xlohi (overflows)

Percentage Accurate: 3.1% → 20.4%

Time: 34.0s

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: 20.4% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \frac{x + \mathsf{expm1}\left(\sqrt[3]{{\left(\mathsf{fma}\left(\frac{lo}{hi}, \frac{lo - x}{lo + -1}, \mathsf{log1p}\left(-lo\right)\right)\right)}^{3}}\right)}{hi} \end{array} \]

FPCore

(FPCore (lo hi x)
 :precision binary64
 (/
  (+
   x
   (expm1
    (cbrt (pow (fma (/ lo hi) (/ (- lo x) (+ lo -1.0)) (log1p (- lo))) 3.0))))
  hi))

C

double code(double lo, double hi, double x) {
	return (x + expm1(cbrt(pow(fma((lo / hi), ((lo - x) / (lo + -1.0)), log1p(-lo)), 3.0)))) / hi;
}

Julia

function code(lo, hi, x)
	return Float64(Float64(x + expm1(cbrt((fma(Float64(lo / hi), Float64(Float64(lo - x) / Float64(lo + -1.0)), log1p(Float64(-lo))) ^ 3.0)))) / hi)
end

Wolfram

code[lo_, hi_, x_] := N[(N[(x + N[(Exp[N[Power[N[Power[N[(N[(lo / hi), $MachinePrecision] * N[(N[(lo - x), $MachinePrecision] / N[(lo + -1.0), $MachinePrecision]), $MachinePrecision] + N[Log[1 + (-lo)], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision], 1/3], $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision] / hi), $MachinePrecision]

Derivation

1. Initial program 3.1%

   \[\frac{x - lo}{hi - lo} \]
2. Add Preprocessing

3. Taylor expanded in hi around inf 0.7%

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

   1. +-commutative 0.7%

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

   2. associate--l+ 0.7%

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

   3. +-commutative 0.7%

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

   4. +-commutative 0.7%

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

   5. associate--l+ 0.7%

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

   6. +-commutative 0.7%

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

   7. associate--l+ 0.7%

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

   8. associate-/l* 9.6%

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

5. Simplified 9.6%

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

   1. expm1-log1p-u 9.0%

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

7. Applied egg-rr 9.0%

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

8. Taylor expanded in hi around inf 0.0%

   \[\leadsto \frac{x + \mathsf{expm1}\left(\color{blue}{\log \left(1 - lo\right) + \frac{lo \cdot \left(x - lo\right)}{hi \cdot \left(1 - lo\right)}}\right)}{hi} \]
9. Step-by-step derivation

   1. +-commutative 0.0%

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

   2. times-frac 20.4%

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

   3. fma-define 20.4%

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

   4. sub-neg 20.4%

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

   5. log1p-undefine 20.4%

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

10. Simplified 20.4%

    \[\leadsto \frac{x + \mathsf{expm1}\left(\color{blue}{\mathsf{fma}\left(\frac{lo}{hi}, \frac{x - lo}{1 - lo}, \mathsf{log1p}\left(-lo\right)\right)}\right)}{hi} \]
11. Step-by-step derivation

    1. add-cbrt-cube 20.4%

       \[\leadsto \frac{x + \mathsf{expm1}\left(\color{blue}{\sqrt[3]{\left(\mathsf{fma}\left(\frac{lo}{hi}, \frac{x - lo}{1 - lo}, \mathsf{log1p}\left(-lo\right)\right) \cdot \mathsf{fma}\left(\frac{lo}{hi}, \frac{x - lo}{1 - lo}, \mathsf{log1p}\left(-lo\right)\right)\right) \cdot \mathsf{fma}\left(\frac{lo}{hi}, \frac{x - lo}{1 - lo}, \mathsf{log1p}\left(-lo\right)\right)}}\right)}{hi} \]

    2. pow3 20.4%

       \[\leadsto \frac{x + \mathsf{expm1}\left(\sqrt[3]{\color{blue}{{\left(\mathsf{fma}\left(\frac{lo}{hi}, \frac{x - lo}{1 - lo}, \mathsf{log1p}\left(-lo\right)\right)\right)}^{3}}}\right)}{hi} \]

