xlohi (overflows)

Percentage Accurate: 3.1% → 18.9%

Time: 47.2s

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: 18.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ 1 - \left(hi \cdot \frac{-1 - \frac{hi}{lo}}{lo} + x \cdot \left(\frac{hi}{{lo}^{2}} + \frac{1}{lo}\right)\right) \end{array} \]

FPCore

(FPCore (lo hi x)
 :precision binary64
 (-
  1.0
  (+
   (* hi (/ (- -1.0 (/ hi lo)) lo))
   (* x (+ (/ hi (pow lo 2.0)) (/ 1.0 lo))))))

C

double code(double lo, double hi, double x) {
	return 1.0 - ((hi * ((-1.0 - (hi / lo)) / lo)) + (x * ((hi / pow(lo, 2.0)) + (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 - ((hi * (((-1.0d0) - (hi / lo)) / lo)) + (x * ((hi / (lo ** 2.0d0)) + (1.0d0 / lo))))
end function

Java

public static double code(double lo, double hi, double x) {
	return 1.0 - ((hi * ((-1.0 - (hi / lo)) / lo)) + (x * ((hi / Math.pow(lo, 2.0)) + (1.0 / lo))));
}

Python

def code(lo, hi, x):
	return 1.0 - ((hi * ((-1.0 - (hi / lo)) / lo)) + (x * ((hi / math.pow(lo, 2.0)) + (1.0 / lo))))

Julia

function code(lo, hi, x)
	return Float64(1.0 - Float64(Float64(hi * Float64(Float64(-1.0 - Float64(hi / lo)) / lo)) + Float64(x * Float64(Float64(hi / (lo ^ 2.0)) + Float64(1.0 / lo)))))
end

MATLAB

function tmp = code(lo, hi, x)
	tmp = 1.0 - ((hi * ((-1.0 - (hi / lo)) / lo)) + (x * ((hi / (lo ^ 2.0)) + (1.0 / lo))));
end

Wolfram

code[lo_, hi_, x_] := N[(1.0 - N[(N[(hi * N[(N[(-1.0 - N[(hi / lo), $MachinePrecision]), $MachinePrecision] / lo), $MachinePrecision]), $MachinePrecision] + N[(x * N[(N[(hi / N[Power[lo, 2.0], $MachinePrecision]), $MachinePrecision] + N[(1.0 / lo), $MachinePrecision]), $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. expm1-log1p-u 14.7%

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

7. Applied egg-rr 14.7%

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

9. Simplified 18.9%

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

10. Taylor expanded in x around 0 18.9%

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

    1. associate--l+ 18.9%

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

    2. sub-neg 18.9%

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

    3. mul-1-neg 18.9%

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

    4. distribute-rgt-neg-in 18.9%

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

    5. distribute-neg-in 18.9%

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

    6. unsub-neg 18.9%

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

    7. distribute-neg-frac 18.9%

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

    8. metadata-eval 18.9%

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

    9. mul-1-neg 18.9%

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

    10. remove-double-neg 18.9%

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

    11. associate-/l* 18.9%

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

12. Simplified 18.9%

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

13. Final simplification 18.9%

    \[\leadsto 1 - \left(hi \cdot \frac{-1 - \frac{hi}{lo}}{lo} + x \cdot \left(\frac{hi}{{lo}^{2}} + \frac{1}{lo}\right)\right) \]
14. Add Preprocessing

Alternative 2: 18.9% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 3.1%

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

3. Taylor expanded in hi around 0 18.9%

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

4. Taylor expanded in x around inf 18.9%

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

5. Taylor expanded in lo around inf 3.1%

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

   1. associate-/l* 14.7%

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

7. Simplified 14.7%

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

8. Taylor expanded in hi around 0 18.9%

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

9. Final simplification 18.9%

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

Alternative 3: 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* 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. expm1-log1p-u 14.7%

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

7. Applied egg-rr 14.7%

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

9. Simplified 18.9%

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

10. Taylor expanded in x around 0 18.9%

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

    1. associate-*r/ 18.9%

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

    2. mul-1-neg 18.9%

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

    3. distribute-rgt-neg-in 18.9%

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

    4. +-commutative 18.9%

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

    5. distribute-neg-in 18.9%

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

    6. mul-1-neg 18.9%

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

    7. sub-neg 18.9%

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

    8. associate-/l* 18.9%

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

    9. sub-neg 18.9%

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

    10. mul-1-neg 18.9%

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

    11. distribute-neg-in 18.9%

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

    12. +-commutative 18.9%

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

    13. distribute-neg-in 18.9%

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

    14. metadata-eval 18.9%

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

    15. unsub-neg 18.9%

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

12. Simplified 18.9%

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

13. Final simplification 18.9%

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

Alternative 4: 18.8% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   1. associate-/r* 18.8%

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

6. Simplified 18.8%

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

7. Final simplification 18.8%

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

Reproduce

herbie shell --seed 2024152 
(FPCore (lo hi x)
  :name "xlohi (overflows)"
  :precision binary64
  :pre (and (< lo -1e+308) (> hi 1e+308))
  (/ (- x lo) (- hi lo)))