xlohi (overflows)

Percentage Accurate: 3.1% → 18.8%

Time: 7.8s

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: 18.8% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \frac{\sqrt[3]{x - lo}}{hi} \cdot {\left({\left(x - lo\right)}^{0.3333333333333333}\right)}^{2} \end{array} \]

FPCore

(FPCore (lo hi x)
 :precision binary64
 (* (/ (cbrt (- x lo)) hi) (pow (pow (- x lo) 0.3333333333333333) 2.0)))

C

double code(double lo, double hi, double x) {
	return (cbrt((x - lo)) / hi) * pow(pow((x - lo), 0.3333333333333333), 2.0);
}

Java

public static double code(double lo, double hi, double x) {
	return (Math.cbrt((x - lo)) / hi) * Math.pow(Math.pow((x - lo), 0.3333333333333333), 2.0);
}

Julia

function code(lo, hi, x)
	return Float64(Float64(cbrt(Float64(x - lo)) / hi) * ((Float64(x - lo) ^ 0.3333333333333333) ^ 2.0))
end

Wolfram

code[lo_, hi_, x_] := N[(N[(N[Power[N[(x - lo), $MachinePrecision], 1/3], $MachinePrecision] / hi), $MachinePrecision] * N[Power[N[Power[N[(x - lo), $MachinePrecision], 0.3333333333333333], $MachinePrecision], 2.0], $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. Taylor expanded in hi around inf 18.8%

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

   1. div-inv 18.8%

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

   2. div-sub 18.8%

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

   3. add-cube-cbrt 18.8%

      \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{x - lo} \cdot \sqrt[3]{x - lo}\right) \cdot \sqrt[3]{x - lo}}}{hi} \]

   4. *-un-lft-identity 18.8%

      \[\leadsto \frac{\left(\sqrt[3]{x - lo} \cdot \sqrt[3]{x - lo}\right) \cdot \sqrt[3]{x - lo}}{\color{blue}{1 \cdot hi}} \]

   5. times-frac 18.8%

      \[\leadsto \color{blue}{\frac{\sqrt[3]{x - lo} \cdot \sqrt[3]{x - lo}}{1} \cdot \frac{\sqrt[3]{x - lo}}{hi}} \]

   6. pow2 18.8%

      \[\leadsto \frac{\color{blue}{{\left(\sqrt[3]{x - lo}\right)}^{2}}}{1} \cdot \frac{\sqrt[3]{x - lo}}{hi} \]

7. Applied egg-rr 18.8%

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

   1. pow1/3 18.8%

      \[\leadsto \frac{{\color{blue}{\left({\left(x - lo\right)}^{0.3333333333333333}\right)}}^{2}}{1} \cdot \frac{\sqrt[3]{x - lo}}{hi} \]

9. Applied egg-rr 18.8%

   \[\leadsto \frac{{\color{blue}{\left({\left(x - lo\right)}^{0.3333333333333333}\right)}}^{2}}{1} \cdot \frac{\sqrt[3]{x - lo}}{hi} \]

10. Final simplification 18.8%

    \[\leadsto \frac{\sqrt[3]{x - lo}}{hi} \cdot {\left({\left(x - lo\right)}^{0.3333333333333333}\right)}^{2} \]

Alternative 2: 18.8% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ {\left(e^{\log \left(x - lo\right) \cdot 0.3333333333333333}\right)}^{2} \cdot \frac{\sqrt[3]{x - lo}}{hi} \end{array} \]

FPCore

(FPCore (lo hi x)
 :precision binary64
 (*
  (pow (exp (* (log (- x lo)) 0.3333333333333333)) 2.0)
  (/ (cbrt (- x lo)) hi)))

C

double code(double lo, double hi, double x) {
	return pow(exp((log((x - lo)) * 0.3333333333333333)), 2.0) * (cbrt((x - lo)) / hi);
}

Java

public static double code(double lo, double hi, double x) {
	return Math.pow(Math.exp((Math.log((x - lo)) * 0.3333333333333333)), 2.0) * (Math.cbrt((x - lo)) / hi);
}

Julia

function code(lo, hi, x)
	return Float64((exp(Float64(log(Float64(x - lo)) * 0.3333333333333333)) ^ 2.0) * Float64(cbrt(Float64(x - lo)) / hi))
end

Wolfram

code[lo_, hi_, x_] := N[(N[Power[N[Exp[N[(N[Log[N[(x - lo), $MachinePrecision]], $MachinePrecision] * 0.3333333333333333), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Power[N[(x - lo), $MachinePrecision], 1/3], $MachinePrecision] / 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. Taylor expanded in hi around inf 18.8%

