xlohi (overflows)

Percentage Accurate: 3.1% → 99.2%

Time: 13.6s

Alternatives: 4

Speedup: 7.0×

Specification

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

Math

\[\begin{array}{l} \\ \frac{x - lo}{hi - lo} \end{array} \]

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

real(8) function code(lo, hi, x)
    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

\[\begin{array}{l} \\ \frac{x - lo}{hi - lo} \end{array} \]

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

real(8) function code(lo, hi, x)
    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: 99.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ 1 + \frac{\frac{x - hi}{lo}}{\frac{hi}{lo} + -1} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 3.1%

   \[\frac{x - lo}{hi - lo} \]
2. Add Preprocessing

3. Taylor expanded in lo around inf 0.0%

   \[\leadsto \color{blue}{\left(1 + \left(-1 \cdot \frac{x}{lo} + \frac{hi \cdot \left(-1 \cdot x - -1 \cdot hi\right)}{{lo}^{2}}\right)\right) - -1 \cdot \frac{hi}{lo}} \]

5. Simplified 18.9%

   \[\leadsto \color{blue}{1 + \left(\frac{hi}{lo} + 1\right) \cdot \frac{hi - x}{lo}} \]
6. Step-by-step derivation

   1. flip-+ 18.9%

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

   2. frac-times 17.6%

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

   3. metadata-eval 17.6%

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

   4. sub-neg 17.6%

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

   5. pow2 17.6%

      \[\leadsto 1 + \frac{\left(\color{blue}{{\left(\frac{hi}{lo}\right)}^{2}} + \left(-1\right)\right) \cdot \left(hi - x\right)}{\left(\frac{hi}{lo} - 1\right) \cdot lo} \]

   6. metadata-eval 17.6%

      \[\leadsto 1 + \frac{\left({\left(\frac{hi}{lo}\right)}^{2} + \color{blue}{-1}\right) \cdot \left(hi - x\right)}{\left(\frac{hi}{lo} - 1\right) \cdot lo} \]

   7. sub-neg 17.6%

      \[\leadsto 1 + \frac{\left({\left(\frac{hi}{lo}\right)}^{2} + -1\right) \cdot \left(hi - x\right)}{\color{blue}{\left(\frac{hi}{lo} + \left(-1\right)\right)} \cdot lo} \]

   8. metadata-eval 17.6%

      \[\leadsto 1 + \frac{\left({\left(\frac{hi}{lo}\right)}^{2} + -1\right) \cdot \left(hi - x\right)}{\left(\frac{hi}{lo} + \color{blue}{-1}\right) \cdot lo} \]

7. Applied egg-rr 17.6%

   \[\leadsto 1 + \color{blue}{\frac{\left({\left(\frac{hi}{lo}\right)}^{2} + -1\right) \cdot \left(hi - x\right)}{\left(\frac{hi}{lo} + -1\right) \cdot lo}} \]
8. Step-by-step derivation

   1. *-commutative 17.6%

      \[\leadsto 1 + \frac{\color{blue}{\left(hi - x\right) \cdot \left({\left(\frac{hi}{lo}\right)}^{2} + -1\right)}}{\left(\frac{hi}{lo} + -1\right) \cdot lo} \]

   2. *-commutative 17.6%

      \[\leadsto 1 + \frac{\left(hi - x\right) \cdot \left({\left(\frac{hi}{lo}\right)}^{2} + -1\right)}{\color{blue}{lo \cdot \left(\frac{hi}{lo} + -1\right)}} \]

   3. associate-/r* 18.1%

      \[\leadsto 1 + \color{blue}{\frac{\frac{\left(hi - x\right) \cdot \left({\left(\frac{hi}{lo}\right)}^{2} + -1\right)}{lo}}{\frac{hi}{lo} + -1}} \]

   4. *-rgt-identity 18.1%

      \[\leadsto 1 + \frac{\frac{\left(hi - x\right) \cdot \left({\left(\frac{hi}{lo}\right)}^{2} + -1\right)}{\color{blue}{lo \cdot 1}}}{\frac{hi}{lo} + -1} \]

