xlohi (overflows)

Percentage Accurate: 3.1% → 21.3%

Time: 10.6s

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: 21.3% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \frac{x}{hi} - \log \log \left(e^{1 + \mathsf{fma}\left({\left(\frac{lo}{hi}\right)}^{2}, 0.5, \frac{lo - \frac{x}{hi} \cdot lo}{hi}\right)}\right) \end{array} \]

FPCore

(FPCore (lo hi x)
 :precision binary64
 (-
  (/ x hi)
  (log
   (log
    (exp
     (+ 1.0 (fma (pow (/ lo hi) 2.0) 0.5 (/ (- lo (* (/ x hi) lo)) hi))))))))

C

double code(double lo, double hi, double x) {
	return (x / hi) - log(log(exp((1.0 + fma(pow((lo / hi), 2.0), 0.5, ((lo - ((x / hi) * lo)) / hi))))));
}

Julia

function code(lo, hi, x)
	return Float64(Float64(x / hi) - log(log(exp(Float64(1.0 + fma((Float64(lo / hi) ^ 2.0), 0.5, Float64(Float64(lo - Float64(Float64(x / hi) * lo)) / hi)))))))
end

Wolfram

code[lo_, hi_, x_] := N[(N[(x / hi), $MachinePrecision] - N[Log[N[Log[N[Exp[N[(1.0 + N[(N[Power[N[(lo / hi), $MachinePrecision], 2.0], $MachinePrecision] * 0.5 + N[(N[(lo - N[(N[(x / hi), $MachinePrecision] * lo), $MachinePrecision]), $MachinePrecision] / hi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]], $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. Step-by-step derivation

   1. log1p-expm1-u 18.8%

      \[\leadsto \frac{x}{hi} - \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(lo \cdot \left(\frac{1}{hi} - \frac{x}{hi \cdot hi}\right)\right)\right)} \]

   2. log1p-udef 18.8%

      \[\leadsto \frac{x}{hi} - \color{blue}{\log \left(1 + \mathsf{expm1}\left(lo \cdot \left(\frac{1}{hi} - \frac{x}{hi \cdot hi}\right)\right)\right)} \]

   3. associate-/r* 18.8%

      \[\leadsto \frac{x}{hi} - \log \left(1 + \mathsf{expm1}\left(lo \cdot \left(\frac{1}{hi} - \color{blue}{\frac{\frac{x}{hi}}{hi}}\right)\right)\right) \]

   4. sub-div 18.8%

      \[\leadsto \frac{x}{hi} - \log \left(1 + \mathsf{expm1}\left(lo \cdot \color{blue}{\frac{1 - \frac{x}{hi}}{hi}}\right)\right) \]

6. Applied egg-rr 18.8%

   \[\leadsto \frac{x}{hi} - \color{blue}{\log \left(1 + \mathsf{expm1}\left(lo \cdot \frac{1 - \frac{x}{hi}}{hi}\right)\right)} \]

7. Taylor expanded in hi around inf 0.0%

   \[\leadsto \frac{x}{hi} - \log \color{blue}{\left(1 + \left(0.5 \cdot \frac{{lo}^{2}}{{hi}^{2}} + \left(-1 \cdot \frac{lo \cdot x}{{hi}^{2}} + \frac{lo}{hi}\right)\right)\right)} \]
8. Step-by-step derivation

   1. *-commutative 0.0%

      \[\leadsto \frac{x}{hi} - \log \left(1 + \left(\color{blue}{\frac{{lo}^{2}}{{hi}^{2}} \cdot 0.5} + \left(-1 \cdot \frac{lo \cdot x}{{hi}^{2}} + \frac{lo}{hi}\right)\right)\right) \]

   2. fma-def 0.0%

      \[\leadsto \frac{x}{hi} - \log \left(1 + \color{blue}{\mathsf{fma}\left(\frac{{lo}^{2}}{{hi}^{2}}, 0.5, -1 \cdot \frac{lo \cdot x}{{hi}^{2}} + \frac{lo}{hi}\right)}\right) \]

