xlohi (overflows)

Percentage Accurate: 3.1% → 18.9%

Time: 31.5s

Alternatives: 8

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: 18.9% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[lo_, hi_, x_] := N[(N[(1.0 - N[(x / lo), $MachinePrecision]), $MachinePrecision] + N[(hi * N[(N[(N[(1.0 / lo), $MachinePrecision] - N[(N[(N[(N[(N[(x / lo), $MachinePrecision] + -1.0), $MachinePrecision] * hi), $MachinePrecision] / lo), $MachinePrecision] / lo), $MachinePrecision]), $MachinePrecision] - N[(x / N[Power[lo, 2.0], $MachinePrecision]), $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 9.4%

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

   1. mul-1-neg 9.4%

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

   2. distribute-neg-frac2 9.4%

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

   3. +-commutative 9.4%

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

   4. associate-/l* 18.9%

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

   5. *-commutative 18.9%

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

   6. distribute-lft-out 18.9%

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

6. Simplified 18.9%

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

   1. mul-1-neg 18.9%

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

   2. div-sub 18.9%

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

   3. add-sqr-sqrt 13.1%

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

   4. fma-neg 13.1%

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

   5. pow1 13.1%

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

   6. pow1 13.1%

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

   7. pow-div 13.1%

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

   8. metadata-eval 13.1%

      \[\leadsto \left(-\mathsf{fma}\left(\sqrt{\frac{x}{lo}}, \sqrt{\frac{x}{lo}}, -{lo}^{\color{blue}{0}}\right)\right) + hi \cdot \left(\left(\frac{1}{lo} + \frac{\frac{hi \cdot \left(\frac{x}{lo} + -1\right)}{-lo}}{lo}\right) - \frac{x}{{lo}^{2}}\right) \]

   9. metadata-eval 13.1%

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

   10. metadata-eval 13.1%

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

   11. fma-define 13.1%

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

   12. add-sqr-sqrt 18.9%

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

8. Applied egg-rr 18.9%

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

9. Final simplification 18.9%

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

Alternative 2: 18.9% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[lo_, hi_, x_] := N[(1.0 + N[(hi * N[(N[(N[(1.0 / lo), $MachinePrecision] - N[(N[(N[(N[(N[(x / lo), $MachinePrecision] + -1.0), $MachinePrecision] * hi), $MachinePrecision] / lo), $MachinePrecision] / lo), $MachinePrecision]), $MachinePrecision] - N[(x / N[Power[lo, 2.0], $MachinePrecision]), $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 9.4%

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

   1. mul-1-neg 9.4%

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

   2. distribute-neg-frac2 9.4%

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

   3. +-commutative 9.4%

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

   4. associate-/l* 18.9%

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

   5. *-commutative 18.9%

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

   6. distribute-lft-out 18.9%

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

6. Simplified 18.9%

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

7. Taylor expanded in x around 0 18.9%

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

8. Final simplification 18.9%

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

Alternative 3: 18.9% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(lo, hi, x)
    real(8), intent (in) :: lo
    real(8), intent (in) :: hi
    real(8), intent (in) :: x
    code = ((lo - x) / lo) + (hi * (((1.0d0 + (hi / lo)) / lo) - (x / (lo ** 2.0d0))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[lo_, hi_, x_] := N[(N[(N[(lo - x), $MachinePrecision] / lo), $MachinePrecision] + N[(hi * N[(N[(N[(1.0 + N[(hi / lo), $MachinePrecision]), $MachinePrecision] / lo), $MachinePrecision] - N[(x / N[Power[lo, 2.0], $MachinePrecision]), $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 18.9%

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

5. Final simplification 18.9%

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

Alternative 4: 18.9% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[lo_, hi_, x_] := N[(N[(N[(lo - x), $MachinePrecision] / lo), $MachinePrecision] + N[(N[(hi * N[(1.0 + N[(N[(hi - x), $MachinePrecision] / lo), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / lo), $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 3.1%

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

   1. +-commutative 3.1%

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

   2. associate-/l* 15.4%

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

   3. fma-define 18.9%

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

6. Simplified 18.9%

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

7. Taylor expanded in hi around 0 18.9%

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

   1. +-commutative 18.9%

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

   2. mul-1-neg 18.9%

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

   3. sub-neg 18.9%

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

   4. div-sub 18.9%

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

9. Simplified 18.9%

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

10. Final simplification 18.9%

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

Alternative 5: 18.9% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[lo_, hi_, x_] := N[(N[(N[(lo - x), $MachinePrecision] / lo), $MachinePrecision] + N[(hi * N[(N[(1.0 + N[(N[(hi - x), $MachinePrecision] / lo), $MachinePrecision]), $MachinePrecision] / lo), $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 18.9%

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

   1. associate--l+ 18.9%

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

   2. div-sub 18.9%

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

6. Simplified 18.9%

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

7. Final simplification 18.9%

   \[\leadsto \frac{lo - x}{lo} + hi \cdot \frac{1 + \frac{hi - x}{lo}}{lo} \]
8. Add Preprocessing

Alternative 6: 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 7: 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. 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-1 18.8%

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

6. Simplified 18.8%

   \[\leadsto \frac{\color{blue}{-lo}}{hi} \]
7. Add Preprocessing

Alternative 8: 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. Add Preprocessing

Reproduce

herbie shell --seed 2024135 
(FPCore (lo hi x)
  :name "xlohi (overflows)"
  :precision binary64
  :pre (and (< lo -1e+308) (> hi 1e+308))
  (/ (- x lo) (- hi lo)))