xlohi (overflows)

Percentage Accurate: 3.1% → 99.2%

Time: 12.2s

Alternatives: 10

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

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x - lo}{hi}\\ t_1 := {t_0}^{2}\\ t_2 := \frac{lo}{\frac{hi \cdot hi}{x - lo}}\\ \frac{{t_0}^{3} + {t_2}^{3}}{{t_2}^{2} + \left(t_1 - t_1 \cdot \frac{lo}{hi}\right)} \end{array} \end{array} \]

FPCore

(FPCore (lo hi x)
 :precision binary64
 (let* ((t_0 (/ (- x lo) hi))
        (t_1 (pow t_0 2.0))
        (t_2 (/ lo (/ (* hi hi) (- x lo)))))
   (/
    (+ (pow t_0 3.0) (pow t_2 3.0))
    (+ (pow t_2 2.0) (- t_1 (* t_1 (/ lo hi)))))))

C

double code(double lo, double hi, double x) {
	double t_0 = (x - lo) / hi;
	double t_1 = pow(t_0, 2.0);
	double t_2 = lo / ((hi * hi) / (x - lo));
	return (pow(t_0, 3.0) + pow(t_2, 3.0)) / (pow(t_2, 2.0) + (t_1 - (t_1 * (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
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    t_0 = (x - lo) / hi
    t_1 = t_0 ** 2.0d0
    t_2 = lo / ((hi * hi) / (x - lo))
    code = ((t_0 ** 3.0d0) + (t_2 ** 3.0d0)) / ((t_2 ** 2.0d0) + (t_1 - (t_1 * (lo / hi))))
end function

Java

public static double code(double lo, double hi, double x) {
	double t_0 = (x - lo) / hi;
	double t_1 = Math.pow(t_0, 2.0);
	double t_2 = lo / ((hi * hi) / (x - lo));
	return (Math.pow(t_0, 3.0) + Math.pow(t_2, 3.0)) / (Math.pow(t_2, 2.0) + (t_1 - (t_1 * (lo / hi))));
}

Python

def code(lo, hi, x):
	t_0 = (x - lo) / hi
	t_1 = math.pow(t_0, 2.0)
	t_2 = lo / ((hi * hi) / (x - lo))
	return (math.pow(t_0, 3.0) + math.pow(t_2, 3.0)) / (math.pow(t_2, 2.0) + (t_1 - (t_1 * (lo / hi))))

Julia

function code(lo, hi, x)
	t_0 = Float64(Float64(x - lo) / hi)
	t_1 = t_0 ^ 2.0
	t_2 = Float64(lo / Float64(Float64(hi * hi) / Float64(x - lo)))
	return Float64(Float64((t_0 ^ 3.0) + (t_2 ^ 3.0)) / Float64((t_2 ^ 2.0) + Float64(t_1 - Float64(t_1 * Float64(lo / hi)))))
end

MATLAB

function tmp = code(lo, hi, x)
	t_0 = (x - lo) / hi;
	t_1 = t_0 ^ 2.0;
	t_2 = lo / ((hi * hi) / (x - lo));
	tmp = ((t_0 ^ 3.0) + (t_2 ^ 3.0)) / ((t_2 ^ 2.0) + (t_1 - (t_1 * (lo / hi))));
end

Wolfram

code[lo_, hi_, x_] := Block[{t$95$0 = N[(N[(x - lo), $MachinePrecision] / hi), $MachinePrecision]}, Block[{t$95$1 = N[Power[t$95$0, 2.0], $MachinePrecision]}, Block[{t$95$2 = N[(lo / N[(N[(hi * hi), $MachinePrecision] / N[(x - lo), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[Power[t$95$0, 3.0], $MachinePrecision] + N[Power[t$95$2, 3.0], $MachinePrecision]), $MachinePrecision] / N[(N[Power[t$95$2, 2.0], $MachinePrecision] + N[(t$95$1 - N[(t$95$1 * N[(lo / hi), $MachinePrecision]), $MachinePrecision]), $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. div-inv 9.4%

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

   2. inv-pow 9.4%

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

6. Applied egg-rr 9.4%

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

   1. unpow-1 9.4%

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

8. Simplified 9.4%

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

   1. clear-num 9.4%

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

   2. inv-pow 9.4%

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

10. Applied egg-rr 9.4%

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

    1. unpow-1 9.4%

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

12. Simplified 9.4%

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

    1. flip3-+ 9.4%

       \[\leadsto \color{blue}{\frac{{\left(\left(\left(x - lo\right) \cdot \frac{1}{hi}\right) \cdot \frac{lo}{hi}\right)}^{3} + {\left(\frac{1}{\frac{hi}{x - lo}}\right)}^{3}}{\left(\left(\left(x - lo\right) \cdot \frac{1}{hi}\right) \cdot \frac{lo}{hi}\right) \cdot \left(\left(\left(x - lo\right) \cdot \frac{1}{hi}\right) \cdot \frac{lo}{hi}\right) + \left(\frac{1}{\frac{hi}{x - lo}} \cdot \frac{1}{\frac{hi}{x - lo}} - \left(\left(\left(x - lo\right) \cdot \frac{1}{hi}\right) \cdot \frac{lo}{hi}\right) \cdot \frac{1}{\frac{hi}{x - lo}}\right)}} \]

    2. div-inv 9.4%

       \[\leadsto \frac{{\left(\color{blue}{\frac{x - lo}{hi}} \cdot \frac{lo}{hi}\right)}^{3} + {\left(\frac{1}{\frac{hi}{x - lo}}\right)}^{3}}{\left(\left(\left(x - lo\right) \cdot \frac{1}{hi}\right) \cdot \frac{lo}{hi}\right) \cdot \left(\left(\left(x - lo\right) \cdot \frac{1}{hi}\right) \cdot \frac{lo}{hi}\right) + \left(\frac{1}{\frac{hi}{x - lo}} \cdot \frac{1}{\frac{hi}{x - lo}} - \left(\left(\left(x - lo\right) \cdot \frac{1}{hi}\right) \cdot \frac{lo}{hi}\right) \cdot \frac{1}{\frac{hi}{x - lo}}\right)} \]

