Cubic critical, wide range

Percentage Accurate: 17.7% → 97.7%

Time: 4.4s

Alternatives: 10

Speedup: 2.9×

Specification

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

Math

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

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: 17.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.7% accurate, 0.2× speedup

Math

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

FPCore

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

C

double code(double a, double b, double c) {
	return fma((((a * a) * ((c * c) * c)) / ((b * b) * (b * b))), -0.5625, fma(-0.5, c, (fma(((pow(a, 3.0) * (c * c)) / pow(b, 6.0)), -1.0546875, ((a / (b * b)) * -0.375)) * (c * c)))) / b;
}

Julia

function code(a, b, c)
	return Float64(fma(Float64(Float64(Float64(a * a) * Float64(Float64(c * c) * c)) / Float64(Float64(b * b) * Float64(b * b))), -0.5625, fma(-0.5, c, Float64(fma(Float64(Float64((a ^ 3.0) * Float64(c * c)) / (b ^ 6.0)), -1.0546875, Float64(Float64(a / Float64(b * b)) * -0.375)) * Float64(c * c)))) / b)
end

Wolfram

code[a_, b_, c_] := N[(N[(N[(N[(N[(a * a), $MachinePrecision] * N[(N[(c * c), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision] / N[(N[(b * b), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.5625 + N[(-0.5 * c + N[(N[(N[(N[(N[Power[a, 3.0], $MachinePrecision] * N[(c * c), $MachinePrecision]), $MachinePrecision] / N[Power[b, 6.0], $MachinePrecision]), $MachinePrecision] * -1.0546875 + N[(N[(a / N[(b * b), $MachinePrecision]), $MachinePrecision] * -0.375), $MachinePrecision]), $MachinePrecision] * N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]

Derivation

1. Initial program 17.7%

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

2. Taylor expanded in b around inf

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

4. Applied rewrites 97.7%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot {c}^{3}}{{b}^{4}}, -0.5625, \mathsf{fma}\left(-0.5, c, \mathsf{fma}\left(\frac{\frac{{\left(c \cdot a\right)}^{4} \cdot 6.328125}{a}}{{b}^{6}}, -0.16666666666666666, \frac{-0.375 \cdot \left(\left(c \cdot c\right) \cdot a\right)}{b \cdot b}\right)\right)\right)}{b}} \]
5. 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 97.7

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

6. Applied rewrites 97.7%

   \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{{b}^{4}}, -0.5625, \mathsf{fma}\left(-0.5, c, \mathsf{fma}\left(\frac{\frac{{\left(c \cdot a\right)}^{4} \cdot 6.328125}{a}}{{b}^{6}}, -0.16666666666666666, \frac{-0.375 \cdot \left(\left(c \cdot c\right) \cdot a\right)}{b \cdot b}\right)\right)\right)}{b} \]
7. 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 97.7

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

8. Applied rewrites 97.7%

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

9. Taylor expanded in c around 0

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

    1. *-commutative N/A

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

    2. lower-*.f64 N/A

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

11. Applied rewrites 97.7%

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

Alternative 2: 97.6% accurate, 0.2× speedup

Math

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

FPCore

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

C

double code(double a, double b, double c) {
	return fma((((a * a) * ((c * c) * c)) / ((b * b) * (b * b))), -0.5625, (((fma(((pow(a, 3.0) * (c * c)) / pow(b, 6.0)), -1.0546875, ((a / (b * b)) * -0.375)) * c) - 0.5) * c)) / b;
}

Julia

function code(a, b, c)
	return Float64(fma(Float64(Float64(Float64(a * a) * Float64(Float64(c * c) * c)) / Float64(Float64(b * b) * Float64(b * b))), -0.5625, Float64(Float64(Float64(fma(Float64(Float64((a ^ 3.0) * Float64(c * c)) / (b ^ 6.0)), -1.0546875, Float64(Float64(a / Float64(b * b)) * -0.375)) * c) - 0.5) * c)) / b)
end

Wolfram

code[a_, b_, c_] := N[(N[(N[(N[(N[(a * a), $MachinePrecision] * N[(N[(c * c), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision] / N[(N[(b * b), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.5625 + N[(N[(N[(N[(N[(N[(N[Power[a, 3.0], $MachinePrecision] * N[(c * c), $MachinePrecision]), $MachinePrecision] / N[Power[b, 6.0], $MachinePrecision]), $MachinePrecision] * -1.0546875 + N[(N[(a / N[(b * b), $MachinePrecision]), $MachinePrecision] * -0.375), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision] - 0.5), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]

