xlohi (overflows)

Percentage Accurate: 3.1% → 19.6%

Time: 20.9s

Alternatives: 5

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: 19.6% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ 1 - \left|hi \cdot \frac{-1 - \frac{hi}{lo}}{lo}\right| \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[lo_, hi_, x_] := N[(1.0 - N[Abs[N[(hi * N[(N[(-1.0 - N[(hi / lo), $MachinePrecision]), $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 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. unsub-neg 3.1%

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

   3. +-commutative 3.1%

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

   4. associate-/l* 14.7%

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

   5. fma-define 14.7%

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

5. Simplified 14.7%

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

   1. add-sqr-sqrt 9.5%

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

   2. sqrt-unprod 14.1%

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

   3. pow2 14.1%

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

7. Applied egg-rr 14.1%

   \[\leadsto 1 - \color{blue}{\sqrt{{\left(\frac{\mathsf{fma}\left(hi, \frac{x - hi}{lo}, x\right) - hi}{lo}\right)}^{2}}} \]
8. Step-by-step derivation

   1. unpow2 14.1%

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

   2. rem-sqrt-square 14.1%

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

9. Simplified 19.8%

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

10. Taylor expanded in x around 0 19.8%

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

    1. associate-*r/ 19.8%

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

    2. mul-1-neg 19.8%

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

    3. distribute-rgt-neg-in 19.8%

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

    4. distribute-neg-in 19.8%

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

    5. metadata-eval 19.8%

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

    6. unsub-neg 19.8%

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

12. Simplified 19.8%

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

    1. associate-/l* 19.8%

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

14. Applied egg-rr 19.8%

    \[\leadsto 1 - \left|\color{blue}{hi \cdot \frac{-1 - \frac{hi}{lo}}{lo}}\right| \]
15. Add Preprocessing

Alternative 2: 18.9% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   3. +-commutative 3.1%

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

   4. associate-/l* 14.7%

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

   5. fma-define 14.7%

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

5. Simplified 14.7%

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

   1. add-sqr-sqrt 9.5%

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

   2. sqrt-unprod 14.1%

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

   3. pow2 14.1%

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

7. Applied egg-rr 14.1%

   \[\leadsto 1 - \color{blue}{\sqrt{{\left(\frac{\mathsf{fma}\left(hi, \frac{x - hi}{lo}, x\right) - hi}{lo}\right)}^{2}}} \]
8. Step-by-step derivation

   1. unpow2 14.1%

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

   2. rem-sqrt-square 14.1%

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

9. Simplified 19.8%

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

10. Taylor expanded in x around 0 19.8%

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

    1. associate-*r/ 19.8%

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

    2. mul-1-neg 19.8%

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

    3. distribute-rgt-neg-in 19.8%

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

    4. distribute-neg-in 19.8%

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

    5. metadata-eval 19.8%

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

    6. unsub-neg 19.8%

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

12. Simplified 19.8%

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

    1. sub-neg 19.8%

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

    2. add-sqr-sqrt 9.5%

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

    3. fabs-sqr 9.5%

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

    4. add-sqr-sqrt 18.9%

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

    5. associate-/l* 18.9%

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

14. Applied egg-rr 18.9%

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

15. Final simplification 18.9%

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

Alternative 3: 18.8% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[lo_, hi_, x_] := N[(lo * N[(N[(-1.0 + N[(x / hi), $MachinePrecision]), $MachinePrecision] / hi), $MachinePrecision]), $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 lo around inf 18.8%

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

   1. rem-square-sqrt 9.8%

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

   2. unpow2 9.8%

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

   3. times-frac 9.8%

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

   4. associate-*r/ 9.8%

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

   5. *-commutative 9.8%

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

   6. associate-*r/ 9.8%

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

   7. rem-square-sqrt 18.8%

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

   8. div-sub 18.8%

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

   9. sub-neg 18.8%

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

   10. metadata-eval 18.8%

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

8. Simplified 18.8%

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

9. Final simplification 18.8%

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

Alternative 4: 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 hi around inf 18.7%

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

4. Taylor expanded in x around 0 18.8%

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

   1. neg-mul-1 18.8%

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

   2. distribute-neg-frac 18.8%

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

6. Simplified 18.8%

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

7. Final simplification 18.8%

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

Alternative 5: 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 2024129 
(FPCore (lo hi x)
  :name "xlohi (overflows)"
  :precision binary64
  :pre (and (< lo -1e+308) (> hi 1e+308))
  (/ (- x lo) (- hi lo)))