xlohi (overflows)

Percentage Accurate: 3.1% → 99.0%

Time: 14.7s

Alternatives: 8

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: 99.0% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -1 - \frac{hi}{lo}\\ \left({\left(t\_0 \cdot \frac{x}{lo}\right)}^{3} + 1\right) \cdot \frac{1}{\left({\left(\frac{x - hi}{lo}\right)}^{2} + t\_0 \cdot \frac{hi - x}{lo}\right) + 1} \end{array} \end{array} \]

FPCore

(FPCore (lo hi x)
 :precision binary64
 (let* ((t_0 (- -1.0 (/ hi lo))))
   (*
    (+ (pow (* t_0 (/ x lo)) 3.0) 1.0)
    (/ 1.0 (+ (+ (pow (/ (- x hi) lo) 2.0) (* t_0 (/ (- hi x) lo))) 1.0)))))

C

double code(double lo, double hi, double x) {
	double t_0 = -1.0 - (hi / lo);
	return (pow((t_0 * (x / lo)), 3.0) + 1.0) * (1.0 / ((pow(((x - hi) / lo), 2.0) + (t_0 * ((hi - x) / lo))) + 1.0));
}

Fortran

real(8) function code(lo, hi, x)
    real(8), intent (in) :: lo
    real(8), intent (in) :: hi
    real(8), intent (in) :: x
    real(8) :: t_0
    t_0 = (-1.0d0) - (hi / lo)
    code = (((t_0 * (x / lo)) ** 3.0d0) + 1.0d0) * (1.0d0 / (((((x - hi) / lo) ** 2.0d0) + (t_0 * ((hi - x) / lo))) + 1.0d0))
end function

Java

public static double code(double lo, double hi, double x) {
	double t_0 = -1.0 - (hi / lo);
	return (Math.pow((t_0 * (x / lo)), 3.0) + 1.0) * (1.0 / ((Math.pow(((x - hi) / lo), 2.0) + (t_0 * ((hi - x) / lo))) + 1.0));
}

Python

def code(lo, hi, x):
	t_0 = -1.0 - (hi / lo)
	return (math.pow((t_0 * (x / lo)), 3.0) + 1.0) * (1.0 / ((math.pow(((x - hi) / lo), 2.0) + (t_0 * ((hi - x) / lo))) + 1.0))

Julia

function code(lo, hi, x)
	t_0 = Float64(-1.0 - Float64(hi / lo))
	return Float64(Float64((Float64(t_0 * Float64(x / lo)) ^ 3.0) + 1.0) * Float64(1.0 / Float64(Float64((Float64(Float64(x - hi) / lo) ^ 2.0) + Float64(t_0 * Float64(Float64(hi - x) / lo))) + 1.0)))
end

MATLAB

function tmp = code(lo, hi, x)
	t_0 = -1.0 - (hi / lo);
	tmp = (((t_0 * (x / lo)) ^ 3.0) + 1.0) * (1.0 / (((((x - hi) / lo) ^ 2.0) + (t_0 * ((hi - x) / lo))) + 1.0));
end

Wolfram

code[lo_, hi_, x_] := Block[{t$95$0 = N[(-1.0 - N[(hi / lo), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[Power[N[(t$95$0 * N[(x / lo), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] + 1.0), $MachinePrecision] * N[(1.0 / N[(N[(N[Power[N[(N[(x - hi), $MachinePrecision] / lo), $MachinePrecision], 2.0], $MachinePrecision] + N[(t$95$0 * N[(N[(hi - x), $MachinePrecision] / lo), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 3.1%

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

3. Taylor expanded in lo around inf 0.0%

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

   1. associate--l+ 0.0%

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

   2. +-commutative 0.0%

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

   3. associate--l+ 0.0%

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

   4. *-commutative 0.0%

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

   5. distribute-lft-out-- 0.0%

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

   6. associate-*r* 0.0%

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

   7. *-commutative 0.0%

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

   8. associate-*r/ 0.0%

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

   9. distribute-lft-out-- 0.0%

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

   10. div-sub 0.0%

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

   11. +-commutative 0.0%

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

   12. mul-1-neg 0.0%

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

5. Simplified 18.8%

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

   1. flip3-+ 18.8%

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

   2. div-inv 18.8%

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

   3. metadata-eval 18.8%

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

   4. *-commutative 18.8%

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

   5. metadata-eval 18.8%

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

7. Applied egg-rr 18.8%

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

8. Taylor expanded in lo around inf 0.0%

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

   1. unpow2 0.0%

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

   2. unpow2 0.0%

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

   3. times-frac 32.2%

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

   4. unpow2 32.2%

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

10. Simplified 32.2%

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

11. Taylor expanded in x around inf 99.4%

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

12. Final simplification 99.4%

    \[\leadsto \left({\left(\left(-1 - \frac{hi}{lo}\right) \cdot \frac{x}{lo}\right)}^{3} + 1\right) \cdot \frac{1}{\left({\left(\frac{x - hi}{lo}\right)}^{2} + \left(-1 - \frac{hi}{lo}\right) \cdot \frac{hi - x}{lo}\right) + 1} \]
13. Add Preprocessing

