xlohi (overflows)

Percentage Accurate: 3.1% → 20.6%

Time: 11.5s

Alternatives: 10

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: 20.6% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{log1p}\left(x \cdot \left(0.5 \cdot \frac{x}{{hi}^{2}} + \frac{1}{hi}\right) - lo \cdot \frac{e^{\frac{x}{hi}}}{hi}\right) \end{array} \]

FPCore

(FPCore (lo hi x)
 :precision binary64
 (log1p
  (-
   (* x (+ (* 0.5 (/ x (pow hi 2.0))) (/ 1.0 hi)))
   (* lo (/ (exp (/ x hi)) hi)))))

C

double code(double lo, double hi, double x) {
	return log1p(((x * ((0.5 * (x / pow(hi, 2.0))) + (1.0 / hi))) - (lo * (exp((x / hi)) / hi))));
}

Java

public static double code(double lo, double hi, double x) {
	return Math.log1p(((x * ((0.5 * (x / Math.pow(hi, 2.0))) + (1.0 / hi))) - (lo * (Math.exp((x / hi)) / hi))));
}

Python

def code(lo, hi, x):
	return math.log1p(((x * ((0.5 * (x / math.pow(hi, 2.0))) + (1.0 / hi))) - (lo * (math.exp((x / hi)) / hi))))

Julia

function code(lo, hi, x)
	return log1p(Float64(Float64(x * Float64(Float64(0.5 * Float64(x / (hi ^ 2.0))) + Float64(1.0 / hi))) - Float64(lo * Float64(exp(Float64(x / hi)) / hi))))
end

Wolfram

code[lo_, hi_, x_] := N[Log[1 + N[(N[(x * N[(N[(0.5 * N[(x / N[Power[hi, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / hi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(lo * N[(N[Exp[N[(x / hi), $MachinePrecision]], $MachinePrecision] / hi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $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. Step-by-step derivation

   1. log1p-expm1-u 18.8%

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

5. Applied egg-rr 18.8%

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

6. Taylor expanded in lo around 0 20.6%

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

   1. +-commutative 20.6%

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

   2. associate--l+ 20.6%

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

   3. mul-1-neg 20.6%

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

   4. associate-/l* 20.6%

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

   5. distribute-rgt-neg-in 20.6%

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

   6. distribute-neg-frac2 20.6%

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

   7. expm1-define 20.6%

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

8. Simplified 20.6%

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

9. Taylor expanded in x around 0 20.6%

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

10. Final simplification 20.6%

    \[\leadsto \mathsf{log1p}\left(x \cdot \left(0.5 \cdot \frac{x}{{hi}^{2}} + \frac{1}{hi}\right) - lo \cdot \frac{e^{\frac{x}{hi}}}{hi}\right) \]
11. Add Preprocessing

Alternative 2: 20.6% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{x}{hi}\right) - lo \cdot \frac{e^{\frac{x}{hi}}}{hi}\right) \end{array} \]

FPCore

(FPCore (lo hi x)
 :precision binary64
 (log1p (- (expm1 (/ x hi)) (* lo (/ (exp (/ x hi)) hi)))))

C

double code(double lo, double hi, double x) {
	return log1p((expm1((x / hi)) - (lo * (exp((x / hi)) / hi))));
}

Java

public static double code(double lo, double hi, double x) {
	return Math.log1p((Math.expm1((x / hi)) - (lo * (Math.exp((x / hi)) / hi))));
}

Python

def code(lo, hi, x):
	return math.log1p((math.expm1((x / hi)) - (lo * (math.exp((x / hi)) / hi))))

Julia

function code(lo, hi, x)
	return log1p(Float64(expm1(Float64(x / hi)) - Float64(lo * Float64(exp(Float64(x / hi)) / hi))))
end

Wolfram

code[lo_, hi_, x_] := N[Log[1 + N[(N[(Exp[N[(x / hi), $MachinePrecision]] - 1), $MachinePrecision] - N[(lo * N[(N[Exp[N[(x / hi), $MachinePrecision]], $MachinePrecision] / hi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $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. Step-by-step derivation

   1. log1p-expm1-u 18.8%

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

5. Applied egg-rr 18.8%

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

6. Taylor expanded in lo around 0 20.6%

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

   1. +-commutative 20.6%

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

   2. associate--l+ 20.6%

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

   3. mul-1-neg 20.6%

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

   4. associate-/l* 20.6%

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

   5. distribute-rgt-neg-in 20.6%

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

   6. distribute-neg-frac2 20.6%

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

   7. expm1-define 20.6%

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

8. Simplified 20.6%

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

   1. +-commutative 20.6%

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

   2. *-un-lft-identity 20.6%

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

   3. fma-define 20.6%

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

10. Applied egg-rr 20.6%

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

    1. associate-*r/ 20.6%

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

    2. distribute-frac-neg2 20.6%

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

    3. fmm-undef 20.6%

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

    4. *-lft-identity 20.6%

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

    5. associate-/l* 20.6%

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

12. Simplified 20.6%

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

13. Final simplification 20.6%

    \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{x}{hi}\right) - lo \cdot \frac{e^{\frac{x}{hi}}}{hi}\right) \]
14. Add Preprocessing

Alternative 3: 20.6% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{x}{hi}\right) - \frac{\mathsf{fma}\left(lo, \frac{x}{hi}, lo\right)}{hi}\right) \end{array} \]

