xlohi (overflows)

Percentage Accurate: 3.1% → 99.2%

Time: 9.4s

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: 99.2% accurate, 0.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[lo_, hi_, x_] := N[(N[(N[Power[N[(N[(lo / N[(hi * hi), $MachinePrecision]), $MachinePrecision] * N[(x - lo), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] - N[Power[N[(N[(x / hi), $MachinePrecision] - N[(lo / hi), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(N[(N[(x - lo), $MachinePrecision] / hi), $MachinePrecision] * 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 9.2%

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

   6. div-sub 9.2%

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

4. Simplified 9.2%

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

   1. flip-+ 9.2%

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

   2. div-sub 9.2%

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

   3. pow2 9.2%

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

   4. *-commutative 9.2%

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

   5. clear-num 9.2%

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

   6. frac-times 30.1%

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

   7. *-un-lft-identity 30.1%

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

   8. associate-*r/ 57.7%

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

   9. sub-div 99.5%

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

6. Applied egg-rr 6.5%

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

   1. div-sub 6.5%

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

   2. associate-*l/ 6.5%

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

   3. associate-/l* 0.0%

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

   4. *-commutative 0.0%

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

   5. associate-/l* 6.5%

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

   6. associate-/r/ 6.5%

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

   7. div-sub 72.2%

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

   8. associate-*r/ 99.5%

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

   9. *-rgt-identity 99.5%

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

   10. distribute-lft-out-- 99.2%

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

8. Simplified 99.2%

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

   1. div-sub 99.2%

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

10. Applied egg-rr 99.2%

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

11. Final simplification 99.2%

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

Alternative 2: 99.2% accurate, 0.1× speedup

Math

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

FPCore

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

Python

def code(lo, hi, x):
	t_0 = (x - lo) / hi
	return -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(-(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 = -(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[Power[t$95$0, 2.0], $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 9.2%

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

   6. div-sub 9.2%

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

4. Simplified 9.2%

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

   1. flip-+ 9.2%

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

   2. div-sub 9.2%

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

   3. pow2 9.2%

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

   4. *-commutative 9.2%

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

   5. clear-num 9.2%

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

   6. frac-times 30.1%

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

   7. *-un-lft-identity 30.1%

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

   8. associate-*r/ 57.7%

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

   9. sub-div 99.5%

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

6. Applied egg-rr 6.5%

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

   1. div-sub 6.5%

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

   2. associate-*l/ 6.5%

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

   3. associate-/l* 0.0%

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

   4. *-commutative 0.0%

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

   5. associate-/l* 6.5%

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

   6. associate-/r/ 6.5%

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

   7. div-sub 72.2%

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

   8. associate-*r/ 99.5%

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

   9. *-rgt-identity 99.5%

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

   10. distribute-lft-out-- 99.2%

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

8. Simplified 99.2%

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

   1. div-sub 99.2%

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

10. Applied egg-rr 99.2%

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

11. Taylor expanded in hi around inf 0.0%

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

    1. mul-1-neg 0.0%

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

    2. unpow2 0.0%

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

    3. unpow2 0.0%

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

    4. times-frac 99.2%

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

    5. unpow2 99.2%

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

13. Simplified 99.2%

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

14. Final simplification 99.2%

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

Alternative 3: 21.1% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

4. Applied egg-rr 18.8%

   \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x - lo}{hi}\right)\right)} \]
5. Step-by-step derivation

   1. expm1-udef 18.8%

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

   2. flip-- 18.8%

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

   3. log1p-udef 18.8%

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

   4. add-exp-log 18.8%

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

   5. log1p-udef 18.8%

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

   6. add-exp-log 18.8%

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

   7. metadata-eval 18.8%

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

   8. log1p-udef 18.8%

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

   9. add-exp-log 18.8%

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

6. Applied egg-rr 18.8%

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

7. Taylor expanded in hi around 0 0.0%

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

   1. associate--l+ 0.0%

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

   2. associate-*r/ 0.0%

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

   3. associate-*r/ 0.0%

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

   4. div-sub 0.0%

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

   5. +-commutative 0.0%

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

   6. distribute-lft-out-- 0.0%

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

   7. associate-*r/ 0.0%

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

   8. unpow2 0.0%

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

   9. unpow2 0.0%

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

   10. times-frac 21.1%

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

   11. distribute-rgt-in 21.1%

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

9. Simplified 21.1%

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

10. Final simplification 21.1%

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

Alternative 4: 20.9% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

4. Applied egg-rr 18.8%

   \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x - lo}{hi}\right)\right)} \]
5. Step-by-step derivation

   1. expm1-udef 18.8%

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

   2. flip-- 18.8%

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

   3. log1p-udef 18.8%

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

   4. add-exp-log 18.8%

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

   5. log1p-udef 18.8%

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

   6. add-exp-log 18.8%

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

   7. metadata-eval 18.8%

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

   8. log1p-udef 18.8%

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

   9. add-exp-log 18.8%

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

6. Applied egg-rr 18.8%

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

7. Taylor expanded in hi around inf 3.1%

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

   1. distribute-lft-out-- 3.1%

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

   2. associate-*r/ 20.9%

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

9. Simplified 20.9%

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

10. Final simplification 20.9%

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

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. 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 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. Taylor expanded in lo around 0 18.8%

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

   1. mul-1-neg 18.8%

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

   2. unsub-neg 18.8%

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

   3. mul-1-neg 18.8%

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

   4. unsub-neg 18.8%

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

   5. unpow2 18.8%

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

4. Simplified 18.8%

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

5. Taylor expanded in x around 0 18.8%

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

   1. neg-mul-1 18.8%

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

   2. distribute-neg-frac 18.8%

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

7. Simplified 18.8%

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

8. Final simplification 18.8%

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

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. Taylor expanded in lo around inf 18.7%

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

3. Final simplification 18.7%

   \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023182 
(FPCore (lo hi x)
  :name "xlohi (overflows)"
  :precision binary64
  :pre (and (< lo -1e+308) (> hi 1e+308))
  (/ (- x lo) (- hi lo)))