xlohi (overflows)

Percentage Accurate: 3.1% → 21.3%

Time: 7.3s

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: 21.3% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \frac{x}{hi} - \mathsf{log1p}\left(\mathsf{fma}\left(0.5, \frac{lo}{hi} \cdot \frac{lo}{hi}, \frac{lo}{hi}\right) - \frac{x}{hi} \cdot \frac{lo}{hi}\right) \end{array} \]

FPCore

(FPCore (lo hi x)
 :precision binary64
 (-
  (/ x hi)
  (log1p
   (- (fma 0.5 (* (/ lo hi) (/ lo hi)) (/ lo hi)) (* (/ x hi) (/ lo hi))))))

C

double code(double lo, double hi, double x) {
	return (x / hi) - log1p((fma(0.5, ((lo / hi) * (lo / hi)), (lo / hi)) - ((x / hi) * (lo / hi))));
}

Julia

function code(lo, hi, x)
	return Float64(Float64(x / hi) - log1p(Float64(fma(0.5, Float64(Float64(lo / hi) * Float64(lo / hi)), Float64(lo / hi)) - Float64(Float64(x / hi) * Float64(lo / hi)))))
end

Wolfram

code[lo_, hi_, x_] := N[(N[(x / hi), $MachinePrecision] - N[Log[1 + N[(N[(0.5 * N[(N[(lo / hi), $MachinePrecision] * N[(lo / hi), $MachinePrecision]), $MachinePrecision] + N[(lo / hi), $MachinePrecision]), $MachinePrecision] - N[(N[(x / hi), $MachinePrecision] * N[(lo / hi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 3.1%

   \[\frac{x - lo}{hi - lo} \]

2. Taylor expanded in lo around 0 18.8%

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

   1. mul-1-neg 18.8%

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

   2. unsub-neg 18.8%

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

   3. mul-1-neg 18.8%

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

   4. unsub-neg 18.8%

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

   5. unpow2 18.8%

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

4. Simplified 18.8%

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

   1. log1p-expm1-u 18.8%

      \[\leadsto \frac{x}{hi} - \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(lo \cdot \left(\frac{1}{hi} - \frac{x}{hi \cdot hi}\right)\right)\right)} \]

   2. inv-pow 18.8%

      \[\leadsto \frac{x}{hi} - \mathsf{log1p}\left(\mathsf{expm1}\left(lo \cdot \left(\color{blue}{{hi}^{-1}} - \frac{x}{hi \cdot hi}\right)\right)\right) \]

   3. div-inv 18.8%

      \[\leadsto \frac{x}{hi} - \mathsf{log1p}\left(\mathsf{expm1}\left(lo \cdot \left({hi}^{-1} - \color{blue}{x \cdot \frac{1}{hi \cdot hi}}\right)\right)\right) \]

   4. pow2 18.8%

      \[\leadsto \frac{x}{hi} - \mathsf{log1p}\left(\mathsf{expm1}\left(lo \cdot \left({hi}^{-1} - x \cdot \frac{1}{\color{blue}{{hi}^{2}}}\right)\right)\right) \]

   5. pow-flip 18.8%

      \[\leadsto \frac{x}{hi} - \mathsf{log1p}\left(\mathsf{expm1}\left(lo \cdot \left({hi}^{-1} - x \cdot \color{blue}{{hi}^{\left(-2\right)}}\right)\right)\right) \]

   6. metadata-eval 18.8%

      \[\leadsto \frac{x}{hi} - \mathsf{log1p}\left(\mathsf{expm1}\left(lo \cdot \left({hi}^{-1} - x \cdot {hi}^{\color{blue}{-2}}\right)\right)\right) \]

6. Applied egg-rr 18.8%

   \[\leadsto \frac{x}{hi} - \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(lo \cdot \left({hi}^{-1} - x \cdot {hi}^{-2}\right)\right)\right)} \]
7. Step-by-step derivation

   1. add-cbrt-cube 18.8%

      \[\leadsto \frac{x}{hi} - \mathsf{log1p}\left(\color{blue}{\sqrt[3]{\left(\mathsf{expm1}\left(lo \cdot \left({hi}^{-1} - x \cdot {hi}^{-2}\right)\right) \cdot \mathsf{expm1}\left(lo \cdot \left({hi}^{-1} - x \cdot {hi}^{-2}\right)\right)\right) \cdot \mathsf{expm1}\left(lo \cdot \left({hi}^{-1} - x \cdot {hi}^{-2}\right)\right)}}\right) \]

   2. pow3 18.8%

      \[\leadsto \frac{x}{hi} - \mathsf{log1p}\left(\sqrt[3]{\color{blue}{{\left(\mathsf{expm1}\left(lo \cdot \left({hi}^{-1} - x \cdot {hi}^{-2}\right)\right)\right)}^{3}}}\right) \]

   3. unpow-1 18.8%

      \[\leadsto \frac{x}{hi} - \mathsf{log1p}\left(\sqrt[3]{{\left(\mathsf{expm1}\left(lo \cdot \left(\color{blue}{\frac{1}{hi}} - x \cdot {hi}^{-2}\right)\right)\right)}^{3}}\right) \]

8. Applied egg-rr 18.8%

   \[\leadsto \frac{x}{hi} - \mathsf{log1p}\left(\color{blue}{\sqrt[3]{{\left(\mathsf{expm1}\left(lo \cdot \left(\frac{1}{hi} - x \cdot {hi}^{-2}\right)\right)\right)}^{3}}}\right) \]

