xlohi (overflows)

Percentage Accurate: 3.1% → 26.7%

Time: 38.2s

Alternatives: 8

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: 26.7% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{log1p}\left(\left|\mathsf{expm1}\left(\frac{lo}{hi}\right)\right|\right) \end{array} \]

FPCore

(FPCore (lo hi x) :precision binary64 (log1p (fabs (expm1 (/ lo hi)))))

C

double code(double lo, double hi, double x) {
	return log1p(fabs(expm1((lo / hi))));
}

Java

public static double code(double lo, double hi, double x) {
	return Math.log1p(Math.abs(Math.expm1((lo / hi))));
}

Python

def code(lo, hi, x):
	return math.log1p(math.fabs(math.expm1((lo / hi))))

Julia

function code(lo, hi, x)
	return log1p(abs(expm1(Float64(lo / hi))))
end

Wolfram

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

Derivation

1. Initial program 3.1%

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

3. Taylor expanded in hi around inf 18.8%

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

   1. log1p-expm1-u 18.8%

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

5. Applied egg-rr 18.8%

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

   1. add-sqr-sqrt 18.8%

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

   2. sqrt-unprod 18.8%

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

   3. pow2 18.8%

      \[\leadsto \mathsf{log1p}\left(\sqrt{\color{blue}{{\left(\mathsf{expm1}\left(\frac{x - lo}{hi}\right)\right)}^{2}}}\right) \]

   4. expm1-log1p-u 18.8%

      \[\leadsto \mathsf{log1p}\left(\sqrt{{\color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{x - lo}{hi}\right)\right)\right)\right)}}^{2}}\right) \]

   5. log1p-expm1-u 18.8%

      \[\leadsto \mathsf{log1p}\left(\sqrt{{\left(\mathsf{expm1}\left(\color{blue}{\frac{x - lo}{hi}}\right)\right)}^{2}}\right) \]

   6. sub-neg 18.8%

      \[\leadsto \mathsf{log1p}\left(\sqrt{{\left(\mathsf{expm1}\left(\frac{\color{blue}{x + \left(-lo\right)}}{hi}\right)\right)}^{2}}\right) \]

   7. add-sqr-sqrt 18.8%

      \[\leadsto \mathsf{log1p}\left(\sqrt{{\left(\mathsf{expm1}\left(\frac{x + \color{blue}{\sqrt{-lo} \cdot \sqrt{-lo}}}{hi}\right)\right)}^{2}}\right) \]

   8. sqrt-unprod 3.1%

      \[\leadsto \mathsf{log1p}\left(\sqrt{{\left(\mathsf{expm1}\left(\frac{x + \color{blue}{\sqrt{\left(-lo\right) \cdot \left(-lo\right)}}}{hi}\right)\right)}^{2}}\right) \]

   9. sqr-neg 3.1%

      \[\leadsto \mathsf{log1p}\left(\sqrt{{\left(\mathsf{expm1}\left(\frac{x + \sqrt{\color{blue}{lo \cdot lo}}}{hi}\right)\right)}^{2}}\right) \]

   10. sqrt-unprod 0.0%

       \[\leadsto \mathsf{log1p}\left(\sqrt{{\left(\mathsf{expm1}\left(\frac{x + \color{blue}{\sqrt{lo} \cdot \sqrt{lo}}}{hi}\right)\right)}^{2}}\right) \]

   11. add-sqr-sqrt 26.7%

       \[\leadsto \mathsf{log1p}\left(\sqrt{{\left(\mathsf{expm1}\left(\frac{x + \color{blue}{lo}}{hi}\right)\right)}^{2}}\right) \]

7. Applied egg-rr 26.7%

   \[\leadsto \mathsf{log1p}\left(\color{blue}{\sqrt{{\left(\mathsf{expm1}\left(\frac{x + lo}{hi}\right)\right)}^{2}}}\right) \]
8. Step-by-step derivation

   1. unpow2 26.7%

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

   2. rem-sqrt-square 26.7%

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

9. Simplified 26.7%

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

10. Taylor expanded in x around 0 26.7%

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

    1. expm1-define 26.7%

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

12. Simplified 26.7%

    \[\leadsto \mathsf{log1p}\left(\left|\color{blue}{\mathsf{expm1}\left(\frac{lo}{hi}\right)}\right|\right) \]
13. Add Preprocessing

Alternative 2: 19.5% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 3.1%

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

3. Taylor expanded in hi around 0 18.9%

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

4. Taylor expanded in lo around inf 18.9%

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

   1. associate--l+ 18.9%

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

   2. div-sub 18.9%

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

6. Simplified 18.9%

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

   1. associate-*r/ 18.9%

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

   2. clear-num 18.9%

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

8. Applied egg-rr 18.9%

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

   1. associate-/r/ 18.9%

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

10. Simplified 18.9%

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

    1. add-sqr-sqrt 9.8%

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

    2. sqrt-unprod 19.7%

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

    3. pow2 19.7%

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

12. Applied egg-rr 19.7%

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

    1. unpow2 19.7%

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

    2. rem-sqrt-square 19.7%

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

14. Simplified 19.7%

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

15. Final simplification 19.7%

    \[\leadsto \frac{lo - x}{lo} + \frac{1}{lo} \cdot \left(hi \cdot \left|1 + \frac{hi - x}{lo}\right|\right) \]
16. Add Preprocessing