   3. unpow2 0.0%

      \[\leadsto \frac{x}{hi} - \log \left(1 + \mathsf{fma}\left(\frac{\color{blue}{lo \cdot lo}}{{hi}^{2}}, 0.5, -1 \cdot \frac{lo \cdot x}{{hi}^{2}} + \frac{lo}{hi}\right)\right) \]

   4. unpow2 0.0%

      \[\leadsto \frac{x}{hi} - \log \left(1 + \mathsf{fma}\left(\frac{lo \cdot lo}{\color{blue}{hi \cdot hi}}, 0.5, -1 \cdot \frac{lo \cdot x}{{hi}^{2}} + \frac{lo}{hi}\right)\right) \]

   5. times-frac 10.1%

      \[\leadsto \frac{x}{hi} - \log \left(1 + \mathsf{fma}\left(\color{blue}{\frac{lo}{hi} \cdot \frac{lo}{hi}}, 0.5, -1 \cdot \frac{lo \cdot x}{{hi}^{2}} + \frac{lo}{hi}\right)\right) \]

   6. +-commutative 10.1%

      \[\leadsto \frac{x}{hi} - \log \left(1 + \mathsf{fma}\left(\frac{lo}{hi} \cdot \frac{lo}{hi}, 0.5, \color{blue}{\frac{lo}{hi} + -1 \cdot \frac{lo \cdot x}{{hi}^{2}}}\right)\right) \]

   7. mul-1-neg 10.1%

      \[\leadsto \frac{x}{hi} - \log \left(1 + \mathsf{fma}\left(\frac{lo}{hi} \cdot \frac{lo}{hi}, 0.5, \frac{lo}{hi} + \color{blue}{\left(-\frac{lo \cdot x}{{hi}^{2}}\right)}\right)\right) \]

   8. unsub-neg 10.1%

      \[\leadsto \frac{x}{hi} - \log \left(1 + \mathsf{fma}\left(\frac{lo}{hi} \cdot \frac{lo}{hi}, 0.5, \color{blue}{\frac{lo}{hi} - \frac{lo \cdot x}{{hi}^{2}}}\right)\right) \]

   9. *-commutative 10.1%

      \[\leadsto \frac{x}{hi} - \log \left(1 + \mathsf{fma}\left(\frac{lo}{hi} \cdot \frac{lo}{hi}, 0.5, \frac{lo}{hi} - \frac{\color{blue}{x \cdot lo}}{{hi}^{2}}\right)\right) \]

   10. unpow2 10.1%

       \[\leadsto \frac{x}{hi} - \log \left(1 + \mathsf{fma}\left(\frac{lo}{hi} \cdot \frac{lo}{hi}, 0.5, \frac{lo}{hi} - \frac{x \cdot lo}{\color{blue}{hi \cdot hi}}\right)\right) \]

   11. times-frac 21.4%

       \[\leadsto \frac{x}{hi} - \log \left(1 + \mathsf{fma}\left(\frac{lo}{hi} \cdot \frac{lo}{hi}, 0.5, \frac{lo}{hi} - \color{blue}{\frac{x}{hi} \cdot \frac{lo}{hi}}\right)\right) \]

9. Simplified 21.4%

   \[\leadsto \frac{x}{hi} - \log \color{blue}{\left(1 + \mathsf{fma}\left(\frac{lo}{hi} \cdot \frac{lo}{hi}, 0.5, \frac{lo}{hi} - \frac{x}{hi} \cdot \frac{lo}{hi}\right)\right)} \]
10. Step-by-step derivation

    1. add-log-exp 21.4%

       \[\leadsto \frac{x}{hi} - \log \color{blue}{\log \left(e^{1 + \mathsf{fma}\left(\frac{lo}{hi} \cdot \frac{lo}{hi}, 0.5, \frac{lo}{hi} - \frac{x}{hi} \cdot \frac{lo}{hi}\right)}\right)} \]

