xlohi (overflows)

Percentage Accurate: 3.1% → 18.8%

Time: 7.7s

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: 18.8% accurate, 0.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[lo_, hi_, x_] := N[(N[(x / hi), $MachinePrecision] - N[(lo * N[Exp[N[Log[N[(N[Power[hi, -1.0], $MachinePrecision] - N[(x / N[(hi * hi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $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. Step-by-step derivation

   1. add-exp-log 18.8%

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

   2. inv-pow 18.8%

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

   3. div-inv 18.8%

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

   4. pow2 18.8%

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

   5. pow-flip 18.8%

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

   6. metadata-eval 18.8%

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

6. Applied egg-rr 18.8%

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

7. Taylor expanded in x around 0 18.8%

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

   1. unpow2 18.8%

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

9. Simplified 18.8%

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

10. Final simplification 18.8%

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

Alternative 2: 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. Taylor expanded in hi around 0 18.7%

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

   1. associate-*r/ 18.7%

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

   2. neg-mul-1 18.7%

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

   3. sub-neg 18.7%

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

   4. distribute-neg-in 18.7%

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

   5. remove-double-neg 18.7%

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

4. Simplified 18.7%

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

5. Taylor expanded in x around 0 18.7%

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

   1. +-commutative 18.7%

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

   2. mul-1-neg 18.7%

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

   3. unsub-neg 18.7%

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

7. Simplified 18.7%

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

8. Final simplification 18.7%

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

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. 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 4: 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 18.7%

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

   1. neg-mul-1 18.7%

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

   2. distribute-neg-frac 18.7%

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

7. Simplified 18.7%

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

8. Final simplification 18.7%

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

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. Taylor expanded in lo around inf 18.6%

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

3. Final simplification 18.6%

   \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023222 
(FPCore (lo hi x)
  :name "xlohi (overflows)"
  :precision binary64
  :pre (and (< lo -1e+308) (> hi 1e+308))
  (/ (- x lo) (- hi lo)))