xlohi (overflows)

Percentage Accurate: 3.1% → 18.8%

Time: 11.1s

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: 18.8% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{log1p}\left(\left(e - x \cdot \frac{e}{lo}\right) + -1\right) - hi \cdot \left(\frac{x}{{lo}^{2}} - \left(\frac{1}{lo} + \frac{\frac{hi - hi \cdot \frac{x}{lo}}{lo}}{lo}\right)\right) \end{array} \]

FPCore

(FPCore (lo hi x)
 :precision binary64
 (-
  (log1p (+ (- E (* x (/ E lo))) -1.0))
  (*
   hi
   (-
    (/ x (pow lo 2.0))
    (+ (/ 1.0 lo) (/ (/ (- hi (* hi (/ x lo))) lo) lo))))))

C

double code(double lo, double hi, double x) {
	return log1p(((((double) M_E) - (x * (((double) M_E) / lo))) + -1.0)) - (hi * ((x / pow(lo, 2.0)) - ((1.0 / lo) + (((hi - (hi * (x / lo))) / lo) / lo))));
}

Java

public static double code(double lo, double hi, double x) {
	return Math.log1p(((Math.E - (x * (Math.E / lo))) + -1.0)) - (hi * ((x / Math.pow(lo, 2.0)) - ((1.0 / lo) + (((hi - (hi * (x / lo))) / lo) / lo))));
}

Python

def code(lo, hi, x):
	return math.log1p(((math.e - (x * (math.e / lo))) + -1.0)) - (hi * ((x / math.pow(lo, 2.0)) - ((1.0 / lo) + (((hi - (hi * (x / lo))) / lo) / lo))))

Julia

function code(lo, hi, x)
	return Float64(log1p(Float64(Float64(exp(1) - Float64(x * Float64(exp(1) / lo))) + -1.0)) - Float64(hi * Float64(Float64(x / (lo ^ 2.0)) - Float64(Float64(1.0 / lo) + Float64(Float64(Float64(hi - Float64(hi * Float64(x / lo))) / lo) / lo)))))
end

Wolfram

code[lo_, hi_, x_] := N[(N[Log[1 + N[(N[(E - N[(x * N[(E / lo), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]], $MachinePrecision] - N[(hi * N[(N[(x / N[Power[lo, 2.0], $MachinePrecision]), $MachinePrecision] - N[(N[(1.0 / lo), $MachinePrecision] + N[(N[(N[(hi - N[(hi * N[(x / lo), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / lo), $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 10.7%

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

   1. mul-1-neg 10.7%

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

   2. unsub-neg 10.7%

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

   3. associate-/l* 18.9%

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

6. Simplified 18.9%

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

   1. log1p-expm1-u 18.9%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(-1 \cdot \frac{x - lo}{lo}\right)\right)} + hi \cdot \left(\left(\frac{1}{lo} + \frac{\frac{hi - hi \cdot \frac{x}{lo}}{lo}}{lo}\right) - \frac{x}{{lo}^{2}}\right) \]

   2. mul-1-neg 18.9%

      \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\color{blue}{-\frac{x - lo}{lo}}\right)\right) + hi \cdot \left(\left(\frac{1}{lo} + \frac{\frac{hi - hi \cdot \frac{x}{lo}}{lo}}{lo}\right) - \frac{x}{{lo}^{2}}\right) \]

   3. div-sub 18.9%

      \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(-\color{blue}{\left(\frac{x}{lo} - \frac{lo}{lo}\right)}\right)\right) + hi \cdot \left(\left(\frac{1}{lo} + \frac{\frac{hi - hi \cdot \frac{x}{lo}}{lo}}{lo}\right) - \frac{x}{{lo}^{2}}\right) \]

   4. pow1 18.9%

      \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(-\left(\frac{x}{lo} - \frac{\color{blue}{{lo}^{1}}}{lo}\right)\right)\right) + hi \cdot \left(\left(\frac{1}{lo} + \frac{\frac{hi - hi \cdot \frac{x}{lo}}{lo}}{lo}\right) - \frac{x}{{lo}^{2}}\right) \]

   5. pow1 18.9%

      \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(-\left(\frac{x}{lo} - \frac{{lo}^{1}}{\color{blue}{{lo}^{1}}}\right)\right)\right) + hi \cdot \left(\left(\frac{1}{lo} + \frac{\frac{hi - hi \cdot \frac{x}{lo}}{lo}}{lo}\right) - \frac{x}{{lo}^{2}}\right) \]

   6. pow-div 18.9%

      \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(-\left(\frac{x}{lo} - \color{blue}{{lo}^{\left(1 - 1\right)}}\right)\right)\right) + hi \cdot \left(\left(\frac{1}{lo} + \frac{\frac{hi - hi \cdot \frac{x}{lo}}{lo}}{lo}\right) - \frac{x}{{lo}^{2}}\right) \]

