xlohi (overflows)

Percentage Accurate: 3.1% → 19.5%

Time: 11.0s

Alternatives: 4

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: 19.5% accurate, 0.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 3.1%

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

2. Taylor expanded in lo around inf 0.0%

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

   1. associate--l+ 0.0%

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

   2. +-commutative 0.0%

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

   3. associate--l+ 0.0%

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

   4. distribute-lft-out-- 0.0%

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

   5. div-sub 0.0%

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

   6. mul-1-neg 0.0%

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

   7. sub-neg 0.0%

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

   8. unpow2 0.0%

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

   9. times-frac 19.0%

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

   10. distribute-lft-out-- 19.0%

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

   11. associate-*r/ 19.0%

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

   12. fma-neg 19.0%

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

4. Simplified 19.0%

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

   1. add-cbrt-cube 19.0%

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

   2. pow3 19.0%

      \[\leadsto \sqrt[3]{\color{blue}{{\left(1 + \mathsf{fma}\left(\frac{hi}{lo}, \frac{hi - x}{lo}, \frac{hi - x}{lo}\right)\right)}^{3}}} \]

   3. +-commutative 19.0%

      \[\leadsto \sqrt[3]{{\color{blue}{\left(\mathsf{fma}\left(\frac{hi}{lo}, \frac{hi - x}{lo}, \frac{hi - x}{lo}\right) + 1\right)}}^{3}} \]

6. Applied egg-rr 19.0%

   \[\leadsto \color{blue}{\sqrt[3]{{\left(\mathsf{fma}\left(\frac{hi}{lo}, \frac{hi - x}{lo}, \frac{hi - x}{lo}\right) + 1\right)}^{3}}} \]

7. Taylor expanded in lo around 0 0.0%

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

   1. unpow2 0.0%

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

   2. times-frac 19.7%

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

9. Simplified 19.7%

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

    1. expm1-log1p-u 19.7%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{hi}{lo} \cdot \frac{hi - x}{lo}\right)\right)} \]

    2. expm1-udef 19.7%

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

11. Applied egg-rr 19.7%

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

12. Final simplification 19.7%

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

Alternative 2: 19.5% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 3.1%

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

2. Taylor expanded in lo around inf 0.0%

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

   1. associate--l+ 0.0%

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

   2. +-commutative 0.0%

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

   3. associate--l+ 0.0%

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

   4. distribute-lft-out-- 0.0%

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

   5. div-sub 0.0%

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

   6. mul-1-neg 0.0%

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

   7. sub-neg 0.0%

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

   8. unpow2 0.0%

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

   9. times-frac 19.0%

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

   10. distribute-lft-out-- 19.0%

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

   11. associate-*r/ 19.0%

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

   12. fma-neg 19.0%

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

4. Simplified 19.0%

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

   1. add-cbrt-cube 19.0%

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

   2. pow3 19.0%

      \[\leadsto \sqrt[3]{\color{blue}{{\left(1 + \mathsf{fma}\left(\frac{hi}{lo}, \frac{hi - x}{lo}, \frac{hi - x}{lo}\right)\right)}^{3}}} \]

   3. +-commutative 19.0%

      \[\leadsto \sqrt[3]{{\color{blue}{\left(\mathsf{fma}\left(\frac{hi}{lo}, \frac{hi - x}{lo}, \frac{hi - x}{lo}\right) + 1\right)}}^{3}} \]

6. Applied egg-rr 19.0%

   \[\leadsto \color{blue}{\sqrt[3]{{\left(\mathsf{fma}\left(\frac{hi}{lo}, \frac{hi - x}{lo}, \frac{hi - x}{lo}\right) + 1\right)}^{3}}} \]

7. Taylor expanded in lo around 0 0.0%

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

   1. unpow2 0.0%

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

   2. times-frac 19.7%

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

9. Simplified 19.7%

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

10. Final simplification 19.7%

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

Alternative 3: 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 hi around inf 18.7%

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

3. Taylor expanded in x around 0 18.8%

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

   1. associate-*r/ 18.8%

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

   2. neg-mul-1 18.8%

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

5. Simplified 18.8%

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

6. Final simplification 18.8%

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

Alternative 4: 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 2023275 
(FPCore (lo hi x)
  :name "xlohi (overflows)"
  :precision binary64
  :pre (and (< lo -1e+308) (> hi 1e+308))
  (/ (- x lo) (- hi lo)))