xlohi (overflows)

Percentage Accurate: 3.1% → 19.4%

Time: 16.6s

Alternatives: 5

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: 19.4% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \sqrt[3]{{\left({\left(\frac{hi}{lo}\right)}^{2}\right)}^{3}} \end{array} \]

FPCore

(FPCore (lo hi x) :precision binary64 (cbrt (pow (pow (/ hi lo) 2.0) 3.0)))

C

double code(double lo, double hi, double x) {
	return cbrt(pow(pow((hi / lo), 2.0), 3.0));
}

Java

public static double code(double lo, double hi, double x) {
	return Math.cbrt(Math.pow(Math.pow((hi / lo), 2.0), 3.0));
}

Julia

function code(lo, hi, x)
	return cbrt(((Float64(hi / lo) ^ 2.0) ^ 3.0))
end

Wolfram

code[lo_, hi_, x_] := N[Power[N[Power[N[Power[N[(hi / lo), $MachinePrecision], 2.0], $MachinePrecision], 3.0], $MachinePrecision], 1/3], $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%

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

   1. +-commutative 18.9%

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

   2. associate-/l* 18.9%

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

   3. fma-define 18.9%

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

8. Simplified 18.9%

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

9. Taylor expanded in hi around inf 0.0%

   \[\leadsto \color{blue}{\frac{{hi}^{2}}{{lo}^{2}}} \]
10. Step-by-step derivation

    1. unpow2 0.0%

       \[\leadsto \frac{\color{blue}{hi \cdot hi}}{{lo}^{2}} \]

    2. unpow2 0.0%

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

    3. times-frac 19.6%

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

    4. unpow2 19.6%

       \[\leadsto \color{blue}{{\left(\frac{hi}{lo}\right)}^{2}} \]

11. Simplified 19.6%

    \[\leadsto \color{blue}{{\left(\frac{hi}{lo}\right)}^{2}} \]
12. Step-by-step derivation

    1. add-cbrt-cube 19.6%

       \[\leadsto \color{blue}{\sqrt[3]{\left({\left(\frac{hi}{lo}\right)}^{2} \cdot {\left(\frac{hi}{lo}\right)}^{2}\right) \cdot {\left(\frac{hi}{lo}\right)}^{2}}} \]

    2. pow3 19.6%

       \[\leadsto \sqrt[3]{\color{blue}{{\left({\left(\frac{hi}{lo}\right)}^{2}\right)}^{3}}} \]

13. Applied egg-rr 19.6%

    \[\leadsto \color{blue}{\sqrt[3]{{\left({\left(\frac{hi}{lo}\right)}^{2}\right)}^{3}}} \]
14. Add Preprocessing

Alternative 2: 19.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double lo, double hi, double x) {
	return (hi / 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 = (hi / lo) * (hi / lo)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   1. +-commutative 18.9%

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

   2. associate-/l* 18.9%

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

   3. fma-define 18.9%

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

8. Simplified 18.9%

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

9. Taylor expanded in hi around inf 0.0%

   \[\leadsto \color{blue}{\frac{{hi}^{2}}{{lo}^{2}}} \]
10. Step-by-step derivation

    1. unpow2 0.0%

       \[\leadsto \frac{\color{blue}{hi \cdot hi}}{{lo}^{2}} \]

    2. unpow2 0.0%

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

    3. times-frac 19.6%

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

    4. unpow2 19.6%

       \[\leadsto \color{blue}{{\left(\frac{hi}{lo}\right)}^{2}} \]

11. Simplified 19.6%

    \[\leadsto \color{blue}{{\left(\frac{hi}{lo}\right)}^{2}} \]
12. Step-by-step derivation

    1. unpow2 19.6%

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

13. Applied egg-rr 19.6%

    \[\leadsto \color{blue}{\frac{hi}{lo} \cdot \frac{hi}{lo}} \]
14. Add Preprocessing

Alternative 3: 18.8% accurate, 1.4× speedup

Math

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

FPCore

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

C

double code(double lo, double hi, double x) {
	return (x - 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 = (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. Add Preprocessing

3. Taylor expanded in hi around inf 18.8%

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

Alternative 4: 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. associate-*r/ 18.8%

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

   2. neg-mul-1 18.8%

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

6. Simplified 18.8%

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

7. Final simplification 18.8%

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

Alternative 5: 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. Add Preprocessing

Reproduce

herbie shell --seed 2024096 
(FPCore (lo hi x)
  :name "xlohi (overflows)"
  :precision binary64
  :pre (and (< lo -1e+308) (> hi 1e+308))
  (/ (- x lo) (- hi lo)))