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

   1. div-inv 18.8%

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

   2. div-sub 18.8%

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

   3. add-cube-cbrt 18.8%

      \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{x - lo} \cdot \sqrt[3]{x - lo}\right) \cdot \sqrt[3]{x - lo}}}{hi} \]

   4. *-un-lft-identity 18.8%

      \[\leadsto \frac{\left(\sqrt[3]{x - lo} \cdot \sqrt[3]{x - lo}\right) \cdot \sqrt[3]{x - lo}}{\color{blue}{1 \cdot hi}} \]

   5. times-frac 18.8%

      \[\leadsto \color{blue}{\frac{\sqrt[3]{x - lo} \cdot \sqrt[3]{x - lo}}{1} \cdot \frac{\sqrt[3]{x - lo}}{hi}} \]

   6. pow2 18.8%

      \[\leadsto \frac{\color{blue}{{\left(\sqrt[3]{x - lo}\right)}^{2}}}{1} \cdot \frac{\sqrt[3]{x - lo}}{hi} \]

7. Applied egg-rr 18.8%

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

   1. pow1/3 18.8%

      \[\leadsto \frac{{\color{blue}{\left({\left(x - lo\right)}^{0.3333333333333333}\right)}}^{2}}{1} \cdot \frac{\sqrt[3]{x - lo}}{hi} \]

   2. pow-to-exp 18.8%

      \[\leadsto \frac{{\color{blue}{\left(e^{\log \left(x - lo\right) \cdot 0.3333333333333333}\right)}}^{2}}{1} \cdot \frac{\sqrt[3]{x - lo}}{hi} \]

9. Applied egg-rr 18.8%

   \[\leadsto \frac{{\color{blue}{\left(e^{\log \left(x - lo\right) \cdot 0.3333333333333333}\right)}}^{2}}{1} \cdot \frac{\sqrt[3]{x - lo}}{hi} \]

10. Final simplification 18.8%

    \[\leadsto {\left(e^{\log \left(x - lo\right) \cdot 0.3333333333333333}\right)}^{2} \cdot \frac{\sqrt[3]{x - lo}}{hi} \]

Alternative 3: 18.8% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{x - lo}{\sqrt[3]{hi}}}{{\left(\sqrt[3]{hi}\right)}^{2}} \end{array} \]

FPCore

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

C

double code(double lo, double hi, double x) {
	return ((x - lo) / cbrt(hi)) / pow(cbrt(hi), 2.0);
}

Java

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

Julia

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

Wolfram

code[lo_, hi_, x_] := N[(N[(N[(x - lo), $MachinePrecision] / N[Power[hi, 1/3], $MachinePrecision]), $MachinePrecision] / N[Power[N[Power[hi, 1/3], $MachinePrecision], 2.0], $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. Taylor expanded in hi around inf 18.8%

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

   1. div-inv 18.8%

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

   2. div-sub 18.8%

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

   3. *-un-lft-identity 18.8%

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

   4. add-cube-cbrt 18.8%

      \[\leadsto \frac{1 \cdot \left(x - lo\right)}{\color{blue}{\left(\sqrt[3]{hi} \cdot \sqrt[3]{hi}\right) \cdot \sqrt[3]{hi}}} \]

   5. times-frac 18.8%

      \[\leadsto \color{blue}{\frac{1}{\sqrt[3]{hi} \cdot \sqrt[3]{hi}} \cdot \frac{x - lo}{\sqrt[3]{hi}}} \]

   6. pow2 18.8%

      \[\leadsto \frac{1}{\color{blue}{{\left(\sqrt[3]{hi}\right)}^{2}}} \cdot \frac{x - lo}{\sqrt[3]{hi}} \]

7. Applied egg-rr 18.8%

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

   1. associate-*l/ 18.8%

      \[\leadsto \color{blue}{\frac{1 \cdot \frac{x - lo}{\sqrt[3]{hi}}}{{\left(\sqrt[3]{hi}\right)}^{2}}} \]

   2. *-lft-identity 18.8%

      \[\leadsto \frac{\color{blue}{\frac{x - lo}{\sqrt[3]{hi}}}}{{\left(\sqrt[3]{hi}\right)}^{2}} \]

9. Simplified 18.8%

   \[\leadsto \color{blue}{\frac{\frac{x - lo}{\sqrt[3]{hi}}}{{\left(\sqrt[3]{hi}\right)}^{2}}} \]

10. Final simplification 18.8%

    \[\leadsto \frac{\frac{x - lo}{\sqrt[3]{hi}}}{{\left(\sqrt[3]{hi}\right)}^{2}} \]

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 18.6%

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

3. Final simplification 18.6%

   \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023238 
(FPCore (lo hi x)
  :name "xlohi (overflows)"
  :precision binary64
  :pre (and (< lo -1e+308) (> hi 1e+308))
  (/ (- x lo) (- hi lo)))