Alternative 2: 31.8% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{hi - x}{lo}\\ t_1 := \left(\frac{hi}{lo} + 1\right) \cdot t\_0\\ \left({t\_1}^{3} + 1\right) \cdot \frac{1}{\left(t\_0 \cdot t\_0 - t\_1\right) + 1} \end{array} \end{array} \]

FPCore

(FPCore (lo hi x)
 :precision binary64
 (let* ((t_0 (/ (- hi x) lo)) (t_1 (* (+ (/ hi lo) 1.0) t_0)))
   (* (+ (pow t_1 3.0) 1.0) (/ 1.0 (+ (- (* t_0 t_0) t_1) 1.0)))))

C

double code(double lo, double hi, double x) {
	double t_0 = (hi - x) / lo;
	double t_1 = ((hi / lo) + 1.0) * t_0;
	return (pow(t_1, 3.0) + 1.0) * (1.0 / (((t_0 * t_0) - t_1) + 1.0));
}

Fortran

real(8) function code(lo, hi, x)
    real(8), intent (in) :: lo
    real(8), intent (in) :: hi
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    t_0 = (hi - x) / lo
    t_1 = ((hi / lo) + 1.0d0) * t_0
    code = ((t_1 ** 3.0d0) + 1.0d0) * (1.0d0 / (((t_0 * t_0) - t_1) + 1.0d0))
end function

Java

public static double code(double lo, double hi, double x) {
	double t_0 = (hi - x) / lo;
	double t_1 = ((hi / lo) + 1.0) * t_0;
	return (Math.pow(t_1, 3.0) + 1.0) * (1.0 / (((t_0 * t_0) - t_1) + 1.0));
}

Python

def code(lo, hi, x):
	t_0 = (hi - x) / lo
	t_1 = ((hi / lo) + 1.0) * t_0
	return (math.pow(t_1, 3.0) + 1.0) * (1.0 / (((t_0 * t_0) - t_1) + 1.0))

Julia

function code(lo, hi, x)
	t_0 = Float64(Float64(hi - x) / lo)
	t_1 = Float64(Float64(Float64(hi / lo) + 1.0) * t_0)
	return Float64(Float64((t_1 ^ 3.0) + 1.0) * Float64(1.0 / Float64(Float64(Float64(t_0 * t_0) - t_1) + 1.0)))
end

MATLAB

function tmp = code(lo, hi, x)
	t_0 = (hi - x) / lo;
	t_1 = ((hi / lo) + 1.0) * t_0;
	tmp = ((t_1 ^ 3.0) + 1.0) * (1.0 / (((t_0 * t_0) - t_1) + 1.0));
end

Wolfram

code[lo_, hi_, x_] := Block[{t$95$0 = N[(N[(hi - x), $MachinePrecision] / lo), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(hi / lo), $MachinePrecision] + 1.0), $MachinePrecision] * t$95$0), $MachinePrecision]}, N[(N[(N[Power[t$95$1, 3.0], $MachinePrecision] + 1.0), $MachinePrecision] * N[(1.0 / N[(N[(N[(t$95$0 * t$95$0), $MachinePrecision] - t$95$1), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 3.1%

