xlohi (overflows)

Percentage Accurate: 3.1% → 98.5%

Time: 14.6s

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: 98.5% accurate, 0.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 3.1%

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

2. Taylor expanded in x around 0 3.1%

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

   1. associate-*r/ 3.1%

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

   2. associate-/l* 3.1%

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

   3. div-sub 98.1%

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

   4. *-inverses 98.1%

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

4. Simplified 98.1%

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

   1. add-sqr-sqrt 97.6%

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

   2. sqrt-unprod 98.1%

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

   3. frac-times 98.1%

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

   4. metadata-eval 98.1%

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

   5. pow2 98.1%

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

   6. sub-neg 98.1%

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

   7. metadata-eval 98.1%

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

   8. +-commutative 98.1%

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

6. Applied egg-rr 98.1%

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

   1. +-commutative 98.1%

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

8. Simplified 98.1%

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

   1. metadata-eval 98.1%

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

   2. +-commutative 98.1%

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

   3. pow2 98.1%

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

   4. frac-times 98.1%

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

   5. sqrt-unprod 97.6%

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

   6. add-sqr-sqrt 98.1%

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

   7. add-exp-log 98.1%

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

   8. frac-2neg 98.1%

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

   9. metadata-eval 98.1%

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

   10. log-rec 98.0%

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

   11. distribute-neg-in 98.0%

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

   12. metadata-eval 98.0%

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

   13. neg-mul-1 98.0%

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

   14. log1p-udef 98.3%

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

   15. neg-mul-1 98.3%

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

   16. distribute-neg-frac 98.3%

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

10. Applied egg-rr 98.3%

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

11. Final simplification 98.3%

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

Alternative 2: 98.3% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -1 + \frac{hi}{lo}\\ \sqrt{\frac{1}{t_0 \cdot t_0}} \end{array} \end{array} \]

FPCore

(FPCore (lo hi x)
 :precision binary64
 (let* ((t_0 (+ -1.0 (/ hi lo)))) (sqrt (/ 1.0 (* t_0 t_0)))))

C

double code(double lo, double hi, double x) {
	double t_0 = -1.0 + (hi / lo);
	return sqrt((1.0 / (t_0 * t_0)));
}

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 = (-1.0d0) + (hi / lo)
    code = sqrt((1.0d0 / (t_0 * t_0)))
end function

Java

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

Python

def code(lo, hi, x):
	t_0 = -1.0 + (hi / lo)
	return math.sqrt((1.0 / (t_0 * t_0)))

Julia

function code(lo, hi, x)
	t_0 = Float64(-1.0 + Float64(hi / lo))
	return sqrt(Float64(1.0 / Float64(t_0 * t_0)))
end

MATLAB

function tmp = code(lo, hi, x)
	t_0 = -1.0 + (hi / lo);
	tmp = sqrt((1.0 / (t_0 * t_0)));
end

Wolfram

code[lo_, hi_, x_] := Block[{t$95$0 = N[(-1.0 + N[(hi / lo), $MachinePrecision]), $MachinePrecision]}, N[Sqrt[N[(1.0 / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Initial program 3.1%

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

2. Taylor expanded in x around 0 3.1%

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

   1. associate-*r/ 3.1%

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

   2. associate-/l* 3.1%

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

   3. div-sub 98.1%

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

   4. *-inverses 98.1%

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

4. Simplified 98.1%

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

   1. add-sqr-sqrt 97.6%

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

   2. sqrt-unprod 98.1%

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

   3. frac-times 98.1%

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

   4. metadata-eval 98.1%

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

   5. pow2 98.1%

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

   6. sub-neg 98.1%

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

   7. metadata-eval 98.1%

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

   8. +-commutative 98.1%

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

6. Applied egg-rr 98.1%

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

   1. +-commutative 98.1%

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

8. Simplified 98.1%

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

   1. unpow2 98.1%

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

10. Applied egg-rr 98.1%

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

11. Final simplification 98.1%

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

Alternative 3: 98.4% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \frac{-1 - \frac{hi}{lo}}{\mathsf{fma}\left(\frac{hi}{lo}, \frac{hi}{lo}, -1\right)} \end{array} \]

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 3.1%

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

2. Taylor expanded in x around 0 3.1%

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

   1. associate-*r/ 3.1%

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

   2. associate-/l* 3.1%

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

   3. div-sub 98.1%

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

   4. *-inverses 98.1%

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

4. Simplified 98.1%

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

   1. flip-- 97.0%

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

   2. associate-/r/ 97.0%

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

   3. metadata-eval 97.0%

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

   4. sub-neg 97.0%

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

   5. pow2 97.0%

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

   6. metadata-eval 97.0%

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

6. Applied egg-rr 97.0%

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

   1. associate-*l/ 97.0%

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

   2. neg-mul-1 97.0%

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

   3. neg-sub0 97.0%

      \[\leadsto \frac{\color{blue}{0 - \left(\frac{hi}{lo} + 1\right)}}{{\left(\frac{hi}{lo}\right)}^{2} + -1} \]

   4. +-commutative 97.0%

      \[\leadsto \frac{0 - \color{blue}{\left(1 + \frac{hi}{lo}\right)}}{{\left(\frac{hi}{lo}\right)}^{2} + -1} \]

   5. associate--r+ 97.0%

      \[\leadsto \frac{\color{blue}{\left(0 - 1\right) - \frac{hi}{lo}}}{{\left(\frac{hi}{lo}\right)}^{2} + -1} \]

   6. metadata-eval 97.0%

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

   7. +-commutative 97.0%

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

8. Simplified 97.0%

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

   1. +-commutative 97.0%

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

   2. unpow2 97.0%

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

   3. fma-def 98.2%

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

10. Applied egg-rr 98.2%

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

11. Final simplification 98.2%

    \[\leadsto \frac{-1 - \frac{hi}{lo}}{\mathsf{fma}\left(\frac{hi}{lo}, \frac{hi}{lo}, -1\right)} \]

Alternative 4: 98.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 3.1%

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

2. Taylor expanded in x around 0 3.1%

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

   1. associate-*r/ 3.1%

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

   2. associate-/l* 3.1%

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

   3. div-sub 98.1%

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

   4. *-inverses 98.1%

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

4. Simplified 98.1%

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

5. Final simplification 98.1%

   \[\leadsto \frac{-1}{-1 + \frac{hi}{lo}} \]

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. 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 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(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 x around 0 3.1%

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

   1. associate-*r/ 3.1%

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

   2. associate-/l* 3.1%

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

   3. div-sub 98.1%

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

   4. *-inverses 98.1%

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

4. Simplified 98.1%

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

5. Taylor expanded in hi around inf 18.8%

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

   1. neg-mul-1 18.8%

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

   2. distribute-neg-frac 18.8%

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

7. Simplified 18.8%

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

8. Final simplification 18.8%

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

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. Taylor expanded in lo around inf 18.7%

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

3. Final simplification 18.7%

   \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023334 
(FPCore (lo hi x)
  :name "xlohi (overflows)"
  :precision binary64
  :pre (and (< lo -1e+308) (> hi 1e+308))
  (/ (- x lo) (- hi lo)))