{-1}{6}, \frac{-1}{2} \cdot c\right)\right)\right)}{b} \]

   3. associate-*r* N/A

      \[\leadsto \frac{\mathsf{fma}\left(\left({b}^{-4} \cdot \left(\left(a \cdot a\right) \cdot c\right)\right) \cdot \left(c \cdot c\right), \frac{-9}{16}, \mathsf{fma}\left(\frac{-3}{8} \cdot a, c \cdot \frac{c}{b \cdot b}, \mathsf{fma}\left({\left(c \cdot a\right)}^{4} \cdot \frac{\frac{405}{64}}{{b}^{6} \cdot a}, \frac{-1}{6}, \frac{-1}{2} \cdot c\right)\right)\right)}{b} \]

   4. lift-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\left({b}^{-4} \cdot \left(\left(a \cdot a\right) \cdot c\right)\right) \cdot \left(c \cdot c\right), \frac{-9}{16}, \mathsf{fma}\left(\frac{-3}{8} \cdot a, c \cdot \frac{c}{b \cdot b}, \mathsf{fma}\left({\left(c \cdot a\right)}^{4} \cdot \frac{\frac{405}{64}}{{b}^{6} \cdot a}, \frac{-1}{6}, \frac{-1}{2} \cdot c\right)\right)\right)}{b} \]

   5. associate-*r* N/A

      \[\leadsto \frac{\mathsf{fma}\left(\left(\left({b}^{-4} \cdot \left(\left(a \cdot a\right) \cdot c\right)\right) \cdot c\right) \cdot c, \frac{-9}{16}, \mathsf{fma}\left(\frac{-3}{8} \cdot a, c \cdot \frac{c}{b \cdot b}, \mathsf{fma}\left({\left(c \cdot a\right)}^{4} \cdot \frac{\frac{405}{64}}{{b}^{6} \cdot a}, \frac{-1}{6}, \frac{-1}{2} \cdot c\right)\right)\right)}{b} \]

   6. lower-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\left(\left({b}^{-4} \cdot \left(\left(a \cdot a\right) \cdot c\right)\right) \cdot c\right) \cdot c, \frac{-9}{16}, \mathsf{fma}\left(\frac{-3}{8} \cdot a, c \cdot \frac{c}{b \cdot b}, \mathsf{fma}\left({\left(c \cdot a\right)}^{4} \cdot \frac{\frac{405}{64}}{{b}^{6} \cdot a}, \frac{-1}{6}, \frac{-1}{2} \cdot c\right)\right)\right)}{b} \]

   7. lower-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\left(\left({b}^{-4} \cdot \left(\left(a \cdot a\right) \cdot c\right)\right) \cdot c\right) \cdot c, \frac{-9}{16}, \mathsf{fma}\left(\frac{-3}{8} \cdot a, c \cdot \frac{c}{b \cdot b}, \mathsf{fma}\left({\left(c \cdot a\right)}^{4} \cdot \frac{\frac{405}{64}}{{b}^{6} \cdot a}, \frac{-1}{6}, \frac{-1}{2} \cdot c\right)\right)\right)}{b} \]

   8. lift-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\left(\left({b}^{-4} \cdot \left(\left(a \cdot a\right) \cdot c\right)\right) \cdot c\right) \cdot c, \frac{-9}{16}, \mathsf{fma}\left(\frac{-3}{8} \cdot a, c \cdot \frac{c}{b \cdot b}, \mathsf{fma}\left({\left(c \cdot a\right)}^{4} \cdot \frac{\frac{405}{64}}{{b}^{6} \cdot a}, \frac{-1}{6}, \frac{-1}{2} \cdot c\right)\right)\right)}{b} \]

   9. *-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left(\left(\left({b}^{-4} \cdot \left(c \cdot \left(a \cdot a\right)\right)\right) \cdot c\right) \cdot c, \frac{-9}{16}, \mathsf{fma}\left(\frac{-3}{8} \cdot a, c \cdot \frac{c}{b \cdot b}, \mathsf{fma}\left({\left(c \cdot a\right)}^{4} \cdot \frac{\frac{405}{64}}{{b}^{6} \cdot a}, \frac{-1}{6}, \frac{-1}{2} \cdot c\right)\right)\right)}{b} \]

