From Rump in a 1983 paper, rewritten

Percentage Accurate: 3.1% → 100.0%
Time: 2.9s
Alternatives: 1
Speedup: 128.0×

Specification

?
\[x = 10864 \land y = 18817\]
\[\begin{array}{l} \\ 9 \cdot {x}^{4} - \left(y \cdot y\right) \cdot \left(y \cdot y - 2\right) \end{array} \]
(FPCore (x y)
 :precision binary64
 (- (* 9.0 (pow x 4.0)) (* (* y y) (- (* y y) 2.0))))
double code(double x, double y) {
	return (9.0 * pow(x, 4.0)) - ((y * y) * ((y * y) - 2.0));
}

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(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (9.0d0 * (x ** 4.0d0)) - ((y * y) * ((y * y) - 2.0d0))
end function

public static double code(double x, double y) {
	return (9.0 * Math.pow(x, 4.0)) - ((y * y) * ((y * y) - 2.0));
}

def code(x, y):
	return (9.0 * math.pow(x, 4.0)) - ((y * y) * ((y * y) - 2.0))

function code(x, y)
	return Float64(Float64(9.0 * (x ^ 4.0)) - Float64(Float64(y * y) * Float64(Float64(y * y) - 2.0)))
end

function tmp = code(x, y)
	tmp = (9.0 * (x ^ 4.0)) - ((y * y) * ((y * y) - 2.0));
end

code[x_, y_] := N[(N[(9.0 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision] - N[(N[(y * y), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

\begin{array}{l}

\\
9 \cdot {x}^{4} - \left(y \cdot y\right) \cdot \left(y \cdot y - 2\right)
\end{array}

Sampling outcomes in binary64 precision:

Local Percentage Accuracy vs ?

The average percentage accuracy by input value. Horizontal axis shows value of an input variable; the variable is choosen in the title. Vertical axis is accuracy; higher is better. Red represent the original program, while blue represents Herbie's suggestion. These can be toggled with buttons below the plot. The line is an average while dots represent individual samples.

Accuracy vs Speed?

Herbie found 1 alternatives:

AlternativeAccuracySpeedup
The accuracy (vertical axis) and speed (horizontal axis) of each alternatives. Up and to the right is better. The red square shows the initial program, and each blue circle shows an alternative.The line shows the best available speed-accuracy tradeoffs.

Initial Program: 3.1% accurate, 1.0× speedup?

\[\begin{array}{l} \\ 9 \cdot {x}^{4} - \left(y \cdot y\right) \cdot \left(y \cdot y - 2\right) \end{array} \]
(FPCore (x y)
 :precision binary64
 (- (* 9.0 (pow x 4.0)) (* (* y y) (- (* y y) 2.0))))
double code(double x, double y) {
	return (9.0 * pow(x, 4.0)) - ((y * y) * ((y * y) - 2.0));
}

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(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (9.0d0 * (x ** 4.0d0)) - ((y * y) * ((y * y) - 2.0d0))
end function

public static double code(double x, double y) {
	return (9.0 * Math.pow(x, 4.0)) - ((y * y) * ((y * y) - 2.0));
}

def code(x, y):
	return (9.0 * math.pow(x, 4.0)) - ((y * y) * ((y * y) - 2.0))

function code(x, y)
	return Float64(Float64(9.0 * (x ^ 4.0)) - Float64(Float64(y * y) * Float64(Float64(y * y) - 2.0)))
end

function tmp = code(x, y)
	tmp = (9.0 * (x ^ 4.0)) - ((y * y) * ((y * y) - 2.0));
end

code[x_, y_] := N[(N[(9.0 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision] - N[(N[(y * y), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

\begin{array}{l}

\\
9 \cdot {x}^{4} - \left(y \cdot y\right) \cdot \left(y \cdot y - 2\right)
\end{array}

Alternative 1: 100.0% accurate, 128.0× speedup?

\[\begin{array}{l} \\ 1 \end{array} \]
(FPCore (x y) :precision binary64 1.0)
double code(double x, double y) {
	return 1.0;
}

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(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 1.0d0
end function

public static double code(double x, double y) {
	return 1.0;
}

def code(x, y):
	return 1.0

function code(x, y)
	return 1.0
end

function tmp = code(x, y)
	tmp = 1.0;
end

code[x_, y_] := 1.0

\begin{array}{l}

\\
1
\end{array}
Derivation

  1. Initial program 3.1%

    \[9 \cdot {x}^{4} - \left(y \cdot y\right) \cdot \left(y \cdot y - 2\right) \]
  2. Add Preprocessing

  3. Taylor expanded in x around 0

    \[\leadsto \color{blue}{-1 \cdot \left({y}^{2} \cdot \left({y}^{2} - 2\right)\right)} \]
  4. Step-by-step derivation

    1. mul-1-negN/A

      \[\leadsto \mathsf{neg}\left({y}^{2} \cdot \left({y}^{2} - 2\right)\right) \]

    2. pow2N/A

      \[\leadsto \mathsf{neg}\left(\left(y \cdot y\right) \cdot \left({y}^{2} - 2\right)\right) \]

