xlohi (overflows)

Percentage Accurate: 3.1% → 99.4%

Time: 3.2s

Alternatives: 8

Speedup: 9.6×

Specification

\[lo < -1 \cdot 10^{+308} \land hi > 10^{+308}\]

Math

\[\frac{x - lo}{hi - lo} \]

FPCore

(FPCore (lo hi x) :precision binary64 (/ (- x lo) (- hi lo)))

C

double code(double lo, double hi, double x) {
	return (x - lo) / (hi - lo);
}

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(lo, hi, x)
use fmin_fmax_functions
    real(8), intent (in) :: lo
    real(8), intent (in) :: hi
    real(8), intent (in) :: x
    code = (x - lo) / (hi - lo)
end function

Java

public static double code(double lo, double hi, double x) {
	return (x - lo) / (hi - lo);
}

Python

def code(lo, hi, x):
	return (x - lo) / (hi - lo)

Julia

function code(lo, hi, x)
	return Float64(Float64(x - lo) / Float64(hi - lo))
end

MATLAB

function tmp = code(lo, hi, x)
	tmp = (x - lo) / (hi - lo);
end

Wolfram

code[lo_, hi_, x_] := N[(N[(x - lo), $MachinePrecision] / N[(hi - lo), $MachinePrecision]), $MachinePrecision]

Initial Program: 3.1% accurate, 1.0× speedup

Math

\[\frac{x - lo}{hi - lo} \]

FPCore

(FPCore (lo hi x) :precision binary64 (/ (- x lo) (- hi lo)))

C

double code(double lo, double hi, double x) {
	return (x - lo) / (hi - lo);
}

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(lo, hi, x)
use fmin_fmax_functions
    real(8), intent (in) :: lo
    real(8), intent (in) :: hi
    real(8), intent (in) :: x
    code = (x - lo) / (hi - lo)
end function

Java

public static double code(double lo, double hi, double x) {
	return (x - lo) / (hi - lo);
}

Python

def code(lo, hi, x):
	return (x - lo) / (hi - lo)

Julia

function code(lo, hi, x)
	return Float64(Float64(x - lo) / Float64(hi - lo))
end

MATLAB

function tmp = code(lo, hi, x)
	tmp = (x - lo) / (hi - lo);
end

Wolfram

code[lo_, hi_, x_] := N[(N[(x - lo), $MachinePrecision] / N[(hi - lo), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.4% accurate, 0.4× speedup

Math

\[\frac{1}{\mathsf{fma}\left(-1, \frac{lo}{x - lo}, \frac{1}{\frac{x - lo}{hi}}\right)} \]

FPCore

(FPCore (lo hi x)
 :precision binary64
 (/ 1.0 (fma -1.0 (/ lo (- x lo)) (/ 1.0 (/ (- x lo) hi)))))

C

double code(double lo, double hi, double x) {
	return 1.0 / fma(-1.0, (lo / (x - lo)), (1.0 / ((x - lo) / hi)));
}

Julia

function code(lo, hi, x)
	return Float64(1.0 / fma(-1.0, Float64(lo / Float64(x - lo)), Float64(1.0 / Float64(Float64(x - lo) / hi))))
end

Wolfram

code[lo_, hi_, x_] := N[(1.0 / N[(-1.0 * N[(lo / N[(x - lo), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(N[(x - lo), $MachinePrecision] / hi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 3.1%

   \[\frac{x - lo}{hi - lo} \]

2. Taylor expanded in lo around 0

   \[\leadsto \frac{x - lo}{\color{blue}{hi}} \]
3. Step-by-step derivation

4. Applied rewrites 18.8%

   \[\leadsto \frac{x - lo}{\color{blue}{hi}} \]
5. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{x - lo}{hi}} \]

   2. div-flip N/A

      \[\leadsto \color{blue}{\frac{1}{\frac{hi}{x - lo}}} \]

   3. lower-unsound-/.f64 N/A

      \[\leadsto \color{blue}{\frac{1}{\frac{hi}{x - lo}}} \]