   7. metadata-eval 18.9%

      \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(-\left(\frac{x}{lo} - {lo}^{\color{blue}{0}}\right)\right)\right) + hi \cdot \left(\left(\frac{1}{lo} + \frac{\frac{hi - hi \cdot \frac{x}{lo}}{lo}}{lo}\right) - \frac{x}{{lo}^{2}}\right) \]

   8. metadata-eval 18.9%

      \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(-\left(\frac{x}{lo} - \color{blue}{1}\right)\right)\right) + hi \cdot \left(\left(\frac{1}{lo} + \frac{\frac{hi - hi \cdot \frac{x}{lo}}{lo}}{lo}\right) - \frac{x}{{lo}^{2}}\right) \]

8. Applied egg-rr 18.9%

   \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(-\left(\frac{x}{lo} - 1\right)\right)\right)} + hi \cdot \left(\left(\frac{1}{lo} + \frac{\frac{hi - hi \cdot \frac{x}{lo}}{lo}}{lo}\right) - \frac{x}{{lo}^{2}}\right) \]

9. Taylor expanded in x around 0 18.9%

   \[\leadsto \mathsf{log1p}\left(\color{blue}{\left(e^{1} + -1 \cdot \frac{x \cdot e^{1}}{lo}\right) - 1}\right) + hi \cdot \left(\left(\frac{1}{lo} + \frac{\frac{hi - hi \cdot \frac{x}{lo}}{lo}}{lo}\right) - \frac{x}{{lo}^{2}}\right) \]
10. Step-by-step derivation

    1. sub-neg 18.9%

       \[\leadsto \mathsf{log1p}\left(\color{blue}{\left(e^{1} + -1 \cdot \frac{x \cdot e^{1}}{lo}\right) + \left(-1\right)}\right) + hi \cdot \left(\left(\frac{1}{lo} + \frac{\frac{hi - hi \cdot \frac{x}{lo}}{lo}}{lo}\right) - \frac{x}{{lo}^{2}}\right) \]

    2. mul-1-neg 18.9%

       \[\leadsto \mathsf{log1p}\left(\left(e^{1} + \color{blue}{\left(-\frac{x \cdot e^{1}}{lo}\right)}\right) + \left(-1\right)\right) + hi \cdot \left(\left(\frac{1}{lo} + \frac{\frac{hi - hi \cdot \frac{x}{lo}}{lo}}{lo}\right) - \frac{x}{{lo}^{2}}\right) \]

    3. unsub-neg 18.9%

       \[\leadsto \mathsf{log1p}\left(\color{blue}{\left(e^{1} - \frac{x \cdot e^{1}}{lo}\right)} + \left(-1\right)\right) + hi \cdot \left(\left(\frac{1}{lo} + \frac{\frac{hi - hi \cdot \frac{x}{lo}}{lo}}{lo}\right) - \frac{x}{{lo}^{2}}\right) \]

    4. exp-1-e 18.9%

       \[\leadsto \mathsf{log1p}\left(\left(\color{blue}{e} - \frac{x \cdot e^{1}}{lo}\right) + \left(-1\right)\right) + hi \cdot \left(\left(\frac{1}{lo} + \frac{\frac{hi - hi \cdot \frac{x}{lo}}{lo}}{lo}\right) - \frac{x}{{lo}^{2}}\right) \]

    5. associate-/l* 18.9%

       \[\leadsto \mathsf{log1p}\left(\left(e - \color{blue}{x \cdot \frac{e^{1}}{lo}}\right) + \left(-1\right)\right) + hi \cdot \left(\left(\frac{1}{lo} + \frac{\frac{hi - hi \cdot \frac{x}{lo}}{lo}}{lo}\right) - \frac{x}{{lo}^{2}}\right) \]

    6. exp-1-e 18.9%

       \[\leadsto \mathsf{log1p}\left(\left(e - x \cdot \frac{\color{blue}{e}}{lo}\right) + \left(-1\right)\right) + hi \cdot \left(\left(\frac{1}{lo} + \frac{\frac{hi - hi \cdot \frac{x}{lo}}{lo}}{lo}\right) - \frac{x}{{lo}^{2}}\right) \]

    7. metadata-eval 18.9%

       \[\leadsto \mathsf{log1p}\left(\left(e - x \cdot \frac{e}{lo}\right) + \color{blue}{-1}\right) + hi \cdot \left(\left(\frac{1}{lo} + \frac{\frac{hi - hi \cdot \frac{x}{lo}}{lo}}{lo}\right) - \frac{x}{{lo}^{2}}\right) \]

11. Simplified 18.9%

    \[\leadsto \mathsf{log1p}\left(\color{blue}{\left(e - x \cdot \frac{e}{lo}\right) + -1}\right) + hi \cdot \left(\left(\frac{1}{lo} + \frac{\frac{hi - hi \cdot \frac{x}{lo}}{lo}}{lo}\right) - \frac{x}{{lo}^{2}}\right) \]