FPCore

(FPCore (lo hi x)
 :precision binary64
 (log1p (- (expm1 (/ x hi)) (/ (fma lo (/ x hi) lo) hi))))

C

double code(double lo, double hi, double x) {
	return log1p((expm1((x / hi)) - (fma(lo, (x / hi), lo) / hi)));
}

Julia

function code(lo, hi, x)
	return log1p(Float64(expm1(Float64(x / hi)) - Float64(fma(lo, Float64(x / hi), lo) / hi)))
end

Wolfram

code[lo_, hi_, x_] := N[Log[1 + N[(N[(Exp[N[(x / hi), $MachinePrecision]] - 1), $MachinePrecision] - N[(N[(lo * N[(x / hi), $MachinePrecision] + lo), $MachinePrecision] / hi), $MachinePrecision]), $MachinePrecision]], $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. Step-by-step derivation

   1. log1p-expm1-u 18.8%

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

5. Applied egg-rr 18.8%

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

6. Taylor expanded in lo around 0 20.6%

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

   1. +-commutative 20.6%

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

   2. associate--l+ 20.6%

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

   3. mul-1-neg 20.6%

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

   4. associate-/l* 20.6%

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

   5. distribute-rgt-neg-in 20.6%

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

   6. distribute-neg-frac2 20.6%

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

   7. expm1-define 20.6%

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

8. Simplified 20.6%

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

9. Taylor expanded in hi around inf 11.3%

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

    1. distribute-lft-out 11.3%

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

    2. associate-*r/ 11.3%

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

    3. mul-1-neg 11.3%

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

    4. distribute-neg-frac2 11.3%

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

    5. +-commutative 11.3%

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

    6. associate-/l* 20.6%

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

    7. fma-define 20.6%

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

11. Simplified 20.6%

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

12. Final simplification 20.6%

    \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{x}{hi}\right) - \frac{\mathsf{fma}\left(lo, \frac{x}{hi}, lo\right)}{hi}\right) \]
13. Add Preprocessing

Alternative 4: 20.6% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Java

public static double code(double lo, double hi, double x) {
	return Math.log1p(((x - lo) / hi));
}

Python

def code(lo, hi, x):
	return math.log1p(((x - lo) / hi))

Julia

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

Wolfram

code[lo_, hi_, x_] := N[Log[1 + N[(N[(x - lo), $MachinePrecision] / hi), $MachinePrecision]], $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. Step-by-step derivation

   1. log1p-expm1-u 18.8%

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

5. Applied egg-rr 18.8%

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

6. Taylor expanded in hi around inf 20.6%

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

7. Final simplification 20.6%

   \[\leadsto \mathsf{log1p}\left(\frac{x - lo}{hi}\right) \]
8. Add Preprocessing

Alternative 5: 20.6% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \mathsf{log1p}\left(\frac{lo}{-hi}\right) \end{array} \]

FPCore

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

C

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

Java

public static double code(double lo, double hi, double x) {
	return Math.log1p((lo / -hi));
}

Python

def code(lo, hi, x):
	return math.log1p((lo / -hi))

Julia

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

Wolfram

code[lo_, hi_, x_] := N[Log[1 + N[(lo / (-hi)), $MachinePrecision]], $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. Step-by-step derivation

   1. log1p-expm1-u 18.8%

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

5. Applied egg-rr 18.8%

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

6. Taylor expanded in lo around 0 20.6%

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

   1. +-commutative 20.6%

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

   2. associate--l+ 20.6%

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

   3. mul-1-neg 20.6%

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

   4. associate-/l* 20.6%

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

   5. distribute-rgt-neg-in 20.6%

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

   6. distribute-neg-frac2 20.6%

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

   7. expm1-define 20.6%

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

8. Simplified 20.6%

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

9. Taylor expanded in x around 0 20.6%

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

    1. log1p-define 20.6%

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

    2. associate-*r/ 20.6%

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

    3. mul-1-neg 20.6%

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

11. Simplified 20.6%

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

12. Final simplification 20.6%

    \[\leadsto \mathsf{log1p}\left(\frac{lo}{-hi}\right) \]
13. Add Preprocessing

Alternative 6: 18.9% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[lo_, hi_, x_] := N[(N[(N[(lo - x), $MachinePrecision] / lo), $MachinePrecision] + N[(hi * N[(N[(N[(1.0 / lo), $MachinePrecision] + N[(N[(N[(hi - N[(hi * N[(x / lo), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / lo), $MachinePrecision] / lo), $MachinePrecision]), $MachinePrecision] - N[(N[(x / lo), $MachinePrecision] / lo), $MachinePrecision]), $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 10.6%

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

   1. mul-1-neg 10.6%

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

   2. unsub-neg 10.6%

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

   3. associate-/l* 18.9%

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

6. Simplified 18.9%

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

   1. *-un-lft-identity 18.9%

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

   2. unpow2 18.9%

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

   3. times-frac 18.9%

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

8. Applied egg-rr 18.9%

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

   1. associate-*l/ 18.9%

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

   2. *-lft-identity 18.9%

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

10. Simplified 18.9%

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

11. Final simplification 18.9%

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

Alternative 7: 18.9% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

Alternative 8: 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 9: 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 10: 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 2024115 
(FPCore (lo hi x)
  :name "xlohi (overflows)"
  :precision binary64
  :pre (and (< lo -1e+308) (> hi 1e+308))
  (/ (- x lo) (- hi lo)))