   4. lower-unsound-/.f64 18.8%

      \[\leadsto \frac{1}{\color{blue}{\frac{hi}{x - lo}}} \]

6. Applied rewrites 18.8%

   \[\leadsto \color{blue}{\frac{1}{\frac{hi}{x - lo}}} \]

7. Taylor expanded in hi around 0

   \[\leadsto \frac{1}{\color{blue}{-1 \cdot \frac{lo}{x - lo} + \frac{hi}{x - lo}}} \]
8. Step-by-step derivation

   1. lower-fma.f64 N/A

      \[\leadsto \frac{1}{\mathsf{fma}\left(-1, \color{blue}{\frac{lo}{x - lo}}, \frac{hi}{x - lo}\right)} \]

   2. lower-/.f64 N/A

      \[\leadsto \frac{1}{\mathsf{fma}\left(-1, \frac{lo}{\color{blue}{x - lo}}, \frac{hi}{x - lo}\right)} \]

   3. lower--.f64 N/A

      \[\leadsto \frac{1}{\mathsf{fma}\left(-1, \frac{lo}{x - \color{blue}{lo}}, \frac{hi}{x - lo}\right)} \]

   4. lower-/.f64 N/A

      \[\leadsto \frac{1}{\mathsf{fma}\left(-1, \frac{lo}{x - lo}, \frac{hi}{x - lo}\right)} \]

   5. lower--.f64 99.4%

      \[\leadsto \frac{1}{\mathsf{fma}\left(-1, \frac{lo}{x - lo}, \frac{hi}{x - lo}\right)} \]

9. Applied rewrites 99.4%

   \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(-1, \frac{lo}{x - lo}, \frac{hi}{x - lo}\right)}} \]
10. Step-by-step derivation

    1. lift-/.f64 N/A

       \[\leadsto \frac{1}{\mathsf{fma}\left(-1, \frac{lo}{x - lo}, \frac{hi}{x - lo}\right)} \]

    2. div-flip N/A

       \[\leadsto \frac{1}{\mathsf{fma}\left(-1, \frac{lo}{x - lo}, \frac{1}{\frac{x - lo}{hi}}\right)} \]

    3. lower-unsound-/.f64 N/A

       \[\leadsto \frac{1}{\mathsf{fma}\left(-1, \frac{lo}{x - lo}, \frac{1}{\frac{x - lo}{hi}}\right)} \]

    4. lower-unsound-/.f64 99.4%

       \[\leadsto \frac{1}{\mathsf{fma}\left(-1, \frac{lo}{x - lo}, \frac{1}{\frac{x - lo}{hi}}\right)} \]

11. Applied rewrites 99.4%

    \[\leadsto \frac{1}{\mathsf{fma}\left(-1, \frac{lo}{x - lo}, \frac{1}{\frac{x - lo}{hi}}\right)} \]
12. Add Preprocessing

Alternative 2: 99.4% accurate, 0.5× speedup

Math

\[\frac{1}{\frac{lo}{lo - x} - \frac{hi}{lo - x}} \]

FPCore

(FPCore (lo hi x)
 :precision binary64
 (/ 1.0 (- (/ lo (- lo x)) (/ hi (- lo x)))))

C

double code(double lo, double hi, double x) {
	return 1.0 / ((lo / (lo - x)) - (hi / (lo - x)));
}

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(lo, hi, x)
use fmin_fmax_functions
    real(8), intent (in) :: lo
    real(8), intent (in) :: hi
    real(8), intent (in) :: x
    code = 1.0d0 / ((lo / (lo - x)) - (hi / (lo - x)))
end function

Java

public static double code(double lo, double hi, double x) {
	return 1.0 / ((lo / (lo - x)) - (hi / (lo - x)));
}

Python

def code(lo, hi, x):
	return 1.0 / ((lo / (lo - x)) - (hi / (lo - x)))

Julia

function code(lo, hi, x)
	return Float64(1.0 / Float64(Float64(lo / Float64(lo - x)) - Float64(hi / Float64(lo - x))))
end