    3. pow2N/A

      \[\leadsto \mathsf{neg}\left(\left(y \cdot y\right) \cdot \left(y \cdot y - 2\right)\right) \]

    4. associate-*l*N/A

      \[\leadsto \mathsf{neg}\left(y \cdot \left(y \cdot \left(y \cdot y - 2\right)\right)\right) \]

    5. distribute-lft-neg-inN/A

      \[\leadsto \left(\mathsf{neg}\left(y\right)\right) \cdot \color{blue}{\left(y \cdot \left(y \cdot y - 2\right)\right)} \]

    6. lower-*.f64N/A

      \[\leadsto \left(\mathsf{neg}\left(y\right)\right) \cdot \color{blue}{\left(y \cdot \left(y \cdot y - 2\right)\right)} \]

    7. lower-neg.f64N/A

      \[\leadsto \left(-y\right) \cdot \left(\color{blue}{y} \cdot \left(y \cdot y - 2\right)\right) \]

    8. *-commutativeN/A

      \[\leadsto \left(-y\right) \cdot \left(\left(y \cdot y - 2\right) \cdot \color{blue}{y}\right) \]

    9. lower-*.f64N/A

      \[\leadsto \left(-y\right) \cdot \left(\left(y \cdot y - 2\right) \cdot \color{blue}{y}\right) \]

    10. pow2N/A

      \[\leadsto \left(-y\right) \cdot \left(\left({y}^{2} - 2\right) \cdot y\right) \]

    11. metadata-evalN/A

      \[\leadsto \left(-y\right) \cdot \left(\left({y}^{2} - 2 \cdot 1\right) \cdot y\right) \]

    12. fp-cancel-sub-sign-invN/A

      \[\leadsto \left(-y\right) \cdot \left(\left({y}^{2} + \left(\mathsf{neg}\left(2\right)\right) \cdot 1\right) \cdot y\right) \]

    13. pow2N/A

      \[\leadsto \left(-y\right) \cdot \left(\left(y \cdot y + \left(\mathsf{neg}\left(2\right)\right) \cdot 1\right) \cdot y\right) \]

    14. metadata-evalN/A

      \[\leadsto \left(-y\right) \cdot \left(\left(y \cdot y + -2 \cdot 1\right) \cdot y\right) \]

    15. metadata-evalN/A

      \[\leadsto \left(-y\right) \cdot \left(\left(y \cdot y + -2\right) \cdot y\right) \]

    16. lower-fma.f641.5

      \[\leadsto \left(-y\right) \cdot \left(\mathsf{fma}\left(y, y, -2\right) \cdot y\right) \]

  5. Applied rewrites1.5%

    \[\leadsto \color{blue}{\left(-y\right) \cdot \left(\mathsf{fma}\left(y, y, -2\right) \cdot y\right)} \]

  6. Taylor expanded in y around 0

    \[\leadsto 2 \cdot \color{blue}{{y}^{2}} \]
  7. Step-by-step derivation

    1. *-commutativeN/A

      \[\leadsto {y}^{2} \cdot 2 \]

    2. lower-*.f64N/A

      \[\leadsto {y}^{2} \cdot 2 \]

    3. pow2N/A

      \[\leadsto \left(y \cdot y\right) \cdot 2 \]

    4. lower-*.f6411.1

      \[\leadsto \left(y \cdot y\right) \cdot 2 \]

  8. Applied rewrites11.1%

    \[\leadsto \left(y \cdot y\right) \cdot \color{blue}{2} \]
  9. Step-by-step derivation

    1. pow2N/A

      \[\leadsto {y}^{2} \cdot 2 \]

    2. *-commutativeN/A

      \[\leadsto 2 \cdot {y}^{\color{blue}{2}} \]

    3. count-2-revN/A

      \[\leadsto {y}^{2} + {y}^{\color{blue}{2}} \]

    4. pow2N/A

      \[\leadsto y \cdot y + {y}^{2} \]

    5. pow2N/A

      \[\leadsto y \cdot y + y \cdot y \]

    6. distribute-lft-outN/A

      \[\leadsto y \cdot \left(y + \color{blue}{y}\right) \]

    7. lower-*.f64N/A

      \[\leadsto y \cdot \left(y + \color{blue}{y}\right) \]

    8. lower-+.f6411.1

      \[\leadsto y \cdot \left(y + y\right) \]

  10. Applied rewrites11.1%

    \[\leadsto y \cdot \left(y + \color{blue}{y}\right) \]

  11. Taylor expanded in y around 0

    \[\leadsto {y}^{2} \cdot \color{blue}{\left(2 + -1 \cdot {y}^{2}\right)} \]

  13. Applied rewrites100.0%

    \[\leadsto 1 \]
  14. Add Preprocessing

Reproduce

?
herbie shell --seed 2025044 
(FPCore (x y)
  :name "From Rump in a 1983 paper, rewritten"
  :precision binary64
  :pre (and (== x 10864.0) (== y 18817.0))
  (- (* 9.0 (pow x 4.0)) (* (* y y) (- (* y y) 2.0))))