xlohi (overflows)

Percentage Accurate: 3.1% → 96.6%

Time: 16.2s

Alternatives: 5

Speedup: 18.0×

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: 96.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(x, \frac{-1 - \frac{hi}{lo}}{lo}, x \cdot \frac{\mathsf{fma}\left(hi, \frac{\frac{\mathsf{fma}\left(hi, \frac{hi}{lo \cdot lo}, -1\right)}{-1 + \frac{hi}{lo}}}{lo}, 1\right)}{x}\right) \end{array} \]

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 3.1%

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

3. Taylor expanded in lo around inf

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

5. Applied rewrites 18.8%

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

6. Taylor expanded in x around -inf

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

8. Applied rewrites 18.8%

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

10. Applied rewrites 96.4%

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

11. Final simplification 96.4%

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

Alternative 2: 18.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\frac{lo + hi}{lo}, \frac{hi - x}{lo}, 1\right) \end{array} \]

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 3.1%

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

3. Taylor expanded in lo around inf

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

5. Applied rewrites 18.8%

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

6. Taylor expanded in lo around 0

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

8. Applied rewrites 18.8%

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

9. Final simplification 18.8%

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

Alternative 3: 18.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ 1 + \frac{\mathsf{fma}\left(hi, \frac{hi}{lo}, hi\right)}{lo} \end{array} \]

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 3.1%

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

3. Taylor expanded in lo around inf

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

5. Applied rewrites 18.8%

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

6. Taylor expanded in x around -inf

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

8. Applied rewrites 18.8%

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

9. Taylor expanded in x around 0

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

11. Applied rewrites 18.8%

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

Alternative 4: 18.8% accurate, 1.3× 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(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. Add Preprocessing

3. Taylor expanded in hi around inf

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

   1. lower-/.f64 N/A

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

   2. lower--.f64 18.8

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

5. Applied rewrites 18.8%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 18.8%

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

Alternative 5: 18.7% accurate, 18.0× 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

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

5. Applied rewrites 18.7%

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

Reproduce

herbie shell --seed 2024226 
(FPCore (lo hi x)
  :name "xlohi (overflows)"
  :precision binary64
  :pre (and (< lo -1e+308) (> hi 1e+308))
  (/ (- x lo) (- hi lo)))