xlohi (overflows)

Percentage Accurate: 3.1% → 18.9%
Time: 10.0s
Alternatives: 5
Speedup: 7.0×

Specification

?
\[lo < -1 \cdot 10^{+308} \land hi > 10^{+308}\]
\[\begin{array}{l} \\ \frac{x - lo}{hi - lo} \end{array} \]
(FPCore (lo hi x) :precision binary64 (/ (- x lo) (- hi lo)))
double code(double lo, double hi, double x) {
	return (x - lo) / (hi - lo);
}

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

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

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

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

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

code[lo_, hi_, x_] := N[(N[(x - lo), $MachinePrecision] / N[(hi - lo), $MachinePrecision]), $MachinePrecision]

\begin{array}{l}

\\
\frac{x - lo}{hi - lo}
\end{array}

Sampling outcomes in binary64 precision:

Local Percentage Accuracy vs ?

The average percentage accuracy by input value. Horizontal axis shows value of an input variable; the variable is choosen in the title. Vertical axis is accuracy; higher is better. Red represent the original program, while blue represents Herbie's suggestion. These can be toggled with buttons below the plot. The line is an average while dots represent individual samples.

Accuracy vs Speed?

Herbie found 5 alternatives:

AlternativeAccuracySpeedup
The accuracy (vertical axis) and speed (horizontal axis) of each alternatives. Up and to the right is better. The red square shows the initial program, and each blue circle shows an alternative.The line shows the best available speed-accuracy tradeoffs.

Initial Program: 3.1% accurate, 1.0× speedup?

\[\begin{array}{l} \\ \frac{x - lo}{hi - lo} \end{array} \]
(FPCore (lo hi x) :precision binary64 (/ (- x lo) (- hi lo)))
double code(double lo, double hi, double x) {
	return (x - lo) / (hi - lo);
}

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

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

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

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

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

code[lo_, hi_, x_] := N[(N[(x - lo), $MachinePrecision] / N[(hi - lo), $MachinePrecision]), $MachinePrecision]

\begin{array}{l}

\\
\frac{x - lo}{hi - lo}
\end{array}

Alternative 1: 18.9% accurate, 0.1× speedup?

\[\begin{array}{l} \\ \frac{lo - x}{lo} + hi \cdot \left(\frac{\frac{hi}{lo} + 1}{lo} - \frac{x}{{lo}^{2}}\right) \end{array} \]
(FPCore (lo hi x)
 :precision binary64
 (+ (/ (- lo x) lo) (* hi (- (/ (+ (/ hi lo) 1.0) lo) (/ x (pow lo 2.0))))))
double code(double lo, double hi, double x) {
	return ((lo - x) / lo) + (hi * ((((hi / lo) + 1.0) / lo) - (x / pow(lo, 2.0))));
}

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 * ((((hi / lo) + 1.0d0) / lo) - (x / (lo ** 2.0d0))))
end function

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

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

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

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

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

\begin{array}{l}

\\
\frac{lo - x}{lo} + hi \cdot \left(\frac{\frac{hi}{lo} + 1}{lo} - \frac{x}{{lo}^{2}}\right)
\end{array}
Derivation

  1. Initial program 3.1%

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

  3. Taylor expanded in hi around 0 18.8%

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

  4. Taylor expanded in lo around inf 18.8%

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

  5. Final simplification18.8%

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

Alternative 2: 18.9% accurate, 0.4× speedup?

\[\begin{array}{l} \\ \frac{lo - x}{lo} + hi \cdot \frac{\left(\frac{hi}{lo} + 1\right) - \frac{x}{lo}}{lo} \end{array} \]
(FPCore (lo hi x)
 :precision binary64
 (+ (/ (- lo x) lo) (* hi (/ (- (+ (/ hi lo) 1.0) (/ x lo)) lo))))
double code(double lo, double hi, double x) {
	return ((lo - x) / lo) + (hi * ((((hi / lo) + 1.0) - (x / lo)) / lo));
}

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 * ((((hi / lo) + 1.0d0) - (x / lo)) / lo))
end function

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

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

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

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

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

\begin{array}{l}

\\
\frac{lo - x}{lo} + hi \cdot \frac{\left(\frac{hi}{lo} + 1\right) - \frac{x}{lo}}{lo}
\end{array}
Derivation

  1. Initial program 3.1%

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

  3. Taylor expanded in hi around 0 18.8%

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

  4. Taylor expanded in lo around inf 18.8%

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

  5. Final simplification18.8%

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

Alternative 3: 18.8% accurate, 1.4× speedup?

\[\begin{array}{l} \\ \frac{x - lo}{hi} \end{array} \]
(FPCore (lo hi x) :precision binary64 (/ (- x lo) hi))
double code(double lo, double hi, double x) {
	return (x - lo) / hi;
}

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

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

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

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

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

code[lo_, hi_, x_] := N[(N[(x - lo), $MachinePrecision] / hi), $MachinePrecision]

\begin{array}{l}

\\
\frac{x - lo}{hi}
\end{array}
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?

\[\begin{array}{l} \\ \frac{lo}{-hi} \end{array} \]
(FPCore (lo hi x) :precision binary64 (/ lo (- hi)))
double code(double lo, double hi, double x) {
	return lo / -hi;
}

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

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

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

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

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

code[lo_, hi_, x_] := N[(lo / (-hi)), $MachinePrecision]

\begin{array}{l}

\\
\frac{lo}{-hi}
\end{array}
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%

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

    1. neg-mul-118.8%

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

  6. Simplified18.8%

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

  7. Final simplification18.8%

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

Alternative 5: 18.7% accurate, 7.0× speedup?

\[\begin{array}{l} \\ 1 \end{array} \]
(FPCore (lo hi x) :precision binary64 1.0)
double code(double lo, double hi, double x) {
	return 1.0;
}

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

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

def code(lo, hi, x):
	return 1.0

function code(lo, hi, x)
	return 1.0
end

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

code[lo_, hi_, x_] := 1.0

\begin{array}{l}

\\
1
\end{array}
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 2024182 
(FPCore (lo hi x)
  :name "xlohi (overflows)"
  :precision binary64
  :pre (and (< lo -1e+308) (> hi 1e+308))
  (/ (- x lo) (- hi lo)))