    3. *-commutative 9.4%

       \[\leadsto \frac{{\color{blue}{\left(\frac{lo}{hi} \cdot \frac{x - lo}{hi}\right)}}^{3} + {\left(\frac{1}{\frac{hi}{x - lo}}\right)}^{3}}{\left(\left(\left(x - lo\right) \cdot \frac{1}{hi}\right) \cdot \frac{lo}{hi}\right) \cdot \left(\left(\left(x - lo\right) \cdot \frac{1}{hi}\right) \cdot \frac{lo}{hi}\right) + \left(\frac{1}{\frac{hi}{x - lo}} \cdot \frac{1}{\frac{hi}{x - lo}} - \left(\left(\left(x - lo\right) \cdot \frac{1}{hi}\right) \cdot \frac{lo}{hi}\right) \cdot \frac{1}{\frac{hi}{x - lo}}\right)} \]

    4. clear-num 9.4%

       \[\leadsto \frac{{\left(\frac{lo}{hi} \cdot \frac{x - lo}{hi}\right)}^{3} + {\color{blue}{\left(\frac{x - lo}{hi}\right)}}^{3}}{\left(\left(\left(x - lo\right) \cdot \frac{1}{hi}\right) \cdot \frac{lo}{hi}\right) \cdot \left(\left(\left(x - lo\right) \cdot \frac{1}{hi}\right) \cdot \frac{lo}{hi}\right) + \left(\frac{1}{\frac{hi}{x - lo}} \cdot \frac{1}{\frac{hi}{x - lo}} - \left(\left(\left(x - lo\right) \cdot \frac{1}{hi}\right) \cdot \frac{lo}{hi}\right) \cdot \frac{1}{\frac{hi}{x - lo}}\right)} \]

14. Applied egg-rr 9.4%

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

    1. times-frac 0.0%

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

    2. +-commutative 0.0%

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

    3. unpow2 0.0%

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

    4. associate-/l* 19.7%

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

    5. unpow2 19.7%

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

16. Simplified 98.9%

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

17. Final simplification 98.9%

    \[\leadsto \frac{{\left(\frac{x - lo}{hi}\right)}^{3} + {\left(\frac{lo}{\frac{hi \cdot hi}{x - lo}}\right)}^{3}}{{\left(\frac{lo}{\frac{hi \cdot hi}{x - lo}}\right)}^{2} + \left({\left(\frac{x - lo}{hi}\right)}^{2} - {\left(\frac{x - lo}{hi}\right)}^{2} \cdot \frac{lo}{hi}\right)} \]

Alternative 2: 99.0% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x - lo}{hi}\\ t_1 := lo \cdot \frac{x - lo}{hi \cdot hi}\\ \frac{{t_0}^{3} + {t_1}^{3}}{{t_1}^{2} + \left({t_0}^{2} - \frac{lo}{hi} \cdot {\left(\frac{hi}{x - lo}\right)}^{-2}\right)} \end{array} \end{array} \]

FPCore

(FPCore (lo hi x)
 :precision binary64
 (let* ((t_0 (/ (- x lo) hi)) (t_1 (* lo (/ (- x lo) (* hi hi)))))
   (/
    (+ (pow t_0 3.0) (pow t_1 3.0))
    (+
     (pow t_1 2.0)
     (- (pow t_0 2.0) (* (/ lo hi) (pow (/ hi (- x lo)) -2.0)))))))

C

double code(double lo, double hi, double x) {
	double t_0 = (x - lo) / hi;
	double t_1 = lo * ((x - lo) / (hi * hi));
	return (pow(t_0, 3.0) + pow(t_1, 3.0)) / (pow(t_1, 2.0) + (pow(t_0, 2.0) - ((lo / hi) * pow((hi / (x - 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
    real(8) :: t_0
    real(8) :: t_1
    t_0 = (x - lo) / hi
    t_1 = lo * ((x - lo) / (hi * hi))
    code = ((t_0 ** 3.0d0) + (t_1 ** 3.0d0)) / ((t_1 ** 2.0d0) + ((t_0 ** 2.0d0) - ((lo / hi) * ((hi / (x - lo)) ** (-2.0d0)))))
end function

Java

public static double code(double lo, double hi, double x) {
	double t_0 = (x - lo) / hi;
	double t_1 = lo * ((x - lo) / (hi * hi));
	return (Math.pow(t_0, 3.0) + Math.pow(t_1, 3.0)) / (Math.pow(t_1, 2.0) + (Math.pow(t_0, 2.0) - ((lo / hi) * Math.pow((hi / (x - lo)), -2.0))));
}

Python

def code(lo, hi, x):
	t_0 = (x - lo) / hi
	t_1 = lo * ((x - lo) / (hi * hi))
	return (math.pow(t_0, 3.0) + math.pow(t_1, 3.0)) / (math.pow(t_1, 2.0) + (math.pow(t_0, 2.0) - ((lo / hi) * math.pow((hi / (x - lo)), -2.0))))

Julia

function code(lo, hi, x)
	t_0 = Float64(Float64(x - lo) / hi)
	t_1 = Float64(lo * Float64(Float64(x - lo) / Float64(hi * hi)))
	return Float64(Float64((t_0 ^ 3.0) + (t_1 ^ 3.0)) / Float64((t_1 ^ 2.0) + Float64((t_0 ^ 2.0) - Float64(Float64(lo / hi) * (Float64(hi / Float64(x - lo)) ^ -2.0)))))
end

MATLAB

function tmp = code(lo, hi, x)
	t_0 = (x - lo) / hi;
	t_1 = lo * ((x - lo) / (hi * hi));
	tmp = ((t_0 ^ 3.0) + (t_1 ^ 3.0)) / ((t_1 ^ 2.0) + ((t_0 ^ 2.0) - ((lo / hi) * ((hi / (x - lo)) ^ -2.0))));
end

Wolfram

code[lo_, hi_, x_] := Block[{t$95$0 = N[(N[(x - lo), $MachinePrecision] / hi), $MachinePrecision]}, Block[{t$95$1 = N[(lo * N[(N[(x - lo), $MachinePrecision] / N[(hi * hi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[Power[t$95$0, 3.0], $MachinePrecision] + N[Power[t$95$1, 3.0], $MachinePrecision]), $MachinePrecision] / N[(N[Power[t$95$1, 2.0], $MachinePrecision] + N[(N[Power[t$95$0, 2.0], $MachinePrecision] - N[(N[(lo / hi), $MachinePrecision] * N[Power[N[(hi / N[(x - lo), $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision]), $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. div-inv 9.4%