Derivation

1. Initial program 17.7%

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

2. Taylor expanded in b around inf

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

4. Applied rewrites 97.7%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot {c}^{3}}{{b}^{4}}, -0.5625, \mathsf{fma}\left(-0.5, c, \mathsf{fma}\left(\frac{\frac{{\left(c \cdot a\right)}^{4} \cdot 6.328125}{a}}{{b}^{6}}, -0.16666666666666666, \frac{-0.375 \cdot \left(\left(c \cdot c\right) \cdot a\right)}{b \cdot b}\right)\right)\right)}{b}} \]
5. 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 97.7

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

6. Applied rewrites 97.7%

   \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{{b}^{4}}, -0.5625, \mathsf{fma}\left(-0.5, c, \mathsf{fma}\left(\frac{\frac{{\left(c \cdot a\right)}^{4} \cdot 6.328125}{a}}{{b}^{6}}, -0.16666666666666666, \frac{-0.375 \cdot \left(\left(c \cdot c\right) \cdot a\right)}{b \cdot b}\right)\right)\right)}{b} \]
7. 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 97.7

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

8. Applied rewrites 97.7%

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

9. Taylor expanded in c around 0

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

    1. *-commutative N/A

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

    2. lower-*.f64 N/A

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

11. Applied rewrites 97.6%

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

Alternative 3: 96.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{c \cdot c}{b \cdot b}, -0.375, \frac{-0.5625 \cdot \left({c}^{3} \cdot a\right)}{{b}^{4}}\right), a, -0.5 \cdot c\right)}{b} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/
  (fma
   (fma
    (/ (* c c) (* b b))
    -0.375
    (/ (* -0.5625 (* (pow c 3.0) a)) (pow b 4.0)))
   a
   (* -0.5 c))
  b))

C

double code(double a, double b, double c) {
	return fma(fma(((c * c) / (b * b)), -0.375, ((-0.5625 * (pow(c, 3.0) * a)) / pow(b, 4.0))), a, (-0.5 * c)) / b;
}

Julia

function code(a, b, c)
	return Float64(fma(fma(Float64(Float64(c * c) / Float64(b * b)), -0.375, Float64(Float64(-0.5625 * Float64((c ^ 3.0) * a)) / (b ^ 4.0))), a, Float64(-0.5 * c)) / b)
end

Wolfram

code[a_, b_, c_] := N[(N[(N[(N[(N[(c * c), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision] * -0.375 + N[(N[(-0.5625 * N[(N[Power[c, 3.0], $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] / N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * a + N[(-0.5 * c), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]

Derivation

1. Initial program 17.7%

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

2. Taylor expanded in b around inf

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

4. Applied rewrites 97.7%

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

5. Taylor expanded in a around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

7. Applied rewrites 96.9%

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

Alternative 4: 96.9% accurate, 0.5× speedup

Math

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

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 b) (/ (* (* c c) a) 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 / b) * (((c * c) * a) / 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(-0.375 / b) * Float64(Float64(Float64(c * c) * a) / 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[(-0.375 / b), $MachinePrecision] * N[(N[(N[(c * c), $MachinePrecision] * a), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]

Derivation

1. Initial program 17.7%

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

2. Taylor expanded in b around inf

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

4. Applied rewrites 97.7%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot {c}^{3}}{{b}^{4}}, -0.5625, \mathsf{fma}\left(-0.5, c, \mathsf{fma}\left(\frac{\frac{{\left(c \cdot a\right)}^{4} \cdot 6.328125}{a}}{{b}^{6}}, -0.16666666666666666, \frac{-0.375 \cdot \left(\left(c \cdot c\right) \cdot a\right)}{b \cdot b}\right)\right)\right)}{b}} \]
5. 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 97.7

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

6. Applied rewrites 97.7%

   \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{{b}^{4}}, -0.5625, \mathsf{fma}\left(-0.5, c, \mathsf{fma}\left(\frac{\frac{{\left(c \cdot a\right)}^{4} \cdot 6.328125}{a}}{{b}^{6}}, -0.16666666666666666, \frac{-0.375 \cdot \left(\left(c \cdot c\right) \cdot a\right)}{b \cdot b}\right)\right)\right)}{b} \]
7. 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 97.7