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

3. Taylor expanded in lo around inf 0.0%

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

   1. associate--l+ 0.0%

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

   2. +-commutative 0.0%

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

   3. associate--l+ 0.0%

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

   4. *-commutative 0.0%

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

   5. distribute-lft-out-- 0.0%

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

   6. associate-*r* 0.0%

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

   7. *-commutative 0.0%

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

   8. associate-*r/ 0.0%

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

   9. distribute-lft-out-- 0.0%

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

   10. div-sub 0.0%

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

   11. +-commutative 0.0%

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

   12. mul-1-neg 0.0%

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

5. Simplified 18.8%

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

   1. flip3-+ 18.8%

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

   2. div-inv 18.8%

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

   3. metadata-eval 18.8%

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

   4. *-commutative 18.8%

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

   5. metadata-eval 18.8%

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

7. Applied egg-rr 18.8%

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

8. Taylor expanded in lo around inf 0.0%

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

   1. unpow2 0.0%

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

   2. unpow2 0.0%

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

   3. times-frac 32.2%

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

   4. unpow2 32.2%

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

10. Simplified 32.2%

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

    1. unpow2 32.2%

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

12. Applied egg-rr 32.2%

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

13. Final simplification 32.2%

    \[\leadsto \left({\left(\left(\frac{hi}{lo} + 1\right) \cdot \frac{hi - x}{lo}\right)}^{3} + 1\right) \cdot \frac{1}{\left(\frac{hi - x}{lo} \cdot \frac{hi - x}{lo} - \left(\frac{hi}{lo} + 1\right) \cdot \frac{hi - x}{lo}\right) + 1} \]
14. Add Preprocessing

Alternative 3: 31.8% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \left({\left(\left(\frac{hi}{lo} + 1\right) \cdot \frac{hi - x}{lo}\right)}^{3} + 1\right) \cdot \frac{1}{\frac{x - hi}{lo} + 1} \end{array} \]

FPCore

(FPCore (lo hi x)
 :precision binary64
 (*
  (+ (pow (* (+ (/ hi lo) 1.0) (/ (- hi x) lo)) 3.0) 1.0)
  (/ 1.0 (+ (/ (- x hi) lo) 1.0))))

C

double code(double lo, double hi, double x) {
	return (pow((((hi / lo) + 1.0) * ((hi - x) / lo)), 3.0) + 1.0) * (1.0 / (((x - hi) / lo) + 1.0));
}

Fortran

real(8) function code(lo, hi, x)
    real(8), intent (in) :: lo
    real(8), intent (in) :: hi
    real(8), intent (in) :: x
    code = (((((hi / lo) + 1.0d0) * ((hi - x) / lo)) ** 3.0d0) + 1.0d0) * (1.0d0 / (((x - hi) / lo) + 1.0d0))
end function

Java

public static double code(double lo, double hi, double x) {
	return (Math.pow((((hi / lo) + 1.0) * ((hi - x) / lo)), 3.0) + 1.0) * (1.0 / (((x - hi) / lo) + 1.0));
}

Python

def code(lo, hi, x):
	return (math.pow((((hi / lo) + 1.0) * ((hi - x) / lo)), 3.0) + 1.0) * (1.0 / (((x - hi) / lo) + 1.0))

Julia

function code(lo, hi, x)
	return Float64(Float64((Float64(Float64(Float64(hi / lo) + 1.0) * Float64(Float64(hi - x) / lo)) ^ 3.0) + 1.0) * Float64(1.0 / Float64(Float64(Float64(x - hi) / lo) + 1.0)))
end

MATLAB

function tmp = code(lo, hi, x)
	tmp = (((((hi / lo) + 1.0) * ((hi - x) / lo)) ^ 3.0) + 1.0) * (1.0 / (((x - hi) / lo) + 1.0));
end

Wolfram

code[lo_, hi_, x_] := N[(N[(N[Power[N[(N[(N[(hi / lo), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(hi - x), $MachinePrecision] / lo), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] + 1.0), $MachinePrecision] * N[(1.0 / N[(N[(N[(x - hi), $MachinePrecision] / lo), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 3.1%

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

3. Taylor expanded in lo around inf 0.0%

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

   1. associate--l+ 0.0%

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

   2. +-commutative 0.0%

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

   3. associate--l+ 0.0%

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

   4. *-commutative 0.0%

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

   5. distribute-lft-out-- 0.0%

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

   6. associate-*r* 0.0%

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

   7. *-commutative 0.0%

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

   8. associate-*r/ 0.0%

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

   9. distribute-lft-out-- 0.0%

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

   10. div-sub 0.0%

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

   11. +-commutative 0.0%

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

   12. mul-1-neg 0.0%

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

5. Simplified 18.8%

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

   1. flip3-+ 18.8%

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

   2. div-inv 18.8%

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

   3. metadata-eval 18.8%

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

   4. *-commutative 18.8%

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

   5. metadata-eval 18.8%

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

7. Applied egg-rr 18.8%

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

8. Taylor expanded in lo around inf 32.1%

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

9. Final simplification 32.1%

   \[\leadsto \left({\left(\left(\frac{hi}{lo} + 1\right) \cdot \frac{hi - x}{lo}\right)}^{3} + 1\right) \cdot \frac{1}{\frac{x - hi}{lo} + 1} \]
10. Add Preprocessing