12. Final simplification 18.9%

    \[\leadsto \mathsf{log1p}\left(\left(e - x \cdot \frac{e}{lo}\right) + -1\right) - hi \cdot \left(\frac{x}{{lo}^{2}} - \left(\frac{1}{lo} + \frac{\frac{hi - hi \cdot \frac{x}{lo}}{lo}}{lo}\right)\right) \]
13. Add Preprocessing

Alternative 2: 18.8% accurate, 0.1× speedup

Math

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

FPCore

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

C

double code(double lo, double hi, double x) {
	return (hi * (((1.0 / lo) + (((hi - (hi * (x / lo))) / lo) / lo)) - (x / pow(lo, 2.0)))) + ((lo - 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 * (((1.0d0 / lo) + (((hi - (hi * (x / lo))) / lo) / lo)) - (x / (lo ** 2.0d0)))) + ((lo - x) / lo)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[lo_, hi_, x_] := N[(N[(hi * N[(N[(N[(1.0 / lo), $MachinePrecision] + N[(N[(N[(hi - N[(hi * N[(x / lo), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / lo), $MachinePrecision] / lo), $MachinePrecision]), $MachinePrecision] - N[(x / N[Power[lo, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(lo - x), $MachinePrecision] / lo), $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 10.7%

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

   1. mul-1-neg 10.7%

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

   2. unsub-neg 10.7%

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

   3. associate-/l* 18.9%

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

6. Simplified 18.9%

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

7. Taylor expanded in lo around 0 18.9%

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

   1. mul-1-neg 18.9%

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

   2. sub-neg 18.9%

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

9. Simplified 18.9%

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

10. Final simplification 18.9%

    \[\leadsto hi \cdot \left(\left(\frac{1}{lo} + \frac{\frac{hi - hi \cdot \frac{x}{lo}}{lo}}{lo}\right) - \frac{x}{{lo}^{2}}\right) + \frac{lo - x}{lo} \]
11. Add Preprocessing

Alternative 3: 18.9% accurate, 0.1× speedup

Math

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

FPCore

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

C

double code(double lo, double hi, double x) {
	return 1.0 - (hi * ((x / pow(lo, 2.0)) - ((1.0 / lo) + (((hi - (hi * (x / lo))) / 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 * ((x / (lo ** 2.0d0)) - ((1.0d0 / lo) + (((hi - (hi * (x / lo))) / lo) / lo))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[lo_, hi_, x_] := N[(1.0 - N[(hi * N[(N[(x / N[Power[lo, 2.0], $MachinePrecision]), $MachinePrecision] - N[(N[(1.0 / lo), $MachinePrecision] + N[(N[(N[(hi - N[(hi * N[(x / lo), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / lo), $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 10.7%

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

   1. mul-1-neg 10.7%

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

   2. unsub-neg 10.7%

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

   3. associate-/l* 18.9%

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

6. Simplified 18.9%

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

7. Taylor expanded in x around 0 18.9%

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

8. Final simplification 18.9%

   \[\leadsto 1 - hi \cdot \left(\frac{x}{{lo}^{2}} - \left(\frac{1}{lo} + \frac{\frac{hi - hi \cdot \frac{x}{lo}}{lo}}{lo}\right)\right) \]
9. Add Preprocessing

Alternative 4: 18.9% accurate, 0.1× speedup

Math

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

FPCore

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

C

double code(double lo, double hi, double x) {
	return ((lo - x) / lo) - (hi * ((x / pow(lo, 2.0)) + ((-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 = ((lo - x) / lo) - (hi * ((x / (lo ** 2.0d0)) + (((-1.0d0) - (hi / lo)) / lo)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[lo_, hi_, x_] := N[(N[(N[(lo - x), $MachinePrecision] / lo), $MachinePrecision] - N[(hi * N[(N[(x / N[Power[lo, 2.0], $MachinePrecision]), $MachinePrecision] + N[(N[(-1.0 - N[(hi / lo), $MachinePrecision]), $MachinePrecision] / lo), $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%

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

5. Final simplification 18.9%

   \[\leadsto \frac{lo - x}{lo} - hi \cdot \left(\frac{x}{{lo}^{2}} + \frac{-1 - \frac{hi}{lo}}{lo}\right) \]
6. Add Preprocessing

Alternative 5: 18.8% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[lo_, hi_, x_] := N[(N[(N[(lo - x), $MachinePrecision] / lo), $MachinePrecision] + N[(hi * N[(N[(1.0 + N[(N[(hi - x), $MachinePrecision] / 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. Final simplification 18.9%

   \[\leadsto \frac{lo - x}{lo} + hi \cdot \frac{1 + \frac{hi - x}{lo}}{lo} \]
8. Add Preprocessing

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. Add Preprocessing

3. Taylor expanded in hi around inf 18.8%

   \[\leadsto \color{blue}{\frac{x - lo}{hi}} \]
4. 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 2024131 
(FPCore (lo hi x)
  :name "xlohi (overflows)"
  :precision binary64
  :pre (and (< lo -1e+308) (> hi 1e+308))
  (/ (- x lo) (- hi lo)))