xlohi (overflows)

Percentage Accurate: 3.1% → 19.5%

Time: 21.4s

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: 19.5% accurate, 0.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 3.1%

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

3. Taylor expanded in hi around 0 18.9%

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

4. Taylor expanded in x around inf 18.9%

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

5. Taylor expanded in lo around inf 3.1%

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

   1. associate-/l* 14.7%

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

7. Simplified 14.7%

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

   1. add-sqr-sqrt 9.8%

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

   2. sqrt-unprod 0.7%

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

   3. pow2 0.7%

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

   4. +-commutative 0.7%

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

   5. fma-define 0.7%

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

9. Applied egg-rr 0.7%

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

    1. unpow2 0.7%

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

    2. rem-sqrt-square 19.6%

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

11. Simplified 19.6%

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

12. Final simplification 19.6%

    \[\leadsto x \cdot \left(\frac{1}{x} + \frac{-1}{lo}\right) + \frac{\left|\mathsf{fma}\left(hi, \frac{hi - x}{lo}, hi\right)\right|}{lo} \]
13. Add Preprocessing

Alternative 2: 19.3% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \left|{\left(1 + -0.3333333333333333 \cdot \frac{x - hi}{lo}\right)}^{3}\right| \end{array} \]

FPCore

(FPCore (lo hi x)
 :precision binary64
 (fabs (pow (+ 1.0 (* -0.3333333333333333 (/ (- x hi) lo))) 3.0)))

C

double code(double lo, double hi, double x) {
	return fabs(pow((1.0 + (-0.3333333333333333 * ((x - hi) / lo))), 3.0));
}

Fortran

real(8) function code(lo, hi, x)
    real(8), intent (in) :: lo
    real(8), intent (in) :: hi
    real(8), intent (in) :: x
    code = abs(((1.0d0 + ((-0.3333333333333333d0) * ((x - hi) / lo))) ** 3.0d0))
end function

Java

public static double code(double lo, double hi, double x) {
	return Math.abs(Math.pow((1.0 + (-0.3333333333333333 * ((x - hi) / lo))), 3.0));
}

Python

def code(lo, hi, x):
	return math.fabs(math.pow((1.0 + (-0.3333333333333333 * ((x - hi) / lo))), 3.0))

Julia

function code(lo, hi, x)
	return abs((Float64(1.0 + Float64(-0.3333333333333333 * Float64(Float64(x - hi) / lo))) ^ 3.0))
end

MATLAB

function tmp = code(lo, hi, x)
	tmp = abs(((1.0 + (-0.3333333333333333 * ((x - hi) / lo))) ^ 3.0));
end

Wolfram

code[lo_, hi_, x_] := N[Abs[N[Power[N[(1.0 + N[(-0.3333333333333333 * N[(N[(x - hi), $MachinePrecision] / lo), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 3.1%

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

3. Taylor expanded in lo around inf 9.3%

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

   1. associate--l+ 9.3%

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

   2. distribute-lft-out-- 9.3%

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

   3. div-sub 9.3%

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

   4. mul-1-neg 9.3%

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

   5. unsub-neg 9.3%

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

5. Simplified 9.3%

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

   1. add-sqr-sqrt 8.5%

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

   2. sqrt-unprod 17.9%

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

   3. pow2 17.9%

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

7. Applied egg-rr 17.9%

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

   1. unpow2 17.9%

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

   2. rem-sqrt-square 17.9%

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

9. Simplified 17.9%

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

    1. add-cube-cbrt 17.9%

       \[\leadsto \left|\color{blue}{\left(\sqrt[3]{1 - \frac{x - hi}{lo}} \cdot \sqrt[3]{1 - \frac{x - hi}{lo}}\right) \cdot \sqrt[3]{1 - \frac{x - hi}{lo}}}\right| \]

    2. pow3 17.9%

       \[\leadsto \left|\color{blue}{{\left(\sqrt[3]{1 - \frac{x - hi}{lo}}\right)}^{3}}\right| \]

11. Applied egg-rr 17.9%

    \[\leadsto \left|\color{blue}{{\left(\sqrt[3]{1 - \frac{x - hi}{lo}}\right)}^{3}}\right| \]

12. Taylor expanded in lo around -inf 19.3%

    \[\leadsto \left|{\color{blue}{\left(1 + -0.3333333333333333 \cdot \frac{x - hi}{lo}\right)}}^{3}\right| \]
13. Step-by-step derivation

    1. *-commutative 19.3%

       \[\leadsto \left|{\left(1 + \color{blue}{\frac{x - hi}{lo} \cdot -0.3333333333333333}\right)}^{3}\right| \]

14. Simplified 19.3%

    \[\leadsto \left|{\color{blue}{\left(1 + \frac{x - hi}{lo} \cdot -0.3333333333333333\right)}}^{3}\right| \]

15. Final simplification 19.3%

    \[\leadsto \left|{\left(1 + -0.3333333333333333 \cdot \frac{x - hi}{lo}\right)}^{3}\right| \]
16. Add Preprocessing

Alternative 3: 19.3% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \left|\frac{hi - x}{lo}\right| \end{array} \]

FPCore

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

C

double code(double lo, double hi, double x) {
	return fabs(((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 = abs(((hi - x) / lo))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 3.1%

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

3. Taylor expanded in lo around inf 9.3%

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

   1. associate--l+ 9.3%

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

   2. distribute-lft-out-- 9.3%

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

   3. div-sub 9.3%

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

   4. mul-1-neg 9.3%

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

   5. unsub-neg 9.3%

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

5. Simplified 9.3%

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

   1. add-sqr-sqrt 8.5%

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

   2. sqrt-unprod 17.9%

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

   3. pow2 17.9%

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

7. Applied egg-rr 17.9%

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

   1. unpow2 17.9%

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

   2. rem-sqrt-square 17.9%

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

9. Simplified 17.9%

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

10. Taylor expanded in lo around 0 19.3%

    \[\leadsto \left|\color{blue}{\frac{hi - x}{lo}}\right| \]
11. Add Preprocessing

Alternative 4: 18.9% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

4. Taylor expanded in lo around inf 18.9%

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

   1. associate--l+ 18.9%

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

   2. div-sub 18.9%

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

6. Simplified 18.9%

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

7. Final simplification 18.9%

   \[\leadsto \frac{lo - x}{lo} + hi \cdot \frac{1 + \frac{hi - x}{lo}}{lo} \]
8. 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. Add Preprocessing

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. Add Preprocessing

3. Taylor expanded in lo around inf 18.7%

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

Reproduce

herbie shell --seed 2024113 
(FPCore (lo hi x)
  :name "xlohi (overflows)"
  :precision binary64
  :pre (and (< lo -1e+308) (> hi 1e+308))
  (/ (- x lo) (- hi lo)))