Cubic critical, medium range

Percentage Accurate: 31.3% → 95.6%

Time: 5.6s

Alternatives: 11

Speedup: 2.9×

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: 31.3% 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: 95.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}, -0.5625, \mathsf{fma}\left(-0.5, c, \left(c \cdot c\right) \cdot \mathsf{fma}\left(-1.0546875, \frac{{a}^{3} \cdot \left(c \cdot c\right)}{{b}^{6}}, -0.375 \cdot \frac{a}{b \cdot b}\right)\right)\right)}{b} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/
  (fma
   (/ (* (* a a) (* (* c c) c)) (* (* b b) (* b b)))
   -0.5625
   (fma
    -0.5
    c
    (*
     (* c c)
     (fma
      -1.0546875
      (/ (* (pow a 3.0) (* c c)) (pow b 6.0))
      (* -0.375 (/ a (* b b)))))))
  b))

C

double code(double a, double b, double c) {
	return fma((((a * a) * ((c * c) * c)) / ((b * b) * (b * b))), -0.5625, fma(-0.5, c, ((c * c) * fma(-1.0546875, ((pow(a, 3.0) * (c * c)) / pow(b, 6.0)), (-0.375 * (a / (b * b))))))) / 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(-0.5, c, Float64(Float64(c * c) * fma(-1.0546875, Float64(Float64((a ^ 3.0) * Float64(c * c)) / (b ^ 6.0)), Float64(-0.375 * Float64(a / Float64(b * b))))))) / 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[(-0.5 * c + N[(N[(c * c), $MachinePrecision] * N[(-1.0546875 * N[(N[(N[Power[a, 3.0], $MachinePrecision] * N[(c * c), $MachinePrecision]), $MachinePrecision] / N[Power[b, 6.0], $MachinePrecision]), $MachinePrecision] + N[(-0.375 * N[(a / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]

Derivation

1. Initial program 31.4%

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

3. Taylor expanded in b around inf

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

5. Applied rewrites 94.0%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot {c}^{3}}{{b}^{4}}, -0.5625, \mathsf{fma}\left(-0.5, c, \mathsf{fma}\left(\frac{\frac{{\left(c \cdot a\right)}^{4} \cdot 6.328125}{a}}{{b}^{6}}, -0.16666666666666666, \frac{-0.375 \cdot \left(\left(c \cdot c\right) \cdot a\right)}{b \cdot b}\right)\right)\right)}{b}} \]
6. Step-by-step derivation

   1. lift-pow.f64 N/A

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

   2. unpow3 N/A

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

   3. pow2 N/A

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

   4. lower-*.f64 N/A

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

   5. pow2 N/A

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

   6. lift-*.f64 94.0

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

7. Applied rewrites 94.0%

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

   1. lift-pow.f64 N/A

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

   2. sqr-pow N/A

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

   3. metadata-eval N/A

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

   4. metadata-eval N/A

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

   5. lower-*.f64 N/A

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

   6. pow2 N/A

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

   7. lift-*.f64 N/A

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

   8. pow2 N/A

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

   9. lift-*.f64 94.0

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

9. Applied rewrites 94.0%

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

10. Taylor expanded in c around 0

    \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}, \frac{-9}{16}, \mathsf{fma}\left(\frac{-1}{2}, c, {c}^{2} \cdot \left(\frac{-135}{128} \cdot \frac{{a}^{3} \cdot {c}^{2}}{{b}^{6}} + \frac{-3}{8} \cdot \frac{a}{{b}^{2}}\right)\right)\right)}{b} \]
11. Step-by-step derivation

    1. lower-*.f64 N/A

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

    2. pow2 N/A

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

    3. lift-*.f64 N/A

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

    4. lower-fma.f64 N/A

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

    5. lower-/.f64 N/A

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

    6. lower-*.f64 N/A

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

    7. lower-pow.f64 N/A

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

    8. pow2 N/A

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

    9. lift-*.f64 N/A

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

    10. lift-pow.f64 N/A

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

    11. lower-*.f64 N/A

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

    12. lower-/.f64 N/A

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

    13. pow2 N/A

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

    14. lift-*.f64 94.0

        \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}, -0.5625, \mathsf{fma}\left(-0.5, c, \left(c \cdot c\right) \cdot \mathsf{fma}\left(-1.0546875, \frac{{a}^{3} \cdot \left(c \cdot c\right)}{{b}^{6}}, -0.375 \cdot \frac{a}{b \cdot b}\right)\right)\right)}{b} \]