MATLAB

function tmp = code(lo, hi, x)
	tmp = 1.0 / ((lo / (lo - x)) - (hi / (lo - x)));
end

Wolfram

code[lo_, hi_, x_] := N[(1.0 / N[(N[(lo / N[(lo - x), $MachinePrecision]), $MachinePrecision] - N[(hi / N[(lo - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 3.1%

   \[\frac{x - lo}{hi - lo} \]

2. Taylor expanded in lo around 0

   \[\leadsto \frac{x - lo}{\color{blue}{hi}} \]
3. Step-by-step derivation

4. Applied rewrites 18.8%

   \[\leadsto \frac{x - lo}{\color{blue}{hi}} \]
5. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{x - lo}{hi}} \]

   2. div-flip N/A

      \[\leadsto \color{blue}{\frac{1}{\frac{hi}{x - lo}}} \]

   3. lower-unsound-/.f64 N/A

      \[\leadsto \color{blue}{\frac{1}{\frac{hi}{x - lo}}} \]

   4. lower-unsound-/.f64 18.8%

      \[\leadsto \frac{1}{\color{blue}{\frac{hi}{x - lo}}} \]

6. Applied rewrites 18.8%

   \[\leadsto \color{blue}{\frac{1}{\frac{hi}{x - lo}}} \]

7. Taylor expanded in hi around 0

   \[\leadsto \frac{1}{\color{blue}{-1 \cdot \frac{lo}{x - lo} + \frac{hi}{x - lo}}} \]
8. Step-by-step derivation

   1. lower-fma.f64 N/A

      \[\leadsto \frac{1}{\mathsf{fma}\left(-1, \color{blue}{\frac{lo}{x - lo}}, \frac{hi}{x - lo}\right)} \]

   2. lower-/.f64 N/A

      \[\leadsto \frac{1}{\mathsf{fma}\left(-1, \frac{lo}{\color{blue}{x - lo}}, \frac{hi}{x - lo}\right)} \]

   3. lower--.f64 N/A

      \[\leadsto \frac{1}{\mathsf{fma}\left(-1, \frac{lo}{x - \color{blue}{lo}}, \frac{hi}{x - lo}\right)} \]

   4. lower-/.f64 N/A

      \[\leadsto \frac{1}{\mathsf{fma}\left(-1, \frac{lo}{x - lo}, \frac{hi}{x - lo}\right)} \]

   5. lower--.f64 99.4%

      \[\leadsto \frac{1}{\mathsf{fma}\left(-1, \frac{lo}{x - lo}, \frac{hi}{x - lo}\right)} \]

9. Applied rewrites 99.4%

   \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(-1, \frac{lo}{x - lo}, \frac{hi}{x - lo}\right)}} \]
10. Step-by-step derivation

    1. lift-fma.f64 N/A

       \[\leadsto \frac{1}{-1 \cdot \frac{lo}{x - lo} + \color{blue}{\frac{hi}{x - lo}}} \]

    2. add-flip N/A

       \[\leadsto \frac{1}{-1 \cdot \frac{lo}{x - lo} - \color{blue}{\left(\mathsf{neg}\left(\frac{hi}{x - lo}\right)\right)}} \]

    3. lower--.f64 N/A

       \[\leadsto \frac{1}{-1 \cdot \frac{lo}{x - lo} - \color{blue}{\left(\mathsf{neg}\left(\frac{hi}{x - lo}\right)\right)}} \]

    4. mul-1-neg N/A

       \[\leadsto \frac{1}{\left(\mathsf{neg}\left(\frac{lo}{x - lo}\right)\right) - \left(\mathsf{neg}\left(\color{blue}{\frac{hi}{x - lo}}\right)\right)} \]

    5. lift-/.f64 N/A

       \[\leadsto \frac{1}{\left(\mathsf{neg}\left(\frac{lo}{x - lo}\right)\right) - \left(\mathsf{neg}\left(\frac{\color{blue}{hi}}{x - lo}\right)\right)} \]

