xlohi (overflows)

Percentage Accurate: 3.1% → 21.1%

Time: 16.3s

Alternatives: 8

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: 21.1% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ 1 + \left(\left(1 + hi \cdot \left(\frac{1}{lo} + \frac{lo}{{hi}^{2}}\right)\right) + \frac{-1}{\frac{hi}{lo} + -1}\right) \cdot \frac{hi - x}{lo} \end{array} \]

FPCore

(FPCore (lo hi x)
 :precision binary64
 (+
  1.0
  (*
   (+
    (+ 1.0 (* hi (+ (/ 1.0 lo) (/ lo (pow hi 2.0)))))
    (/ -1.0 (+ (/ hi lo) -1.0)))
   (/ (- hi x) lo))))

C

double code(double lo, double hi, double x) {
	return 1.0 + (((1.0 + (hi * ((1.0 / lo) + (lo / pow(hi, 2.0))))) + (-1.0 / ((hi / lo) + -1.0))) * ((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 * ((1.0d0 / lo) + (lo / (hi ** 2.0d0))))) + ((-1.0d0) / ((hi / lo) + (-1.0d0)))) * ((hi - x) / lo))
end function

Java

public static double code(double lo, double hi, double x) {
	return 1.0 + (((1.0 + (hi * ((1.0 / lo) + (lo / Math.pow(hi, 2.0))))) + (-1.0 / ((hi / lo) + -1.0))) * ((hi - x) / lo));
}

Python

def code(lo, hi, x):
	return 1.0 + (((1.0 + (hi * ((1.0 / lo) + (lo / math.pow(hi, 2.0))))) + (-1.0 / ((hi / lo) + -1.0))) * ((hi - x) / lo))

Julia

function code(lo, hi, x)
	return Float64(1.0 + Float64(Float64(Float64(1.0 + Float64(hi * Float64(Float64(1.0 / lo) + Float64(lo / (hi ^ 2.0))))) + Float64(-1.0 / Float64(Float64(hi / lo) + -1.0))) * Float64(Float64(hi - x) / lo)))
end

MATLAB

function tmp = code(lo, hi, x)
	tmp = 1.0 + (((1.0 + (hi * ((1.0 / lo) + (lo / (hi ^ 2.0))))) + (-1.0 / ((hi / lo) + -1.0))) * ((hi - x) / lo));
end

Wolfram

code[lo_, hi_, x_] := N[(1.0 + N[(N[(N[(1.0 + N[(hi * N[(N[(1.0 / lo), $MachinePrecision] + N[(lo / N[Power[hi, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / N[(N[(hi / lo), $MachinePrecision] + -1.0), $MachinePrecision]), $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 18.9%

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

   1. flip-+ 18.9%

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

   2. metadata-eval 18.9%

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

   3. div-sub 18.9%

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

   4. pow2 18.9%

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

7. Applied egg-rr 18.9%

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

8. Taylor expanded in hi around inf 21.0%

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

   1. distribute-lft-in 21.0%

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

   2. rgt-mult-inverse 21.0%

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

10. Simplified 21.0%

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

11. Final simplification 21.0%

    \[\leadsto 1 + \left(\left(1 + hi \cdot \left(\frac{1}{lo} + \frac{lo}{{hi}^{2}}\right)\right) + \frac{-1}{\frac{hi}{lo} + -1}\right) \cdot \frac{hi - x}{lo} \]
12. Add Preprocessing

Alternative 2: 19.7% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   1. flip-+ 18.9%

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

   2. metadata-eval 18.9%

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

   3. div-sub 18.9%

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

   4. pow2 18.9%

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

7. Applied egg-rr 18.9%

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

8. Taylor expanded in hi around inf 21.0%

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

   1. distribute-lft-in 21.0%

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

   2. rgt-mult-inverse 21.0%

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

10. Simplified 21.0%

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

11. Taylor expanded in lo around inf 19.8%

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

    1. distribute-lft-in 19.8%

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

    2. fma-define 19.8%

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

    3. rgt-mult-inverse 19.8%

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

13. Simplified 19.8%

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

14. Final simplification 19.8%

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

Alternative 3: 19.7% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   1. flip-+ 18.9%

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

   2. metadata-eval 18.9%

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

   3. div-sub 18.9%

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

   4. pow2 18.9%

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

7. Applied egg-rr 18.9%

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

8. Taylor expanded in hi around inf 21.0%

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

   1. distribute-lft-in 21.0%

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

   2. rgt-mult-inverse 21.0%

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

10. Simplified 21.0%

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

11. Taylor expanded in hi around 0 19.8%

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

    1. +-commutative 19.8%

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

13. Simplified 19.8%

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

14. Final simplification 19.8%

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

Alternative 4: 18.9% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

6. Taylor expanded in hi around inf 18.9%

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

7. Final simplification 18.9%

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

Alternative 5: 18.9% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

6. Final simplification 18.9%

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

Alternative 6: 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 7: 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 8: 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 2024074 
(FPCore (lo hi x)
  :name "xlohi (overflows)"
  :precision binary64
  :pre (and (< lo -1e+308) (> hi 1e+308))
  (/ (- x lo) (- hi lo)))