Alternative 3: 19.4% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(lo, hi, x)
    real(8), intent (in) :: lo
    real(8), intent (in) :: hi
    real(8), intent (in) :: x
    code = (hi / lo) ** 2.0d0
end function

Java

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

Python

def code(lo, hi, x):
	return math.pow((hi / lo), 2.0)

Julia

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

MATLAB

function tmp = code(lo, hi, x)
	tmp = (hi / lo) ^ 2.0;
end

Wolfram

code[lo_, hi_, x_] := N[Power[N[(hi / lo), $MachinePrecision], 2.0], $MachinePrecision]

Derivation

1. Initial program 3.1%

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

3. Taylor expanded in hi around 0 18.9%

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

4. Taylor expanded in lo around inf 18.9%

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

   1. associate--l+ 18.9%

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

   2. div-sub 18.9%

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

6. Simplified 18.9%

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

   1. associate-*r/ 18.9%

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

   2. clear-num 18.9%

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

8. Applied egg-rr 18.9%

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

   1. associate-/r/ 18.9%

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

10. Simplified 18.9%

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

11. Taylor expanded in hi around inf 0.0%

    \[\leadsto \color{blue}{\frac{{hi}^{2}}{{lo}^{2}}} \]
12. Step-by-step derivation

    1. unpow2 0.0%

       \[\leadsto \frac{\color{blue}{hi \cdot hi}}{{lo}^{2}} \]

    2. unpow2 0.0%

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

    3. times-frac 19.3%

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

    4. unpow2 19.3%

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

13. Simplified 19.3%

    \[\leadsto \color{blue}{{\left(\frac{hi}{lo}\right)}^{2}} \]
14. Add Preprocessing

Alternative 4: 18.9% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 3.1%

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

3. Taylor expanded in hi around 0 18.9%

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

4. Taylor expanded in lo around inf 18.9%

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

   1. associate--l+ 18.9%

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

   2. div-sub 18.9%

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

6. Simplified 18.9%

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

   1. *-commutative 18.9%

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

   2. frac-2neg 18.9%

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

   3. distribute-frac-neg 18.9%

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

   4. add-sqr-sqrt 18.9%

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

   5. sqrt-unprod 18.7%

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

   6. sqr-neg 18.7%

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

   7. sqrt-unprod 0.0%

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

   8. add-sqr-sqrt 19.1%

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

   9. cancel-sign-sub-inv 19.1%

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

   10. *-commutative 19.1%

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

   11. fmm-def 19.1%

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

   12. div-sub 19.1%

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

   13. *-inverses 19.1%

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

   14. sub-neg 19.1%

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

   15. metadata-eval 19.1%

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

8. Applied egg-rr 18.5%

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

   1. fmm-undef 18.5%

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

   2. neg-mul-1 18.5%

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

   3. +-commutative 18.5%

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

   4. distribute-neg-in 18.5%

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

   5. metadata-eval 18.5%

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

   6. sub-neg 18.5%

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

   7. associate-/r/ 19.1%

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

10. Simplified 19.1%

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

11. Final simplification 19.1%

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

Alternative 5: 18.9% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 3.1%

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

3. Taylor expanded in hi around 0 18.9%

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

4. Taylor expanded in lo around inf 18.9%

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

   1. associate--l+ 18.9%

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

   2. div-sub 18.9%

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

6. Simplified 18.9%

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

   1. associate-*r/ 18.9%

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

   2. clear-num 18.9%

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

8. Applied egg-rr 18.9%

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

   1. associate-/r/ 18.9%

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

10. Simplified 18.9%

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

11. Taylor expanded in x around 0 18.9%

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

Alternative 6: 18.9% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 3.1%

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

3. Taylor expanded in hi around 0 18.9%

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

4. Taylor expanded in lo around inf 18.9%

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

   1. associate--l+ 18.9%

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

   2. div-sub 18.9%

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

6. Simplified 18.9%

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

7. Taylor expanded in x around 0 18.9%

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

   1. associate-/l* 18.9%

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

9. Simplified 18.9%

   \[\leadsto \color{blue}{1 + hi \cdot \frac{1 + \frac{hi}{lo}}{lo}} \]
10. Add Preprocessing

Alternative 7: 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 hi around inf 18.8%

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

4. Taylor expanded in x around 0 18.8%

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

   1. associate-*r/ 18.8%

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

   2. neg-mul-1 18.8%

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

6. Simplified 18.8%

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

7. Final simplification 18.8%

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

Alternative 8: 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 2024180 
(FPCore (lo hi x)
  :name "xlohi (overflows)"
  :precision binary64
  :pre (and (< lo -1e+308) (> hi 1e+308))
  (/ (- x lo) (- hi lo)))