xlohi (overflows)

Percentage Accurate: 3.1% → 99.4%

Time: 17.8s

Alternatives: 7

Speedup: 18.0×

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: 99.4% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\frac{lo}{lo - x}, 1, \frac{hi}{x - lo}\right)\\ {\left(t\_0 \cdot t\_0\right)}^{-0.5} \end{array} \end{array} \]

FPCore

(FPCore (lo hi x)
 :precision binary64
 (let* ((t_0 (fma (/ lo (- lo x)) 1.0 (/ hi (- x lo)))))
   (pow (* t_0 t_0) -0.5)))

C

double code(double lo, double hi, double x) {
	double t_0 = fma((lo / (lo - x)), 1.0, (hi / (x - lo)));
	return pow((t_0 * t_0), -0.5);
}

Julia

function code(lo, hi, x)
	t_0 = fma(Float64(lo / Float64(lo - x)), 1.0, Float64(hi / Float64(x - lo)))
	return Float64(t_0 * t_0) ^ -0.5
end

Wolfram

code[lo_, hi_, x_] := Block[{t$95$0 = N[(N[(lo / N[(lo - x), $MachinePrecision]), $MachinePrecision] * 1.0 + N[(hi / N[(x - lo), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[Power[N[(t$95$0 * t$95$0), $MachinePrecision], -0.5], $MachinePrecision]]

Derivation

1. Initial program 3.1%

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

3. Taylor expanded in hi around inf

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

   1. remove-double-neg N/A

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

   2. mul-1-neg N/A

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

   3. sub-neg N/A

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

   4. associate--r+ N/A

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

   5. +-commutative N/A

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

   6. associate-+r+ N/A

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

   7. lower-/.f64 N/A

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

5. Applied rewrites 14.4%

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

7. Applied rewrites 14.4%

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

8. Taylor expanded in hi around -inf

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

10. Applied rewrites 98.9%

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

12. Applied rewrites 99.4%

    \[\leadsto {\left(\mathsf{fma}\left(-\frac{lo}{x - lo}, 1, \frac{hi}{x - lo}\right) \cdot \mathsf{fma}\left(-\frac{lo}{x - lo}, 1, \frac{hi}{x - lo}\right)\right)}^{\color{blue}{-0.5}} \]

13. Final simplification 99.4%

    \[\leadsto {\left(\mathsf{fma}\left(\frac{lo}{lo - x}, 1, \frac{hi}{x - lo}\right) \cdot \mathsf{fma}\left(\frac{lo}{lo - x}, 1, \frac{hi}{x - lo}\right)\right)}^{-0.5} \]
14. Add Preprocessing

Alternative 2: 99.4% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   1. remove-double-neg N/A

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

   2. mul-1-neg N/A

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

   3. sub-neg N/A

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

   4. associate--r+ N/A

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

   5. +-commutative N/A

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

   6. associate-+r+ N/A

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

   7. lower-/.f64 N/A

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

5. Applied rewrites 14.4%

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

7. Applied rewrites 14.4%

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

8. Taylor expanded in hi around -inf

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

10. Applied rewrites 98.9%

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

11. Taylor expanded in hi around 0

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

13. Applied rewrites 99.4%

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

14. Final simplification 99.4%

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

Alternative 3: 18.9% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

5. Applied rewrites 18.9%

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

6. Taylor expanded in lo around 0

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

8. Applied rewrites 18.9%

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

9. Final simplification 18.9%

   \[\leadsto \mathsf{fma}\left(\frac{lo + hi}{lo}, \frac{hi - x}{lo}, 1\right) \]
10. Add Preprocessing

Alternative 4: 18.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ 1 + \frac{\mathsf{fma}\left(hi, \frac{hi}{lo}, hi\right)}{lo} \end{array} \]

FPCore

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

C

double code(double lo, double hi, double x) {
	return 1.0 + (fma(hi, (hi / lo), hi) / lo);
}

Julia

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

Wolfram

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

Derivation

1. Initial program 3.1%

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

3. Taylor expanded in lo around inf

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

5. Applied rewrites 18.9%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 18.9%

   \[\leadsto 1 + \color{blue}{\frac{\mathsf{fma}\left(hi, \frac{hi}{lo}, hi\right)}{lo}} \]
9. Add Preprocessing

Alternative 5: 18.8% accurate, 1.2× 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

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

   1. lower-/.f64 N/A

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

   2. lower--.f64 18.8

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

5. Applied rewrites 18.8%

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

Alternative 6: 18.8% accurate, 1.3× 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

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

   1. lower-/.f64 N/A

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

   2. lower--.f64 18.8

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

5. Applied rewrites 18.8%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 18.8%

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

Alternative 7: 18.7% accurate, 18.0× 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

   \[\leadsto \color{blue}{1} \]
4. Step-by-step derivation

5. Applied rewrites 18.7%

   \[\leadsto \color{blue}{1} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024237 
(FPCore (lo hi x)
  :name "xlohi (overflows)"
  :precision binary64
  :pre (and (< lo -1e+308) (> hi 1e+308))
  (/ (- x lo) (- hi lo)))