12. Applied rewrites 94.0%

    \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}, -0.5625, \mathsf{fma}\left(-0.5, c, \left(c \cdot c\right) \cdot \mathsf{fma}\left(-1.0546875, \frac{{a}^{3} \cdot \left(c \cdot c\right)}{{b}^{6}}, -0.375 \cdot \frac{a}{b \cdot b}\right)\right)\right)}{b} \]
13. Add Preprocessing

Alternative 2: 95.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}, -0.5625, c \cdot \left(c \cdot \mathsf{fma}\left(-1.0546875, \frac{{a}^{3} \cdot \left(c \cdot c\right)}{{b}^{6}}, -0.375 \cdot \frac{a}{b \cdot b}\right) - 0.5\right)\right)}{b} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/
  (fma
   (/ (* (* a a) (* (* c c) c)) (* (* b b) (* b b)))
   -0.5625
   (*
    c
    (-
     (*
      c
      (fma
       -1.0546875
       (/ (* (pow a 3.0) (* c c)) (pow b 6.0))
       (* -0.375 (/ a (* b b)))))
     0.5)))
  b))

C

double code(double a, double b, double c) {
	return fma((((a * a) * ((c * c) * c)) / ((b * b) * (b * b))), -0.5625, (c * ((c * fma(-1.0546875, ((pow(a, 3.0) * (c * c)) / pow(b, 6.0)), (-0.375 * (a / (b * b))))) - 0.5))) / 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, Float64(c * Float64(Float64(c * fma(-1.0546875, Float64(Float64((a ^ 3.0) * Float64(c * c)) / (b ^ 6.0)), Float64(-0.375 * Float64(a / Float64(b * b))))) - 0.5))) / 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[(c * N[(N[(c * N[(-1.0546875 * N[(N[(N[Power[a, 3.0], $MachinePrecision] * N[(c * c), $MachinePrecision]), $MachinePrecision] / N[Power[b, 6.0], $MachinePrecision]), $MachinePrecision] + N[(-0.375 * N[(a / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]

Derivation

1. Initial program 31.4%

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

3. Taylor expanded in b around inf

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

5. Applied rewrites 94.0%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot {c}^{3}}{{b}^{4}}, -0.5625, \mathsf{fma}\left(-0.5, c, \mathsf{fma}\left(\frac{\frac{{\left(c \cdot a\right)}^{4} \cdot 6.328125}{a}}{{b}^{6}}, -0.16666666666666666, \frac{-0.375 \cdot \left(\left(c \cdot c\right) \cdot a\right)}{b \cdot b}\right)\right)\right)}{b}} \]
6. Step-by-step derivation

   1. lift-pow.f64 N/A

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

   2. unpow3 N/A

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

   3. pow2 N/A

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

   4. lower-*.f64 N/A

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

   5. pow2 N/A

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

   6. lift-*.f64 94.0

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

7. Applied rewrites 94.0%

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

   1. lift-pow.f64 N/A

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

   2. sqr-pow N/A

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

   3. metadata-eval N/A

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

   4. metadata-eval N/A

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

   5. lower-*.f64 N/A

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

   6. pow2 N/A

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

   7. lift-*.f64 N/A

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

   8. pow2 N/A

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

   9. lift-*.f64 94.0

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

9. Applied rewrites 94.0%

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

10. Taylor expanded in c around 0

    \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}, \frac{-9}{16}, c \cdot \left(c \cdot \left(\frac{-135}{128} \cdot \frac{{a}^{3} \cdot {c}^{2}}{{b}^{6}} + \frac{-3}{8} \cdot \frac{a}{{b}^{2}}\right) - \frac{1}{2}\right)\right)}{b} \]
11. Step-by-step derivation

    1. lower-*.f64 N/A

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

    2. lower--.f64 N/A

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

12. Applied rewrites 93.9%

    \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}, -0.5625, c \cdot \left(c \cdot \mathsf{fma}\left(-1.0546875, \frac{{a}^{3} \cdot \left(c \cdot c\right)}{{b}^{6}}, -0.375 \cdot \frac{a}{b \cdot b}\right) - 0.5\right)\right)}{b} \]
13. Add Preprocessing