12. Applied egg-rr 20.4%

    \[\leadsto \frac{x + \mathsf{expm1}\left(\color{blue}{\sqrt[3]{{\left(\mathsf{fma}\left(\frac{lo}{hi}, \frac{x - lo}{1 - lo}, \mathsf{log1p}\left(-lo\right)\right)\right)}^{3}}}\right)}{hi} \]

13. Final simplification 20.4%

    \[\leadsto \frac{x + \mathsf{expm1}\left(\sqrt[3]{{\left(\mathsf{fma}\left(\frac{lo}{hi}, \frac{lo - x}{lo + -1}, \mathsf{log1p}\left(-lo\right)\right)\right)}^{3}}\right)}{hi} \]
14. Add Preprocessing

Alternative 2: 20.3% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \frac{x + \mathsf{expm1}\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(\frac{lo}{hi}, \frac{lo - x}{lo + -1}, \mathsf{log1p}\left(-lo\right)\right)\right)\right)\right)}{hi} \end{array} \]

FPCore

(FPCore (lo hi x)
 :precision binary64
 (/
  (+
   x
   (expm1
    (expm1 (log1p (fma (/ lo hi) (/ (- lo x) (+ lo -1.0)) (log1p (- lo)))))))
  hi))

C

double code(double lo, double hi, double x) {
	return (x + expm1(expm1(log1p(fma((lo / hi), ((lo - x) / (lo + -1.0)), log1p(-lo)))))) / hi;
}

Julia

function code(lo, hi, x)
	return Float64(Float64(x + expm1(expm1(log1p(fma(Float64(lo / hi), Float64(Float64(lo - x) / Float64(lo + -1.0)), log1p(Float64(-lo))))))) / hi)
end

Wolfram

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

Derivation

1. Initial program 3.1%

   \[\frac{x - lo}{hi - lo} \]
2. Add Preprocessing

3. Taylor expanded in hi around inf 0.7%

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

   1. +-commutative 0.7%

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

   2. associate--l+ 0.7%

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

   3. +-commutative 0.7%

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

   4. +-commutative 0.7%

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

   5. associate--l+ 0.7%

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

   6. +-commutative 0.7%

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

   7. associate--l+ 0.7%

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

   8. associate-/l* 9.6%

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

5. Simplified 9.6%

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

   1. expm1-log1p-u 9.0%

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

7. Applied egg-rr 9.0%

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

8. Taylor expanded in hi around inf 0.0%

   \[\leadsto \frac{x + \mathsf{expm1}\left(\color{blue}{\log \left(1 - lo\right) + \frac{lo \cdot \left(x - lo\right)}{hi \cdot \left(1 - lo\right)}}\right)}{hi} \]
9. Step-by-step derivation

   1. +-commutative 0.0%

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

   2. times-frac 20.4%

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

   3. fma-define 20.4%

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

   4. sub-neg 20.4%

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

   5. log1p-undefine 20.4%

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

10. Simplified 20.4%

    \[\leadsto \frac{x + \mathsf{expm1}\left(\color{blue}{\mathsf{fma}\left(\frac{lo}{hi}, \frac{x - lo}{1 - lo}, \mathsf{log1p}\left(-lo\right)\right)}\right)}{hi} \]
11. Step-by-step derivation

    1. expm1-log1p-u 20.4%

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

    2. expm1-undefine 20.4%

       \[\leadsto \frac{x + \mathsf{expm1}\left(\color{blue}{e^{\mathsf{log1p}\left(\mathsf{fma}\left(\frac{lo}{hi}, \frac{x - lo}{1 - lo}, \mathsf{log1p}\left(-lo\right)\right)\right)} - 1}\right)}{hi} \]

12. Applied egg-rr 20.4%

    \[\leadsto \frac{x + \mathsf{expm1}\left(\color{blue}{e^{\mathsf{log1p}\left(\mathsf{fma}\left(\frac{lo}{hi}, \frac{x - lo}{1 - lo}, \mathsf{log1p}\left(-lo\right)\right)\right)} - 1}\right)}{hi} \]
13. Step-by-step derivation

