xlohi (overflows)

Percentage Accurate: 3.1% → 99.2%

Time: 12.7s

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: 99.2% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x - lo}{hi}\\ \frac{{\left(\frac{lo}{\frac{hi \cdot hi}{x - lo}}\right)}^{2} - {t_0}^{2}}{t_0 \cdot \left(\frac{lo}{hi} + -1\right)} \end{array} \end{array} \]

FPCore

(FPCore (lo hi x)
 :precision binary64
 (let* ((t_0 (/ (- x lo) hi)))
   (/
    (- (pow (/ lo (/ (* hi hi) (- x lo))) 2.0) (pow t_0 2.0))
    (* t_0 (+ (/ lo hi) -1.0)))))

C

double code(double lo, double hi, double x) {
	double t_0 = (x - lo) / hi;
	return (pow((lo / ((hi * hi) / (x - lo))), 2.0) - pow(t_0, 2.0)) / (t_0 * ((lo / hi) + -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
    real(8) :: t_0
    t_0 = (x - lo) / hi
    code = (((lo / ((hi * hi) / (x - lo))) ** 2.0d0) - (t_0 ** 2.0d0)) / (t_0 * ((lo / hi) + (-1.0d0)))
end function

Java

public static double code(double lo, double hi, double x) {
	double t_0 = (x - lo) / hi;
	return (Math.pow((lo / ((hi * hi) / (x - lo))), 2.0) - Math.pow(t_0, 2.0)) / (t_0 * ((lo / hi) + -1.0));
}

Python

def code(lo, hi, x):
	t_0 = (x - lo) / hi
	return (math.pow((lo / ((hi * hi) / (x - lo))), 2.0) - math.pow(t_0, 2.0)) / (t_0 * ((lo / hi) + -1.0))

Julia

function code(lo, hi, x)
	t_0 = Float64(Float64(x - lo) / hi)
	return Float64(Float64((Float64(lo / Float64(Float64(hi * hi) / Float64(x - lo))) ^ 2.0) - (t_0 ^ 2.0)) / Float64(t_0 * Float64(Float64(lo / hi) + -1.0)))
end

MATLAB

function tmp = code(lo, hi, x)
	t_0 = (x - lo) / hi;
	tmp = (((lo / ((hi * hi) / (x - lo))) ^ 2.0) - (t_0 ^ 2.0)) / (t_0 * ((lo / hi) + -1.0));
end

Wolfram

code[lo_, hi_, x_] := Block[{t$95$0 = N[(N[(x - lo), $MachinePrecision] / hi), $MachinePrecision]}, N[(N[(N[Power[N[(lo / N[(N[(hi * hi), $MachinePrecision] / N[(x - lo), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] - N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision] / N[(t$95$0 * N[(N[(lo / hi), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 3.1%

   \[\frac{x - lo}{hi - lo} \]

2. Taylor expanded in hi around inf 0.0%

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

   1. +-commutative 0.0%

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

   2. associate--l+ 0.0%

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

   3. *-commutative 0.0%

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

   4. unpow2 0.0%

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

   5. times-frac 9.4%

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

   6. div-sub 9.4%

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

4. Simplified 9.4%

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

   1. clear-num 9.4%

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

   2. inv-pow 9.4%

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

6. Applied egg-rr 9.4%

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

   1. unpow-1 9.4%

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

8. Simplified 9.4%

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

   1. inv-pow 9.4%

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

   2. div-inv 9.4%

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

   3. unpow-prod-down 9.4%

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

   4. inv-pow 9.4%

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

10. Applied egg-rr 9.4%

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

    1. flip-+ 9.4%

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

12. Applied egg-rr 0.0%

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

    1. unpow2 0.0%

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

    2. *-commutative 0.0%

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

    3. associate-/l* 0.0%

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

    4. unpow2 0.0%

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

    5. times-frac 99.5%

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

    6. *-rgt-identity 99.5%

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

    7. distribute-lft-out-- 99.3%

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

14. Simplified 99.3%

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

15. Final simplification 99.3%

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

Alternative 2: 21.8% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ e^{\frac{x - lo}{hi} + -0.5 \cdot \left(\frac{lo}{hi} \cdot \frac{lo}{hi}\right)} + -1 \end{array} \]

