xlohi (overflows)

Percentage Accurate: 3.1% → 19.2%

Time: 33.3s

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: 19.2% accurate, 0.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 3.1%

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

3. Taylor expanded in hi around inf 0.7%

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

   1. +-commutative 0.7%

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

   2. associate--l+ 0.7%

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

   3. +-commutative 0.7%

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

   4. *-rgt-identity 0.7%

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

   5. *-commutative 0.7%

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

   6. associate-/l* 8.7%

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

   7. distribute-lft-out 8.9%

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

5. Simplified 8.9%

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

   1. add-sqr-sqrt 8.1%

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

   2. sqrt-unprod 18.0%

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

   3. pow2 18.0%

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

   4. associate-/l* 18.0%

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

7. Applied egg-rr 18.0%

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

8. Taylor expanded in lo around inf 19.2%

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

9. Final simplification 19.2%

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

Alternative 2: 19.2% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ {\left(\frac{lo}{hi}\right)}^{2} \end{array} \]

FPCore

(FPCore (lo hi x) :precision binary64 (pow (/ lo hi) 2.0))

C

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

Java

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

Python

def code(lo, hi, x):
	return math.pow((lo / hi), 2.0)

Julia

function code(lo, hi, x)
	return Float64(lo / hi) ^ 2.0
end

MATLAB

function tmp = code(lo, hi, x)
	tmp = (lo / hi) ^ 2.0;
end

Wolfram

code[lo_, hi_, x_] := N[Power[N[(lo / hi), $MachinePrecision], 2.0], $MachinePrecision]

Derivation

1. Initial program 3.1%

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

3. Taylor expanded in hi around inf 0.7%

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

   1. +-commutative 0.7%

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

   2. associate--l+ 0.7%

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

   3. +-commutative 0.7%

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

   4. *-rgt-identity 0.7%

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

   5. *-commutative 0.7%

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

   6. associate-/l* 8.7%

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

   7. distribute-lft-out 8.9%

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

5. Simplified 8.9%

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

   1. add-sqr-sqrt 8.1%

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

   2. sqrt-unprod 18.0%

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

   3. pow2 18.0%

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

   4. associate-/l* 18.0%

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

7. Applied egg-rr 18.0%

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

8. Taylor expanded in lo around inf 19.2%

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

9. Taylor expanded in lo around -inf 0.0%

   \[\leadsto \color{blue}{\frac{{lo}^{2}}{{hi}^{2}}} \]
10. Step-by-step derivation

    1. unpow2 0.0%

       \[\leadsto \frac{\color{blue}{lo \cdot lo}}{{hi}^{2}} \]

    2. unpow2 0.0%

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

    3. times-frac 19.2%

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

    4. unpow2 19.2%

       \[\leadsto \color{blue}{{\left(\frac{lo}{hi}\right)}^{2}} \]

11. Simplified 19.2%

    \[\leadsto \color{blue}{{\left(\frac{lo}{hi}\right)}^{2}} \]

12. Final simplification 19.2%

    \[\leadsto {\left(\frac{lo}{hi}\right)}^{2} \]
13. Add Preprocessing

Alternative 3: 18.9% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

function tmp = code(lo, hi, x)
	tmp = ((lo - x) / lo) + (hi * (((1.0 / lo) + (((hi - (hi * (x / lo))) / lo) / lo)) + ((x / lo) * (-1.0 / lo))));
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[(hi - N[(hi * N[(x / lo), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / lo), $MachinePrecision] / lo), $MachinePrecision]), $MachinePrecision] + N[(N[(x / lo), $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 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 9.8%

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

   1. mul-1-neg 9.8%

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

   2. unsub-neg 9.8%

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

   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. *-un-lft-identity 18.9%

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

   2. unpow2 18.9%

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

   3. times-frac 18.9%

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

8. Applied egg-rr 18.9%

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

9. Final simplification 18.9%

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

Alternative 4: 18.9% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

Alternative 5: 18.7% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 3.1%

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

3. Taylor expanded in hi around 0 18.7%

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

   1. div-sub 18.7%

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

   2. sub-neg 18.7%

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

   3. *-inverses 18.7%

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

   4. metadata-eval 18.7%

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

   5. distribute-lft-in 18.7%

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

   6. metadata-eval 18.7%

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

   7. +-commutative 18.7%

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

   8. mul-1-neg 18.7%

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

   9. unsub-neg 18.7%

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

5. Simplified 18.7%

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

6. Final simplification 18.7%

   \[\leadsto 1 - \frac{x}{lo} \]
7. Add Preprocessing

Alternative 6: 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 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 2024079 
(FPCore (lo hi x)
  :name "xlohi (overflows)"
  :precision binary64
  :pre (and (< lo -1e+308) (> hi 1e+308))
  (/ (- x lo) (- hi lo)))