    2. pow2 21.4%

       \[\leadsto \frac{x}{hi} - \log \log \left(e^{1 + \mathsf{fma}\left(\color{blue}{{\left(\frac{lo}{hi}\right)}^{2}}, 0.5, \frac{lo}{hi} - \frac{x}{hi} \cdot \frac{lo}{hi}\right)}\right) \]

    3. associate-*r/ 21.4%

       \[\leadsto \frac{x}{hi} - \log \log \left(e^{1 + \mathsf{fma}\left({\left(\frac{lo}{hi}\right)}^{2}, 0.5, \frac{lo}{hi} - \color{blue}{\frac{\frac{x}{hi} \cdot lo}{hi}}\right)}\right) \]

    4. sub-div 21.4%

       \[\leadsto \frac{x}{hi} - \log \log \left(e^{1 + \mathsf{fma}\left({\left(\frac{lo}{hi}\right)}^{2}, 0.5, \color{blue}{\frac{lo - \frac{x}{hi} \cdot lo}{hi}}\right)}\right) \]

11. Applied egg-rr 21.4%

    \[\leadsto \frac{x}{hi} - \log \color{blue}{\log \left(e^{1 + \mathsf{fma}\left({\left(\frac{lo}{hi}\right)}^{2}, 0.5, \frac{lo - \frac{x}{hi} \cdot lo}{hi}\right)}\right)} \]

12. Final simplification 21.4%

    \[\leadsto \frac{x}{hi} - \log \log \left(e^{1 + \mathsf{fma}\left({\left(\frac{lo}{hi}\right)}^{2}, 0.5, \frac{lo - \frac{x}{hi} \cdot lo}{hi}\right)}\right) \]

Alternative 2: 21.3% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \frac{x}{hi} - \log \left(1 + \left(\frac{lo - \frac{x}{hi} \cdot lo}{hi} + {\left(\frac{lo}{hi}\right)}^{2} \cdot 0.5\right)\right) \end{array} \]

FPCore

(FPCore (lo hi x)
 :precision binary64
 (-
  (/ x hi)
  (log (+ 1.0 (+ (/ (- lo (* (/ x hi) lo)) hi) (* (pow (/ lo hi) 2.0) 0.5))))))

C

double code(double lo, double hi, double x) {
	return (x / hi) - log((1.0 + (((lo - ((x / hi) * lo)) / hi) + (pow((lo / hi), 2.0) * 0.5))));
}

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 / hi) - log((1.0d0 + (((lo - ((x / hi) * lo)) / hi) + (((lo / hi) ** 2.0d0) * 0.5d0))))
end function

Java

public static double code(double lo, double hi, double x) {
	return (x / hi) - Math.log((1.0 + (((lo - ((x / hi) * lo)) / hi) + (Math.pow((lo / hi), 2.0) * 0.5))));
}

Python

def code(lo, hi, x):
	return (x / hi) - math.log((1.0 + (((lo - ((x / hi) * lo)) / hi) + (math.pow((lo / hi), 2.0) * 0.5))))

Julia

function code(lo, hi, x)
	return Float64(Float64(x / hi) - log(Float64(1.0 + Float64(Float64(Float64(lo - Float64(Float64(x / hi) * lo)) / hi) + Float64((Float64(lo / hi) ^ 2.0) * 0.5)))))
end

MATLAB

function tmp = code(lo, hi, x)
	tmp = (x / hi) - log((1.0 + (((lo - ((x / hi) * lo)) / hi) + (((lo / hi) ^ 2.0) * 0.5))));
end

Wolfram

code[lo_, hi_, x_] := N[(N[(x / hi), $MachinePrecision] - N[Log[N[(1.0 + N[(N[(N[(lo - N[(N[(x / hi), $MachinePrecision] * lo), $MachinePrecision]), $MachinePrecision] / hi), $MachinePrecision] + N[(N[Power[N[(lo / hi), $MachinePrecision], 2.0], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $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. Step-by-step derivation