9. Taylor expanded in hi around inf 0.0%

   \[\leadsto \frac{x}{hi} - \mathsf{log1p}\left(\color{blue}{-1 \cdot \frac{lo \cdot x}{{hi}^{2}} + \left(0.5 \cdot \frac{{lo}^{2}}{{hi}^{2}} + \frac{lo}{hi}\right)}\right) \]
10. Step-by-step derivation

    1. +-commutative 0.0%

       \[\leadsto \frac{x}{hi} - \mathsf{log1p}\left(\color{blue}{\left(0.5 \cdot \frac{{lo}^{2}}{{hi}^{2}} + \frac{lo}{hi}\right) + -1 \cdot \frac{lo \cdot x}{{hi}^{2}}}\right) \]

    2. mul-1-neg 0.0%

       \[\leadsto \frac{x}{hi} - \mathsf{log1p}\left(\left(0.5 \cdot \frac{{lo}^{2}}{{hi}^{2}} + \frac{lo}{hi}\right) + \color{blue}{\left(-\frac{lo \cdot x}{{hi}^{2}}\right)}\right) \]

    3. unsub-neg 0.0%

       \[\leadsto \frac{x}{hi} - \mathsf{log1p}\left(\color{blue}{\left(0.5 \cdot \frac{{lo}^{2}}{{hi}^{2}} + \frac{lo}{hi}\right) - \frac{lo \cdot x}{{hi}^{2}}}\right) \]

    4. fma-def 0.0%

       \[\leadsto \frac{x}{hi} - \mathsf{log1p}\left(\color{blue}{\mathsf{fma}\left(0.5, \frac{{lo}^{2}}{{hi}^{2}}, \frac{lo}{hi}\right)} - \frac{lo \cdot x}{{hi}^{2}}\right) \]

    5. unpow2 0.0%

       \[\leadsto \frac{x}{hi} - \mathsf{log1p}\left(\mathsf{fma}\left(0.5, \frac{\color{blue}{lo \cdot lo}}{{hi}^{2}}, \frac{lo}{hi}\right) - \frac{lo \cdot x}{{hi}^{2}}\right) \]

    6. unpow2 0.0%

       \[\leadsto \frac{x}{hi} - \mathsf{log1p}\left(\mathsf{fma}\left(0.5, \frac{lo \cdot lo}{\color{blue}{hi \cdot hi}}, \frac{lo}{hi}\right) - \frac{lo \cdot x}{{hi}^{2}}\right) \]

    7. times-frac 10.2%

       \[\leadsto \frac{x}{hi} - \mathsf{log1p}\left(\mathsf{fma}\left(0.5, \color{blue}{\frac{lo}{hi} \cdot \frac{lo}{hi}}, \frac{lo}{hi}\right) - \frac{lo \cdot x}{{hi}^{2}}\right) \]

    8. unpow2 10.2%

       \[\leadsto \frac{x}{hi} - \mathsf{log1p}\left(\mathsf{fma}\left(0.5, \frac{lo}{hi} \cdot \frac{lo}{hi}, \frac{lo}{hi}\right) - \frac{lo \cdot x}{\color{blue}{hi \cdot hi}}\right) \]

    9. times-frac 21.3%

       \[\leadsto \frac{x}{hi} - \mathsf{log1p}\left(\mathsf{fma}\left(0.5, \frac{lo}{hi} \cdot \frac{lo}{hi}, \frac{lo}{hi}\right) - \color{blue}{\frac{lo}{hi} \cdot \frac{x}{hi}}\right) \]

11. Simplified 21.3%

    \[\leadsto \frac{x}{hi} - \mathsf{log1p}\left(\color{blue}{\mathsf{fma}\left(0.5, \frac{lo}{hi} \cdot \frac{lo}{hi}, \frac{lo}{hi}\right) - \frac{lo}{hi} \cdot \frac{x}{hi}}\right) \]

12. Final simplification 21.3%

    \[\leadsto \frac{x}{hi} - \mathsf{log1p}\left(\mathsf{fma}\left(0.5, \frac{lo}{hi} \cdot \frac{lo}{hi}, \frac{lo}{hi}\right) - \frac{x}{hi} \cdot \frac{lo}{hi}\right) \]

Alternative 2: 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. Taylor expanded in hi around inf 18.8%

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

3. Final simplification 18.8%

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

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(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. Taylor expanded in lo around 0 18.8%

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

   1. mul-1-neg 18.8%

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

   2. unsub-neg 18.8%

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

   3. mul-1-neg 18.8%

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

   4. unsub-neg 18.8%

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

   5. unpow2 18.8%

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

4. Simplified 18.8%

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

5. Taylor expanded in x around 0 18.8%

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

   1. neg-mul-1 18.8%

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

   2. distribute-neg-frac 18.8%

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

7. Simplified 18.8%

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

8. Final simplification 18.8%

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

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. Taylor expanded in lo around inf 18.7%

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

3. Final simplification 18.7%

   \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023207 
(FPCore (lo hi x)
  :name "xlohi (overflows)"
  :precision binary64
  :pre (and (< lo -1e+308) (> hi 1e+308))
  (/ (- x lo) (- hi lo)))