xlohi (overflows)

Percentage Accurate: 3.1% → 98.7%

Time: 6.4s

Alternatives: 6

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: 98.7% accurate, 0.0× speedup

Math

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

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(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. Taylor expanded in lo around inf 8.7%

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

   1. +-commutative 8.7%

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

   2. associate--l+ 8.7%

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

   3. associate-*r/ 8.7%

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

   4. associate-*r/ 8.7%

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

   5. div-sub 8.7%

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

   6. distribute-lft-out-- 8.7%

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

   7. associate-*r/ 8.7%

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

   8. mul-1-neg 8.7%

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

   9. unsub-neg 8.7%

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

4. Simplified 8.7%

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

   1. flip-- 8.7%

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

   2. metadata-eval 8.7%

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

   3. pow1 8.7%

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

   4. pow1 8.7%

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

   5. pow-prod-up 8.7%

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

   6. metadata-eval 8.7%

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

6. Applied egg-rr 8.7%

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

7. Taylor expanded in lo around inf 99.1%

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

   1. add-exp-log 99.1%

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

   2. log-rec 99.0%

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

   3. log1p-udef 99.3%

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

9. Applied egg-rr 99.3%

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

10. Final simplification 99.3%

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

Alternative 2: 98.5% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 3.1%

   \[\frac{x - lo}{hi - lo} \]

2. Taylor expanded in lo around inf 8.7%

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

   1. +-commutative 8.7%

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

   2. associate--l+ 8.7%

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

   3. associate-*r/ 8.7%

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

   4. associate-*r/ 8.7%

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

   5. div-sub 8.7%

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

   6. distribute-lft-out-- 8.7%

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

   7. associate-*r/ 8.7%

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

   8. mul-1-neg 8.7%

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

   9. unsub-neg 8.7%

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

4. Simplified 8.7%

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

   1. flip-- 8.7%

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

   2. metadata-eval 8.7%

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

   3. pow1 8.7%

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

   4. pow1 8.7%

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

   5. pow-prod-up 8.7%

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

   6. metadata-eval 8.7%

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

6. Applied egg-rr 8.7%

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

7. Taylor expanded in lo around inf 99.1%

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

8. Final simplification 99.1%

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

Alternative 3: 98.5% 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. Taylor expanded in lo around inf 8.7%

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

   1. +-commutative 8.7%

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

   2. associate--l+ 8.7%

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

   3. associate-*r/ 8.7%

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

   4. associate-*r/ 8.7%

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

   5. div-sub 8.7%

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

   6. distribute-lft-out-- 8.7%

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

   7. associate-*r/ 8.7%

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

   8. mul-1-neg 8.7%

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

   9. unsub-neg 8.7%

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

4. Simplified 8.7%

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

   1. flip-- 8.7%

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

   2. metadata-eval 8.7%

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

   3. pow1 8.7%

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

   4. pow1 8.7%

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

   5. pow-prod-up 8.7%

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

   6. metadata-eval 8.7%

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

6. Applied egg-rr 8.7%

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

7. Taylor expanded in lo around inf 99.1%

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

8. Taylor expanded in x around 0 99.1%

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

9. Final simplification 99.1%

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

Alternative 4: 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. Taylor expanded in hi around inf 18.8%

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

3. Final simplification 18.8%

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

Alternative 5: 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(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. Taylor expanded in lo around 0 18.8%

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

   1. mul-1-neg 18.8%

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

   2. unsub-neg 18.8%

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

   3. mul-1-neg 18.8%

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

   4. unsub-neg 18.8%

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

   5. unpow2 18.8%

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

4. Simplified 18.8%

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

5. Taylor expanded in x around 0 18.8%

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

   1. associate-*r/ 18.8%

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

   2. neg-mul-1 18.8%

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

7. Simplified 18.8%

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

8. Final simplification 18.8%

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

Alternative 6: 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. Taylor expanded in lo around inf 18.6%

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

3. Final simplification 18.6%

   \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023171 
(FPCore (lo hi x)
  :name "xlohi (overflows)"
  :precision binary64
  :pre (and (< lo -1e+308) (> hi 1e+308))
  (/ (- x lo) (- hi lo)))