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

   2. inv-pow 9.4%

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

6. Applied egg-rr 9.4%

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

   1. unpow-1 9.4%

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

8. Simplified 9.4%

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

10. Applied egg-rr 18.7%

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

    1. cube-div 0.0%

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

    2. +-commutative 0.0%

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

    3. cube-div 18.7%

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

    4. associate-*r/ 0.0%

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

    5. *-commutative 0.0%

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

    6. times-frac 10.9%

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

    7. associate-*l/ 10.6%

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

    8. associate-*r/ 10.9%

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

    9. associate-/l/ 18.7%

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

12. Simplified 98.3%

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

    1. expm1-log1p-u 49.1%

       \[\leadsto \frac{{\left(\frac{x - lo}{hi}\right)}^{3} + {\left(lo \cdot \frac{x - lo}{hi \cdot hi}\right)}^{3}}{{\left(lo \cdot \frac{x - lo}{hi \cdot hi}\right)}^{2} + \left({\left(\frac{x - lo}{hi}\right)}^{2} - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(x - lo\right) \cdot \left(\frac{\frac{lo}{hi}}{hi} \cdot \frac{x - lo}{hi}\right)\right)\right)}\right)} \]

    2. expm1-udef 49.2%

       \[\leadsto \frac{{\left(\frac{x - lo}{hi}\right)}^{3} + {\left(lo \cdot \frac{x - lo}{hi \cdot hi}\right)}^{3}}{{\left(lo \cdot \frac{x - lo}{hi \cdot hi}\right)}^{2} + \left({\left(\frac{x - lo}{hi}\right)}^{2} - \color{blue}{\left(e^{\mathsf{log1p}\left(\left(x - lo\right) \cdot \left(\frac{\frac{lo}{hi}}{hi} \cdot \frac{x - lo}{hi}\right)\right)} - 1\right)}\right)} \]

    3. associate-/l/ 18.7%

       \[\leadsto \frac{{\left(\frac{x - lo}{hi}\right)}^{3} + {\left(lo \cdot \frac{x - lo}{hi \cdot hi}\right)}^{3}}{{\left(lo \cdot \frac{x - lo}{hi \cdot hi}\right)}^{2} + \left({\left(\frac{x - lo}{hi}\right)}^{2} - \left(e^{\mathsf{log1p}\left(\left(x - lo\right) \cdot \left(\color{blue}{\frac{lo}{hi \cdot hi}} \cdot \frac{x - lo}{hi}\right)\right)} - 1\right)\right)} \]

14. Applied egg-rr 18.7%

    \[\leadsto \frac{{\left(\frac{x - lo}{hi}\right)}^{3} + {\left(lo \cdot \frac{x - lo}{hi \cdot hi}\right)}^{3}}{{\left(lo \cdot \frac{x - lo}{hi \cdot hi}\right)}^{2} + \left({\left(\frac{x - lo}{hi}\right)}^{2} - \color{blue}{\left(e^{\mathsf{log1p}\left(\left(x - lo\right) \cdot \left(\frac{lo}{hi \cdot hi} \cdot \frac{x - lo}{hi}\right)\right)} - 1\right)}\right)} \]
15. Step-by-step derivation

    1. expm1-def 18.7%

       \[\leadsto \frac{{\left(\frac{x - lo}{hi}\right)}^{3} + {\left(lo \cdot \frac{x - lo}{hi \cdot hi}\right)}^{3}}{{\left(lo \cdot \frac{x - lo}{hi \cdot hi}\right)}^{2} + \left({\left(\frac{x - lo}{hi}\right)}^{2} - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(x - lo\right) \cdot \left(\frac{lo}{hi \cdot hi} \cdot \frac{x - lo}{hi}\right)\right)\right)}\right)} \]

    2. expm1-log1p 18.7%

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

    3. associate-*r* 18.7%

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

    4. unpow2 18.7%

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

    5. associate-*r/ 0.0%

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

    6. *-commutative 0.0%

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

    7. unpow2 0.0%

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

    8. times-frac 98.9%

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

    9. associate-*l* 98.9%

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

    10. *-lft-identity 98.9%

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

    11. associate-*r/ 98.9%

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

    12. associate-*l/ 98.5%

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

    13. associate-/r/ 98.7%

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

    14. unpow-1 98.7%

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

    15. *-lft-identity 98.7%

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

    16. associate-*r/ 98.7%

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

    17. associate-*l/ 98.4%

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

    18. associate-/r/ 98.6%

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

    19. unpow-1 98.6%

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

    20. pow-sqr 98.6%

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

    21. metadata-eval 98.6%

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

16. Simplified 98.6%

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

17. Final simplification 98.6%

    \[\leadsto \frac{{\left(\frac{x - lo}{hi}\right)}^{3} + {\left(lo \cdot \frac{x - lo}{hi \cdot hi}\right)}^{3}}{{\left(lo \cdot \frac{x - lo}{hi \cdot hi}\right)}^{2} + \left({\left(\frac{x - lo}{hi}\right)}^{2} - \frac{lo}{hi} \cdot {\left(\frac{hi}{x - lo}\right)}^{-2}\right)} \]

Alternative 3: 98.8% accurate, 0.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[lo_, hi_, x_] := N[(N[(N[Power[N[(N[(x - lo), $MachinePrecision] * N[(lo / N[(hi * hi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] - N[Power[N[(hi / N[(x - lo), $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision] / N[(N[(N[(lo - x), $MachinePrecision] / hi), $MachinePrecision] * N[(1.0 - N[(lo / hi), $MachinePrecision]), $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. div-inv 9.4%

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

   2. inv-pow 9.4%

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

6. Applied egg-rr 9.4%

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

   1. unpow-1 9.4%

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

8. Simplified 9.4%

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

   1. clear-num 9.4%

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

   2. inv-pow 9.4%

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

10. Applied egg-rr 9.4%

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

    1. unpow-1 9.4%

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

12. Simplified 9.4%

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

    1. clear-num 9.4%

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

    2. flip-+ 9.4%

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

    3. pow2 9.4%

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

    4. div-inv 9.4%

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

    5. *-commutative 9.4%

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

    6. pow2 9.4%

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

    7. div-inv 9.4%

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

    8. *-commutative 9.4%

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

14. Applied egg-rr 9.4%

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

16. Simplified 98.5%

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

17. Final simplification 98.5%

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

Alternative 4: 24.0% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x - lo}{hi}\\ t_1 := \frac{hi}{x - lo}\\ \mathbf{if}\;lo \leq -1.06 \cdot 10^{+308}:\\ \;\;\;\;e^{\frac{x}{hi} + \left({t_0}^{2} \cdot -0.5 - \frac{lo}{hi}\right)} + -1\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\frac{x - lo}{hi \cdot \frac{hi}{lo}}\right)}^{2} + \frac{-1}{t_1 \cdot t_1}}{\frac{lo \cdot t_0 + \left(lo - x\right)}{hi}}\\ \end{array} \end{array} \]

