xlohi (overflows)

Percentage Accurate: 3.1% → 18.9%

Time: 17.5s

Alternatives: 4

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: 18.9% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

Alternative 2: 18.8% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[lo_, hi_, x_] := N[(N[(lo * N[(N[(x / hi), $MachinePrecision] + -1.0), $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* 9.8%

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

   7. distribute-lft-out 10.0%

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

5. Simplified 10.0%

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

6. Taylor expanded in lo around 0 18.8%

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

7. Taylor expanded in lo around inf 18.8%

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

8. Final simplification 18.8%

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

Alternative 3: 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 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* 9.8%

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

   7. distribute-lft-out 10.0%

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

5. Simplified 10.0%

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

6. Taylor expanded in lo around 0 18.8%

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

7. Taylor expanded in x around 0 18.8%

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

   1. associate-*r/ 18.8%

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

   2. neg-mul-1 18.8%

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

9. Simplified 18.8%

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

10. Final simplification 18.8%

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

Alternative 4: 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 2024089 
(FPCore (lo hi x)
  :name "xlohi (overflows)"
  :precision binary64
  :pre (and (< lo -1e+308) (> hi 1e+308))
  (/ (- x lo) (- hi lo)))