xlohi (overflows)

Percentage Accurate: 3.1% → 18.9%

Time: 20.8s

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, 0.0× speedup

Math

\[\begin{array}{l} \\ hi \cdot \left(\left(\frac{1}{lo} + \frac{\frac{hi - hi \cdot \frac{x}{lo}}{lo}}{lo}\right) - \frac{x}{{lo}^{2}}\right) - \frac{\mathsf{expm1}\left(\mathsf{log1p}\left(x - lo\right)\right)}{lo} \end{array} \]

FPCore

(FPCore (lo hi x)
 :precision binary64
 (-
  (*
   hi
   (- (+ (/ 1.0 lo) (/ (/ (- hi (* hi (/ x lo))) lo) lo)) (/ x (pow lo 2.0))))
  (/ (expm1 (log1p (- x lo))) lo)))

C

double code(double lo, double hi, double x) {
	return (hi * (((1.0 / lo) + (((hi - (hi * (x / lo))) / lo) / lo)) - (x / pow(lo, 2.0)))) - (expm1(log1p((x - lo))) / lo);
}

Java

public static double code(double lo, double hi, double x) {
	return (hi * (((1.0 / lo) + (((hi - (hi * (x / lo))) / lo) / lo)) - (x / Math.pow(lo, 2.0)))) - (Math.expm1(Math.log1p((x - lo))) / lo);
}

Python

def code(lo, hi, x):
	return (hi * (((1.0 / lo) + (((hi - (hi * (x / lo))) / lo) / lo)) - (x / math.pow(lo, 2.0)))) - (math.expm1(math.log1p((x - lo))) / lo)

Julia

function code(lo, hi, x)
	return Float64(Float64(hi * Float64(Float64(Float64(1.0 / lo) + Float64(Float64(Float64(hi - Float64(hi * Float64(x / lo))) / lo) / lo)) - Float64(x / (lo ^ 2.0)))) - Float64(expm1(log1p(Float64(x - lo))) / lo))
end

Wolfram

code[lo_, hi_, x_] := N[(N[(hi * N[(N[(N[(1.0 / lo), $MachinePrecision] + N[(N[(N[(hi - N[(hi * N[(x / lo), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / lo), $MachinePrecision] / lo), $MachinePrecision]), $MachinePrecision] - N[(x / N[Power[lo, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(Exp[N[Log[1 + N[(x - lo), $MachinePrecision]], $MachinePrecision]] - 1), $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 lo around inf 11.8%

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

   1. mul-1-neg 11.8%

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

   2. unsub-neg 11.8%

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

   3. associate-/l* 18.9%

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

6. Simplified 18.9%

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

   1. expm1-log1p-u 18.9%

      \[\leadsto -1 \cdot \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x - lo\right)\right)}}{lo} + hi \cdot \left(\left(\frac{1}{lo} + \frac{\frac{hi - hi \cdot \frac{x}{lo}}{lo}}{lo}\right) - \frac{x}{{lo}^{2}}\right) \]

8. Applied egg-rr 18.9%

   \[\leadsto -1 \cdot \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x - lo\right)\right)}}{lo} + hi \cdot \left(\left(\frac{1}{lo} + \frac{\frac{hi - hi \cdot \frac{x}{lo}}{lo}}{lo}\right) - \frac{x}{{lo}^{2}}\right) \]

9. Final simplification 18.9%

   \[\leadsto hi \cdot \left(\left(\frac{1}{lo} + \frac{\frac{hi - hi \cdot \frac{x}{lo}}{lo}}{lo}\right) - \frac{x}{{lo}^{2}}\right) - \frac{\mathsf{expm1}\left(\mathsf{log1p}\left(x - lo\right)\right)}{lo} \]
10. Add Preprocessing

Alternative 2: 18.9% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[lo_, hi_, x_] := N[(N[(hi * N[(N[(N[(1.0 / lo), $MachinePrecision] + N[(N[(N[(hi - N[(hi * N[(x / lo), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / lo), $MachinePrecision] / lo), $MachinePrecision]), $MachinePrecision] - N[(x / N[Power[lo, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(lo - x), $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 lo around inf 11.8%

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

   1. mul-1-neg 11.8%

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

   2. unsub-neg 11.8%

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

   3. associate-/l* 18.9%

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

6. Simplified 18.9%

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

7. Final simplification 18.9%

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

Alternative 3: 18.9% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[lo_, hi_, x_] := N[(N[(hi * N[(N[(N[(1.0 / lo), $MachinePrecision] + N[(N[(N[(hi - N[(hi * N[(x / lo), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / lo), $MachinePrecision] / lo), $MachinePrecision]), $MachinePrecision] - N[(x / N[Power[lo, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $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 11.8%

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

   1. mul-1-neg 11.8%

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

   2. unsub-neg 11.8%

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

   3. associate-/l* 18.9%

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

6. Simplified 18.9%

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

7. Taylor expanded in x around 0 18.9%

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

8. Final simplification 18.9%

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

Alternative 4: 18.9% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   1. associate--l+ 18.9%

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

   2. div-sub 18.9%

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

6. Simplified 18.9%

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

7. Final simplification 18.9%

   \[\leadsto \frac{lo - x}{lo} + hi \cdot \frac{\frac{hi - x}{lo} + 1}{lo} \]
8. 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. 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(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 18.8%

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

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. Add Preprocessing

Reproduce

herbie shell --seed 2024113 
(FPCore (lo hi x)
  :name "xlohi (overflows)"
  :precision binary64
  :pre (and (< lo -1e+308) (> hi 1e+308))
  (/ (- x lo) (- hi lo)))