Alternative 3: 95.5% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 31.4%

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

3. Taylor expanded in b around inf

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

5. Applied rewrites 94.0%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot {c}^{3}}{{b}^{4}}, -0.5625, \mathsf{fma}\left(-0.5, c, \mathsf{fma}\left(\frac{\frac{{\left(c \cdot a\right)}^{4} \cdot 6.328125}{a}}{{b}^{6}}, -0.16666666666666666, \frac{-0.375 \cdot \left(\left(c \cdot c\right) \cdot a\right)}{b \cdot b}\right)\right)\right)}{b}} \]

6. 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. Step-by-step derivation

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

8. Applied rewrites 93.9%

   \[\leadsto \frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{a \cdot a}{{b}^{4}}, -0.5625, \frac{-1.0546875 \cdot \left({a}^{3} \cdot c\right)}{{b}^{6}}\right), c, \frac{a}{b \cdot b} \cdot -0.375\right) \cdot c - 0.5\right) \cdot c}{b} \]

9. Taylor expanded in a around 0

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

    1. *-commutative N/A

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

    2. lower-*.f64 N/A

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

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

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

    4. *-commutative N/A

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

    5. lower-fma.f64 N/A

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

    6. associate-/l* N/A

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

    7. lower-*.f64 N/A

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

    8. lower-/.f64 N/A

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

    9. lift-pow.f64 N/A

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

    10. metadata-eval N/A

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

    11. lower-*.f64 N/A

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

    12. pow-flip N/A

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

    13. lower-pow.f64 N/A

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

    14. metadata-eval N/A

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

    15. pow2 N/A

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

11. Applied rewrites 93.9%

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

Alternative 4: 91.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \leq -200:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, -3 \cdot \left(c \cdot a\right)\right)}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{c}{b}, -0.5, \frac{\frac{-0.375}{b} \cdot \frac{\left(c \cdot c\right) \cdot a}{b}}{b}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)) -200.0)
   (/ (+ (- b) (sqrt (fma b b (* -3.0 (* c a))))) (* 3.0 a))
   (fma (/ c b) -0.5 (/ (* (/ -0.375 b) (/ (* (* c c) a) b)) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a)) <= -200.0) {
		tmp = (-b + sqrt(fma(b, b, (-3.0 * (c * a))))) / (3.0 * a);
	} else {
		tmp = fma((c / b), -0.5, (((-0.375 / b) * (((c * c) * a) / b)) / b));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c)))) / Float64(3.0 * a)) <= -200.0)
		tmp = Float64(Float64(Float64(-b) + sqrt(fma(b, b, Float64(-3.0 * Float64(c * a))))) / Float64(3.0 * a));
	else
		tmp = fma(Float64(c / b), -0.5, Float64(Float64(Float64(-0.375 / b) * Float64(Float64(Float64(c * c) * a) / b)) / b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[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], -200.0], N[(N[((-b) + N[Sqrt[N[(b * b + N[(-3.0 * N[(c * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] * -0.5 + N[(N[(N[(-0.375 / b), $MachinePrecision] * N[(N[(N[(c * c), $MachinePrecision] * a), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -200

   1. Initial program 83.6%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

      6. associate-*r* N/A

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

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

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

      8. pow2 N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 84.2

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

   4. Applied rewrites 84.2%

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

   if -200 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a))

   1. Initial program 26.7%

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

   3. Taylor expanded in b around inf

      \[\leadsto \color{blue}{\frac{\frac{-1}{2} \cdot c + \frac{-3}{8} \cdot \frac{a \cdot {c}^{2}}{{b}^{2}}}{b}} \]
   4. 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 92.5

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

   5. Applied rewrites 92.5%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{\left(c \cdot c\right) \cdot a}{b \cdot b}, -0.375, -0.5 \cdot c\right)}{b}} \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. *-commutative N/A

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

      9. pow2 N/A

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

      10. lower-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. +-commutative N/A

