xlohi (overflows)

Percentage Accurate: 3.1% → 19.0%

Time: 14.8s

Alternatives: 7

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: 19.0% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[lo_, hi_, x_] := N[(1.0 + N[(N[(1.0 + N[(hi / lo), $MachinePrecision]), $MachinePrecision] * N[Abs[N[(hi / lo), $MachinePrecision]], $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 18.8%

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

   1. add-sqr-sqrt 0.0%

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

   2. sqrt-unprod 18.9%

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

   3. pow2 18.9%

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

7. Applied egg-rr 18.9%

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

   1. unpow2 18.9%

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

   2. rem-sqrt-square 18.9%

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

9. Simplified 18.9%

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

10. Taylor expanded in hi around inf 18.9%

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

    1. +-commutative 18.9%

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

    2. mul-1-neg 18.9%

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

    3. sub-neg 18.9%

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

12. Simplified 18.9%

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

13. Taylor expanded in hi around inf 18.9%

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

14. Final simplification 18.9%

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

Alternative 2: 18.9% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[lo_, hi_, x_] := N[(1.0 + N[(N[(1.0 + N[(hi / lo), $MachinePrecision]), $MachinePrecision] * N[(N[(hi - x), $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 18.8%

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

6. Final simplification 18.8%

   \[\leadsto 1 + \left(1 + \frac{hi}{lo}\right) \cdot \frac{hi - x}{lo} \]
7. Add Preprocessing

Alternative 3: 18.9% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[lo_, hi_, x_] := N[(1.0 + N[(N[(hi - x), $MachinePrecision] * 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 18.8%

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

6. Taylor expanded in lo around inf 3.1%

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

   1. +-commutative 3.1%

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

   2. associate--l+ 3.1%

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

   3. *-rgt-identity 3.1%

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

   4. times-frac 14.7%

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

   5. rem-square-sqrt 14.7%

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

   6. associate-*r/ 14.7%

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

   7. /-rgt-identity 14.7%

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

   8. rem-square-sqrt 14.7%

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

   9. *-lft-identity 14.7%

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

   10. distribute-rgt-in 18.8%

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

   11. +-commutative 18.8%

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

   12. *-lft-identity 18.8%

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

   13. times-frac 18.8%

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

8. Simplified 18.8%

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

9. Final simplification 18.8%

   \[\leadsto 1 + \left(hi - x\right) \cdot \frac{1 + \frac{hi}{lo}}{lo} \]
10. Add Preprocessing

Alternative 4: 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 18.8%

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

6. Taylor expanded in x around 0 18.8%

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

   1. associate-/l* 18.8%

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

8. Simplified 18.8%

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

9. Final simplification 18.8%

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

Alternative 5: 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 18.8%

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

4. Final simplification 18.8%

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

Alternative 6: 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. 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 7: 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 2024076 
(FPCore (lo hi x)
  :name "xlohi (overflows)"
  :precision binary64
  :pre (and (< lo -1e+308) (> hi 1e+308))
  (/ (- x lo) (- hi lo)))