   5. *-commutative 18.1%

      \[\leadsto 1 + \frac{\frac{\left(hi - x\right) \cdot \left({\left(\frac{hi}{lo}\right)}^{2} + -1\right)}{\color{blue}{1 \cdot lo}}}{\frac{hi}{lo} + -1} \]

   6. times-frac 18.9%

      \[\leadsto 1 + \frac{\color{blue}{\frac{hi - x}{1} \cdot \frac{{\left(\frac{hi}{lo}\right)}^{2} + -1}{lo}}}{\frac{hi}{lo} + -1} \]

   7. /-rgt-identity 18.9%

      \[\leadsto 1 + \frac{\color{blue}{\left(hi - x\right)} \cdot \frac{{\left(\frac{hi}{lo}\right)}^{2} + -1}{lo}}{\frac{hi}{lo} + -1} \]

   8. +-commutative 18.9%

      \[\leadsto 1 + \frac{\left(hi - x\right) \cdot \frac{\color{blue}{-1 + {\left(\frac{hi}{lo}\right)}^{2}}}{lo}}{\frac{hi}{lo} + -1} \]

9. Simplified 18.9%

   \[\leadsto 1 + \color{blue}{\frac{\left(hi - x\right) \cdot \frac{-1 + {\left(\frac{hi}{lo}\right)}^{2}}{lo}}{\frac{hi}{lo} + -1}} \]

10. Taylor expanded in hi around 0 99.3%

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

    1. neg-mul-1 99.3%

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

    2. remove-double-neg 99.3%

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

    3. distribute-frac-neg2 99.3%

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

    4. distribute-neg-frac2 99.3%

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

    5. mul-1-neg 99.3%

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

    6. distribute-neg-in 99.3%

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

    7. mul-1-neg 99.3%

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

    8. sub-neg 99.3%

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

    9. div-sub 99.3%

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

    10. distribute-frac-neg2 99.3%

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

12. Simplified 99.3%

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

13. Final simplification 99.3%

    \[\leadsto 1 + \frac{\frac{x - hi}{lo}}{\frac{hi}{lo} + -1} \]
14. Add Preprocessing

Alternative 2: 18.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ 1 + hi \cdot \frac{1 + \frac{hi}{lo}}{lo} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 3.1%

   \[\frac{x - lo}{hi - lo} \]
2. Add Preprocessing

3. Taylor expanded in lo around inf 0.0%

   \[\leadsto \color{blue}{\left(1 + \left(-1 \cdot \frac{x}{lo} + \frac{hi \cdot \left(-1 \cdot x - -1 \cdot hi\right)}{{lo}^{2}}\right)\right) - -1 \cdot \frac{hi}{lo}} \]

5. Simplified 18.9%

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

6. Taylor expanded in x around 0 18.9%

   \[\leadsto 1 + \color{blue}{\frac{hi \cdot \left(1 + \frac{hi}{lo}\right)}{lo}} \]
7. Step-by-step derivation

   1. associate-/l* 18.9%

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

8. Simplified 18.9%

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

9. Final simplification 18.9%

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

Alternative 3: 18.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \frac{lo}{-hi} \end{array} \]

FPCore

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

C

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

Fortran

real(8) function code(lo, hi, x)
    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(lo / Float64(-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. Add Preprocessing

3. Taylor expanded in hi around inf 18.8%

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

4. Taylor expanded in x around 0 18.8%

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

   1. neg-mul-1 18.8%

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

   2. distribute-neg-frac 18.8%

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

6. Simplified 18.8%

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

7. Final simplification 18.8%

   \[\leadsto \frac{lo}{-hi} \]
8. Add Preprocessing

Alternative 4: 18.7% accurate, 7.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

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

C

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

Fortran

real(8) function code(lo, hi, x)
    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. Add Preprocessing

3. Taylor expanded in lo around inf 18.7%

   \[\leadsto \color{blue}{1} \]

4. Final simplification 18.7%

   \[\leadsto 1 \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024077 
(FPCore (lo hi x)
  :name "xlohi (overflows)"
  :precision binary64
  :pre (and (< lo -1e+308) (> hi 1e+308))
  (/ (- x lo) (- hi lo)))