xlohi (overflows)

Percentage Accurate: 3.1% → 20.4%

Time: 16.9s

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: 20.4% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \frac{e^{lo \cdot \left(\frac{1}{hi} - \frac{\log \left(\frac{-1}{lo}\right)}{lo}\right)}}{hi} \end{array} \]

FPCore

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

C

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

Java

public static double code(double lo, double hi, double x) {
	return Math.exp((lo * ((1.0 / hi) - (Math.log((-1.0 / lo)) / lo)))) / hi;
}

Python

def code(lo, hi, x):
	return math.exp((lo * ((1.0 / hi) - (math.log((-1.0 / lo)) / lo)))) / hi

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 3.1%

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

3. Taylor expanded in hi around inf 0.7%

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

   1. +-commutative 0.7%

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

   2. associate--l+ 0.7%

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

   3. +-commutative 0.7%

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

   4. *-rgt-identity 0.7%

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

   5. *-commutative 0.7%

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

   6. associate-/l* 10.4%

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

   7. distribute-lft-out 10.7%

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

5. Simplified 10.7%

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

   1. add-exp-log 10.0%

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

   2. log-prod 10.0%

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

   3. log1p-define 10.0%

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

7. Applied egg-rr 10.0%

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

8. Taylor expanded in hi around inf 20.5%

   \[\leadsto \frac{e^{\color{blue}{\log \left(x - lo\right) + \frac{lo}{hi}}}}{hi} \]
9. Step-by-step derivation

   1. +-commutative 20.5%

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

10. Simplified 20.5%

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

11. Taylor expanded in lo around -inf 20.5%

    \[\leadsto \frac{e^{\color{blue}{-1 \cdot \left(lo \cdot \left(\frac{\log \left(\frac{-1}{lo}\right)}{lo} - \frac{1}{hi}\right)\right)}}}{hi} \]
12. Step-by-step derivation

    1. associate-*r* 20.5%

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

    2. neg-mul-1 20.5%

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

13. Simplified 20.5%

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

14. Final simplification 20.5%

    \[\leadsto \frac{e^{lo \cdot \left(\frac{1}{hi} - \frac{\log \left(\frac{-1}{lo}\right)}{lo}\right)}}{hi} \]
15. Add Preprocessing

Alternative 2: 20.4% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 3.1%

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

3. Taylor expanded in hi around inf 0.7%

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

   1. +-commutative 0.7%

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

   2. associate--l+ 0.7%

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

   3. +-commutative 0.7%

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

   4. *-rgt-identity 0.7%

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

   5. *-commutative 0.7%

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

   6. associate-/l* 10.4%

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

   7. distribute-lft-out 10.7%

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

5. Simplified 10.7%

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

   1. add-exp-log 10.0%

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

   2. log-prod 10.0%

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

   3. log1p-define 10.0%

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

7. Applied egg-rr 10.0%

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

8. Taylor expanded in hi around inf 20.5%

   \[\leadsto \frac{e^{\color{blue}{\log \left(x - lo\right) + \frac{lo}{hi}}}}{hi} \]
9. Step-by-step derivation

   1. +-commutative 20.5%

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

10. Simplified 20.5%

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

11. Taylor expanded in x around 0 20.5%

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

    1. exp-sum 20.5%

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

    2. rem-exp-log 20.5%

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

13. Simplified 20.5%

    \[\leadsto \frac{\color{blue}{\left(-lo\right) \cdot e^{\frac{lo}{hi}}}}{hi} \]
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 18.9%

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

Alternative 4: 18.7% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 3.1%

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

3. Taylor expanded in hi around 0 18.6%

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

   1. div-sub 18.6%

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

   2. sub-neg 18.6%

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

   3. *-inverses 18.6%

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

   4. metadata-eval 18.6%

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

   5. distribute-lft-in 18.6%

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

   6. metadata-eval 18.6%

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

   7. +-commutative 18.6%

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

   8. mul-1-neg 18.6%

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

   9. unsub-neg 18.6%

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

5. Simplified 18.6%

   \[\leadsto \color{blue}{1 - \frac{x}{lo}} \]
6. 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 18.6%

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

Reproduce

herbie shell --seed 2024101 
(FPCore (lo hi x)
  :name "xlohi (overflows)"
  :precision binary64
  :pre (and (< lo -1e+308) (> hi 1e+308))
  (/ (- x lo) (- hi lo)))