    6. distribute-neg-frac2 N/A

       \[\leadsto \frac{1}{\frac{lo}{\mathsf{neg}\left(\left(x - lo\right)\right)} - \left(\mathsf{neg}\left(\color{blue}{\frac{hi}{x - lo}}\right)\right)} \]

    7. lift--.f64 N/A

       \[\leadsto \frac{1}{\frac{lo}{\mathsf{neg}\left(\left(x - lo\right)\right)} - \left(\mathsf{neg}\left(\frac{hi}{\color{blue}{x} - lo}\right)\right)} \]

    8. sub-negate-rev N/A

       \[\leadsto \frac{1}{\frac{lo}{lo - x} - \left(\mathsf{neg}\left(\frac{hi}{\color{blue}{x - lo}}\right)\right)} \]

    9. lift--.f64 N/A

       \[\leadsto \frac{1}{\frac{lo}{lo - x} - \left(\mathsf{neg}\left(\frac{hi}{\color{blue}{x - lo}}\right)\right)} \]

    10. lower-/.f64 N/A

        \[\leadsto \frac{1}{\frac{lo}{lo - x} - \left(\mathsf{neg}\left(\color{blue}{\frac{hi}{x - lo}}\right)\right)} \]

    11. lift-/.f64 N/A

        \[\leadsto \frac{1}{\frac{lo}{lo - x} - \left(\mathsf{neg}\left(\frac{hi}{x - lo}\right)\right)} \]

    12. distribute-neg-frac2 N/A

        \[\leadsto \frac{1}{\frac{lo}{lo - x} - \frac{hi}{\color{blue}{\mathsf{neg}\left(\left(x - lo\right)\right)}}} \]

    13. lift--.f64 N/A

        \[\leadsto \frac{1}{\frac{lo}{lo - x} - \frac{hi}{\mathsf{neg}\left(\left(x - lo\right)\right)}} \]

    14. sub-negate-rev N/A

        \[\leadsto \frac{1}{\frac{lo}{lo - x} - \frac{hi}{lo - \color{blue}{x}}} \]

    15. lift--.f64 N/A

        \[\leadsto \frac{1}{\frac{lo}{lo - x} - \frac{hi}{lo - \color{blue}{x}}} \]

    16. lower-/.f64 99.4%

        \[\leadsto \frac{1}{\frac{lo}{lo - x} - \frac{hi}{\color{blue}{lo - x}}} \]

11. Applied rewrites 99.4%

    \[\leadsto \frac{1}{\frac{lo}{lo - x} - \color{blue}{\frac{hi}{lo - x}}} \]
12. Add Preprocessing

Alternative 3: 98.4% accurate, 0.6× speedup

Math

\[\frac{1}{1 + -1 \cdot \frac{hi - x}{lo}} \]

FPCore

(FPCore (lo hi x) :precision binary64 (/ 1.0 (+ 1.0 (* -1.0 (/ (- hi x) lo)))))

C

double code(double lo, double hi, double x) {
	return 1.0 / (1.0 + (-1.0 * ((hi - x) / lo)));
}

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(lo, hi, x)
use fmin_fmax_functions
    real(8), intent (in) :: lo
    real(8), intent (in) :: hi
    real(8), intent (in) :: x
    code = 1.0d0 / (1.0d0 + ((-1.0d0) * ((hi - x) / lo)))
end function

Java

public static double code(double lo, double hi, double x) {
	return 1.0 / (1.0 + (-1.0 * ((hi - x) / lo)));
}

Python

def code(lo, hi, x):
	return 1.0 / (1.0 + (-1.0 * ((hi - x) / lo)))

Julia

function code(lo, hi, x)
	return Float64(1.0 / Float64(1.0 + Float64(-1.0 * Float64(Float64(hi - x) / lo))))
end

MATLAB

function tmp = code(lo, hi, x)
	tmp = 1.0 / (1.0 + (-1.0 * ((hi - x) / lo)));
end