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

      13. div-add N/A

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

      14. associate-*r/ N/A

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

      15. *-commutative N/A

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

   7. Applied rewrites 92.5%

      \[\leadsto \mathsf{fma}\left(\frac{c}{b}, \color{blue}{-0.5}, \frac{\frac{-0.375}{b} \cdot \frac{\left(c \cdot c\right) \cdot a}{b}}{b}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 91.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \leq -200:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, -3 \cdot \left(c \cdot a\right)\right)}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{\left(c \cdot c\right) \cdot a}{b \cdot b}, -0.375, -0.5 \cdot c\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)) -200.0)
   (/ (+ (- b) (sqrt (fma b b (* -3.0 (* c a))))) (* 3.0 a))
   (/ (fma (/ (* (* c c) a) (* b b)) -0.375 (* -0.5 c)) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (((-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a)) <= -200.0) {
		tmp = (-b + sqrt(fma(b, b, (-3.0 * (c * a))))) / (3.0 * a);
	} else {
		tmp = fma((((c * c) * a) / (b * b)), -0.375, (-0.5 * c)) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c)))) / Float64(3.0 * a)) <= -200.0)
		tmp = Float64(Float64(Float64(-b) + sqrt(fma(b, b, Float64(-3.0 * Float64(c * a))))) / Float64(3.0 * a));
	else
		tmp = Float64(fma(Float64(Float64(Float64(c * c) * a) / Float64(b * b)), -0.375, Float64(-0.5 * c)) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[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], -200.0], N[(N[((-b) + N[Sqrt[N[(b * b + N[(-3.0 * N[(c * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], 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. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -200

   1. Initial program 83.6%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

      6. associate-*r* N/A

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

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

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

      8. pow2 N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 84.2

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

   4. Applied rewrites 84.2%

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

   if -200 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a))

   1. Initial program 26.7%

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

   3. Taylor expanded in b around inf

      \[\leadsto \color{blue}{\frac{\frac{-1}{2} \cdot c + \frac{-3}{8} \cdot \frac{a \cdot {c}^{2}}{{b}^{2}}}{b}} \]
   4. 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 92.5

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

   5. Applied rewrites 92.5%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{\left(c \cdot c\right) \cdot a}{b \cdot b}, -0.375, -0.5 \cdot c\right)}{b}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 90.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \leq -200:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, -3 \cdot \left(c \cdot a\right)\right)}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(a \cdot \frac{c}{b \cdot b}\right) \cdot -0.375 - 0.5\right) \cdot c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)) -200.0)
   (/ (+ (- b) (sqrt (fma b b (* -3.0 (* c a))))) (* 3.0 a))
   (/ (* (- (* (* a (/ c (* b b))) -0.375) 0.5) c) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (((-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a)) <= -200.0) {
		tmp = (-b + sqrt(fma(b, b, (-3.0 * (c * a))))) / (3.0 * a);
	} else {
		tmp = ((((a * (c / (b * b))) * -0.375) - 0.5) * c) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c)))) / Float64(3.0 * a)) <= -200.0)
		tmp = Float64(Float64(Float64(-b) + sqrt(fma(b, b, Float64(-3.0 * Float64(c * a))))) / Float64(3.0 * a));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(a * Float64(c / Float64(b * b))) * -0.375) - 0.5) * c) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[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], -200.0], N[(N[((-b) + N[Sqrt[N[(b * b + N[(-3.0 * N[(c * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(a * N[(c / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.375), $MachinePrecision] - 0.5), $MachinePrecision] * c), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -200

   1. Initial program 83.6%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

      6. associate-*r* N/A

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

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

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

      8. pow2 N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 84.2

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

   4. Applied rewrites 84.2%

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

   if -200 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a))

   1. Initial program 26.7%

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

   3. Taylor expanded in b around inf

      \[\leadsto \color{blue}{\frac{\frac{-1}{2} \cdot c + \frac{-3}{8} \cdot \frac{a \cdot {c}^{2}}{{b}^{2}}}{b}} \]
   4. 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 92.5

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

   5. Applied rewrites 92.5%

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

   6. Taylor expanded in c around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 92.5

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

   8. Applied rewrites 92.5%

      \[\leadsto \frac{\left(\left(a \cdot \frac{c}{b \cdot b}\right) \cdot -0.375 - 0.5\right) \cdot c}{b} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 94.0% accurate, 0.5× speedup

