xlohi (overflows)

Percentage Accurate: 3.1% → 31.8%

Time: 18.1s

Alternatives: 6

Speedup: 7.0×

Specification

\[lo < -1 \cdot 10^{+308} \land hi > 10^{+308}\]

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 3.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 31.8% accurate, 0.0× speedup

Math

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

FPCore

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

C

double code(double lo, double hi, double x) {
	return (pow(((hi - x) * (((hi / lo) + 1.0) / lo)), 3.0) + 1.0) / ((pow(((hi - x) * (1.0 / lo)), 2.0) + ((hi - x) * ((-1.0 - (hi / lo)) / 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 - x) * (((hi / lo) + 1.0d0) / lo)) ** 3.0d0) + 1.0d0) / (((((hi - x) * (1.0d0 / lo)) ** 2.0d0) + ((hi - x) * (((-1.0d0) - (hi / lo)) / lo))) + 1.0d0)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[lo_, hi_, x_] := N[(N[(N[Power[N[(N[(hi - x), $MachinePrecision] * N[(N[(N[(hi / lo), $MachinePrecision] + 1.0), $MachinePrecision] / lo), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] + 1.0), $MachinePrecision] / N[(N[(N[Power[N[(N[(hi - x), $MachinePrecision] * N[(1.0 / lo), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[(N[(hi - x), $MachinePrecision] * N[(N[(-1.0 - N[(hi / lo), $MachinePrecision]), $MachinePrecision] / lo), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $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%

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

5. Simplified 18.9%

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

   1. flip3-+ 18.9%

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

   2. metadata-eval 18.9%

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

   3. +-commutative 18.9%

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

   4. metadata-eval 18.9%

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

   5. pow2 18.9%

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

   6. +-commutative 18.9%

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

   7. *-un-lft-identity 18.9%

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

   8. +-commutative 18.9%

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

7. Applied egg-rr 18.9%

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

   1. associate-*r/ 18.9%

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

   2. *-commutative 18.9%

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

   3. associate-/l* 18.9%

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

   4. associate-*r/ 18.9%

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

   5. *-commutative 18.9%

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

   6. associate-/l* 18.9%

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

   7. associate-*r/ 18.9%

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

   8. *-commutative 18.9%

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

   9. associate-/l* 18.9%

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

9. Simplified 18.9%

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

10. Taylor expanded in hi around 0 31.7%

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

11. Final simplification 31.7%

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

Alternative 2: 31.8% accurate, 0.0× speedup

Math

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

FPCore

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

C

double code(double lo, double hi, double x) {
	return (pow(((hi - x) * (((hi / lo) + 1.0) / lo)), 3.0) + 1.0) / ((pow(((hi - x) / lo), 2.0) + ((hi - x) * ((-1.0 - (hi / lo)) / 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 - x) * (((hi / lo) + 1.0d0) / lo)) ** 3.0d0) + 1.0d0) / (((((hi - x) / lo) ** 2.0d0) + ((hi - x) * (((-1.0d0) - (hi / lo)) / lo))) + 1.0d0)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[lo_, hi_, x_] := N[(N[(N[Power[N[(N[(hi - x), $MachinePrecision] * N[(N[(N[(hi / lo), $MachinePrecision] + 1.0), $MachinePrecision] / lo), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] + 1.0), $MachinePrecision] / N[(N[(N[Power[N[(N[(hi - x), $MachinePrecision] / lo), $MachinePrecision], 2.0], $MachinePrecision] + N[(N[(hi - x), $MachinePrecision] * N[(N[(-1.0 - N[(hi / lo), $MachinePrecision]), $MachinePrecision] / lo), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $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%