FPCore

(FPCore (lo hi x)
 :precision binary64
 (let* ((t_0 (/ (- x lo) hi)) (t_1 (/ hi (- x lo))))
   (if (<= lo -1.06e+308)
     (+ (exp (+ (/ x hi) (- (* (pow t_0 2.0) -0.5) (/ lo hi)))) -1.0)
     (/
      (+ (pow (/ (- x lo) (* hi (/ hi lo))) 2.0) (/ -1.0 (* t_1 t_1)))
      (/ (+ (* lo t_0) (- lo x)) hi)))))

C

double code(double lo, double hi, double x) {
	double t_0 = (x - lo) / hi;
	double t_1 = hi / (x - lo);
	double tmp;
	if (lo <= -1.06e+308) {
		tmp = exp(((x / hi) + ((pow(t_0, 2.0) * -0.5) - (lo / hi)))) + -1.0;
	} else {
		tmp = (pow(((x - lo) / (hi * (hi / lo))), 2.0) + (-1.0 / (t_1 * t_1))) / (((lo * t_0) + (lo - x)) / hi);
	}
	return tmp;
}

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
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (x - lo) / hi
    t_1 = hi / (x - lo)
    if (lo <= (-1.06d+308)) then
        tmp = exp(((x / hi) + (((t_0 ** 2.0d0) * (-0.5d0)) - (lo / hi)))) + (-1.0d0)
    else
        tmp = ((((x - lo) / (hi * (hi / lo))) ** 2.0d0) + ((-1.0d0) / (t_1 * t_1))) / (((lo * t_0) + (lo - x)) / hi)
    end if
    code = tmp
end function

Java

public static double code(double lo, double hi, double x) {
	double t_0 = (x - lo) / hi;
	double t_1 = hi / (x - lo);
	double tmp;
	if (lo <= -1.06e+308) {
		tmp = Math.exp(((x / hi) + ((Math.pow(t_0, 2.0) * -0.5) - (lo / hi)))) + -1.0;
	} else {
		tmp = (Math.pow(((x - lo) / (hi * (hi / lo))), 2.0) + (-1.0 / (t_1 * t_1))) / (((lo * t_0) + (lo - x)) / hi);
	}
	return tmp;
}

Python

def code(lo, hi, x):
	t_0 = (x - lo) / hi
	t_1 = hi / (x - lo)
	tmp = 0
	if lo <= -1.06e+308:
		tmp = math.exp(((x / hi) + ((math.pow(t_0, 2.0) * -0.5) - (lo / hi)))) + -1.0
	else:
		tmp = (math.pow(((x - lo) / (hi * (hi / lo))), 2.0) + (-1.0 / (t_1 * t_1))) / (((lo * t_0) + (lo - x)) / hi)
	return tmp

Julia

function code(lo, hi, x)
	t_0 = Float64(Float64(x - lo) / hi)
	t_1 = Float64(hi / Float64(x - lo))
	tmp = 0.0
	if (lo <= -1.06e+308)
		tmp = Float64(exp(Float64(Float64(x / hi) + Float64(Float64((t_0 ^ 2.0) * -0.5) - Float64(lo / hi)))) + -1.0);
	else
		tmp = Float64(Float64((Float64(Float64(x - lo) / Float64(hi * Float64(hi / lo))) ^ 2.0) + Float64(-1.0 / Float64(t_1 * t_1))) / Float64(Float64(Float64(lo * t_0) + Float64(lo - x)) / hi));
	end
	return tmp
end

MATLAB

function tmp_2 = code(lo, hi, x)
	t_0 = (x - lo) / hi;
	t_1 = hi / (x - lo);
	tmp = 0.0;
	if (lo <= -1.06e+308)
		tmp = exp(((x / hi) + (((t_0 ^ 2.0) * -0.5) - (lo / hi)))) + -1.0;
	else
		tmp = ((((x - lo) / (hi * (hi / lo))) ^ 2.0) + (-1.0 / (t_1 * t_1))) / (((lo * t_0) + (lo - x)) / hi);
	end
	tmp_2 = tmp;
end

Wolfram

code[lo_, hi_, x_] := Block[{t$95$0 = N[(N[(x - lo), $MachinePrecision] / hi), $MachinePrecision]}, Block[{t$95$1 = N[(hi / N[(x - lo), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[lo, -1.06e+308], N[(N[Exp[N[(N[(x / hi), $MachinePrecision] + N[(N[(N[Power[t$95$0, 2.0], $MachinePrecision] * -0.5), $MachinePrecision] - N[(lo / hi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + -1.0), $MachinePrecision], N[(N[(N[Power[N[(N[(x - lo), $MachinePrecision] / N[(hi * N[(hi / lo), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[(-1.0 / N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(lo * t$95$0), $MachinePrecision] + N[(lo - x), $MachinePrecision]), $MachinePrecision] / hi), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if lo < -1.06e308

   1. Initial program 3.1%

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

   2. Taylor expanded in hi around inf 18.7%

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

      1. expm1-log1p-u 18.7%

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

      2. expm1-udef 18.7%

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

   4. Applied egg-rr 18.7%

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

   5. Taylor expanded in hi around inf 0.0%

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

      1. associate--l+ 0.0%

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

      2. unpow2 0.0%

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

      3. unpow2 0.0%

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

      4. *-rgt-identity 0.0%

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

      5. times-frac 21.8%

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

      6. *-rgt-identity 21.8%

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

      7. unpow2 21.8%

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

   7. Simplified 21.8%

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

   if -1.06e308 < lo

   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 18.3%

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

      6. div-sub 18.3%

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

   4. Simplified 18.3%

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

      1. flip-+ 18.3%

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

      2. pow2 18.3%

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

      3. *-commutative 18.3%

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

      4. clear-num 18.3%

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

      5. frac-times 63.3%

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

      6. *-un-lft-identity 63.3%

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

      7. pow2 63.3%

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

      8. associate-*r/ 63.2%

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

      9. sub-div 50.9%

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

   6. Applied egg-rr 50.9%

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

      1. unpow2 50.9%

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

      2. clear-num 51.0%

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

      3. clear-num 50.9%

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

      4. frac-times 51.1%

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

      5. metadata-eval 51.1%

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

   8. Applied egg-rr 51.1%

      \[\leadsto \frac{{\left(\frac{x - lo}{\frac{hi}{lo} \cdot hi}\right)}^{2} - \color{blue}{\frac{1}{\frac{hi}{x - lo} \cdot \frac{hi}{x - lo}}}}{\frac{\frac{x - lo}{hi} \cdot lo - \left(x - lo\right)}{hi}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 23.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;lo \leq -1.06 \cdot 10^{+308}:\\ \;\;\;\;e^{\frac{x}{hi} + \left({\left(\frac{x - lo}{hi}\right)}^{2} \cdot -0.5 - \frac{lo}{hi}\right)} + -1\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\frac{x - lo}{hi \cdot \frac{hi}{lo}}\right)}^{2} + \frac{-1}{\frac{hi}{x - lo} \cdot \frac{hi}{x - lo}}}{\frac{lo \cdot \frac{x - lo}{hi} + \left(lo - x\right)}{hi}}\\ \end{array} \]