   10. associate-*r* N/A

       \[\leadsto \frac{\mathsf{fma}\left(\left(\left(\left({b}^{-4} \cdot c\right) \cdot \left(a \cdot a\right)\right) \cdot c\right) \cdot c, \frac{-9}{16}, \mathsf{fma}\left(\frac{-3}{8} \cdot a, c \cdot \frac{c}{b \cdot b}, \mathsf{fma}\left({\left(c \cdot a\right)}^{4} \cdot \frac{\frac{405}{64}}{{b}^{6} \cdot a}, \frac{-1}{6}, \frac{-1}{2} \cdot c\right)\right)\right)}{b} \]

   11. lower-*.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\left(\left(\left({b}^{-4} \cdot c\right) \cdot \left(a \cdot a\right)\right) \cdot c\right) \cdot c, \frac{-9}{16}, \mathsf{fma}\left(\frac{-3}{8} \cdot a, c \cdot \frac{c}{b \cdot b}, \mathsf{fma}\left({\left(c \cdot a\right)}^{4} \cdot \frac{\frac{405}{64}}{{b}^{6} \cdot a}, \frac{-1}{6}, \frac{-1}{2} \cdot c\right)\right)\right)}{b} \]

   12. lower-*.f64 97.7

       \[\leadsto \frac{\mathsf{fma}\left(\left(\left(\left({b}^{-4} \cdot c\right) \cdot \left(a \cdot a\right)\right) \cdot c\right) \cdot c, -0.5625, \mathsf{fma}\left(-0.375 \cdot a, c \cdot \frac{c}{b \cdot b}, \mathsf{fma}\left({\left(c \cdot a\right)}^{4} \cdot \frac{6.328125}{{b}^{6} \cdot a}, -0.16666666666666666, -0.5 \cdot c\right)\right)\right)}{b} \]

   13. lift-fma.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\left(\left(\left({b}^{-4} \cdot c\right) \cdot \left(a \cdot a\right)\right) \cdot c\right) \cdot c, \frac{-9}{16}, \left(\frac{-3}{8} \cdot a\right) \cdot \left(c \cdot \frac{c}{b \cdot b}\right) + \mathsf{fma}\left({\left(c \cdot a\right)}^{4} \cdot \frac{\frac{405}{64}}{{b}^{6} \cdot a}, \frac{-1}{6}, \frac{-1}{2} \cdot c\right)\right)}{b} \]

8. Applied rewrites 97.7%

   \[\leadsto \frac{\mathsf{fma}\left(\left(\left(\left({b}^{-4} \cdot c\right) \cdot \left(a \cdot a\right)\right) \cdot c\right) \cdot c, -0.5625, \mathsf{fma}\left(-0.5, c, \mathsf{fma}\left(\left(\frac{c}{b \cdot b} \cdot c\right) \cdot -0.375, a, \frac{-1.0546875}{{b}^{6} \cdot a} \cdot {\left(c \cdot a\right)}^{4}\right)\right)\right)}{b} \]

9. Taylor expanded in a around 0

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

    1. lower-*.f64 N/A

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

    2. lower-/.f64 N/A

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

    3. lower-*.f64 N/A

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

    4. lower-pow.f64 N/A

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

    5. lower-pow.f64 96.9

       \[\leadsto \frac{\mathsf{fma}\left(\left(\left(\left({b}^{-4} \cdot c\right) \cdot \left(a \cdot a\right)\right) \cdot c\right) \cdot c, -0.5625, \mathsf{fma}\left(-0.5, c, -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}}\right)\right)}{b} \]

11. Applied rewrites 96.9%

    \[\leadsto \frac{\mathsf{fma}\left(\left(\left(\left({b}^{-4} \cdot c\right) \cdot \left(a \cdot a\right)\right) \cdot c\right) \cdot c, -0.5625, \mathsf{fma}\left(-0.5, c, -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}}\right)\right)}{b} \]
12. Add Preprocessing

Alternative 5: 96.8% accurate, 0.3× speedup

Math

\[\frac{c \cdot \left(c \cdot \mathsf{fma}\left(-0.5625, \frac{{a}^{2} \cdot c}{{b}^{4}}, -0.375 \cdot \frac{a}{{b}^{2}}\right) - 0.5\right)}{b} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/
  (*
   c
   (-
    (*
     c
     (fma
      -0.5625
      (/ (* (pow a 2.0) c) (pow b 4.0))
      (* -0.375 (/ a (pow b 2.0)))))
    0.5))
  b))