Wolfram

code[lo_, hi_, x_] := N[(1.0 / N[(1.0 + N[(-1.0 * N[(N[(hi - x), $MachinePrecision] / lo), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 3.1%

   \[\frac{x - lo}{hi - lo} \]

2. Taylor expanded in lo around 0

   \[\leadsto \frac{x - lo}{\color{blue}{hi}} \]
3. Step-by-step derivation

4. Applied rewrites 18.8%

   \[\leadsto \frac{x - lo}{\color{blue}{hi}} \]
5. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{x - lo}{hi}} \]

   2. div-flip N/A

      \[\leadsto \color{blue}{\frac{1}{\frac{hi}{x - lo}}} \]

   3. lower-unsound-/.f64 N/A

      \[\leadsto \color{blue}{\frac{1}{\frac{hi}{x - lo}}} \]

   4. lower-unsound-/.f64 18.8%

      \[\leadsto \frac{1}{\color{blue}{\frac{hi}{x - lo}}} \]

6. Applied rewrites 18.8%

   \[\leadsto \color{blue}{\frac{1}{\frac{hi}{x - lo}}} \]

7. Taylor expanded in lo around -inf

   \[\leadsto \frac{1}{\color{blue}{1 + -1 \cdot \frac{hi - x}{lo}}} \]
8. Step-by-step derivation

   1. lower-+.f64 N/A

      \[\leadsto \frac{1}{1 + \color{blue}{-1 \cdot \frac{hi - x}{lo}}} \]

   2. lower-*.f64 N/A

      \[\leadsto \frac{1}{1 + -1 \cdot \color{blue}{\frac{hi - x}{lo}}} \]

   3. lower-/.f64 N/A

      \[\leadsto \frac{1}{1 + -1 \cdot \frac{hi - x}{\color{blue}{lo}}} \]

   4. lower--.f64 98.4%

      \[\leadsto \frac{1}{1 + -1 \cdot \frac{hi - x}{lo}} \]

9. Applied rewrites 98.4%

   \[\leadsto \frac{1}{\color{blue}{1 + -1 \cdot \frac{hi - x}{lo}}} \]
10. Add Preprocessing

Alternative 4: 98.4% accurate, 0.6× speedup

Math

\[\frac{1}{\mathsf{fma}\left(-1, -1, \frac{hi}{x - lo}\right)} \]

FPCore

(FPCore (lo hi x) :precision binary64 (/ 1.0 (fma -1.0 -1.0 (/ hi (- x lo)))))

C

double code(double lo, double hi, double x) {
	return 1.0 / fma(-1.0, -1.0, (hi / (x - lo)));
}

Julia

function code(lo, hi, x)
	return Float64(1.0 / fma(-1.0, -1.0, Float64(hi / Float64(x - lo))))
end

Wolfram

code[lo_, hi_, x_] := N[(1.0 / N[(-1.0 * -1.0 + N[(hi / N[(x - lo), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 3.1%

   \[\frac{x - lo}{hi - lo} \]

2. Taylor expanded in lo around 0

   \[\leadsto \frac{x - lo}{\color{blue}{hi}} \]
3. Step-by-step derivation

4. Applied rewrites 18.8%

   \[\leadsto \frac{x - lo}{\color{blue}{hi}} \]
5. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{x - lo}{hi}} \]

   2. div-flip N/A

      \[\leadsto \color{blue}{\frac{1}{\frac{hi}{x - lo}}} \]

   3. lower-unsound-/.f64 N/A

      \[\leadsto \color{blue}{\frac{1}{\frac{hi}{x - lo}}} \]

   4. lower-unsound-/.f64 18.8%

      \[\leadsto \frac{1}{\color{blue}{\frac{hi}{x - lo}}} \]

6. Applied rewrites 18.8%

   \[\leadsto \color{blue}{\frac{1}{\frac{hi}{x - lo}}} \]

7. Taylor expanded in hi around 0

   \[\leadsto \frac{1}{\color{blue}{-1 \cdot \frac{lo}{x - lo} + \frac{hi}{x - lo}}} \]
8. Step-by-step derivation