FPCore

(FPCore (lo hi x)
 :precision binary64
 (+ (exp (+ (/ (- x lo) hi) (* -0.5 (* (/ lo hi) (/ lo hi))))) -1.0))

C

double code(double lo, double hi, double x) {
	return exp((((x - lo) / hi) + (-0.5 * ((lo / hi) * (lo / hi))))) + -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 = exp((((x - lo) / hi) + ((-0.5d0) * ((lo / hi) * (lo / hi))))) + (-1.0d0)
end function

Java

public static double code(double lo, double hi, double x) {
	return Math.exp((((x - lo) / hi) + (-0.5 * ((lo / hi) * (lo / hi))))) + -1.0;
}

Python

def code(lo, hi, x):
	return math.exp((((x - lo) / hi) + (-0.5 * ((lo / hi) * (lo / hi))))) + -1.0

Julia

function code(lo, hi, x)
	return Float64(exp(Float64(Float64(Float64(x - lo) / hi) + Float64(-0.5 * Float64(Float64(lo / hi) * Float64(lo / hi))))) + -1.0)
end

MATLAB

function tmp = code(lo, hi, x)
	tmp = exp((((x - lo) / hi) + (-0.5 * ((lo / hi) * (lo / hi))))) + -1.0;
end

Wolfram

code[lo_, hi_, x_] := N[(N[Exp[N[(N[(N[(x - lo), $MachinePrecision] / hi), $MachinePrecision] + N[(-0.5 * N[(N[(lo / hi), $MachinePrecision] * N[(lo / hi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + -1.0), $MachinePrecision]

Derivation

1. Initial program 3.1%

   \[\frac{x - lo}{hi - lo} \]

2. Taylor expanded in hi around inf 18.8%

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

   1. expm1-log1p-u 18.8%

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

   2. expm1-udef 18.8%

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

4. Applied egg-rr 18.8%

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

5. Taylor expanded in hi around inf 0.0%

   \[\leadsto e^{\color{blue}{\left(-0.5 \cdot \frac{{\left(x - lo\right)}^{2}}{{hi}^{2}} + \frac{x}{hi}\right) - \frac{lo}{hi}}} - 1 \]
6. Step-by-step derivation

   1. associate--l+ 0.0%

      \[\leadsto e^{\color{blue}{-0.5 \cdot \frac{{\left(x - lo\right)}^{2}}{{hi}^{2}} + \left(\frac{x}{hi} - \frac{lo}{hi}\right)}} - 1 \]

   2. unpow2 0.0%

      \[\leadsto e^{-0.5 \cdot \frac{\color{blue}{\left(x - lo\right) \cdot \left(x - lo\right)}}{{hi}^{2}} + \left(\frac{x}{hi} - \frac{lo}{hi}\right)} - 1 \]

   3. unpow2 0.0%

      \[\leadsto e^{-0.5 \cdot \frac{\left(x - lo\right) \cdot \left(x - lo\right)}{\color{blue}{hi \cdot hi}} + \left(\frac{x}{hi} - \frac{lo}{hi}\right)} - 1 \]

   4. times-frac 21.6%

      \[\leadsto e^{-0.5 \cdot \color{blue}{\left(\frac{x - lo}{hi} \cdot \frac{x - lo}{hi}\right)} + \left(\frac{x}{hi} - \frac{lo}{hi}\right)} - 1 \]

   5. unpow2 21.6%

      \[\leadsto e^{-0.5 \cdot \color{blue}{{\left(\frac{x - lo}{hi}\right)}^{2}} + \left(\frac{x}{hi} - \frac{lo}{hi}\right)} - 1 \]

