xlohi (overflows)

Percentage Accurate: 3.1% → 19.5%

Time: 33.2s

Alternatives: 6

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: 19.5% accurate, 0.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[lo_, hi_, x_] := N[(1.0 - N[Abs[N[(N[(hi * N[(N[(N[(x - hi), $MachinePrecision] / lo), $MachinePrecision] + -1.0), $MachinePrecision] + x), $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* 15.7%

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

   5. fma-define 15.7%

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

5. Simplified 15.7%

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

   1. add-sqr-sqrt 11.0%

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

   2. sqrt-unprod 15.2%

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

   3. pow2 15.2%

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

7. Applied egg-rr 15.2%

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

   1. unpow2 15.2%

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

   2. rem-sqrt-square 15.2%

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

9. Simplified 19.6%

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

10. Final simplification 19.6%

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

Alternative 2: 19.5% accurate, 0.1× speedup

Math

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

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(Float64(hi * 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[(N[(hi * N[(1.0 + N[(hi / lo), $MachinePrecision]), $MachinePrecision]), $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* 15.7%

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

   5. fma-define 15.7%

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

5. Simplified 15.7%

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

   1. add-sqr-sqrt 11.0%

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

   2. sqrt-unprod 15.2%

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

   3. pow2 15.2%

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

7. Applied egg-rr 15.2%

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

   1. unpow2 15.2%

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

   2. rem-sqrt-square 15.2%

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

9. Simplified 19.6%

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

10. Taylor expanded in x around 0 19.6%

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

    1. mul-1-neg 19.6%

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

    2. *-commutative 19.6%

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

    3. distribute-lft-neg-in 19.6%

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

    4. mul-1-neg 19.6%

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

    5. distribute-lft-in 19.6%

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

    6. metadata-eval 19.6%

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

    7. mul-1-neg 19.6%

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

    8. unsub-neg 19.6%

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

12. Simplified 19.6%

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

13. Final simplification 19.6%

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

Alternative 3: 18.9% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[lo_, hi_, x_] := N[(1.0 + N[(N[(N[(hi * N[(N[(N[(hi / lo), $MachinePrecision] - N[(x / lo), $MachinePrecision]), $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision] - x), $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 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* 15.7%

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

   5. fma-define 15.7%

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

5. Simplified 15.7%

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

6. Taylor expanded in hi around 0 19.1%

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

7. Final simplification 19.1%

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

Alternative 4: 18.9% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[lo_, hi_, x_] := N[(1.0 + N[(hi * N[(N[(1.0 + N[(hi / 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 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* 15.7%

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

   5. fma-define 15.7%

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

5. Simplified 15.7%

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

   1. expm1-log1p-u 15.7%

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

7. Applied egg-rr 15.7%

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

9. Simplified 19.1%

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

10. Taylor expanded in x around 0 19.1%

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

    1. sub-neg 19.1%

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

    2. mul-1-neg 19.1%

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

    3. remove-double-neg 19.1%

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

    4. associate-/l* 19.1%

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

12. Simplified 19.1%

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

13. Final simplification 19.1%

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

Alternative 5: 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. associate-*r/ 18.8%

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

   2. neg-mul-1 18.8%

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

6. Simplified 18.8%

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

7. Final simplification 18.8%

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

Alternative 6: 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 2024053 
(FPCore (lo hi x)
  :name "xlohi (overflows)"
  :precision binary64
  :pre (and (< lo -1e+308) (> hi 1e+308))
  (/ (- x lo) (- hi lo)))