xlohi (overflows)

Percentage Accurate: 3.1% → 20.4%

Time: 15.8s

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: 20.4% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \frac{x + \mathsf{expm1}\left(\log \left(1 - lo\right) + \frac{lo}{hi} \cdot \frac{x - lo}{1 - lo}\right)}{hi} \end{array} \]

FPCore

(FPCore (lo hi x)
 :precision binary64
 (/
  (+ x (expm1 (+ (log (- 1.0 lo)) (* (/ lo hi) (/ (- x lo) (- 1.0 lo))))))
  hi))

C

double code(double lo, double hi, double x) {
	return (x + expm1((log((1.0 - lo)) + ((lo / hi) * ((x - lo) / (1.0 - lo)))))) / hi;
}

Java

public static double code(double lo, double hi, double x) {
	return (x + Math.expm1((Math.log((1.0 - lo)) + ((lo / hi) * ((x - lo) / (1.0 - lo)))))) / hi;
}

Python

def code(lo, hi, x):
	return (x + math.expm1((math.log((1.0 - lo)) + ((lo / hi) * ((x - lo) / (1.0 - lo)))))) / hi

Julia

function code(lo, hi, x)
	return Float64(Float64(x + expm1(Float64(log(Float64(1.0 - lo)) + Float64(Float64(lo / hi) * Float64(Float64(x - lo) / Float64(1.0 - lo)))))) / hi)
end

Wolfram

code[lo_, hi_, x_] := N[(N[(x + N[(Exp[N[(N[Log[N[(1.0 - lo), $MachinePrecision]], $MachinePrecision] + N[(N[(lo / hi), $MachinePrecision] * N[(N[(x - lo), $MachinePrecision] / N[(1.0 - lo), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]] - 1), $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. +-commutative 0.7%

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

   5. associate--l+ 0.7%

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

   6. +-commutative 0.7%

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

   7. associate--l+ 0.7%

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

   8. associate-/l* 10.0%

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

5. Simplified 10.0%

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

   1. expm1-log1p-u 9.5%

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

7. Applied egg-rr 9.5%

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

8. Taylor expanded in hi around inf 0.0%

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

   1. times-frac 20.4%

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

10. Simplified 20.4%

    \[\leadsto \frac{x + \mathsf{expm1}\left(\color{blue}{\log \left(1 - lo\right) + \frac{lo}{hi} \cdot \frac{x - lo}{1 - lo}}\right)}{hi} \]
11. Add Preprocessing

Alternative 2: 18.8% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \frac{e^{\log \left(-lo\right)}}{hi} \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   1. neg-mul-1 18.8%

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

   2. distribute-neg-frac 18.8%

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

8. Simplified 18.8%

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

   1. add-exp-log 18.8%

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

10. Applied egg-rr 18.8%

    \[\leadsto \frac{\color{blue}{e^{\log \left(-lo\right)}}}{hi} \]
11. 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 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. Taylor expanded in x around 0 18.8%

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

   1. neg-mul-1 18.8%

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

   2. distribute-neg-frac 18.8%

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

8. Simplified 18.8%

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

9. Final simplification 18.8%

   \[\leadsto \frac{lo}{-hi} \]
10. 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 2024109 
(FPCore (lo hi x)
  :name "xlohi (overflows)"
  :precision binary64
  :pre (and (< lo -1e+308) (> hi 1e+308))
  (/ (- x lo) (- hi lo)))