   1. log1p-expm1-u 18.8%

      \[\leadsto \frac{x}{hi} - \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(lo \cdot \left(\frac{1}{hi} - \frac{x}{hi \cdot hi}\right)\right)\right)} \]

   2. log1p-udef 18.8%

      \[\leadsto \frac{x}{hi} - \color{blue}{\log \left(1 + \mathsf{expm1}\left(lo \cdot \left(\frac{1}{hi} - \frac{x}{hi \cdot hi}\right)\right)\right)} \]

   3. associate-/r* 18.8%

      \[\leadsto \frac{x}{hi} - \log \left(1 + \mathsf{expm1}\left(lo \cdot \left(\frac{1}{hi} - \color{blue}{\frac{\frac{x}{hi}}{hi}}\right)\right)\right) \]

   4. sub-div 18.8%

      \[\leadsto \frac{x}{hi} - \log \left(1 + \mathsf{expm1}\left(lo \cdot \color{blue}{\frac{1 - \frac{x}{hi}}{hi}}\right)\right) \]

6. Applied egg-rr 18.8%

   \[\leadsto \frac{x}{hi} - \color{blue}{\log \left(1 + \mathsf{expm1}\left(lo \cdot \frac{1 - \frac{x}{hi}}{hi}\right)\right)} \]

7. Taylor expanded in hi around inf 0.0%

   \[\leadsto \frac{x}{hi} - \log \color{blue}{\left(1 + \left(0.5 \cdot \frac{{lo}^{2}}{{hi}^{2}} + \left(-1 \cdot \frac{lo \cdot x}{{hi}^{2}} + \frac{lo}{hi}\right)\right)\right)} \]
8. Step-by-step derivation

   1. *-commutative 0.0%

      \[\leadsto \frac{x}{hi} - \log \left(1 + \left(\color{blue}{\frac{{lo}^{2}}{{hi}^{2}} \cdot 0.5} + \left(-1 \cdot \frac{lo \cdot x}{{hi}^{2}} + \frac{lo}{hi}\right)\right)\right) \]

   2. fma-def 0.0%

      \[\leadsto \frac{x}{hi} - \log \left(1 + \color{blue}{\mathsf{fma}\left(\frac{{lo}^{2}}{{hi}^{2}}, 0.5, -1 \cdot \frac{lo \cdot x}{{hi}^{2}} + \frac{lo}{hi}\right)}\right) \]

   3. unpow2 0.0%

      \[\leadsto \frac{x}{hi} - \log \left(1 + \mathsf{fma}\left(\frac{\color{blue}{lo \cdot lo}}{{hi}^{2}}, 0.5, -1 \cdot \frac{lo \cdot x}{{hi}^{2}} + \frac{lo}{hi}\right)\right) \]

   4. unpow2 0.0%

      \[\leadsto \frac{x}{hi} - \log \left(1 + \mathsf{fma}\left(\frac{lo \cdot lo}{\color{blue}{hi \cdot hi}}, 0.5, -1 \cdot \frac{lo \cdot x}{{hi}^{2}} + \frac{lo}{hi}\right)\right) \]

   5. times-frac 10.1%

      \[\leadsto \frac{x}{hi} - \log \left(1 + \mathsf{fma}\left(\color{blue}{\frac{lo}{hi} \cdot \frac{lo}{hi}}, 0.5, -1 \cdot \frac{lo \cdot x}{{hi}^{2}} + \frac{lo}{hi}\right)\right) \]

   6. +-commutative 10.1%

      \[\leadsto \frac{x}{hi} - \log \left(1 + \mathsf{fma}\left(\frac{lo}{hi} \cdot \frac{lo}{hi}, 0.5, \color{blue}{\frac{lo}{hi} + -1 \cdot \frac{lo \cdot x}{{hi}^{2}}}\right)\right) \]

