xlohi (overflows)

Percentage Accurate: 3.1% → 98.5%

Time: 17.7s

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: 98.5% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ e^{-\mathsf{log1p}\left(\frac{\left(-x\right) - hi}{lo}\right)} \end{array} \]

FPCore

(FPCore (lo hi x) :precision binary64 (exp (- (log1p (/ (- (- x) hi) lo)))))

C

double code(double lo, double hi, double x) {
	return exp(-log1p(((-x - hi) / lo)));
}

Java

public static double code(double lo, double hi, double x) {
	return Math.exp(-Math.log1p(((-x - hi) / lo)));
}

Python

def code(lo, hi, x):
	return math.exp(-math.log1p(((-x - hi) / lo)))

Julia

function code(lo, hi, x)
	return exp(Float64(-log1p(Float64(Float64(Float64(-x) - hi) / lo))))
end

Wolfram

code[lo_, hi_, x_] := N[Exp[(-N[Log[1 + N[(N[((-x) - hi), $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. Step-by-step derivation

   1. flip-+ 18.9%

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

   2. clear-num 18.9%

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

   3. sub-neg 18.9%

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

   4. add-sqr-sqrt 9.9%

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

   5. sqrt-unprod 14.5%

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

   6. sqr-neg 14.5%

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

   7. sqrt-unprod 8.9%

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

   8. add-sqr-sqrt 18.8%

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

   9. metadata-eval 18.8%

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

   10. pow2 18.8%

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

7. Applied egg-rr 18.8%

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

8. Taylor expanded in lo around inf 98.2%

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

   1. mul-1-neg 98.2%

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

   2. unsub-neg 98.2%

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

10. Simplified 98.2%

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

    1. add-exp-log 98.2%

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

    2. log-rec 98.1%

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

    3. sub-neg 98.1%

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

    4. log1p-define 98.4%

       \[\leadsto e^{-\color{blue}{\mathsf{log1p}\left(-\frac{hi + x}{lo}\right)}} \]

    5. distribute-neg-frac2 98.4%

       \[\leadsto e^{-\mathsf{log1p}\left(\color{blue}{\frac{hi + x}{-lo}}\right)} \]

12. Applied egg-rr 98.4%

    \[\leadsto \color{blue}{e^{-\mathsf{log1p}\left(\frac{hi + x}{-lo}\right)}} \]

13. Final simplification 98.4%

    \[\leadsto e^{-\mathsf{log1p}\left(\frac{\left(-x\right) - hi}{lo}\right)} \]
14. Add Preprocessing

Alternative 2: 98.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double lo, double hi, double x) {
	return 1.0 / (1.0 - (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 = 1.0d0 / (1.0d0 - (hi / lo))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[lo_, hi_, x_] := N[(1.0 / N[(1.0 - N[(hi / 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. Step-by-step derivation

   1. flip-+ 18.9%

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

   2. clear-num 18.9%

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

   3. sub-neg 18.9%

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

   4. add-sqr-sqrt 9.9%

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

   5. sqrt-unprod 14.5%

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

   6. sqr-neg 14.5%

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

   7. sqrt-unprod 8.9%

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

   8. add-sqr-sqrt 18.8%

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

   9. metadata-eval 18.8%

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

   10. pow2 18.8%

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

7. Applied egg-rr 18.8%

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

8. Taylor expanded in lo around inf 98.2%

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

   1. mul-1-neg 98.2%

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

   2. unsub-neg 98.2%

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

10. Simplified 98.2%

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

11. Taylor expanded in x around 0 98.3%

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

12. Final simplification 98.3%

    \[\leadsto \frac{1}{1 - \frac{hi}{lo}} \]
13. 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 2024096 
(FPCore (lo hi x)
  :name "xlohi (overflows)"
  :precision binary64
  :pre (and (< lo -1e+308) (> hi 1e+308))
  (/ (- x lo) (- hi lo)))