xlohi (overflows)

Percentage Accurate: 3.1% → 20.6%

Time: 7.4s

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: 20.6% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

code[lo_, hi_, x_] := N[(N[Log[1 + N[(lo * N[(N[(x / N[(hi * hi), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / hi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(x / hi), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 3.1%

   \[\frac{x - lo}{hi - lo} \]

2. Taylor expanded in lo around 0 18.8%

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

   1. mul-1-neg 18.8%

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

   2. unsub-neg 18.8%

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

   3. mul-1-neg 18.8%

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

   4. unsub-neg 18.8%

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

   5. unpow2 18.8%

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

4. Simplified 18.8%

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

   1. add-log-exp 18.8%

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

   2. associate-/r* 18.8%

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

   3. sub-div 18.8%

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

6. Applied egg-rr 18.8%

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

7. Taylor expanded in lo around 0 20.6%

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

   1. associate-*r* 20.6%

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

   2. distribute-lft1-in 20.6%

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

   3. *-commutative 20.6%

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

   4. sub-neg 20.6%

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

   5. unpow2 20.6%

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

   6. distribute-neg-frac 20.6%

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

   7. metadata-eval 20.6%

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

9. Simplified 20.6%

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

    1. log-prod 20.6%

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

    2. +-commutative 20.6%

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

    3. log1p-udef 20.6%

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

    4. add-log-exp 20.6%

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

11. Applied egg-rr 20.6%

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

12. Final simplification 20.6%

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

Alternative 2: 20.6% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 3.1%

   \[\frac{x - lo}{hi - lo} \]

2. Taylor expanded in lo around 0 18.8%

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

   1. mul-1-neg 18.8%

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

   2. unsub-neg 18.8%

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

   3. mul-1-neg 18.8%

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

   4. unsub-neg 18.8%

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

   5. unpow2 18.8%

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

4. Simplified 18.8%

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

   1. add-log-exp 18.8%

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

   2. associate-/r* 18.8%

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

   3. sub-div 18.8%

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

6. Applied egg-rr 18.8%

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

7. Taylor expanded in hi around inf 20.6%

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

   1. associate--l+ 20.6%

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

9. Simplified 20.6%

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

10. Final simplification 20.6%

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

Alternative 3: 20.6% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \mathsf{log1p}\left(\frac{-lo}{hi}\right) \end{array} \]

FPCore

(FPCore (lo hi x) :precision binary64 (log1p (/ (- lo) hi)))

C

double code(double lo, double hi, double x) {
	return log1p((-lo / hi));
}

Java

public static double code(double lo, double hi, double x) {
	return Math.log1p((-lo / hi));
}

Python

def code(lo, hi, x):
	return math.log1p((-lo / hi))

Julia

function code(lo, hi, x)
	return log1p(Float64(Float64(-lo) / hi))
end

Wolfram

code[lo_, hi_, x_] := N[Log[1 + N[((-lo) / hi), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 3.1%

   \[\frac{x - lo}{hi - lo} \]

2. Taylor expanded in lo around 0 18.8%

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

   1. mul-1-neg 18.8%

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

   2. unsub-neg 18.8%

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

   3. mul-1-neg 18.8%

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

   4. unsub-neg 18.8%

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

   5. unpow2 18.8%

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

4. Simplified 18.8%

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

   1. add-log-exp 18.8%

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

   2. associate-/r* 18.8%

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

   3. sub-div 18.8%

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

6. Applied egg-rr 18.8%

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

7. Taylor expanded in lo around 0 20.6%

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

   1. associate-*r* 20.6%

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

   2. distribute-lft1-in 20.6%

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

   3. *-commutative 20.6%

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

   4. sub-neg 20.6%

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

   5. unpow2 20.6%

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

   6. distribute-neg-frac 20.6%

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

   7. metadata-eval 20.6%

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

9. Simplified 20.6%

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

10. Taylor expanded in x around 0 20.6%

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

    1. log1p-def 20.6%

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

    2. mul-1-neg 20.6%

       \[\leadsto \mathsf{log1p}\left(\color{blue}{-\frac{lo}{hi}}\right) \]

    3. distribute-frac-neg 20.6%

       \[\leadsto \mathsf{log1p}\left(\color{blue}{\frac{-lo}{hi}}\right) \]

12. Simplified 20.6%

    \[\leadsto \color{blue}{\mathsf{log1p}\left(\frac{-lo}{hi}\right)} \]

13. Final simplification 20.6%

    \[\leadsto \mathsf{log1p}\left(\frac{-lo}{hi}\right) \]

Alternative 4: 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. Taylor expanded in hi around inf 18.8%

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

3. Final simplification 18.8%

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

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(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. Taylor expanded in lo around 0 18.8%

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

   1. mul-1-neg 18.8%

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

   2. unsub-neg 18.8%

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

   3. mul-1-neg 18.8%

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

   4. unsub-neg 18.8%

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

   5. unpow2 18.8%

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

4. Simplified 18.8%

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

5. Taylor expanded in x around 0 18.8%

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

   1. associate-*r/ 18.8%

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

   2. neg-mul-1 18.8%

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

7. Simplified 18.8%

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

8. Final simplification 18.8%

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

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. Taylor expanded in lo around inf 18.7%

   \[\leadsto \color{blue}{1} \]

3. Final simplification 18.7%

   \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023200 
(FPCore (lo hi x)
  :name "xlohi (overflows)"
  :precision binary64
  :pre (and (< lo -1e+308) (> hi 1e+308))
  (/ (- x lo) (- hi lo)))