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

8. Applied rewrites 97.7%

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

9. Taylor expanded in a around 0

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

    1. associate-*r/ N/A

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

    2. *-commutative N/A

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

    3. *-commutative N/A

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

    4. pow2 N/A

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

    5. *-commutative N/A

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

    6. pow2 N/A

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

    7. times-frac N/A

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

    8. lower-*.f64 N/A

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

    9. lower-/.f64 N/A

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

    10. pow2 N/A

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

    11. *-commutative N/A

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

    12. lower-/.f64 N/A

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

    13. *-commutative N/A

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

    14. pow2 N/A

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

    15. lift-*.f64 N/A

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

    16. lift-*.f64 96.9

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

11. Applied rewrites 96.9%

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

Alternative 5: 96.8% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 17.7%

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

2. Taylor expanded in b around inf

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

4. Applied rewrites 97.7%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot {c}^{3}}{{b}^{4}}, -0.5625, \mathsf{fma}\left(-0.5, c, \mathsf{fma}\left(\frac{\frac{{\left(c \cdot a\right)}^{4} \cdot 6.328125}{a}}{{b}^{6}}, -0.16666666666666666, \frac{-0.375 \cdot \left(\left(c \cdot c\right) \cdot a\right)}{b \cdot b}\right)\right)\right)}{b}} \]
5. 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 97.7

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

6. Applied rewrites 97.7%

   \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{{b}^{4}}, -0.5625, \mathsf{fma}\left(-0.5, c, \mathsf{fma}\left(\frac{\frac{{\left(c \cdot a\right)}^{4} \cdot 6.328125}{a}}{{b}^{6}}, -0.16666666666666666, \frac{-0.375 \cdot \left(\left(c \cdot c\right) \cdot a\right)}{b \cdot b}\right)\right)\right)}{b} \]
7. 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 97.7

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

8. Applied rewrites 97.7%

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

9. Taylor expanded in c around 0

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

    1. *-commutative N/A

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

    2. lower-*.f64 N/A

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

    3. lower--.f64 N/A

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

    4. associate-*r/ N/A

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

    5. *-commutative N/A

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

    6. lift-*.f64 N/A

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

    7. *-commutative N/A

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

    8. lift-*.f64 N/A

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

    9. lower-/.f64 N/A

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

    10. lift-*.f64 N/A

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

    11. *-commutative N/A

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

    12. lift-*.f64 N/A

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

    13. *-commutative N/A

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

    14. associate-*r* N/A

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

    15. lower-*.f64 N/A

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

    16. lower-*.f64 N/A

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

    17. pow2 N/A

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

    18. lift-*.f64 96.8

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

11. Applied rewrites 96.8%

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

Alternative 6: 95.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 17.7%

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

2. Taylor expanded in b around inf

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

   1. lower-/.f64 N/A

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

   2. +-commutative N/A

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

   3. *-commutative N/A

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

   4. lower-fma.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f64 N/A

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

   10. pow2 N/A

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

   11. lift-*.f64 N/A

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

   12. lower-*.f64 95.3

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

4. Applied rewrites 95.3%

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

Alternative 7: 95.2% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 17.7%

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

2. Taylor expanded in b around inf

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

   1. lower-/.f64 N/A

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

   2. +-commutative N/A

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

   3. *-commutative N/A

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

   4. lower-fma.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f64 N/A

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

   10. pow2 N/A

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

   11. lift-*.f64 N/A

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

   12. lower-*.f64 95.3

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

4. Applied rewrites 95.3%

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

5. Taylor expanded in c around 0

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

   1. *-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. associate-*r/ N/A

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

   5. lower-/.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lift-*.f64 N/A

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

   10. pow2 N/A

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

   11. lift-*.f64 95.2

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

7. Applied rewrites 95.2%

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

Alternative 8: 94.9% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

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 * a) * (-0.375d0)) / (b * b)) - 0.5d0) / b) * c
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 17.7%

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

2. Taylor expanded in c around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

4. Applied rewrites 96.5%

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

5. Taylor expanded in b around inf

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

   1. lower-/.f64 N/A

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

   2. lower--.f64 N/A

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

   3. associate-*r/ N/A

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

   4. lower-/.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lift-*.f64 N/A

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

   9. pow2 N/A

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

   10. lift-*.f64 94.9

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

7. Applied rewrites 94.9%

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

Alternative 9: 90.4% 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 17.7%

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

2. Taylor expanded in a around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-/.f64 90.4

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

4. Applied rewrites 90.4%

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

Alternative 10: 90.1% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 17.7%

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

2. Taylor expanded in c around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

4. Applied rewrites 96.5%

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

5. Taylor expanded in a around 0

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

   1. lower-/.f64 90.1

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

7. Applied rewrites 90.1%

   \[\leadsto \frac{-0.5}{b} \cdot c \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2025097 
(FPCore (a b c)
  :name "Cubic critical, wide range"
  :precision binary64
  :pre (and (and (and (< 4.930380657631324e-32 a) (< a 2.028240960365167e+31)) (and (< 4.930380657631324e-32 b) (< b 2.028240960365167e+31))) (and (< 4.930380657631324e-32 c) (< c 2.028240960365167e+31)))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))