xlohi (overflows)

Percentage Accurate: 3.1% → 18.8%

Time: 22.8s

Alternatives: 5

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: 18.8% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \frac{x}{hi} - lo \cdot e^{-\sqrt[3]{{\log hi}^{3}}} \end{array} \]

FPCore

(FPCore (lo hi x)
 :precision binary64
 (- (/ x hi) (* lo (exp (- (cbrt (pow (log hi) 3.0)))))))

C

double code(double lo, double hi, double x) {
	return (x / hi) - (lo * exp(-cbrt(pow(log(hi), 3.0))));
}

Java

public static double code(double lo, double hi, double x) {
	return (x / hi) - (lo * Math.exp(-Math.cbrt(Math.pow(Math.log(hi), 3.0))));
}

Julia

function code(lo, hi, x)
	return Float64(Float64(x / hi) - Float64(lo * exp(Float64(-cbrt((log(hi) ^ 3.0))))))
end

Wolfram

code[lo_, hi_, x_] := N[(N[(x / hi), $MachinePrecision] - N[(lo * N[Exp[(-N[Power[N[Power[N[Log[hi], $MachinePrecision], 3.0], $MachinePrecision], 1/3], $MachinePrecision])], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 3.1%

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

3. Taylor expanded in lo around 0 18.8%

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

   1. +-commutative 18.8%

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

   2. mul-1-neg 18.8%

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

   3. unsub-neg 18.8%

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

   4. +-commutative 18.8%

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

   5. mul-1-neg 18.8%

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

   6. unsub-neg 18.8%

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

5. Simplified 18.8%

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

   1. add-exp-log 18.8%

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

   2. log-rec 18.8%

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

7. Applied egg-rr 18.8%

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

8. Taylor expanded in x around 0 18.8%

   \[\leadsto \frac{x}{hi} - \color{blue}{lo \cdot e^{-\log hi}} \]
9. Step-by-step derivation

   1. add-cbrt-cube 18.8%

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

   2. pow3 18.8%

      \[\leadsto \frac{x}{hi} - lo \cdot e^{-\sqrt[3]{\color{blue}{{\log hi}^{3}}}} \]

10. Applied egg-rr 18.8%

    \[\leadsto \frac{x}{hi} - lo \cdot e^{-\color{blue}{\sqrt[3]{{\log hi}^{3}}}} \]
11. Add Preprocessing

Alternative 2: 18.8% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \frac{x}{hi} - lo \cdot e^{-e^{\log \log hi}} \end{array} \]

FPCore

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

C

double code(double lo, double hi, double x) {
	return (x / hi) - (lo * exp(-exp(log(log(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 / hi) - (lo * exp(-exp(log(log(hi)))))
end function

Java

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

Python

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

Julia

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

MATLAB

function tmp = code(lo, hi, x)
	tmp = (x / hi) - (lo * exp(-exp(log(log(hi)))));
end

Wolfram

code[lo_, hi_, x_] := N[(N[(x / hi), $MachinePrecision] - N[(lo * N[Exp[(-N[Exp[N[Log[N[Log[hi], $MachinePrecision]], $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 0 18.8%

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

   1. +-commutative 18.8%

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

   2. mul-1-neg 18.8%

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

   3. unsub-neg 18.8%

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

   4. +-commutative 18.8%

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

   5. mul-1-neg 18.8%

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

   6. unsub-neg 18.8%

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

5. Simplified 18.8%

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

   1. add-exp-log 18.8%

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

   2. log-rec 18.8%

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

7. Applied egg-rr 18.8%

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

8. Taylor expanded in x around 0 18.8%

   \[\leadsto \frac{x}{hi} - \color{blue}{lo \cdot e^{-\log hi}} \]
9. Step-by-step derivation

   1. add-exp-log 18.8%

      \[\leadsto \frac{x}{hi} - lo \cdot e^{-\color{blue}{e^{\log \log hi}}} \]

10. Applied egg-rr 18.8%

    \[\leadsto \frac{x}{hi} - lo \cdot e^{-\color{blue}{e^{\log \log hi}}} \]
11. Add Preprocessing

Alternative 3: 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 4: 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 18.7%

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

   1. neg-mul-1 18.7%

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

   2. distribute-neg-frac 18.7%

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

6. Simplified 18.7%

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

7. Final simplification 18.7%

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

Alternative 5: 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 2024130 
(FPCore (lo hi x)
  :name "xlohi (overflows)"
  :precision binary64
  :pre (and (< lo -1e+308) (> hi 1e+308))
  (/ (- x lo) (- hi lo)))