xlohi (overflows)

Percentage Accurate: 3.1% → 18.9%

Time: 15.4s

Alternatives: 7

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

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

FPCore

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

C

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

Fortran

real(8) function code(lo, hi, x)
    real(8), intent (in) :: lo
    real(8), intent (in) :: hi
    real(8), intent (in) :: x
    code = ((lo - x) / lo) + (hi * (((1.0d0 / lo) - ((((x * (hi / lo)) - hi) / lo) / lo)) - (x / (lo ** 2.0d0))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[lo_, hi_, x_] := N[(N[(N[(lo - x), $MachinePrecision] / lo), $MachinePrecision] + N[(hi * N[(N[(N[(1.0 / lo), $MachinePrecision] - N[(N[(N[(N[(x * N[(hi / lo), $MachinePrecision]), $MachinePrecision] - hi), $MachinePrecision] / lo), $MachinePrecision] / lo), $MachinePrecision]), $MachinePrecision] - N[(x / N[Power[lo, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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 lo around inf 10.4%

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

   1. mul-1-neg 10.4%

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

   2. unsub-neg 10.4%

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

   3. *-commutative 10.4%

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

   4. *-lft-identity 10.4%

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

   5. times-frac 18.9%

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

   6. /-rgt-identity 18.9%

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

6. Simplified 18.9%

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

7. Final simplification 18.9%

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

Alternative 2: 18.9% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[lo_, hi_, x_] := N[(N[(N[(lo - x), $MachinePrecision] / lo), $MachinePrecision] - N[(hi * N[(N[(x / N[Power[lo, 2.0], $MachinePrecision]), $MachinePrecision] + 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 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 lo around inf 18.9%

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

5. Final simplification 18.9%

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

Alternative 3: 18.9% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

   1. *-un-lft-identity 18.9%

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

   2. unpow2 18.9%

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

   3. times-frac 18.9%

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

5. Applied egg-rr 18.9%

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

   1. associate-*l/ 18.9%

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

   2. *-lft-identity 18.9%

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

7. Simplified 18.9%

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

   1. pow1 18.9%

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

   2. associate--l+ 18.9%

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

   3. unpow2 18.9%

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

   4. associate-/l/ 18.9%

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

   5. sub-div 18.9%

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

   6. sub-div 18.9%

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

9. Applied egg-rr 18.9%

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

    1. unpow1 18.9%

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

    2. associate-*r/ 18.9%

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

11. Simplified 18.9%

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

12. Final simplification 18.9%

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

Alternative 4: 18.9% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 3.1%

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

3. Taylor expanded in lo around inf 0.0%

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

5. Simplified 18.9%

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

6. Final simplification 18.9%

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

Alternative 5: 18.8% accurate, 1.4× speedup

Math

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

FPCore

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

C

double code(double lo, double hi, double x) {
	return (x - 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 = (x - lo) / hi
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 3.1%

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

3. Taylor expanded in hi around inf 18.8%

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

4. Final simplification 18.8%

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

Alternative 6: 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(-Float64(lo / 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 18.8%

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

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 7: 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 2024076 
(FPCore (lo hi x)
  :name "xlohi (overflows)"
  :precision binary64
  :pre (and (< lo -1e+308) (> hi 1e+308))
  (/ (- x lo) (- hi lo)))