xlohi (overflows)

Percentage Accurate: 3.1% → 26.7%

Time: 17.4s

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: 26.7% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 3.1%

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

3. Taylor expanded in lo around -inf

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

   1. mul-1-neg N/A

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

   2. unsub-neg N/A

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

   3. lower--.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. +-commutative N/A

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

   6. associate--l+ N/A

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

   7. associate-/l* N/A

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

   8. lower-fma.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lower--.f64 N/A

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

   11. lower--.f64 18.9

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

5. Applied rewrites 18.9%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 18.9%

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

10. Applied rewrites 18.9%

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

11. Taylor expanded in lo around inf

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

13. Applied rewrites 27.2%

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

Alternative 2: 18.9% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 3.1%

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

3. Taylor expanded in lo around -inf

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

   1. mul-1-neg N/A

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

   2. unsub-neg N/A

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

   3. lower--.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. +-commutative N/A

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

   6. associate--l+ N/A

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

   7. associate-/l* N/A

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

   8. lower-fma.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lower--.f64 N/A

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

   11. lower--.f64 18.9

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

5. Applied rewrites 18.9%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 18.9%

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

9. Taylor expanded in hi around 0

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

11. Applied rewrites 18.9%

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

12. Final simplification 18.9%

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

Reproduce

herbie shell --seed 2024228 
(FPCore (lo hi x)
  :name "xlohi (overflows)"
  :precision binary64
  :pre (and (< lo -1e+308) (> hi 1e+308))
  (/ (- x lo) (- hi lo)))