xlohi (overflows)

Percentage Accurate: 3.1% → 98.6%

Time: 8.3s

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: 98.6% accurate, 0.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[lo_, hi_, x_] := N[Sqrt[N[(1.0 / N[(N[(1.0 - N[(hi / lo), $MachinePrecision]), $MachinePrecision] + N[(N[Power[N[(hi / lo), $MachinePrecision], 2.0], $MachinePrecision] - 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 x around 0 3.1%

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

   1. associate-*r/ 3.1%

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

   2. associate-/l* 3.1%

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

   3. div-sub 98.4%

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

   4. *-inverses 98.4%

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

5. Simplified 98.4%

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

   1. add-sqr-sqrt 98.1%

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

   2. sqrt-unprod 98.4%

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

   3. frac-times 98.4%

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

   4. metadata-eval 98.4%

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

   5. pow2 98.4%

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

   6. sub-neg 98.4%

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

   7. metadata-eval 98.4%

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

7. Applied egg-rr 98.4%

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

   1. unpow2 98.4%

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

   2. +-commutative 98.4%

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

   3. distribute-rgt-in 98.5%

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

   4. neg-mul-1 98.5%

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

   5. +-commutative 98.5%

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

   6. distribute-neg-in 98.5%

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

   7. metadata-eval 98.5%

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

   8. *-rgt-identity 98.5%

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

   9. sub-neg 98.5%

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

   10. *-rgt-identity 98.5%

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

9. Applied egg-rr 98.5%

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

    1. distribute-rgt-in 98.6%

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

    2. unpow2 98.6%

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

    3. neg-mul-1 98.6%

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

    4. distribute-neg-frac 98.6%

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

11. Applied egg-rr 98.6%

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

12. Final simplification 98.6%

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

Alternative 2: 98.6% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 3.1%

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

3. Taylor expanded in x around 0 3.1%

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

   1. associate-*r/ 3.1%

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

   2. associate-/l* 3.1%

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

   3. div-sub 98.4%

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

   4. *-inverses 98.4%

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

5. Simplified 98.4%

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

   1. add-sqr-sqrt 98.1%

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

   2. sqrt-unprod 98.4%

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

   3. frac-times 98.4%

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

   4. metadata-eval 98.4%

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

   5. pow2 98.4%

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

   6. sub-neg 98.4%

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

   7. metadata-eval 98.4%

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

7. Applied egg-rr 98.4%

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

   1. unpow2 98.4%

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

   2. +-commutative 98.4%

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

   3. distribute-rgt-in 98.5%

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

   4. neg-mul-1 98.5%

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

   5. +-commutative 98.5%

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

   6. distribute-neg-in 98.5%

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

   7. metadata-eval 98.5%

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

   8. *-rgt-identity 98.5%

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

   9. sub-neg 98.5%

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

   10. *-rgt-identity 98.5%

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

9. Applied egg-rr 98.5%

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

    1. expm1-log1p-u 98.5%

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

    2. expm1-udef 98.0%

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

    3. associate-+l- 98.0%

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

    4. *-un-lft-identity 98.0%

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

    5. *-commutative 98.0%

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

    6. distribute-rgt-out-- 98.0%

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

11. Applied egg-rr 98.0%

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

    1. expm1-def 98.5%

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

    2. expm1-log1p 98.6%

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

    3. +-commutative 98.6%

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

    4. associate--r+ 98.6%

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

    5. metadata-eval 98.6%

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

13. Simplified 98.6%

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

14. Final simplification 98.6%

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

Alternative 3: 98.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 3.1%

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

3. Taylor expanded in x around 0 3.1%

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

   1. associate-*r/ 3.1%

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

   2. associate-/l* 3.1%

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

   3. div-sub 98.4%

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

   4. *-inverses 98.4%

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

5. Simplified 98.4%

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

6. Final simplification 98.4%

   \[\leadsto \frac{-1}{\frac{hi}{lo} + -1} \]
7. Add Preprocessing

Alternative 4: 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. Final simplification 18.8%

   \[\leadsto \frac{x - lo}{hi} \]
5. 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(Float64(-lo) / 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 x around 0 3.1%

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

   1. associate-*r/ 3.1%

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

   2. associate-/l* 3.1%

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

   3. div-sub 98.4%

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

   4. *-inverses 98.4%

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

5. Simplified 98.4%

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

6. Taylor expanded in hi around inf 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. Final simplification 18.7%

   \[\leadsto 1 \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024010 
(FPCore (lo hi x)
  :name "xlohi (overflows)"
  :precision binary64
  :pre (and (< lo -1e+308) (> hi 1e+308))
  (/ (- x lo) (- hi lo)))