    1. expm1-define 20.4%

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

14. Simplified 20.4%

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

15. Final simplification 20.4%

    \[\leadsto \frac{x + \mathsf{expm1}\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(\frac{lo}{hi}, \frac{lo - x}{lo + -1}, \mathsf{log1p}\left(-lo\right)\right)\right)\right)\right)}{hi} \]
16. Add Preprocessing

Alternative 3: 20.4% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \frac{x + \mathsf{expm1}\left(lo \cdot \left(\frac{1}{hi} + \frac{\left(\frac{1}{hi} - \frac{x}{hi}\right) - \log \left(\frac{-1}{lo}\right)}{lo}\right)\right)}{hi} \end{array} \]

FPCore

(FPCore (lo hi x)
 :precision binary64
 (/
  (+
   x
   (expm1
    (*
     lo
     (+ (/ 1.0 hi) (/ (- (- (/ 1.0 hi) (/ x hi)) (log (/ -1.0 lo))) lo)))))
  hi))

C

double code(double lo, double hi, double x) {
	return (x + expm1((lo * ((1.0 / hi) + ((((1.0 / hi) - (x / hi)) - log((-1.0 / lo))) / lo))))) / hi;
}

Java

public static double code(double lo, double hi, double x) {
	return (x + Math.expm1((lo * ((1.0 / hi) + ((((1.0 / hi) - (x / hi)) - Math.log((-1.0 / lo))) / lo))))) / hi;
}

Python

def code(lo, hi, x):
	return (x + math.expm1((lo * ((1.0 / hi) + ((((1.0 / hi) - (x / hi)) - math.log((-1.0 / lo))) / lo))))) / hi

Julia

function code(lo, hi, x)
	return Float64(Float64(x + expm1(Float64(lo * Float64(Float64(1.0 / hi) + Float64(Float64(Float64(Float64(1.0 / hi) - Float64(x / hi)) - log(Float64(-1.0 / lo))) / lo))))) / hi)
end

Wolfram

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

Derivation

1. Initial program 3.1%

   \[\frac{x - lo}{hi - lo} \]
2. Add Preprocessing

3. Taylor expanded in hi around inf 0.7%

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

   1. +-commutative 0.7%

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

   2. associate--l+ 0.7%

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

   3. +-commutative 0.7%

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

   4. +-commutative 0.7%

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

   5. associate--l+ 0.7%

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

   6. +-commutative 0.7%

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

   7. associate--l+ 0.7%

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

   8. associate-/l* 9.6%

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

5. Simplified 9.6%

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

   1. expm1-log1p-u 9.0%

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

7. Applied egg-rr 9.0%

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

8. Taylor expanded in hi around inf 0.0%

   \[\leadsto \frac{x + \mathsf{expm1}\left(\color{blue}{\log \left(1 - lo\right) + \frac{lo \cdot \left(x - lo\right)}{hi \cdot \left(1 - lo\right)}}\right)}{hi} \]
9. Step-by-step derivation

   1. +-commutative 0.0%

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

   2. times-frac 20.4%

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

   3. fma-define 20.4%

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

   4. sub-neg 20.4%

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

   5. log1p-undefine 20.4%

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

10. Simplified 20.4%

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

11. Taylor expanded in lo around -inf 20.4%

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

12. Final simplification 20.4%

    \[\leadsto \frac{x + \mathsf{expm1}\left(lo \cdot \left(\frac{1}{hi} + \frac{\left(\frac{1}{hi} - \frac{x}{hi}\right) - \log \left(\frac{-1}{lo}\right)}{lo}\right)\right)}{hi} \]
13. Add Preprocessing

Alternative 4: 20.4% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \frac{x + \mathsf{expm1}\left(\mathsf{log1p}\left(-lo\right) + \frac{lo}{hi} \cdot \frac{lo - x}{lo + -1}\right)}{hi} \end{array} \]

