xlohi (overflows)

Percentage Accurate: 3.1% → 98.7%

Time: 8.8s

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: 98.7% accurate, 0.0× speedup

Math

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

FPCore

(FPCore (lo hi x) :precision binary64 (exp (- (log1p (/ (- hi) lo)))))

C

double code(double lo, double hi, double x) {
	return exp(-log1p((-hi / lo)));
}

Java

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

Python

def code(lo, hi, x):
	return math.exp(-math.log1p((-hi / lo)))

Julia

function code(lo, hi, x)
	return exp(Float64(-log1p(Float64(Float64(-hi) / lo))))
end

Wolfram

code[lo_, hi_, x_] := N[Exp[(-N[Log[1 + N[((-hi) / lo), $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. flip-- 97.2%

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

   2. associate-/r/ 97.3%

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

   3. metadata-eval 97.3%

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

   4. sub-neg 97.3%

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

   5. pow2 97.3%

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

   6. metadata-eval 97.3%

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

7. Applied egg-rr 97.3%

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

   1. associate-*l/ 97.3%

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

   2. neg-mul-1 97.3%

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

   3. neg-sub0 97.3%

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

   4. +-commutative 97.3%

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

   5. associate--r+ 97.3%

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

   6. metadata-eval 97.3%

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

   7. +-commutative 97.3%

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

9. Simplified 97.3%

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

    1. +-commutative 97.3%

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

    2. unpow2 97.3%

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

    3. fma-def 98.5%

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

11. Applied egg-rr 98.5%

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

    1. div-inv 98.4%

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

    2. *-commutative 98.4%

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

    3. frac-2neg 98.4%

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

    4. metadata-eval 98.4%

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

    5. fma-udef 97.3%

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

    6. unpow2 97.3%

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

    7. +-commutative 97.3%

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

    8. distribute-neg-in 97.3%

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

    9. metadata-eval 97.3%

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

    10. sub-neg 97.3%

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

    11. associate-/r/ 97.2%

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

    12. metadata-eval 97.2%

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

    13. unpow2 97.2%

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

    14. flip-+ 98.4%

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

    15. frac-2neg 98.4%

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

    16. metadata-eval 98.4%

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

    17. add-exp-log 98.4%

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

    18. log-rec 98.4%

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

13. Applied egg-rr 98.5%

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

14. Final simplification 98.5%

    \[\leadsto e^{-\mathsf{log1p}\left(\frac{-hi}{lo}\right)} \]
15. Add Preprocessing

Alternative 2: 98.6% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[lo_, hi_, x_] := N[(N[(-1.0 - N[(hi / lo), $MachinePrecision]), $MachinePrecision] / N[(N[(hi / lo), $MachinePrecision] * 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. Step-by-step derivation

   1. flip-- 97.2%

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

   2. associate-/r/ 97.3%

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

   3. metadata-eval 97.3%

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

   4. sub-neg 97.3%

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

   5. pow2 97.3%

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

   6. metadata-eval 97.3%

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

7. Applied egg-rr 97.3%

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

   1. associate-*l/ 97.3%

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

   2. neg-mul-1 97.3%

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

   3. neg-sub0 97.3%

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

   4. +-commutative 97.3%

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

   5. associate--r+ 97.3%

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

   6. metadata-eval 97.3%

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

   7. +-commutative 97.3%

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

9. Simplified 97.3%

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

    1. +-commutative 97.3%

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

    2. unpow2 97.3%

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

    3. fma-def 98.5%

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

11. Applied egg-rr 98.5%

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

12. Final simplification 98.5%

    \[\leadsto \frac{-1 - \frac{hi}{lo}}{\mathsf{fma}\left(\frac{hi}{lo}, \frac{hi}{lo}, -1\right)} \]
13. Add Preprocessing

Alternative 3: 98.5% accurate, 0.1× speedup

Math

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

FPCore

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

C

double code(double lo, double hi, double x) {
	return pow((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 Math.pow((1.0 - (hi / lo)), -1.0);
}

Python

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

Julia

function code(lo, hi, x)
	return Float64(1.0 - 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[Power[N[(1.0 - N[(hi / lo), $MachinePrecision]), $MachinePrecision], -1.0], $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. frac-2neg 98.4%

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

   2. metadata-eval 98.4%

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

   3. inv-pow 98.4%

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

   4. sub-neg 98.4%

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

   5. metadata-eval 98.4%

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

   6. distribute-neg-in 98.4%

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

   7. metadata-eval 98.4%

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

7. Applied egg-rr 98.4%

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

8. Taylor expanded in hi around 0 98.4%

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

   1. mul-1-neg 98.4%

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

   2. sub-neg 98.4%

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

10. Simplified 98.4%

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

11. Final simplification 98.4%

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

Alternative 4: 98.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[lo_, hi_, x_] := N[(-1.0 / N[(-1.0 + N[(hi / lo), $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. Final simplification 98.4%

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

Alternative 5: 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 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(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 18.7%

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

   1. mul-1-neg 18.7%

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

   2. distribute-frac-neg 18.7%

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

8. Simplified 18.7%

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

9. Final simplification 18.7%

   \[\leadsto \frac{-lo}{hi} \]
10. 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 18.6%

   \[\leadsto \color{blue}{1} \]

4. Final simplification 18.6%

   \[\leadsto 1 \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024012 
(FPCore (lo hi x)
  :name "xlohi (overflows)"
  :precision binary64
  :pre (and (< lo -1e+308) (> hi 1e+308))
  (/ (- x lo) (- hi lo)))