   7. mul-1-neg 10.1%

      \[\leadsto \frac{x}{hi} - \log \left(1 + \mathsf{fma}\left(\frac{lo}{hi} \cdot \frac{lo}{hi}, 0.5, \frac{lo}{hi} + \color{blue}{\left(-\frac{lo \cdot x}{{hi}^{2}}\right)}\right)\right) \]

   8. unsub-neg 10.1%

      \[\leadsto \frac{x}{hi} - \log \left(1 + \mathsf{fma}\left(\frac{lo}{hi} \cdot \frac{lo}{hi}, 0.5, \color{blue}{\frac{lo}{hi} - \frac{lo \cdot x}{{hi}^{2}}}\right)\right) \]

   9. *-commutative 10.1%

      \[\leadsto \frac{x}{hi} - \log \left(1 + \mathsf{fma}\left(\frac{lo}{hi} \cdot \frac{lo}{hi}, 0.5, \frac{lo}{hi} - \frac{\color{blue}{x \cdot lo}}{{hi}^{2}}\right)\right) \]

   10. unpow2 10.1%

       \[\leadsto \frac{x}{hi} - \log \left(1 + \mathsf{fma}\left(\frac{lo}{hi} \cdot \frac{lo}{hi}, 0.5, \frac{lo}{hi} - \frac{x \cdot lo}{\color{blue}{hi \cdot hi}}\right)\right) \]

   11. times-frac 21.4%

       \[\leadsto \frac{x}{hi} - \log \left(1 + \mathsf{fma}\left(\frac{lo}{hi} \cdot \frac{lo}{hi}, 0.5, \frac{lo}{hi} - \color{blue}{\frac{x}{hi} \cdot \frac{lo}{hi}}\right)\right) \]

9. Simplified 21.4%

   \[\leadsto \frac{x}{hi} - \log \color{blue}{\left(1 + \mathsf{fma}\left(\frac{lo}{hi} \cdot \frac{lo}{hi}, 0.5, \frac{lo}{hi} - \frac{x}{hi} \cdot \frac{lo}{hi}\right)\right)} \]
10. Step-by-step derivation

    1. fma-udef 21.4%

       \[\leadsto \frac{x}{hi} - \log \left(1 + \color{blue}{\left(\left(\frac{lo}{hi} \cdot \frac{lo}{hi}\right) \cdot 0.5 + \left(\frac{lo}{hi} - \frac{x}{hi} \cdot \frac{lo}{hi}\right)\right)}\right) \]

    2. pow2 21.4%

       \[\leadsto \frac{x}{hi} - \log \left(1 + \left(\color{blue}{{\left(\frac{lo}{hi}\right)}^{2}} \cdot 0.5 + \left(\frac{lo}{hi} - \frac{x}{hi} \cdot \frac{lo}{hi}\right)\right)\right) \]

    3. associate-*r/ 21.4%

       \[\leadsto \frac{x}{hi} - \log \left(1 + \left({\left(\frac{lo}{hi}\right)}^{2} \cdot 0.5 + \left(\frac{lo}{hi} - \color{blue}{\frac{\frac{x}{hi} \cdot lo}{hi}}\right)\right)\right) \]

    4. sub-div 21.4%

       \[\leadsto \frac{x}{hi} - \log \left(1 + \left({\left(\frac{lo}{hi}\right)}^{2} \cdot 0.5 + \color{blue}{\frac{lo - \frac{x}{hi} \cdot lo}{hi}}\right)\right) \]

11. Applied egg-rr 21.4%

    \[\leadsto \frac{x}{hi} - \log \left(1 + \color{blue}{\left({\left(\frac{lo}{hi}\right)}^{2} \cdot 0.5 + \frac{lo - \frac{x}{hi} \cdot lo}{hi}\right)}\right) \]