Alternative 5: 21.1% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{hi}{x - lo}\\ \mathbf{if}\;lo \leq -1.06 \cdot 10^{+308}:\\ \;\;\;\;\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x}{hi} - lo \cdot \frac{1 - \frac{x}{hi}}{hi}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\frac{x - lo}{hi \cdot \frac{hi}{lo}}\right)}^{2} + \frac{-1}{t_0 \cdot t_0}}{\frac{lo \cdot \frac{x - lo}{hi} + \left(lo - x\right)}{hi}}\\ \end{array} \end{array} \]

FPCore

(FPCore (lo hi x)
 :precision binary64
 (let* ((t_0 (/ hi (- x lo))))
   (if (<= lo -1.06e+308)
     (expm1 (log1p (- (/ x hi) (* lo (/ (- 1.0 (/ x hi)) hi)))))
     (/
      (+ (pow (/ (- x lo) (* hi (/ hi lo))) 2.0) (/ -1.0 (* t_0 t_0)))
      (/ (+ (* lo (/ (- x lo) hi)) (- lo x)) hi)))))

C

double code(double lo, double hi, double x) {
	double t_0 = hi / (x - lo);
	double tmp;
	if (lo <= -1.06e+308) {
		tmp = expm1(log1p(((x / hi) - (lo * ((1.0 - (x / hi)) / hi)))));
	} else {
		tmp = (pow(((x - lo) / (hi * (hi / lo))), 2.0) + (-1.0 / (t_0 * t_0))) / (((lo * ((x - lo) / hi)) + (lo - x)) / hi);
	}
	return tmp;
}

Java

public static double code(double lo, double hi, double x) {
	double t_0 = hi / (x - lo);
	double tmp;
	if (lo <= -1.06e+308) {
		tmp = Math.expm1(Math.log1p(((x / hi) - (lo * ((1.0 - (x / hi)) / hi)))));
	} else {
		tmp = (Math.pow(((x - lo) / (hi * (hi / lo))), 2.0) + (-1.0 / (t_0 * t_0))) / (((lo * ((x - lo) / hi)) + (lo - x)) / hi);
	}
	return tmp;
}

Python

def code(lo, hi, x):
	t_0 = hi / (x - lo)
	tmp = 0
	if lo <= -1.06e+308:
		tmp = math.expm1(math.log1p(((x / hi) - (lo * ((1.0 - (x / hi)) / hi)))))
	else:
		tmp = (math.pow(((x - lo) / (hi * (hi / lo))), 2.0) + (-1.0 / (t_0 * t_0))) / (((lo * ((x - lo) / hi)) + (lo - x)) / hi)
	return tmp

Julia

function code(lo, hi, x)
	t_0 = Float64(hi / Float64(x - lo))
	tmp = 0.0
	if (lo <= -1.06e+308)
		tmp = expm1(log1p(Float64(Float64(x / hi) - Float64(lo * Float64(Float64(1.0 - Float64(x / hi)) / hi)))));
	else
		tmp = Float64(Float64((Float64(Float64(x - lo) / Float64(hi * Float64(hi / lo))) ^ 2.0) + Float64(-1.0 / Float64(t_0 * t_0))) / Float64(Float64(Float64(lo * Float64(Float64(x - lo) / hi)) + Float64(lo - x)) / hi));
	end
	return tmp
end

Wolfram

code[lo_, hi_, x_] := Block[{t$95$0 = N[(hi / N[(x - lo), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[lo, -1.06e+308], N[(Exp[N[Log[1 + N[(N[(x / hi), $MachinePrecision] - N[(lo * N[(N[(1.0 - N[(x / hi), $MachinePrecision]), $MachinePrecision] / hi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]] - 1), $MachinePrecision], N[(N[(N[Power[N[(N[(x - lo), $MachinePrecision] / N[(hi * N[(hi / lo), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[(-1.0 / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(lo * N[(N[(x - lo), $MachinePrecision] / hi), $MachinePrecision]), $MachinePrecision] + N[(lo - x), $MachinePrecision]), $MachinePrecision] / hi), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if lo < -1.06e308

   1. Initial program 3.1%

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

   2. Taylor expanded in lo around 0 18.7%

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

      1. mul-1-neg 18.7%

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

      2. unsub-neg 18.7%

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

      3. mul-1-neg 18.7%

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

      4. unsub-neg 18.7%

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

      5. unpow2 18.7%

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

   4. Simplified 18.7%

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

      1. expm1-log1p-u 18.7%

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

      2. associate-/r* 18.7%

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

      3. sub-div 18.7%

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

   6. Applied egg-rr 18.7%

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

   if -1.06e308 < lo

   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 18.3%

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

      6. div-sub 18.3%

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

   4. Simplified 18.3%

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

      1. flip-+ 18.3%

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

      2. pow2 18.3%

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

      3. *-commutative 18.3%

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

      4. clear-num 18.3%

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

      5. frac-times 63.3%

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

      6. *-un-lft-identity 63.3%

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

      7. pow2 63.3%

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

      8. associate-*r/ 63.2%

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

      9. sub-div 50.9%

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

   6. Applied egg-rr 50.9%

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

      1. unpow2 50.9%

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

      2. clear-num 51.0%

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

      3. clear-num 50.9%

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

      4. frac-times 51.1%

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

      5. metadata-eval 51.1%

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

   8. Applied egg-rr 51.1%

      \[\leadsto \frac{{\left(\frac{x - lo}{\frac{hi}{lo} \cdot hi}\right)}^{2} - \color{blue}{\frac{1}{\frac{hi}{x - lo} \cdot \frac{hi}{x - lo}}}}{\frac{\frac{x - lo}{hi} \cdot lo - \left(x - lo\right)}{hi}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 20.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;lo \leq -1.06 \cdot 10^{+308}:\\ \;\;\;\;\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x}{hi} - lo \cdot \frac{1 - \frac{x}{hi}}{hi}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\frac{x - lo}{hi \cdot \frac{hi}{lo}}\right)}^{2} + \frac{-1}{\frac{hi}{x - lo} \cdot \frac{hi}{x - lo}}}{\frac{lo \cdot \frac{x - lo}{hi} + \left(lo - x\right)}{hi}}\\ \end{array} \]