   1. lower-fma.f64 N/A

      \[\leadsto \frac{1}{\mathsf{fma}\left(-1, \color{blue}{\frac{lo}{x - lo}}, \frac{hi}{x - lo}\right)} \]

   2. lower-/.f64 N/A

      \[\leadsto \frac{1}{\mathsf{fma}\left(-1, \frac{lo}{\color{blue}{x - lo}}, \frac{hi}{x - lo}\right)} \]

   3. lower--.f64 N/A

      \[\leadsto \frac{1}{\mathsf{fma}\left(-1, \frac{lo}{x - \color{blue}{lo}}, \frac{hi}{x - lo}\right)} \]

   4. lower-/.f64 N/A

      \[\leadsto \frac{1}{\mathsf{fma}\left(-1, \frac{lo}{x - lo}, \frac{hi}{x - lo}\right)} \]

   5. lower--.f64 99.4%

      \[\leadsto \frac{1}{\mathsf{fma}\left(-1, \frac{lo}{x - lo}, \frac{hi}{x - lo}\right)} \]

9. Applied rewrites 99.4%

   \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(-1, \frac{lo}{x - lo}, \frac{hi}{x - lo}\right)}} \]

10. Taylor expanded in lo around inf

    \[\leadsto \frac{1}{\mathsf{fma}\left(-1, -1, \frac{hi}{x - lo}\right)} \]
11. Step-by-step derivation

12. Applied rewrites 98.4%

    \[\leadsto \frac{1}{\mathsf{fma}\left(-1, -1, \frac{hi}{x - lo}\right)} \]
13. Add Preprocessing

Alternative 5: 98.4% accurate, 0.7× speedup

Math

\[\frac{1}{1 + -1 \cdot \frac{hi}{lo}} \]

FPCore

(FPCore (lo hi x) :precision binary64 (/ 1.0 (+ 1.0 (* -1.0 (/ hi lo)))))

C

double code(double lo, double hi, double x) {
	return 1.0 / (1.0 + (-1.0 * (hi / lo)));
}

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(lo, hi, x)
use fmin_fmax_functions
    real(8), intent (in) :: lo
    real(8), intent (in) :: hi
    real(8), intent (in) :: x
    code = 1.0d0 / (1.0d0 + ((-1.0d0) * (hi / lo)))
end function

Java

public static double code(double lo, double hi, double x) {
	return 1.0 / (1.0 + (-1.0 * (hi / lo)));
}

Python

def code(lo, hi, x):
	return 1.0 / (1.0 + (-1.0 * (hi / lo)))

Julia

function code(lo, hi, x)
	return Float64(1.0 / Float64(1.0 + Float64(-1.0 * Float64(hi / lo))))
end

MATLAB

function tmp = code(lo, hi, x)
	tmp = 1.0 / (1.0 + (-1.0 * (hi / lo)));
end

Wolfram

code[lo_, hi_, x_] := N[(1.0 / N[(1.0 + N[(-1.0 * N[(hi / lo), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 3.1%

   \[\frac{x - lo}{hi - lo} \]

2. Taylor expanded in lo around 0

   \[\leadsto \frac{x - lo}{\color{blue}{hi}} \]
3. Step-by-step derivation

4. Applied rewrites 18.8%

   \[\leadsto \frac{x - lo}{\color{blue}{hi}} \]
5. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{x - lo}{hi}} \]

   2. div-flip N/A

      \[\leadsto \color{blue}{\frac{1}{\frac{hi}{x - lo}}} \]

   3. lower-unsound-/.f64 N/A

      \[\leadsto \color{blue}{\frac{1}{\frac{hi}{x - lo}}} \]

   4. lower-unsound-/.f64 18.8%

      \[\leadsto \frac{1}{\color{blue}{\frac{hi}{x - lo}}} \]

6. Applied rewrites 18.8%

   \[\leadsto \color{blue}{\frac{1}{\frac{hi}{x - lo}}} \]

7. Taylor expanded in hi around 0

   \[\leadsto \frac{1}{\color{blue}{-1 \cdot \frac{lo}{x - lo} + \frac{hi}{x - lo}}} \]
8. Step-by-step derivation