12. Final simplification 21.4%

    \[\leadsto \frac{x}{hi} - \log \left(1 + \left(\frac{lo - \frac{x}{hi} \cdot lo}{hi} + {\left(\frac{lo}{hi}\right)}^{2} \cdot 0.5\right)\right) \]

Alternative 3: 19.3% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \left|\frac{hi}{lo}\right| \end{array} \]

FPCore

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

C

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

Java

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

Python

def code(lo, hi, x):
	return math.fabs((hi / lo))

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 3.1%

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

2. Taylor expanded in lo around inf 10.1%

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

   1. +-commutative 10.1%

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

   2. associate--l+ 10.1%

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

   3. associate-*r/ 10.1%

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

   4. associate-*r/ 10.1%

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

   5. div-sub 10.1%

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

   6. distribute-lft-out-- 10.1%

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

   7. associate-*r/ 10.1%

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

   8. mul-1-neg 10.1%

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

   9. unsub-neg 10.1%

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

4. Simplified 10.1%

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

5. Taylor expanded in x around 0 10.1%

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

   1. add-sqr-sqrt 9.4%

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

   2. sqrt-unprod 18.1%

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

   3. pow2 18.1%

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

   4. +-commutative 18.1%

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

7. Applied egg-rr 18.1%

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

   1. unpow2 18.1%

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

   2. rem-sqrt-square 18.1%

      \[\leadsto \color{blue}{\left|1 + \frac{hi}{lo}\right|} \]

9. Simplified 18.1%

   \[\leadsto \color{blue}{\left|1 + \frac{hi}{lo}\right|} \]

10. Taylor expanded in hi around inf 19.5%

    \[\leadsto \left|\color{blue}{\frac{hi}{lo}}\right| \]

11. Final simplification 19.5%

    \[\leadsto \left|\frac{hi}{lo}\right| \]

Alternative 4: 18.8% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[lo_, hi_, x_] := N[(N[(x / hi), $MachinePrecision] + N[(lo * N[(N[(N[(x / hi), $MachinePrecision] + -1.0), $MachinePrecision] / hi), $MachinePrecision]), $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 lo around 0 18.8%

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

   1. unpow2 18.8%

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

   2. associate-/r* 18.8%

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

   3. div-sub 18.8%

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

   4. *-commutative 18.8%

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

7. Simplified 18.8%

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

8. Final simplification 18.8%

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

Alternative 5: 18.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \frac{x + lo}{lo} \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 3.1%

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

2. Taylor expanded in hi around 0 18.7%

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

   1. associate-*r/ 18.7%

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

   2. neg-mul-1 18.7%

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

   3. sub-neg 18.7%

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

   4. distribute-neg-in 18.7%

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

   5. remove-double-neg 18.7%

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

4. Simplified 18.7%

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

   1. expm1-log1p-u 18.7%

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

   2. expm1-udef 18.7%

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

   3. +-commutative 18.7%

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

   4. add-sqr-sqrt 9.1%

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

   5. sqrt-unprod 14.1%

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

   6. sqr-neg 14.1%

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

   7. sqrt-unprod 9.6%

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

   8. add-sqr-sqrt 18.7%

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

6. Applied egg-rr 18.7%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{lo + x}{lo}\right)} - 1} \]
7. Step-by-step derivation

   1. expm1-def 18.7%

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

   2. expm1-log1p 18.7%

      \[\leadsto \color{blue}{\frac{lo + x}{lo}} \]

   3. +-commutative 18.7%

      \[\leadsto \frac{\color{blue}{x + lo}}{lo} \]

8. Simplified 18.7%

   \[\leadsto \color{blue}{\frac{x + lo}{lo}} \]

9. Final simplification 18.7%

   \[\leadsto \frac{x + lo}{lo} \]

Alternative 6: 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 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 2023187 
(FPCore (lo hi x)
  :name "xlohi (overflows)"
  :precision binary64
  :pre (and (< lo -1e+308) (> hi 1e+308))
  (/ (- x lo) (- hi lo)))