xlohi (overflows)

Percentage Accurate: 3.1% → 20.6%

Time: 36.3s

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: 20.6% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \mathsf{log1p}\left(\frac{lo}{-hi}\right) \end{array} \]

FPCore

(FPCore (lo hi x) :precision binary64 (log1p (/ lo (- hi))))

C

double code(double lo, double hi, double x) {
	return log1p((lo / -hi));
}

Java

public static double code(double lo, double hi, double x) {
	return Math.log1p((lo / -hi));
}

Python

def code(lo, hi, x):
	return math.log1p((lo / -hi))

Julia

function code(lo, hi, x)
	return log1p(Float64(lo / Float64(-hi)))
end

Wolfram

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

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

   5. *-commutative 0.7%

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

   6. associate-/l* 9.7%

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

   7. distribute-lft-out 9.9%

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

5. Simplified 9.9%

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

   1. log1p-expm1-u 9.9%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\left(x - lo\right) \cdot \left(1 + \frac{lo}{hi}\right)}{hi}\right)\right)} \]

   2. log1p-undefine 9.9%

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

   3. associate-/l* 9.9%

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

7. Applied egg-rr 9.9%

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

8. Taylor expanded in lo around 0 20.6%

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

   1. sub-neg 20.6%

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

   2. +-commutative 20.6%

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

   3. metadata-eval 20.6%

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

   4. associate-+l+ 20.6%

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

   5. associate-*r* 20.6%

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

   6. sub-neg 20.6%

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

   7. distribute-neg-frac 20.6%

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

   8. metadata-eval 20.6%

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

   9. metadata-eval 20.6%

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

   10. sub-neg 20.6%

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

   11. expm1-define 20.6%

       \[\leadsto \log \left(1 + \left(\left(lo \cdot e^{\frac{x}{hi}}\right) \cdot \left(\frac{x}{{hi}^{2}} + \frac{-1}{hi}\right) + \color{blue}{\mathsf{expm1}\left(\frac{x}{hi}\right)}\right)\right) \]

10. Simplified 20.6%

    \[\leadsto \log \left(1 + \color{blue}{\left(\left(lo \cdot e^{\frac{x}{hi}}\right) \cdot \left(\frac{x}{{hi}^{2}} + \frac{-1}{hi}\right) + \mathsf{expm1}\left(\frac{x}{hi}\right)\right)}\right) \]

11. Taylor expanded in x around 0 20.7%

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

    1. log1p-define 20.7%

       \[\leadsto \color{blue}{\mathsf{log1p}\left(-1 \cdot \frac{lo}{hi}\right)} \]

    2. associate-*r/ 20.7%

       \[\leadsto \mathsf{log1p}\left(\color{blue}{\frac{-1 \cdot lo}{hi}}\right) \]

    3. neg-mul-1 20.7%

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

13. Simplified 20.7%

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

14. Final simplification 20.7%

    \[\leadsto \mathsf{log1p}\left(\frac{lo}{-hi}\right) \]
15. Add Preprocessing

Alternative 2: 18.9% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   1. mul-1-neg 3.1%

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

   2. associate--l+ 3.1%

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

   3. associate-/l* 14.9%

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

5. Simplified 14.9%

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

   1. add-sqr-sqrt 9.3%

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

   2. sqrt-unprod 14.5%

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

   3. sqr-neg 14.5%

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

   4. sqrt-unprod 5.1%

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

   5. add-sqr-sqrt 13.9%

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

   6. distribute-neg-frac2 13.9%

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

   7. div-inv 13.9%

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

7. Applied egg-rr 13.9%

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

8. Taylor expanded in hi around 0 18.9%

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

   1. sub-neg 18.9%

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

   2. +-commutative 18.9%

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

   3. mul-1-neg 18.9%

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

   4. sub-neg 18.9%

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

   5. div-sub 18.9%

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

   6. metadata-eval 18.9%

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

10. Simplified 18.9%

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

11. Final simplification 18.9%

    \[\leadsto 1 + \left(x + hi \cdot \left(\frac{x - hi}{lo} + -1\right)\right) \cdot \frac{1}{lo} \]
12. 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. associate-*r/ 18.8%

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

   2. neg-mul-1 18.8%

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

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 18.7%

   \[\leadsto \color{blue}{1} \]

4. Final simplification 18.7%

   \[\leadsto 1 \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024079 
(FPCore (lo hi x)
  :name "xlohi (overflows)"
  :precision binary64
  :pre (and (< lo -1e+308) (> hi 1e+308))
  (/ (- x lo) (- hi lo)))