Math

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

FPCore

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

C

double code(double a, double b, double c) {
	return fma((((a * a) * ((c * c) * c)) / ((b * b) * (b * b))), -0.5625, fma(-0.5, c, (-0.375 * ((a * (c * c)) / (b * b))))) / 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(-0.5, c, Float64(-0.375 * Float64(Float64(a * Float64(c * c)) / Float64(b * b))))) / 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[(-0.5 * c + N[(-0.375 * N[(N[(a * N[(c * c), $MachinePrecision]), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]

Derivation

1. Initial program 31.4%

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

3. Taylor expanded in b around inf

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

5. Applied rewrites 94.0%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot {c}^{3}}{{b}^{4}}, -0.5625, \mathsf{fma}\left(-0.5, c, \mathsf{fma}\left(\frac{\frac{{\left(c \cdot a\right)}^{4} \cdot 6.328125}{a}}{{b}^{6}}, -0.16666666666666666, \frac{-0.375 \cdot \left(\left(c \cdot c\right) \cdot a\right)}{b \cdot b}\right)\right)\right)}{b}} \]
6. Step-by-step derivation

   1. lift-pow.f64 N/A

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

   2. unpow3 N/A

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

   3. pow2 N/A

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

   4. lower-*.f64 N/A

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

   5. pow2 N/A

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

   6. lift-*.f64 94.0

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

7. Applied rewrites 94.0%

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

   1. lift-pow.f64 N/A

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

   2. sqr-pow N/A

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

   3. metadata-eval N/A

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

   4. metadata-eval N/A

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

   5. lower-*.f64 N/A

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

   6. pow2 N/A

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

   7. lift-*.f64 N/A

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

   8. pow2 N/A

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

   9. lift-*.f64 94.0

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

9. Applied rewrites 94.0%

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

10. Taylor expanded in a around 0

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

    1. lower-*.f64 N/A

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

    2. lower-/.f64 N/A

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

    3. lower-*.f64 N/A

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

    4. pow2 N/A

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

    5. lift-*.f64 N/A

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

    6. pow2 N/A

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

    7. lift-*.f64 92.3

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

12. Applied rewrites 92.3%

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

Alternative 8: 94.0% accurate, 0.5× speedup

Math

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

FPCore

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

C

double code(double a, double b, double c) {
	return fma((((a * a) * ((c * c) * c)) / ((b * b) * (b * b))), -0.5625, (c * ((-0.375 * ((a * c) / (b * b))) - 0.5))) / 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, Float64(c * Float64(Float64(-0.375 * Float64(Float64(a * c) / Float64(b * b))) - 0.5))) / 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[(c * N[(N[(-0.375 * N[(N[(a * c), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]

Derivation

1. Initial program 31.4%

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

3. Taylor expanded in b around inf

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

5. Applied rewrites 94.0%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot {c}^{3}}{{b}^{4}}, -0.5625, \mathsf{fma}\left(-0.5, c, \mathsf{fma}\left(\frac{\frac{{\left(c \cdot a\right)}^{4} \cdot 6.328125}{a}}{{b}^{6}}, -0.16666666666666666, \frac{-0.375 \cdot \left(\left(c \cdot c\right) \cdot a\right)}{b \cdot b}\right)\right)\right)}{b}} \]
6. Step-by-step derivation

   1. lift-pow.f64 N/A

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

   2. unpow3 N/A

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

   3. pow2 N/A

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

   4. lower-*.f64 N/A

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

   5. pow2 N/A

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

   6. lift-*.f64 94.0

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

7. Applied rewrites 94.0%

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

   1. lift-pow.f64 N/A

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

   2. sqr-pow N/A

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

   3. metadata-eval N/A

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

   4. metadata-eval N/A

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

   5. lower-*.f64 N/A

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

   6. pow2 N/A

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

   7. lift-*.f64 N/A

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

   8. pow2 N/A

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

   9. lift-*.f64 94.0

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

9. Applied rewrites 94.0%

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

10. Taylor expanded in c around 0

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

    1. lower-*.f64 N/A

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

    2. lower--.f64 N/A

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

    3. lower-*.f64 N/A

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

    4. lower-/.f64 N/A

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

    5. lower-*.f64 N/A

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

    6. pow2 N/A

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

    7. lift-*.f64 92.2

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

12. Applied rewrites 92.2%

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

Alternative 9: 94.0% accurate, 0.7× speedup

Math

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

FPCore

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

C

double code(double a, double b, double c) {
	return (((((a * ((-0.5625 * ((a * c) / (b * b))) - 0.375)) / (b * b)) * c) - 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 = (((((a * (((-0.5625d0) * ((a * c) / (b * b))) - 0.375d0)) / (b * b)) * c) - 0.5d0) * c) / b
end function

