xlohi (overflows)

Percentage Accurate: 3.1% → 19.5%

Time: 9.2s

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: 19.5% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 3.1%

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

3. Taylor expanded in lo around -inf 3.1%

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

   1. mul-1-neg 3.1%

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

   2. unsub-neg 3.1%

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

   3. associate--l+ 3.1%

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

   4. sub-neg 3.1%

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

   5. associate-/l* 15.2%

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

   6. mul-1-neg 15.2%

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

   7. *-commutative 15.2%

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

   8. distribute-lft-out 18.9%

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

5. Simplified 18.9%

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

   1. add-sqr-sqrt 10.5%

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

   2. sqrt-unprod 19.7%

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

   3. pow2 19.7%

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

   4. +-commutative 19.7%

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

   5. fma-define 19.7%

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

7. Applied egg-rr 19.7%

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

   1. unpow2 19.7%

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

   2. rem-sqrt-square 19.7%

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

   3. +-commutative 19.7%

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

9. Simplified 19.7%

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

    1. +-commutative 19.7%

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

    2. add-sqr-sqrt 19.7%

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

    3. fma-define 19.7%

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

11. Applied egg-rr 19.7%

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

12. Taylor expanded in lo around inf 0.7%

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

    1. neg-mul-1 0.7%

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

    2. associate-+r+ 0.7%

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

    3. sub-neg 0.7%

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

    4. associate-*l/ 14.7%

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

    5. distribute-rgt1-in 19.7%

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

    6. +-commutative 19.7%

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

    7. associate-*r/ 19.7%

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

14. Simplified 19.7%

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

15. Final simplification 19.7%

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

Alternative 2: 19.5% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 3.1%

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

3. Taylor expanded in lo around -inf 3.1%

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

   1. mul-1-neg 3.1%

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

   2. unsub-neg 3.1%

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

   3. associate--l+ 3.1%

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

   4. sub-neg 3.1%

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

   5. associate-/l* 15.2%

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

   6. mul-1-neg 15.2%

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

   7. *-commutative 15.2%

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

   8. distribute-lft-out 18.9%

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

5. Simplified 18.9%

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

   1. add-sqr-sqrt 10.5%

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

   2. sqrt-unprod 19.7%

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

   3. pow2 19.7%

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

   4. +-commutative 19.7%

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

   5. fma-define 19.7%

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

7. Applied egg-rr 19.7%

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

   1. unpow2 19.7%

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

   2. rem-sqrt-square 19.7%

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

   3. +-commutative 19.7%

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

9. Simplified 19.7%

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

10. Taylor expanded in x around 0 19.7%

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

    1. associate-*r/ 19.7%

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

    2. neg-mul-1 19.7%

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

    3. distribute-rgt-neg-in 19.7%

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

    4. +-commutative 19.7%

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

    5. distribute-neg-in 19.7%

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

    6. mul-1-neg 19.7%

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

    7. sub-neg 19.7%

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

    8. sub-neg 19.7%

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

    9. mul-1-neg 19.7%

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

    10. distribute-neg-in 19.7%

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

    11. +-commutative 19.7%

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

    12. distribute-neg-in 19.7%

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

    13. metadata-eval 19.7%

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

    14. unsub-neg 19.7%

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

12. Simplified 19.7%

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

13. Final simplification 19.7%

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

Alternative 3: 19.5% 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 lo around -inf 3.1%

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

   1. mul-1-neg 3.1%

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

   2. unsub-neg 3.1%

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

   3. associate--l+ 3.1%

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

   4. sub-neg 3.1%

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

   5. associate-/l* 15.2%

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

   6. mul-1-neg 15.2%

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

   7. *-commutative 15.2%

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

   8. distribute-lft-out 18.9%

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

5. Simplified 18.9%

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

6. Taylor expanded in x around inf 18.8%

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

7. Taylor expanded in hi around inf 0.0%

   \[\leadsto \color{blue}{\frac{{hi}^{2}}{{lo}^{2}}} \]
8. 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 19.5%

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

   4. unpow2 19.5%

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

9. Simplified 19.5%

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

Alternative 4: 18.9% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 3.1%

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

3. Taylor expanded in lo around -inf 3.1%

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

   1. mul-1-neg 3.1%

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

   2. unsub-neg 3.1%

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

   3. associate--l+ 3.1%

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

   4. sub-neg 3.1%

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

   5. associate-/l* 15.2%

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

   6. mul-1-neg 15.2%

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

   7. *-commutative 15.2%

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

   8. distribute-lft-out 18.9%

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

5. Simplified 18.9%

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

6. Taylor expanded in x around 0 18.9%

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

   1. distribute-lft-in 18.9%

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

   2. metadata-eval 18.9%

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

   3. mul-1-neg 18.9%

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

   4. unsub-neg 18.9%

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

8. Simplified 18.9%

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

9. Final simplification 18.9%

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

Alternative 5: 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 lo around -inf 3.1%

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

   1. mul-1-neg 3.1%

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

   2. unsub-neg 3.1%

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

   3. associate--l+ 3.1%

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

   4. sub-neg 3.1%

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

   5. associate-/l* 15.2%

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

   6. mul-1-neg 15.2%

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

   7. *-commutative 15.2%

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

   8. distribute-lft-out 18.9%

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

5. Simplified 18.9%

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

6. Taylor expanded in x around 0 18.9%

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

   1. associate-*r/ 18.9%

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

   2. mul-1-neg 18.9%

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

   3. distribute-rgt-neg-in 18.9%

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

   4. +-commutative 18.9%

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

   5. distribute-neg-in 18.9%

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

   6. mul-1-neg 18.9%

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

   7. sub-neg 18.9%

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

   8. associate-/l* 18.9%

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

   9. sub-neg 18.9%

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

   10. mul-1-neg 18.9%

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

   11. distribute-neg-in 18.9%

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

   12. +-commutative 18.9%

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

   13. distribute-neg-in 18.9%

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

   14. metadata-eval 18.9%

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

   15. unsub-neg 18.9%

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

8. Simplified 18.9%

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

9. Final simplification 18.9%

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

Alternative 6: 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. mul-1-neg 18.8%

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

   2. distribute-neg-frac2 18.8%

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

6. Simplified 18.8%

   \[\leadsto \color{blue}{\frac{lo}{-hi}} \]
7. Add Preprocessing

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. Add Preprocessing

3. Taylor expanded in lo around inf 18.7%

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

Reproduce

herbie shell --seed 2024095 
(FPCore (lo hi x)
  :name "xlohi (overflows)"
  :precision binary64
  :pre (and (< lo -1e+308) (> hi 1e+308))
  (/ (- x lo) (- hi lo)))