C

double code(double a, double b, double c) {
	return (c * ((c * fma(-0.5625, ((pow(a, 2.0) * c) / pow(b, 4.0)), (-0.375 * (a / pow(b, 2.0))))) - 0.5)) / b;
}

Julia

function code(a, b, c)
	return Float64(Float64(c * Float64(Float64(c * fma(-0.5625, Float64(Float64((a ^ 2.0) * c) / (b ^ 4.0)), Float64(-0.375 * Float64(a / (b ^ 2.0))))) - 0.5)) / b)
end

Wolfram

code[a_, b_, c_] := N[(N[(c * N[(N[(c * N[(-0.5625 * N[(N[(N[Power[a, 2.0], $MachinePrecision] * c), $MachinePrecision] / N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision] + N[(-0.375 * N[(a / N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]

Derivation

1. Initial program 18.1%

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

2. Taylor expanded in b around inf

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

   1. lower-/.f64 N/A

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

4. Applied rewrites 97.7%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.5625, \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}}, \mathsf{fma}\left(-0.5, c, \mathsf{fma}\left(-0.375, \frac{a \cdot {c}^{2}}{{b}^{2}}, -0.16666666666666666 \cdot \frac{\mathsf{fma}\left(1.265625, {a}^{4} \cdot {c}^{4}, 5.0625 \cdot \left({a}^{4} \cdot {c}^{4}\right)\right)}{a \cdot {b}^{6}}\right)\right)\right)}{b}} \]

5. Taylor expanded in c around 0

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

   1. lower-*.f64 N/A

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

   2. lower--.f64 N/A

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

7. Applied rewrites 96.8%

   \[\leadsto \frac{c \cdot \left(c \cdot \mathsf{fma}\left(-0.5625, \frac{{a}^{2} \cdot c}{{b}^{4}}, -0.375 \cdot \frac{a}{{b}^{2}}\right) - 0.5\right)}{b} \]
8. Add Preprocessing

Alternative 6: 95.2% accurate, 0.5× speedup

Math

\[\mathsf{fma}\left(-0.5, \frac{c}{b}, -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}\right) \]

FPCore

(FPCore (a b c)
 :precision binary64
 (fma -0.5 (/ c b) (* -0.375 (/ (* a (pow c 2.0)) (pow b 3.0)))))

C

double code(double a, double b, double c) {
	return fma(-0.5, (c / b), (-0.375 * ((a * pow(c, 2.0)) / pow(b, 3.0))));
}

Julia

function code(a, b, c)
	return fma(-0.5, Float64(c / b), Float64(-0.375 * Float64(Float64(a * (c ^ 2.0)) / (b ^ 3.0))))
end

Wolfram

code[a_, b_, c_] := N[(-0.5 * N[(c / b), $MachinePrecision] + N[(-0.375 * N[(N[(a * N[Power[c, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 18.1%

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

2. Taylor expanded in b around inf

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

   1. lower-/.f64 N/A

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

4. Applied rewrites 97.7%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.5625, \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}}, \mathsf{fma}\left(-0.5, c, \mathsf{fma}\left(-0.375, \frac{a \cdot {c}^{2}}{{b}^{2}}, -0.16666666666666666 \cdot \frac{\mathsf{fma}\left(1.265625, {a}^{4} \cdot {c}^{4}, 5.0625 \cdot \left({a}^{4} \cdot {c}^{4}\right)\right)}{a \cdot {b}^{6}}\right)\right)\right)}{b}} \]

5. Taylor expanded in a around 0

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

   1. lower-fma.f64 N/A

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

   2. lower-/.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. lower-pow.f64 N/A

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

   7. lower-pow.f64 95.2

      \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{b}, -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}\right) \]

7. Applied rewrites 95.2%

   \[\leadsto \mathsf{fma}\left(-0.5, \color{blue}{\frac{c}{b}}, -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}\right) \]
8. Add Preprocessing

Alternative 7: 95.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 18.1%

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

2. Taylor expanded in b around inf

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

   1. lower-/.f64 N/A

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

4. Applied rewrites 97.7%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.5625, \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}}, \mathsf{fma}\left(-0.5, c, \mathsf{fma}\left(-0.375, \frac{a \cdot {c}^{2}}{{b}^{2}}, -0.16666666666666666 \cdot \frac{\mathsf{fma}\left(1.265625, {a}^{4} \cdot {c}^{4}, 5.0625 \cdot \left({a}^{4} \cdot {c}^{4}\right)\right)}{a \cdot {b}^{6}}\right)\right)\right)}{b}} \]