Java

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

Python

def code(a, b, c):
	return (((((a * ((-0.5625 * ((a * c) / (b * b))) - 0.375)) / (b * b)) * c) - 0.5) * c) / b

Julia

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

MATLAB

function tmp = code(a, b, c)
	tmp = (((((a * ((-0.5625 * ((a * c) / (b * b))) - 0.375)) / (b * b)) * c) - 0.5) * c) / b;
end

Wolfram

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

Derivation

1. Initial program 31.4%

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

3. Taylor expanded in b around inf

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

5. Applied rewrites 94.0%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot {c}^{3}}{{b}^{4}}, -0.5625, \mathsf{fma}\left(-0.5, c, \mathsf{fma}\left(\frac{\frac{{\left(c \cdot a\right)}^{4} \cdot 6.328125}{a}}{{b}^{6}}, -0.16666666666666666, \frac{-0.375 \cdot \left(\left(c \cdot c\right) \cdot a\right)}{b \cdot b}\right)\right)\right)}{b}} \]

6. 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. Step-by-step derivation

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

8. Applied rewrites 93.9%

   \[\leadsto \frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{a \cdot a}{{b}^{4}}, -0.5625, \frac{-1.0546875 \cdot \left({a}^{3} \cdot c\right)}{{b}^{6}}\right), c, \frac{a}{b \cdot b} \cdot -0.375\right) \cdot c - 0.5\right) \cdot c}{b} \]

9. Taylor expanded in b around inf

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

    1. lower-/.f64 N/A

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

    2. *-commutative N/A

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

    3. lower-fma.f64 N/A

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

    4. associate-/l* N/A

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

    5. lower-*.f64 N/A

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

    6. pow2 N/A

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

    7. lift-*.f64 N/A

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

    8. lower-/.f64 N/A

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

    9. pow2 N/A

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

    10. lift-*.f64 N/A

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

    11. lower-*.f64 N/A

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

    12. pow2 N/A

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

    13. lift-*.f64 92.2

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

11. Applied rewrites 92.2%

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

12. Taylor expanded in a around 0

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

    1. lower-*.f64 N/A

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

    2. lower--.f64 N/A

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

    3. lower-*.f64 N/A

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

    4. lower-/.f64 N/A

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

    5. lower-*.f64 N/A

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

    6. pow2 N/A

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

    7. lift-*.f64 92.2

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

14. Applied rewrites 92.2%

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

Alternative 10: 90.8% accurate, 1.1× speedup

Math

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

FPCore

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

C

double code(double a, double b, double c) {
	return ((((a * (c / (b * b))) * -0.375) - 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 = ((((a * (c / (b * b))) * (-0.375d0)) - 0.5d0) * c) / b
end function

Java

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

Python

def code(a, b, c):
	return ((((a * (c / (b * b))) * -0.375) - 0.5) * c) / b

Julia

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

MATLAB

function tmp = code(a, b, c)
	tmp = ((((a * (c / (b * b))) * -0.375) - 0.5) * c) / b;
end

Wolfram

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

Derivation

1. Initial program 31.4%

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

3. Taylor expanded in b around inf

   \[\leadsto \color{blue}{\frac{\frac{-1}{2} \cdot c + \frac{-3}{8} \cdot \frac{a \cdot {c}^{2}}{{b}^{2}}}{b}} \]
4. 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 89.5

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

5. Applied rewrites 89.5%

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

6. Taylor expanded in c around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower--.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. associate-/l* N/A

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

   7. lower-*.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. pow2 N/A

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

   10. lift-*.f64 89.5

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

8. Applied rewrites 89.5%

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

Alternative 11: 81.3% accurate, 2.9× 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 31.4%

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

3. Taylor expanded in a around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-/.f64 80.7

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

5. Applied rewrites 80.7%

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

Reproduce

herbie shell --seed 2025082 
(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)))