xlohi (overflows)

Percentage Accurate: 3.1% → 24.3%

Time: 13.4s

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: 24.3% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[lo_, hi_, x_] := N[(N[(-1.0 / N[(lo * N[(N[(1.0 / N[(x - hi), $MachinePrecision]), $MachinePrecision] + N[(N[(N[Power[N[(hi / lo), $MachinePrecision], 2.0], $MachinePrecision] / N[(x - hi), $MachinePrecision]), $MachinePrecision] - N[(hi / N[(lo * N[(x - hi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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 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. associate--l+ 3.1%

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

   3. associate-/l* 14.6%

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

5. Simplified 14.6%

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

   1. neg-mul-1 14.6%

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

   2. clear-num 14.6%

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

   3. un-div-inv 14.6%

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

   4. clear-num 14.6%

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

   5. un-div-inv 14.6%

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

7. Applied egg-rr 14.6%

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

8. Taylor expanded in lo around inf 0.0%

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

   1. associate--l+ 0.0%

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

   2. associate-/r* 0.0%

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

   3. unpow2 0.0%

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

   4. unpow2 0.0%

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

   5. times-frac 24.0%

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

   6. unpow2 24.0%

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

10. Simplified 24.0%

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

11. Final simplification 24.0%

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

Alternative 2: 19.5% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[lo_, hi_, x_] := N[Power[N[(hi / lo), $MachinePrecision], 2.0], $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. associate--l+ 3.1%

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

   3. associate-/l* 14.6%

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

5. Simplified 14.6%

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

   1. neg-mul-1 14.6%

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

   2. clear-num 14.6%

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

   3. un-div-inv 14.6%

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

   4. clear-num 14.6%

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

   5. un-div-inv 14.6%

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

7. Applied egg-rr 14.6%

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

8. Taylor expanded in hi around inf 0.0%

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

   1. unpow2 0.0%

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

   2. unpow2 0.0%

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

   3. times-frac 19.6%

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

   4. unpow2 19.6%

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

10. Simplified 19.6%

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

Alternative 3: 18.9% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[lo_, hi_, x_] := N[(1.0 - N[(N[(x + N[(hi * N[(-1.0 + N[(N[(x - hi), $MachinePrecision] / 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 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. associate--l+ 3.1%

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

   3. associate-/l* 14.6%

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

5. Simplified 14.6%

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

6. Taylor expanded in hi around 0 18.8%

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

   1. sub-neg 18.8%

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

   2. +-commutative 18.8%

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

   3. neg-mul-1 18.8%

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

   4. sub-neg 18.8%

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

   5. div-sub 18.8%

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

   6. metadata-eval 18.8%

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

8. Simplified 18.8%

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

9. Final simplification 18.8%

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

Alternative 4: 18.9% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[lo_, hi_, x_] := N[(N[(hi * N[(N[(N[(hi / lo), $MachinePrecision] + 1.0), $MachinePrecision] / lo), $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 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. associate--l+ 3.1%

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

   3. associate-/l* 14.6%

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

5. Simplified 14.6%

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

6. Taylor expanded in hi around 0 18.8%

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

   1. sub-neg 18.8%

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

   2. +-commutative 18.8%

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

   3. neg-mul-1 18.8%

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

   4. sub-neg 18.8%

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

   5. div-sub 18.8%

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

   6. metadata-eval 18.8%

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

8. Simplified 18.8%

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

9. Taylor expanded in x around 0 18.8%

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

    1. associate-/l* 18.8%

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

11. Simplified 18.8%

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

12. Final simplification 18.8%

    \[\leadsto hi \cdot \frac{\frac{hi}{lo} + 1}{lo} + 1 \]
13. 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 lo around 0 18.8%

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

   1. +-commutative 18.8%

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

   2. mul-1-neg 18.8%

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

   3. unsub-neg 18.8%

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

   4. +-commutative 18.8%

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

   5. mul-1-neg 18.8%

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

   6. unsub-neg 18.8%

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

5. Simplified 18.8%

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

6. Taylor expanded in x around 0 18.8%

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

   1. associate-*r/ 18.8%

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

   2. neg-mul-1 18.8%

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

8. Simplified 18.8%

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

9. Final simplification 18.8%

   \[\leadsto \frac{lo}{-hi} \]
10. 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 2024151 
(FPCore (lo hi x)
  :name "xlohi (overflows)"
  :precision binary64
  :pre (and (< lo -1e+308) (> hi 1e+308))
  (/ (- x lo) (- hi lo)))