Alternative 6: 21.1% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{hi}{x - lo}\\ \mathbf{if}\;lo \leq -1.06 \cdot 10^{+308}:\\ \;\;\;\;\frac{x}{hi} - lo \cdot e^{\log \left(\frac{1 - \frac{x}{hi}}{hi}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\frac{x - lo}{hi \cdot \frac{hi}{lo}}\right)}^{2} + \frac{-1}{t_0 \cdot t_0}}{\frac{lo \cdot \frac{x - lo}{hi} + \left(lo - x\right)}{hi}}\\ \end{array} \end{array} \]

FPCore

(FPCore (lo hi x)
 :precision binary64
 (let* ((t_0 (/ hi (- x lo))))
   (if (<= lo -1.06e+308)
     (- (/ x hi) (* lo (exp (log (/ (- 1.0 (/ x hi)) hi)))))
     (/
      (+ (pow (/ (- x lo) (* hi (/ hi lo))) 2.0) (/ -1.0 (* t_0 t_0)))
      (/ (+ (* lo (/ (- x lo) hi)) (- lo x)) hi)))))

C

double code(double lo, double hi, double x) {
	double t_0 = hi / (x - lo);
	double tmp;
	if (lo <= -1.06e+308) {
		tmp = (x / hi) - (lo * exp(log(((1.0 - (x / hi)) / hi))));
	} else {
		tmp = (pow(((x - lo) / (hi * (hi / lo))), 2.0) + (-1.0 / (t_0 * t_0))) / (((lo * ((x - lo) / hi)) + (lo - x)) / hi);
	}
	return tmp;
}

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
    real(8) :: tmp
    t_0 = hi / (x - lo)
    if (lo <= (-1.06d+308)) then
        tmp = (x / hi) - (lo * exp(log(((1.0d0 - (x / hi)) / hi))))
    else
        tmp = ((((x - lo) / (hi * (hi / lo))) ** 2.0d0) + ((-1.0d0) / (t_0 * t_0))) / (((lo * ((x - lo) / hi)) + (lo - x)) / hi)
    end if
    code = tmp
end function

Java

public static double code(double lo, double hi, double x) {
	double t_0 = hi / (x - lo);
	double tmp;
	if (lo <= -1.06e+308) {
		tmp = (x / hi) - (lo * Math.exp(Math.log(((1.0 - (x / hi)) / hi))));
	} else {
		tmp = (Math.pow(((x - lo) / (hi * (hi / lo))), 2.0) + (-1.0 / (t_0 * t_0))) / (((lo * ((x - lo) / hi)) + (lo - x)) / hi);
	}
	return tmp;
}

Python

def code(lo, hi, x):
	t_0 = hi / (x - lo)
	tmp = 0
	if lo <= -1.06e+308:
		tmp = (x / hi) - (lo * math.exp(math.log(((1.0 - (x / hi)) / hi))))
	else:
		tmp = (math.pow(((x - lo) / (hi * (hi / lo))), 2.0) + (-1.0 / (t_0 * t_0))) / (((lo * ((x - lo) / hi)) + (lo - x)) / hi)
	return tmp

Julia

function code(lo, hi, x)
	t_0 = Float64(hi / Float64(x - lo))
	tmp = 0.0
	if (lo <= -1.06e+308)
		tmp = Float64(Float64(x / hi) - Float64(lo * exp(log(Float64(Float64(1.0 - Float64(x / hi)) / hi)))));
	else
		tmp = Float64(Float64((Float64(Float64(x - lo) / Float64(hi * Float64(hi / lo))) ^ 2.0) + Float64(-1.0 / Float64(t_0 * t_0))) / Float64(Float64(Float64(lo * Float64(Float64(x - lo) / hi)) + Float64(lo - x)) / hi));
	end
	return tmp
end

MATLAB

function tmp_2 = code(lo, hi, x)
	t_0 = hi / (x - lo);
	tmp = 0.0;
	if (lo <= -1.06e+308)
		tmp = (x / hi) - (lo * exp(log(((1.0 - (x / hi)) / hi))));
	else
		tmp = ((((x - lo) / (hi * (hi / lo))) ^ 2.0) + (-1.0 / (t_0 * t_0))) / (((lo * ((x - lo) / hi)) + (lo - x)) / hi);
	end
	tmp_2 = tmp;
end

Wolfram

code[lo_, hi_, x_] := Block[{t$95$0 = N[(hi / N[(x - lo), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[lo, -1.06e+308], N[(N[(x / hi), $MachinePrecision] - N[(lo * N[Exp[N[Log[N[(N[(1.0 - N[(x / hi), $MachinePrecision]), $MachinePrecision] / hi), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Power[N[(N[(x - lo), $MachinePrecision] / N[(hi * N[(hi / lo), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[(-1.0 / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(lo * N[(N[(x - lo), $MachinePrecision] / hi), $MachinePrecision]), $MachinePrecision] + N[(lo - x), $MachinePrecision]), $MachinePrecision] / hi), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if lo < -1.06e308