   1. lower-fma.f64 N/A

      \[\leadsto \frac{1}{\mathsf{fma}\left(-1, \color{blue}{\frac{lo}{x - lo}}, \frac{hi}{x - lo}\right)} \]

   2. lower-/.f64 N/A

      \[\leadsto \frac{1}{\mathsf{fma}\left(-1, \frac{lo}{\color{blue}{x - lo}}, \frac{hi}{x - lo}\right)} \]

   3. lower--.f64 N/A

      \[\leadsto \frac{1}{\mathsf{fma}\left(-1, \frac{lo}{x - \color{blue}{lo}}, \frac{hi}{x - lo}\right)} \]

   4. lower-/.f64 N/A

      \[\leadsto \frac{1}{\mathsf{fma}\left(-1, \frac{lo}{x - lo}, \frac{hi}{x - lo}\right)} \]

   5. lower--.f64 99.4%

      \[\leadsto \frac{1}{\mathsf{fma}\left(-1, \frac{lo}{x - lo}, \frac{hi}{x - lo}\right)} \]

9. Applied rewrites 99.4%

   \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(-1, \frac{lo}{x - lo}, \frac{hi}{x - lo}\right)}} \]

10. Taylor expanded in x around 0

    \[\leadsto \frac{1}{1 + \color{blue}{-1 \cdot \frac{hi}{lo}}} \]
11. Step-by-step derivation

    1. lower-+.f64 N/A

       \[\leadsto \frac{1}{1 + -1 \cdot \color{blue}{\frac{hi}{lo}}} \]

    2. lower-*.f64 N/A

       \[\leadsto \frac{1}{1 + -1 \cdot \frac{hi}{\color{blue}{lo}}} \]

    3. lower-/.f64 98.4%

       \[\leadsto \frac{1}{1 + -1 \cdot \frac{hi}{lo}} \]

12. Applied rewrites 98.4%

    \[\leadsto \frac{1}{1 + \color{blue}{-1 \cdot \frac{hi}{lo}}} \]
13. Add Preprocessing

Alternative 6: 18.8% accurate, 1.4× speedup

Math

\[\frac{x - lo}{hi} \]

FPCore

(FPCore (lo hi x) :precision binary64 (/ (- x lo) hi))

C

double code(double lo, double hi, double x) {
	return (x - lo) / hi;
}

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(lo, hi, x)
use fmin_fmax_functions
    real(8), intent (in) :: lo
    real(8), intent (in) :: hi
    real(8), intent (in) :: x
    code = (x - lo) / hi
end function

Java

public static double code(double lo, double hi, double x) {
	return (x - lo) / hi;
}

Python

def code(lo, hi, x):
	return (x - lo) / hi

Julia

function code(lo, hi, x)
	return Float64(Float64(x - lo) / hi)
end

MATLAB

function tmp = code(lo, hi, x)
	tmp = (x - lo) / hi;
end

Wolfram

code[lo_, hi_, x_] := N[(N[(x - lo), $MachinePrecision] / hi), $MachinePrecision]

Derivation

1. Initial program 3.1%

   \[\frac{x - lo}{hi - lo} \]

2. Taylor expanded in lo around 0

   \[\leadsto \frac{x - lo}{\color{blue}{hi}} \]
3. Step-by-step derivation

4. Applied rewrites 18.8%

   \[\leadsto \frac{x - lo}{\color{blue}{hi}} \]
5. Add Preprocessing

Alternative 7: 18.8% accurate, 1.8× speedup

Math

\[\frac{-lo}{hi} \]

FPCore

(FPCore (lo hi x) :precision binary64 (/ (- lo) hi))

C

double code(double lo, double hi, double x) {
	return -lo / hi;
}

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(lo, hi, x)
use fmin_fmax_functions
    real(8), intent (in) :: lo
    real(8), intent (in) :: hi
    real(8), intent (in) :: x
    code = -lo / hi
end function