   6. div-sub 21.6%

      \[\leadsto e^{-0.5 \cdot {\left(\frac{x - lo}{hi}\right)}^{2} + \color{blue}{\frac{x - lo}{hi}}} - 1 \]

7. Simplified 21.6%

   \[\leadsto e^{\color{blue}{-0.5 \cdot {\left(\frac{x - lo}{hi}\right)}^{2} + \frac{x - lo}{hi}}} - 1 \]

8. Taylor expanded in x around 0 0.0%

   \[\leadsto e^{-0.5 \cdot \color{blue}{\frac{{lo}^{2}}{{hi}^{2}}} + \frac{x - lo}{hi}} - 1 \]
9. Step-by-step derivation

   1. unpow2 0.0%

      \[\leadsto e^{-0.5 \cdot \frac{\color{blue}{lo \cdot lo}}{{hi}^{2}} + \frac{x - lo}{hi}} - 1 \]

   2. unpow2 0.0%

      \[\leadsto e^{-0.5 \cdot \frac{lo \cdot lo}{\color{blue}{hi \cdot hi}} + \frac{x - lo}{hi}} - 1 \]

   3. times-frac 21.6%

      \[\leadsto e^{-0.5 \cdot \color{blue}{\left(\frac{lo}{hi} \cdot \frac{lo}{hi}\right)} + \frac{x - lo}{hi}} - 1 \]

10. Simplified 21.6%

    \[\leadsto e^{-0.5 \cdot \color{blue}{\left(\frac{lo}{hi} \cdot \frac{lo}{hi}\right)} + \frac{x - lo}{hi}} - 1 \]

11. Final simplification 21.6%

    \[\leadsto e^{\frac{x - lo}{hi} + -0.5 \cdot \left(\frac{lo}{hi} \cdot \frac{lo}{hi}\right)} + -1 \]

Alternative 3: 18.9% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 3.1%

   \[\frac{x - lo}{hi - lo} \]

2. Taylor expanded in lo around inf 0.0%

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

   1. associate--l+ 0.0%

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

   2. +-commutative 0.0%

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

   3. associate--l+ 0.0%

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

   4. distribute-lft-out-- 0.0%

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

   5. div-sub 0.0%

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

   6. mul-1-neg 0.0%

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

   7. sub-neg 0.0%

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

   8. unpow2 0.0%

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

   9. times-frac 18.9%

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

   10. distribute-lft-out-- 18.9%

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

   11. associate-*r/ 18.9%

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

   12. fma-neg 18.9%

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

4. Simplified 18.9%

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

5. Taylor expanded in hi around -inf 0.0%

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

   1. +-commutative 0.0%

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

   2. mul-1-neg 0.0%

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

   3. unsub-neg 0.0%

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

   4. unpow2 0.0%

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

   5. unpow2 0.0%

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

   6. times-frac 18.9%

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

   7. sub-neg 18.9%

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

   8. unpow2 18.9%

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

   9. distribute-neg-frac 18.9%

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

   10. metadata-eval 18.9%

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

7. Simplified 18.9%

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

8. Taylor expanded in x around 0 0.0%

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

   1. unpow2 0.0%

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

   2. unpow2 0.0%

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

   3. times-frac 18.9%

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

   4. *-commutative 18.9%

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

   5. distribute-lft-out-- 18.9%

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

10. Simplified 18.9%

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

11. Final simplification 18.9%

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

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. Taylor expanded in hi around inf 18.8%

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

3. Final simplification 18.8%

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

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. Taylor expanded in hi around inf 18.8%

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

3. Taylor expanded in x around 0 18.8%

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

   1. neg-mul-1 18.8%

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

   2. distribute-neg-frac 18.8%

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

5. Simplified 18.8%

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

6. Final simplification 18.8%

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

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. Taylor expanded in lo around inf 18.7%

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

3. Final simplification 18.7%

   \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023285 
(FPCore (lo hi x)
  :name "xlohi (overflows)"
  :precision binary64
  :pre (and (< lo -1e+308) (> hi 1e+308))
  (/ (- x lo) (- hi lo)))