FPCore

(FPCore (lo hi x)
 :precision binary64
 (/
  (+ x (expm1 (+ (log1p (- lo)) (* (/ lo hi) (/ (- lo x) (+ lo -1.0))))))
  hi))

C

double code(double lo, double hi, double x) {
	return (x + expm1((log1p(-lo) + ((lo / hi) * ((lo - x) / (lo + -1.0)))))) / hi;
}

Java

public static double code(double lo, double hi, double x) {
	return (x + Math.expm1((Math.log1p(-lo) + ((lo / hi) * ((lo - x) / (lo + -1.0)))))) / hi;
}

Python

def code(lo, hi, x):
	return (x + math.expm1((math.log1p(-lo) + ((lo / hi) * ((lo - x) / (lo + -1.0)))))) / hi

Julia

function code(lo, hi, x)
	return Float64(Float64(x + expm1(Float64(log1p(Float64(-lo)) + Float64(Float64(lo / hi) * Float64(Float64(lo - x) / Float64(lo + -1.0)))))) / hi)
end

Wolfram

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

Derivation

1. Initial program 3.1%

   \[\frac{x - lo}{hi - lo} \]
2. Add Preprocessing

3. Taylor expanded in hi around inf 0.7%

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

   1. +-commutative 0.7%

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

   2. associate--l+ 0.7%

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

   3. +-commutative 0.7%

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

   4. +-commutative 0.7%

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

   5. associate--l+ 0.7%

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

   6. +-commutative 0.7%

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

   7. associate--l+ 0.7%

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

   8. associate-/l* 9.6%

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

5. Simplified 9.6%

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

   1. expm1-log1p-u 9.0%

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

7. Applied egg-rr 9.0%

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

8. Taylor expanded in hi around inf 0.0%

   \[\leadsto \frac{x + \mathsf{expm1}\left(\color{blue}{\log \left(1 - lo\right) + \frac{lo \cdot \left(x - lo\right)}{hi \cdot \left(1 - lo\right)}}\right)}{hi} \]
9. Step-by-step derivation

   1. +-commutative 0.0%

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

   2. times-frac 20.4%

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

   3. fma-define 20.4%

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

   4. sub-neg 20.4%

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

   5. log1p-undefine 20.4%

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

10. Simplified 20.4%

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

11. Taylor expanded in x around -inf 0.0%

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

    1. expm1-define 0.0%

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

    2. mul-1-neg 0.0%

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

    3. unsub-neg 0.0%

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

    4. sub-neg 0.0%

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

    5. log1p-undefine 0.0%

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

    6. times-frac 20.4%

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

    7. mul-1-neg 20.4%

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

    8. unsub-neg 20.4%

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

13. Simplified 20.4%

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

14. Final simplification 20.4%

    \[\leadsto \frac{x + \mathsf{expm1}\left(\mathsf{log1p}\left(-lo\right) + \frac{lo}{hi} \cdot \frac{lo - x}{lo + -1}\right)}{hi} \]
15. Add Preprocessing

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(lo / Float64(-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. Add Preprocessing

3. Taylor expanded in lo around 0 18.8%

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

   1. +-commutative 18.8%

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

   2. mul-1-neg 18.8%

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

   3. unsub-neg 18.8%

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

   4. +-commutative 18.8%

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

   5. mul-1-neg 18.8%

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

   6. unsub-neg 18.8%

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

5. Simplified 18.8%

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

6. Taylor expanded in x around 0 18.8%

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

   1. neg-mul-1 18.8%

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

   2. distribute-neg-frac 18.8%

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

8. Simplified 18.8%

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

9. Final simplification 18.8%

   \[\leadsto \frac{lo}{-hi} \]
10. Add Preprocessing

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. Add Preprocessing

3. Taylor expanded in lo around inf 18.7%

   \[\leadsto \color{blue}{1} \]
4. Add Preprocessing

Reproduce

herbie shell --seed 2024110 
(FPCore (lo hi x)
  :name "xlohi (overflows)"
  :precision binary64
  :pre (and (< lo -1e+308) (> hi 1e+308))
  (/ (- x lo) (- hi lo)))