   1. Initial program 3.1%

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

   2. Taylor expanded in lo around 0 18.7%

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

      1. mul-1-neg 18.7%

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

      2. unsub-neg 18.7%

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

      3. mul-1-neg 18.7%

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

      4. unsub-neg 18.7%

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

      5. unpow2 18.7%

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

   4. Simplified 18.7%

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

      1. add-exp-log 18.7%

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

      2. associate-/r* 18.7%

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

      3. sub-div 18.7%

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

   6. Applied egg-rr 18.7%

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

   if -1.06e308 < lo

   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 18.3%

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

      6. div-sub 18.3%

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

   4. Simplified 18.3%

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

      1. flip-+ 18.3%

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

      2. pow2 18.3%

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

      3. *-commutative 18.3%

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

      4. clear-num 18.3%

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

      5. frac-times 63.3%

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

      6. *-un-lft-identity 63.3%

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

      7. pow2 63.3%

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

      8. associate-*r/ 63.2%

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

      9. sub-div 50.9%

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

   6. Applied egg-rr 50.9%

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

      1. unpow2 50.9%

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

      2. clear-num 51.0%

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

      3. clear-num 50.9%

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

      4. frac-times 51.1%

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

      5. metadata-eval 51.1%

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

   8. Applied egg-rr 51.1%

      \[\leadsto \frac{{\left(\frac{x - lo}{\frac{hi}{lo} \cdot hi}\right)}^{2} - \color{blue}{\frac{1}{\frac{hi}{x - lo} \cdot \frac{hi}{x - lo}}}}{\frac{\frac{x - lo}{hi} \cdot lo - \left(x - lo\right)}{hi}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 20.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;lo \leq -1.06 \cdot 10^{+308}:\\ \;\;\;\;\frac{x}{hi} - lo \cdot e^{\log \left(\frac{1 - \frac{x}{hi}}{hi}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\frac{x - lo}{hi \cdot \frac{hi}{lo}}\right)}^{2} + \frac{-1}{\frac{hi}{x - lo} \cdot \frac{hi}{x - lo}}}{\frac{lo \cdot \frac{x - lo}{hi} + \left(lo - x\right)}{hi}}\\ \end{array} \]

Alternative 7: 21.1% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{hi}{x - lo}\\ \mathbf{if}\;lo \leq -1.06 \cdot 10^{+308}:\\ \;\;\;\;\frac{-lo}{hi}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\frac{x - lo}{hi \cdot \frac{hi}{lo}}\right)}^{2} + \frac{-1}{t_0 \cdot t_0}}{\frac{lo \cdot \frac{x - lo}{hi} + \left(lo - x\right)}{hi}}\\ \end{array} \end{array} \]

FPCore

(FPCore (lo hi x)
 :precision binary64
 (let* ((t_0 (/ hi (- x lo))))
   (if (<= lo -1.06e+308)
     (/ (- lo) hi)
     (/
      (+ (pow (/ (- x lo) (* hi (/ hi lo))) 2.0) (/ -1.0 (* t_0 t_0)))
      (/ (+ (* lo (/ (- x lo) hi)) (- lo x)) hi)))))

C

double code(double lo, double hi, double x) {
	double t_0 = hi / (x - lo);
	double tmp;
	if (lo <= -1.06e+308) {
		tmp = -lo / hi;
	} else {
		tmp = (pow(((x - lo) / (hi * (hi / lo))), 2.0) + (-1.0 / (t_0 * t_0))) / (((lo * ((x - lo) / hi)) + (lo - x)) / hi);
	}
	return tmp;
}

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
    real(8) :: tmp
    t_0 = hi / (x - lo)
    if (lo <= (-1.06d+308)) then
        tmp = -lo / hi
    else
        tmp = ((((x - lo) / (hi * (hi / lo))) ** 2.0d0) + ((-1.0d0) / (t_0 * t_0))) / (((lo * ((x - lo) / hi)) + (lo - x)) / hi)
    end if
    code = tmp
end function

Java

public static double code(double lo, double hi, double x) {
	double t_0 = hi / (x - lo);
	double tmp;
	if (lo <= -1.06e+308) {
		tmp = -lo / hi;
	} else {
		tmp = (Math.pow(((x - lo) / (hi * (hi / lo))), 2.0) + (-1.0 / (t_0 * t_0))) / (((lo * ((x - lo) / hi)) + (lo - x)) / hi);
	}
	return tmp;
}

Python

def code(lo, hi, x):
	t_0 = hi / (x - lo)
	tmp = 0
	if lo <= -1.06e+308:
		tmp = -lo / hi
	else:
		tmp = (math.pow(((x - lo) / (hi * (hi / lo))), 2.0) + (-1.0 / (t_0 * t_0))) / (((lo * ((x - lo) / hi)) + (lo - x)) / hi)
	return tmp

Julia

function code(lo, hi, x)
	t_0 = Float64(hi / Float64(x - lo))
	tmp = 0.0
	if (lo <= -1.06e+308)
		tmp = Float64(Float64(-lo) / hi);
	else
		tmp = Float64(Float64((Float64(Float64(x - lo) / Float64(hi * Float64(hi / lo))) ^ 2.0) + Float64(-1.0 / Float64(t_0 * t_0))) / Float64(Float64(Float64(lo * Float64(Float64(x - lo) / hi)) + Float64(lo - x)) / hi));
	end
	return tmp
end

MATLAB

function tmp_2 = code(lo, hi, x)
	t_0 = hi / (x - lo);
	tmp = 0.0;
	if (lo <= -1.06e+308)
		tmp = -lo / hi;
	else
		tmp = ((((x - lo) / (hi * (hi / lo))) ^ 2.0) + (-1.0 / (t_0 * t_0))) / (((lo * ((x - lo) / hi)) + (lo - x)) / hi);
	end
	tmp_2 = tmp;
end

Wolfram

code[lo_, hi_, x_] := Block[{t$95$0 = N[(hi / N[(x - lo), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[lo, -1.06e+308], N[((-lo) / hi), $MachinePrecision], N[(N[(N[Power[N[(N[(x - lo), $MachinePrecision] / N[(hi * N[(hi / lo), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[(-1.0 / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(lo * N[(N[(x - lo), $MachinePrecision] / hi), $MachinePrecision]), $MachinePrecision] + N[(lo - x), $MachinePrecision]), $MachinePrecision] / hi), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if lo < -1.06e308

   1. Initial program 3.1%

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

   2. Taylor expanded in lo around 0 18.7%

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

      1. mul-1-neg 18.7%

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

      2. unsub-neg 18.7%

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

      3. mul-1-neg 18.7%

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

      4. unsub-neg 18.7%

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

      5. unpow2 18.7%

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

   4. Simplified 18.7%

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

   5. Taylor expanded in x around 0 18.7%

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

      1. neg-mul-1 18.7%

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

      2. distribute-neg-frac 18.7%

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

   7. Simplified 18.7%

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

   if -1.06e308 < lo

   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 18.3%

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

      6. div-sub 18.3%

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

   4. Simplified 18.3%

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

      1. flip-+ 18.3%

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

      2. pow2 18.3%

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

      3. *-commutative 18.3%

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

      4. clear-num 18.3%

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

      5. frac-times 63.3%

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

      6. *-un-lft-identity 63.3%

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

      7. pow2 63.3%

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

      8. associate-*r/ 63.2%

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

      9. sub-div 50.9%

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

   6. Applied egg-rr 50.9%

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

      1. unpow2 50.9%

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

      2. clear-num 51.0%

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

      3. clear-num 50.9%

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

      4. frac-times 51.1%

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

      5. metadata-eval 51.1%

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

   8. Applied egg-rr 51.1%

      \[\leadsto \frac{{\left(\frac{x - lo}{\frac{hi}{lo} \cdot hi}\right)}^{2} - \color{blue}{\frac{1}{\frac{hi}{x - lo} \cdot \frac{hi}{x - lo}}}}{\frac{\frac{x - lo}{hi} \cdot lo - \left(x - lo\right)}{hi}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 20.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;lo \leq -1.06 \cdot 10^{+308}:\\ \;\;\;\;\frac{-lo}{hi}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\frac{x - lo}{hi \cdot \frac{hi}{lo}}\right)}^{2} + \frac{-1}{\frac{hi}{x - lo} \cdot \frac{hi}{x - lo}}}{\frac{lo \cdot \frac{x - lo}{hi} + \left(lo - x\right)}{hi}}\\ \end{array} \]

