xlohi (overflows)

Percentage Accurate: 3.1% → 19.6%

Time: 17.3s

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: 19.6% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ 1 + hi \cdot \frac{\left|1 + \frac{hi}{lo}\right|}{lo} \end{array} \]

FPCore

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

C

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

Java

public static double code(double lo, double hi, double x) {
	return 1.0 + (hi * (Math.abs((1.0 + (hi / lo))) / lo));
}

Python

def code(lo, hi, x):
	return 1.0 + (hi * (math.fabs((1.0 + (hi / lo))) / lo))

Julia

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

MATLAB

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

Wolfram

code[lo_, hi_, x_] := N[(1.0 + N[(hi * N[(N[Abs[N[(1.0 + N[(hi / lo), $MachinePrecision]), $MachinePrecision]], $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. add-sqr-sqrt 10.1%

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

   2. sqrt-unprod 19.5%

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

   3. pow2 19.5%

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

7. Applied egg-rr 19.5%

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

   1. unpow2 19.5%

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

   2. rem-sqrt-square 19.5%

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

   3. +-commutative 19.5%

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

9. Simplified 19.5%

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

10. Taylor expanded in x around 0 19.5%

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

    1. associate-/l* 19.5%

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

12. Simplified 19.5%

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

13. Final simplification 19.5%

    \[\leadsto 1 + hi \cdot \frac{\left|1 + \frac{hi}{lo}\right|}{lo} \]
14. Add Preprocessing

Alternative 2: 19.5% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ {\left(\frac{hi}{lo}\right)}^{2} \end{array} \]

FPCore

(FPCore (lo hi x) :precision binary64 (pow (/ hi lo) 2.0))

C

double code(double lo, double hi, double x) {
	return pow((hi / lo), 2.0);
}

Fortran

real(8) function code(lo, hi, x)
    real(8), intent (in) :: lo
    real(8), intent (in) :: hi
    real(8), intent (in) :: x
    code = (hi / lo) ** 2.0d0
end function

Java

public static double code(double lo, double hi, double x) {
	return Math.pow((hi / lo), 2.0);
}

Python

def code(lo, hi, x):
	return math.pow((hi / lo), 2.0)

Julia

function code(lo, hi, x)
	return Float64(hi / lo) ^ 2.0
end

MATLAB

function tmp = code(lo, hi, x)
	tmp = (hi / lo) ^ 2.0;
end

Wolfram

code[lo_, hi_, x_] := N[Power[N[(hi / lo), $MachinePrecision], 2.0], $MachinePrecision]

Derivation

1. Initial program 3.1%

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

3. Taylor expanded in lo around -inf 3.1%

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

   1. mul-1-neg 3.1%

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

   2. associate--l+ 3.1%

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

   3. associate-/l* 14.6%

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

5. Simplified 14.6%

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

   1. div-inv 14.6%

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

7. Applied egg-rr 14.6%

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

8. Taylor expanded in hi around inf 0.0%

   \[\leadsto \color{blue}{\frac{{hi}^{2}}{{lo}^{2}}} \]
9. Step-by-step derivation

   1. unpow2 0.0%

      \[\leadsto \frac{\color{blue}{hi \cdot hi}}{{lo}^{2}} \]

   2. unpow2 0.0%

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

   3. times-frac 19.4%

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

   4. unpow2 19.4%

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

10. Simplified 19.4%

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

11. Final simplification 19.4%

    \[\leadsto {\left(\frac{hi}{lo}\right)}^{2} \]
12. Add Preprocessing

Alternative 3: 18.9% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[lo_, hi_, x_] := N[(1.0 + N[(hi * N[(N[(1.0 + N[(hi / lo), $MachinePrecision]), $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. Taylor expanded in x around 0 18.9%

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

   1. associate-/l* 18.9%

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

8. Simplified 18.9%

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

9. Final simplification 18.9%

   \[\leadsto 1 + hi \cdot \frac{1 + \frac{hi}{lo}}{lo} \]
10. Add Preprocessing

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. 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. associate-*r/ 18.8%

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

   2. neg-mul-1 18.8%

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

6. Simplified 18.8%

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

7. Final simplification 18.8%

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

Alternative 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 18.7%

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

4. Final simplification 18.7%

   \[\leadsto 1 \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024085 
(FPCore (lo hi x)
  :name "xlohi (overflows)"
  :precision binary64
  :pre (and (< lo -1e+308) (> hi 1e+308))
  (/ (- x lo) (- hi lo)))