6. Applied rewrites 97.7%

   \[\leadsto \frac{\mathsf{fma}\left({b}^{-4} \cdot \left(\left(\left(a \cdot a\right) \cdot c\right) \cdot \left(c \cdot c\right)\right), -0.5625, \mathsf{fma}\left(-0.375 \cdot a, c \cdot \frac{c}{b \cdot b}, \mathsf{fma}\left({\left(c \cdot a\right)}^{4} \cdot \frac{6.328125}{{b}^{6} \cdot a}, -0.16666666666666666, -0.5 \cdot c\right)\right)\right)}{b} \]

7. Taylor expanded in c around 0

   \[\leadsto \frac{c \cdot \left(\frac{-3}{8} \cdot \frac{a \cdot c}{{b}^{2}} - \frac{1}{2}\right)}{b} \]
8. 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. lower--.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. lower-pow.f64 95.2

      \[\leadsto \frac{c \cdot \left(-0.375 \cdot \frac{a \cdot c}{{b}^{2}} - 0.5\right)}{b} \]

9. Applied rewrites 95.2%

   \[\leadsto \frac{c \cdot \left(-0.375 \cdot \frac{a \cdot c}{{b}^{2}} - 0.5\right)}{b} \]
10. Step-by-step derivation

    1. lift-*.f64 N/A

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

    2. lift--.f64 N/A

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

    3. sub-flip N/A

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

    4. metadata-eval N/A

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

    5. distribute-rgt-in N/A

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

11. Applied rewrites 95.2%

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

Alternative 8: 95.2% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 18.1%

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

2. Taylor expanded in b around inf

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

   1. lower-/.f64 N/A

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

4. Applied rewrites 97.7%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.5625, \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}}, \mathsf{fma}\left(-0.5, c, \mathsf{fma}\left(-0.375, \frac{a \cdot {c}^{2}}{{b}^{2}}, -0.16666666666666666 \cdot \frac{\mathsf{fma}\left(1.265625, {a}^{4} \cdot {c}^{4}, 5.0625 \cdot \left({a}^{4} \cdot {c}^{4}\right)\right)}{a \cdot {b}^{6}}\right)\right)\right)}{b}} \]

6. Applied rewrites 97.7%

   \[\leadsto \frac{\mathsf{fma}\left({b}^{-4} \cdot \left(\left(\left(a \cdot a\right) \cdot c\right) \cdot \left(c \cdot c\right)\right), -0.5625, \mathsf{fma}\left(-0.375 \cdot a, c \cdot \frac{c}{b \cdot b}, \mathsf{fma}\left({\left(c \cdot a\right)}^{4} \cdot \frac{6.328125}{{b}^{6} \cdot a}, -0.16666666666666666, -0.5 \cdot c\right)\right)\right)}{b} \]

7. Taylor expanded in c around 0

   \[\leadsto \frac{c \cdot \left(\frac{-3}{8} \cdot \frac{a \cdot c}{{b}^{2}} - \frac{1}{2}\right)}{b} \]
8. 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. lower--.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. lower-pow.f64 95.2

      \[\leadsto \frac{c \cdot \left(-0.375 \cdot \frac{a \cdot c}{{b}^{2}} - 0.5\right)}{b} \]

9. Applied rewrites 95.2%

   \[\leadsto \frac{c \cdot \left(-0.375 \cdot \frac{a \cdot c}{{b}^{2}} - 0.5\right)}{b} \]
10. Step-by-step derivation

11. Applied rewrites 95.2%

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

Alternative 9: 90.2% accurate, 3.3× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 18.1%

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

2. Taylor expanded in b around inf

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

   1. lower-*.f64 N/A

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

   2. lower-/.f64 90.2

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

4. Applied rewrites 90.2%

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

Reproduce

herbie shell --seed 2025173 
(FPCore (a b c)
  :name "Cubic critical, wide range"
  :precision binary64
  :pre (and (and (and (< 4.930380657631324e-32 a) (< a 2.028240960365167e+31)) (and (< 4.930380657631324e-32 b) (< b 2.028240960365167e+31))) (and (< 4.930380657631324e-32 c) (< c 2.028240960365167e+31)))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))