Alternative 4: 18.9% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(lo, hi, x)
    real(8), intent (in) :: lo
    real(8), intent (in) :: hi
    real(8), intent (in) :: x
    code = ((hi * (((hi / lo) + 1.0d0) / lo)) - (x / lo)) + 1.0d0
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 3.1%

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

3. Taylor expanded in lo around inf 0.0%

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

   1. associate--l+ 0.0%

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

   2. +-commutative 0.0%

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

   3. associate--l+ 0.0%

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

   4. *-commutative 0.0%

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

   5. distribute-lft-out-- 0.0%

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

   6. associate-*r* 0.0%

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

   7. *-commutative 0.0%

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

   8. associate-*r/ 0.0%

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

   9. distribute-lft-out-- 0.0%

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

   10. div-sub 0.0%

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

   11. +-commutative 0.0%

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

   12. mul-1-neg 0.0%

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

5. Simplified 18.8%

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

6. Taylor expanded in x around 0 18.8%

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

   1. +-commutative 18.8%

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

   2. mul-1-neg 18.8%

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

   3. unsub-neg 18.8%

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

   4. associate-/l* 18.8%

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

   5. +-commutative 18.8%

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

8. Simplified 18.8%

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

9. Taylor expanded in lo around inf 18.8%

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

10. Final simplification 18.8%

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

Alternative 5: 18.9% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(lo, hi, x)
    real(8), intent (in) :: lo
    real(8), intent (in) :: hi
    real(8), intent (in) :: x
    code = (hi * (((hi / lo) + 1.0d0) / lo)) + 1.0d0
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 3.1%

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

3. Taylor expanded in lo around inf 0.0%

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

   1. associate--l+ 0.0%

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

   2. +-commutative 0.0%

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

   3. associate--l+ 0.0%

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

   4. *-commutative 0.0%

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

   5. distribute-lft-out-- 0.0%

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

   6. associate-*r* 0.0%

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

   7. *-commutative 0.0%

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

   8. associate-*r/ 0.0%

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

   9. distribute-lft-out-- 0.0%

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

   10. div-sub 0.0%

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

   11. +-commutative 0.0%

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

   12. mul-1-neg 0.0%

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

5. Simplified 18.8%

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

6. Taylor expanded in x around 0 18.8%

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

   1. associate-/l* 18.8%

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

   2. +-commutative 18.8%

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

8. Simplified 18.8%

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

9. Final simplification 18.8%

   \[\leadsto hi \cdot \frac{\frac{hi}{lo} + 1}{lo} + 1 \]
10. Add Preprocessing

Alternative 6: 18.7% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(lo, hi, x)
    real(8), intent (in) :: lo
    real(8), intent (in) :: hi
    real(8), intent (in) :: x
    code = 1.0d0 - (x / lo)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 3.1%

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

3. Taylor expanded in hi around 0 18.7%

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

   1. div-sub 18.7%

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

   2. sub-neg 18.7%

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

   3. *-inverses 18.7%

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

   4. metadata-eval 18.7%

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

   5. distribute-lft-in 18.7%

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

   6. metadata-eval 18.7%

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

   7. +-commutative 18.7%

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

   8. mul-1-neg 18.7%

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

   9. unsub-neg 18.7%

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

5. Simplified 18.7%

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

6. Final simplification 18.7%

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

Alternative 7: 18.8% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(lo, hi, x)
    real(8), intent (in) :: lo
    real(8), intent (in) :: hi
    real(8), intent (in) :: x
    code = (x - lo) / hi
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 3.1%

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

3. Taylor expanded in hi around inf 18.8%

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

4. Final simplification 18.8%

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

Alternative 8: 18.7% accurate, 7.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (lo hi x) :precision binary64 1.0)

C

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

Fortran

real(8) function code(lo, hi, x)
    real(8), intent (in) :: lo
    real(8), intent (in) :: hi
    real(8), intent (in) :: x
    code = 1.0d0
end function

Java

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

Python

def code(lo, hi, x):
	return 1.0

Julia

function code(lo, hi, x)
	return 1.0
end

MATLAB

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

Wolfram

code[lo_, hi_, x_] := 1.0

Derivation

1. Initial program 3.1%

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

3. Taylor expanded in lo around inf 18.7%

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

4. Final simplification 18.7%

   \[\leadsto 1 \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024047 
(FPCore (lo hi x)
  :name "xlohi (overflows)"
  :precision binary64
  :pre (and (< lo -1e+308) (> hi 1e+308))
  (/ (- x lo) (- hi lo)))