xlohi (overflows)

Percentage Accurate: 3.1% → 20.6%

Time: 21.4s

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: 20.6% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\frac{x}{hi}}\\ \log \left(t\_0 + lo \cdot \left(t\_0 \cdot \left(\frac{x}{{hi}^{2}} + \frac{-1}{hi}\right)\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (lo hi x)
 :precision binary64
 (let* ((t_0 (exp (/ x hi))))
   (log (+ t_0 (* lo (* t_0 (+ (/ x (pow hi 2.0)) (/ -1.0 hi))))))))

C

double code(double lo, double hi, double x) {
	double t_0 = exp((x / hi));
	return log((t_0 + (lo * (t_0 * ((x / pow(hi, 2.0)) + (-1.0 / hi))))));
}

Fortran

real(8) function code(lo, hi, x)
    real(8), intent (in) :: lo
    real(8), intent (in) :: hi
    real(8), intent (in) :: x
    real(8) :: t_0
    t_0 = exp((x / hi))
    code = log((t_0 + (lo * (t_0 * ((x / (hi ** 2.0d0)) + ((-1.0d0) / hi))))))
end function

Java

public static double code(double lo, double hi, double x) {
	double t_0 = Math.exp((x / hi));
	return Math.log((t_0 + (lo * (t_0 * ((x / Math.pow(hi, 2.0)) + (-1.0 / hi))))));
}

Python

def code(lo, hi, x):
	t_0 = math.exp((x / hi))
	return math.log((t_0 + (lo * (t_0 * ((x / math.pow(hi, 2.0)) + (-1.0 / hi))))))

Julia

function code(lo, hi, x)
	t_0 = exp(Float64(x / hi))
	return log(Float64(t_0 + Float64(lo * Float64(t_0 * Float64(Float64(x / (hi ^ 2.0)) + Float64(-1.0 / hi))))))
end

MATLAB

function tmp = code(lo, hi, x)
	t_0 = exp((x / hi));
	tmp = log((t_0 + (lo * (t_0 * ((x / (hi ^ 2.0)) + (-1.0 / hi))))));
end

Wolfram

code[lo_, hi_, x_] := Block[{t$95$0 = N[Exp[N[(x / hi), $MachinePrecision]], $MachinePrecision]}, N[Log[N[(t$95$0 + N[(lo * N[(t$95$0 * N[(N[(x / N[Power[hi, 2.0], $MachinePrecision]), $MachinePrecision] + N[(-1.0 / hi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $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. Step-by-step derivation

   1. add-log-exp 18.8%

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

   2. div-inv 18.8%

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

   3. pow-flip 18.8%

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

   4. metadata-eval 18.8%

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

7. Applied egg-rr 18.8%

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

8. Taylor expanded in lo around 0 20.6%

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

   1. *-commutative 20.6%

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

   2. sub-neg 20.6%

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

   3. distribute-neg-frac 20.6%

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

   4. metadata-eval 20.6%

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

10. Simplified 20.6%

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

11. Final simplification 20.6%

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

Alternative 2: 20.6% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

   1. add-log-exp 18.8%

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

   2. div-inv 18.8%

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

   3. pow-flip 18.8%

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

   4. metadata-eval 18.8%

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

7. Applied egg-rr 18.8%

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

8. Taylor expanded in hi around inf 20.6%

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

   1. associate--l+ 20.6%

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

   2. div-sub 20.6%

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

10. Simplified 20.6%

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

Alternative 3: 19.5% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[lo_, hi_, x_] := N[Power[N[(hi * N[(1.0 / lo), $MachinePrecision]), $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* 15.0%

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

5. Simplified 15.0%

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

6. Taylor expanded in x around 0 15.0%

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

   1. associate-*r/ 15.0%

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

   2. neg-mul-1 15.0%

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

8. Simplified 15.0%

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

9. Taylor expanded in hi around inf 0.0%

   \[\leadsto \color{blue}{\frac{{hi}^{2}}{{lo}^{2}}} \]
10. 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 19.7%

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

    4. unpow2 19.7%

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

11. Simplified 19.7%

    \[\leadsto \color{blue}{{\left(\frac{hi}{lo}\right)}^{2}} \]
12. Step-by-step derivation

    1. div-inv 19.7%

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

13. Applied egg-rr 19.7%

    \[\leadsto {\color{blue}{\left(hi \cdot \frac{1}{lo}\right)}}^{2} \]
14. Add Preprocessing

Alternative 4: 19.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[lo_, hi_, x_] := N[(N[(hi / lo), $MachinePrecision] * N[(hi / 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* 15.0%

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

5. Simplified 15.0%

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

6. Taylor expanded in x around 0 15.0%

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

   1. associate-*r/ 15.0%

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

   2. neg-mul-1 15.0%

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

8. Simplified 15.0%

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

9. Taylor expanded in hi around inf 0.0%

   \[\leadsto \color{blue}{\frac{{hi}^{2}}{{lo}^{2}}} \]
10. 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 19.7%

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

    4. unpow2 19.7%

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

11. Simplified 19.7%

    \[\leadsto \color{blue}{{\left(\frac{hi}{lo}\right)}^{2}} \]
12. Step-by-step derivation

    1. unpow2 19.7%

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

13. Applied egg-rr 19.7%

    \[\leadsto \color{blue}{\frac{hi}{lo} \cdot \frac{hi}{lo}} \]
14. 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 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 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 2024191 
(FPCore (lo hi x)
  :name "xlohi (overflows)"
  :precision binary64
  :pre (and (< lo -1e+308) (> hi 1e+308))
  (/ (- x lo) (- hi lo)))