xlohi (overflows)

Percentage Accurate: 3.1% → 98.5%

Time: 12.3s

Alternatives: 5

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 98.6%

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

   4. *-inverses 98.6%

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

4. Simplified 98.6%

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

   1. add-sqr-sqrt 98.2%

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

   2. sqrt-unprod 98.6%

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

   3. frac-times 98.5%

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

   4. metadata-eval 98.5%

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

   5. pow2 98.5%

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

   6. sub-neg 98.5%

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

   7. metadata-eval 98.5%

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

   8. +-commutative 98.5%

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

6. Applied egg-rr 98.5%

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

   1. +-commutative 98.5%

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

8. Simplified 98.5%

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

   1. metadata-eval 98.5%

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

   2. +-commutative 98.5%

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

   3. pow2 98.5%

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

   4. frac-times 98.6%

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

   5. sqrt-unprod 98.2%

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

   6. add-sqr-sqrt 98.6%

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

   7. add-exp-log 98.6%

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

   8. frac-2neg 98.6%

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

   9. metadata-eval 98.6%

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

   10. log-rec 98.5%

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

   11. distribute-neg-in 98.5%

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

   12. metadata-eval 98.5%

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

   13. mul-1-neg 98.5%

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

   14. log1p-udef 98.7%

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

   15. associate-*r/ 98.7%

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

   16. neg-mul-1 98.7%

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

10. Applied egg-rr 98.7%

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

11. Final simplification 98.7%

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

Alternative 2: 98.4% accurate, 0.1× speedup

Math

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

FPCore

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

C

double code(double lo, double hi, double x) {
	double t_0 = (hi / lo) + -1.0;
	return sqrt((1.0 / (((hi / lo) * 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 = (hi / lo) + (-1.0d0)
    code = sqrt((1.0d0 / (((hi / lo) * t_0) - t_0)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[lo_, hi_, x_] := Block[{t$95$0 = N[(N[(hi / lo), $MachinePrecision] + -1.0), $MachinePrecision]}, N[Sqrt[N[(1.0 / N[(N[(N[(hi / lo), $MachinePrecision] * t$95$0), $MachinePrecision] - 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 98.6%

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

   4. *-inverses 98.6%

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

4. Simplified 98.6%

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

   1. add-sqr-sqrt 98.2%

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

   2. sqrt-unprod 98.6%

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

   3. frac-times 98.5%

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

   4. metadata-eval 98.5%

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

   5. pow2 98.5%

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

   6. sub-neg 98.5%

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

   7. metadata-eval 98.5%

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

   8. +-commutative 98.5%

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

6. Applied egg-rr 98.5%

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

   1. +-commutative 98.5%

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

8. Simplified 98.5%

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

   1. unpow2 98.5%

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

   2. +-commutative 98.5%

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

   3. distribute-rgt-in 98.6%

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

   4. +-commutative 98.6%

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

   5. neg-mul-1 98.6%

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

   6. +-commutative 98.6%

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

10. Applied egg-rr 98.6%

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

11. Final simplification 98.6%

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

Alternative 3: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[lo_, hi_, x_] := N[(-1.0 / N[(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 98.6%

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

   4. *-inverses 98.6%

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

4. Simplified 98.6%

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

5. Final simplification 98.6%

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

Alternative 4: 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 98.6%

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

   4. *-inverses 98.6%

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

4. Simplified 98.6%

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

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 5: 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 2023331 
(FPCore (lo hi x)
  :name "xlohi (overflows)"
  :precision binary64
  :pre (and (< lo -1e+308) (> hi 1e+308))
  (/ (- x lo) (- hi lo)))