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

5. Simplified 18.9%

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

   1. flip3-+ 18.9%

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

   2. metadata-eval 18.9%

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

   3. +-commutative 18.9%

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

   4. metadata-eval 18.9%

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

   5. pow2 18.9%

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

   6. +-commutative 18.9%

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

   7. *-un-lft-identity 18.9%

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

   8. +-commutative 18.9%

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

7. Applied egg-rr 18.9%

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

   1. associate-*r/ 18.9%

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

   2. *-commutative 18.9%

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

   3. associate-/l* 18.9%

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

   4. associate-*r/ 18.9%

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

   5. *-commutative 18.9%

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

   6. associate-/l* 18.9%

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

   7. associate-*r/ 18.9%

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

   8. *-commutative 18.9%

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

   9. associate-/l* 18.9%

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

9. Simplified 18.9%

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

10. Taylor expanded in lo around inf 31.7%

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

11. Final simplification 31.7%

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

Alternative 3: 29.4% accurate, 0.0× speedup

Math

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

FPCore

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

C

double code(double lo, double hi, double x) {
	double t_0 = (hi - x) * (((hi / lo) + 1.0) / lo);
	return (pow(t_0, 3.0) + 1.0) / ((pow(t_0, 2.0) + ((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
    real(8) :: t_0
    t_0 = (hi - x) * (((hi / lo) + 1.0d0) / lo)
    code = ((t_0 ** 3.0d0) + 1.0d0) / (((t_0 ** 2.0d0) + ((x - hi) / lo)) + 1.0d0)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

5. Simplified 18.9%

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

   1. flip3-+ 18.9%

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

   2. metadata-eval 18.9%

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

   3. +-commutative 18.9%

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

   4. metadata-eval 18.9%

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

   5. pow2 18.9%

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

   6. +-commutative 18.9%

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

   7. *-un-lft-identity 18.9%

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

   8. +-commutative 18.9%

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

7. Applied egg-rr 18.9%

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

   1. associate-*r/ 18.9%

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

   2. *-commutative 18.9%

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

   3. associate-/l* 18.9%

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

   4. associate-*r/ 18.9%

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

   5. *-commutative 18.9%

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

   6. associate-/l* 18.9%

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

   7. associate-*r/ 18.9%

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

   8. *-commutative 18.9%

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

   9. associate-/l* 18.9%

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

9. Simplified 18.9%

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

10. Taylor expanded in lo around inf 29.3%

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

11. Final simplification 29.3%

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

Alternative 4: 19.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double lo, double hi, double x) {
	return (hi / 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 = (hi / lo) * (hi / lo)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 3.1%

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

3. Taylor expanded in lo around -inf 3.1%

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

   1. mul-1-neg 3.1%

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

   2. associate--l+ 3.1%

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

   3. associate-/l* 14.7%

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

5. Simplified 14.7%

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

   1. unsub-neg 14.7%

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

7. Applied egg-rr 14.7%

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

8. Taylor expanded in hi around inf 0.0%

   \[\leadsto \color{blue}{\frac{{hi}^{2}}{{lo}^{2}}} \]
9. Step-by-step derivation

   1. unpow2 0.0%

      \[\leadsto \frac{\color{blue}{hi \cdot hi}}{{lo}^{2}} \]

   2. unpow2 0.0%

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

   3. times-frac 19.5%

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

   4. unpow2 19.5%

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

10. Simplified 19.5%

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

    1. unpow2 19.5%

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

12. Applied egg-rr 19.5%

    \[\leadsto \color{blue}{\frac{hi}{lo} \cdot \frac{hi}{lo}} \]
13. Add Preprocessing

Alternative 5: 18.8% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 3.1%

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

3. Taylor expanded in hi around inf 18.8%

   \[\leadsto \color{blue}{\frac{x - lo}{hi}} \]
4. Add Preprocessing

Alternative 6: 18.7% accurate, 7.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[lo_, hi_, x_] := 1.0

Derivation

1. Initial program 3.1%

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

3. Taylor expanded in lo around inf 18.7%

   \[\leadsto \color{blue}{1} \]
4. Add Preprocessing

Reproduce

herbie shell --seed 2024085 
(FPCore (lo hi x)
  :name "xlohi (overflows)"
  :precision binary64
  :pre (and (< lo -1e+308) (> hi 1e+308))
  (/ (- x lo) (- hi lo)))