xlohi (overflows)

Percentage Accurate: 3.1% → 18.9%

Time: 18.7s

Alternatives: 4

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: 18.9% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[lo_, hi_, x_] := N[(N[(1.0 + N[(hi / lo), $MachinePrecision]), $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. Simplified 18.8%

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

Alternative 2: 18.9% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   1. mul-1-neg N/A

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

   2. unsub-neg N/A

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

   3. --lowering--.f64 N/A

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

   4. /-lowering-/.f64 N/A

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

   5. associate--l+ N/A

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

   6. +-commutative N/A

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

   7. associate-/l* N/A

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

   8. *-rgt-identity N/A

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

   9. distribute-lft-out-- N/A

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

   10. accelerator-lowering-fma.f64 N/A

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

   11. sub-neg N/A

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

   12. metadata-eval N/A

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

   13. +-lowering-+.f64 N/A

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

   14. /-lowering-/.f64 N/A

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

   15. --lowering--.f64 18.8

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

5. Simplified 18.8%

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

6. Taylor expanded in x around 0

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

   1. cancel-sign-sub-inv N/A

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

   2. metadata-eval N/A

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

   3. *-lft-identity N/A

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

   4. +-commutative N/A

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

   5. associate-/l* N/A

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

   6. accelerator-lowering-fma.f64 N/A

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

   7. /-lowering-/.f64 N/A

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

   8. +-lowering-+.f64 N/A

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

   9. /-lowering-/.f64 18.8

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

8. Simplified 18.8%

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

Alternative 3: 18.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ 0 - \frac{lo}{hi} \end{array} \]

FPCore

(FPCore (lo hi x) :precision binary64 (- 0.0 (/ lo hi)))

C

double code(double lo, double hi, double x) {
	return 0.0 - (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 = 0.0d0 - (lo / hi)
end function

Java

public static double code(double lo, double hi, double x) {
	return 0.0 - (lo / hi);
}

Python

def code(lo, hi, x):
	return 0.0 - (lo / hi)

Julia

function code(lo, hi, x)
	return Float64(0.0 - Float64(lo / hi))
end

MATLAB

function tmp = code(lo, hi, x)
	tmp = 0.0 - (lo / hi);
end

Wolfram

code[lo_, hi_, x_] := N[(0.0 - N[(lo / hi), $MachinePrecision]), $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. /-lowering-/.f64 N/A

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

   2. --lowering--.f64 18.8

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

5. Simplified 18.8%

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

6. Taylor expanded in x around 0

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

   1. mul-1-neg N/A

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

   2. neg-sub0 N/A

      \[\leadsto \frac{\color{blue}{0 - lo}}{hi} \]

   3. --lowering--.f64 18.8

      \[\leadsto \frac{\color{blue}{0 - lo}}{hi} \]

8. Simplified 18.8%

   \[\leadsto \frac{\color{blue}{0 - lo}}{hi} \]
9. Step-by-step derivation

   1. sub0-neg N/A

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

   2. neg-lowering-neg.f64 18.8

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

10. Applied egg-rr 18.8%

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

11. Final simplification 18.8%

    \[\leadsto 0 - \frac{lo}{hi} \]
12. Add Preprocessing

Alternative 4: 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. Simplified 18.7%

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

Reproduce

herbie shell --seed 2024198 
(FPCore (lo hi x)
  :name "xlohi (overflows)"
  :precision binary64
  :pre (and (< lo -1e+308) (> hi 1e+308))
  (/ (- x lo) (- hi lo)))