Alternative 8: 21.3% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x - lo}{hi}\\ \mathbf{if}\;lo \leq -1.06 \cdot 10^{+308}:\\ \;\;\;\;\frac{-lo}{hi}\\ \mathbf{else}:\\ \;\;\;\;\frac{-{t_0}^{2}}{\frac{lo \cdot t_0 + \left(lo - x\right)}{hi}}\\ \end{array} \end{array} \]

FPCore

(FPCore (lo hi x)
 :precision binary64
 (let* ((t_0 (/ (- x lo) hi)))
   (if (<= lo -1.06e+308)
     (/ (- lo) hi)
     (/ (- (pow t_0 2.0)) (/ (+ (* lo t_0) (- lo x)) hi)))))

C

double code(double lo, double hi, double x) {
	double t_0 = (x - lo) / hi;
	double tmp;
	if (lo <= -1.06e+308) {
		tmp = -lo / hi;
	} else {
		tmp = -pow(t_0, 2.0) / (((lo * t_0) + (lo - x)) / hi);
	}
	return tmp;
}

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
    real(8) :: tmp
    t_0 = (x - lo) / hi
    if (lo <= (-1.06d+308)) then
        tmp = -lo / hi
    else
        tmp = -(t_0 ** 2.0d0) / (((lo * t_0) + (lo - x)) / hi)
    end if
    code = tmp
end function

Java

public static double code(double lo, double hi, double x) {
	double t_0 = (x - lo) / hi;
	double tmp;
	if (lo <= -1.06e+308) {
		tmp = -lo / hi;
	} else {
		tmp = -Math.pow(t_0, 2.0) / (((lo * t_0) + (lo - x)) / hi);
	}
	return tmp;
}

Python

def code(lo, hi, x):
	t_0 = (x - lo) / hi
	tmp = 0
	if lo <= -1.06e+308:
		tmp = -lo / hi
	else:
		tmp = -math.pow(t_0, 2.0) / (((lo * t_0) + (lo - x)) / hi)
	return tmp

Julia

function code(lo, hi, x)
	t_0 = Float64(Float64(x - lo) / hi)
	tmp = 0.0
	if (lo <= -1.06e+308)
		tmp = Float64(Float64(-lo) / hi);
	else
		tmp = Float64(Float64(-(t_0 ^ 2.0)) / Float64(Float64(Float64(lo * t_0) + Float64(lo - x)) / hi));
	end
	return tmp
end

MATLAB

function tmp_2 = code(lo, hi, x)
	t_0 = (x - lo) / hi;
	tmp = 0.0;
	if (lo <= -1.06e+308)
		tmp = -lo / hi;
	else
		tmp = -(t_0 ^ 2.0) / (((lo * t_0) + (lo - x)) / hi);
	end
	tmp_2 = tmp;
end

Wolfram

code[lo_, hi_, x_] := Block[{t$95$0 = N[(N[(x - lo), $MachinePrecision] / hi), $MachinePrecision]}, If[LessEqual[lo, -1.06e+308], N[((-lo) / hi), $MachinePrecision], N[((-N[Power[t$95$0, 2.0], $MachinePrecision]) / N[(N[(N[(lo * t$95$0), $MachinePrecision] + N[(lo - x), $MachinePrecision]), $MachinePrecision] / hi), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if lo < -1.06e308

   1. Initial program 3.1%

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

   2. Taylor expanded in lo around 0 18.7%

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

      1. mul-1-neg 18.7%

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

      2. unsub-neg 18.7%

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

      3. mul-1-neg 18.7%

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

      4. unsub-neg 18.7%

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

      5. unpow2 18.7%

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

   4. Simplified 18.7%

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

   5. Taylor expanded in x around 0 18.7%

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

      1. neg-mul-1 18.7%

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

      2. distribute-neg-frac 18.7%

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

   7. Simplified 18.7%

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

   if -1.06e308 < lo

   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 18.3%

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

      6. div-sub 18.3%

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

   4. Simplified 18.3%

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

      1. flip-+ 18.3%

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

      2. pow2 18.3%

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

      3. *-commutative 18.3%

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

      4. clear-num 18.3%

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

      5. frac-times 63.3%

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

      6. *-un-lft-identity 63.3%

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

      7. pow2 63.3%

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

      8. associate-*r/ 63.2%

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

      9. sub-div 50.9%

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

   6. Applied egg-rr 50.9%

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

   7. Taylor expanded in hi around inf 0.0%

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

      1. mul-1-neg 0.0%

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

      2. unpow2 0.0%

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

      3. unpow2 0.0%

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

      4. *-rgt-identity 0.0%

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

      5. times-frac 50.9%

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

      6. *-rgt-identity 50.9%

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

      7. unpow2 50.9%

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

   9. Simplified 50.9%

      \[\leadsto \frac{\color{blue}{-{\left(\frac{x - lo}{hi}\right)}^{2}}}{\frac{\frac{x - lo}{hi} \cdot lo - \left(x - lo\right)}{hi}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 20.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;lo \leq -1.06 \cdot 10^{+308}:\\ \;\;\;\;\frac{-lo}{hi}\\ \mathbf{else}:\\ \;\;\;\;\frac{-{\left(\frac{x - lo}{hi}\right)}^{2}}{\frac{lo \cdot \frac{x - lo}{hi} + \left(lo - x\right)}{hi}}\\ \end{array} \]

Alternative 9: 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 10: 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 2023203 
(FPCore (lo hi x)
  :name "xlohi (overflows)"
  :precision binary64
  :pre (and (< lo -1e+308) (> hi 1e+308))
  (/ (- x lo) (- hi lo)))