Java

public static double code(double lo, double hi, double x) {
	return -lo / hi;
}

Python

def code(lo, hi, x):
	return -lo / hi

Julia

function code(lo, hi, x)
	return Float64(Float64(-lo) / hi)
end

MATLAB

function tmp = code(lo, hi, x)
	tmp = -lo / hi;
end

Wolfram

code[lo_, hi_, x_] := N[((-lo) / hi), $MachinePrecision]

Derivation

1. Initial program 3.1%

   \[\frac{x - lo}{hi - lo} \]

2. Taylor expanded in lo around 0

   \[\leadsto \frac{x - lo}{\color{blue}{hi}} \]
3. Step-by-step derivation

4. Applied rewrites 18.8%

   \[\leadsto \frac{x - lo}{\color{blue}{hi}} \]
5. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{x - lo}{hi}} \]

   2. div-flip N/A

      \[\leadsto \color{blue}{\frac{1}{\frac{hi}{x - lo}}} \]

   3. lower-unsound-/.f64 N/A

      \[\leadsto \color{blue}{\frac{1}{\frac{hi}{x - lo}}} \]

   4. lower-unsound-/.f64 18.8%

      \[\leadsto \frac{1}{\color{blue}{\frac{hi}{x - lo}}} \]

6. Applied rewrites 18.8%

   \[\leadsto \color{blue}{\frac{1}{\frac{hi}{x - lo}}} \]

7. Taylor expanded in lo around inf

   \[\leadsto \frac{1}{\frac{hi}{\color{blue}{-1 \cdot lo}}} \]
8. Step-by-step derivation

   1. lower-*.f64 18.8%

      \[\leadsto \frac{1}{\frac{hi}{-1 \cdot \color{blue}{lo}}} \]

9. Applied rewrites 18.8%

   \[\leadsto \frac{1}{\frac{hi}{\color{blue}{-1 \cdot lo}}} \]
10. Step-by-step derivation

    1. lift-/.f64 N/A

       \[\leadsto \color{blue}{\frac{1}{\frac{hi}{-1 \cdot lo}}} \]

    2. lift-/.f64 N/A

       \[\leadsto \frac{1}{\color{blue}{\frac{hi}{-1 \cdot lo}}} \]

    3. div-flip-rev N/A

       \[\leadsto \color{blue}{\frac{-1 \cdot lo}{hi}} \]

    4. lower-/.f64 18.8%

       \[\leadsto \color{blue}{\frac{-1 \cdot lo}{hi}} \]

    5. lift-*.f64 N/A

       \[\leadsto \frac{-1 \cdot \color{blue}{lo}}{hi} \]

    6. mul-1-neg N/A

       \[\leadsto \frac{\mathsf{neg}\left(lo\right)}{hi} \]

    7. lower-neg.f64 18.8%

       \[\leadsto \frac{-lo}{hi} \]

11. Applied rewrites 18.8%

    \[\leadsto \color{blue}{\frac{-lo}{hi}} \]
12. Add Preprocessing

Alternative 8: 18.7% accurate, 9.6× speedup

Math

\[1 \]

FPCore

(FPCore (lo hi x) :precision binary64 1.0)

C

double code(double lo, double hi, double x) {
	return 1.0;
}

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

Java

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

Python

def code(lo, hi, x):
	return 1.0

Julia

function code(lo, hi, x)
	return 1.0
end

MATLAB

function tmp = code(lo, hi, x)
	tmp = 1.0;
end

Wolfram

code[lo_, hi_, x_] := 1.0

Derivation

1. Initial program 3.1%

   \[\frac{x - lo}{hi - lo} \]

2. Taylor expanded in lo around inf

   \[\leadsto \color{blue}{1} \]
3. Step-by-step derivation

4. Applied rewrites 18.7%

   \[\leadsto \color{blue}{1} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2025191 
(FPCore (lo hi x)
  :name "xlohi (overflows)"
  :precision binary64
  :pre (and (< lo -1e+308) (> hi 1e+308))
  (/ (- x lo) (- hi lo)))