xlohi (overflows)

Percentage Accurate: 3.1% → 98.5%

Time: 22.4s

Alternatives: 9

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: 98.5% accurate, 0.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

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. frac-2neg 18.9%

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

   3. metadata-eval 18.9%

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

   4. +-commutative 18.9%

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

   5. metadata-eval 18.9%

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

7. Applied egg-rr 18.9%

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

   1. distribute-neg-in 18.9%

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

   2. metadata-eval 18.9%

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

   3. unsub-neg 18.9%

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

   4. associate-*r/ 18.9%

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

   5. *-commutative 18.9%

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

   6. *-lft-identity 18.9%

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

   7. times-frac 18.9%

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

   8. rem-square-sqrt 18.9%

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

   9. associate-*r/ 18.9%

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

   10. /-rgt-identity 18.9%

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

   11. rem-square-sqrt 18.9%

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

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 32.8%

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

11. Taylor expanded in hi around 0 98.4%

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

    1. associate-*r/ 98.4%

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

    2. mul-1-neg 98.4%

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

13. Simplified 98.4%

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

14. Final simplification 98.4%

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

Alternative 2: 32.0% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

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. frac-2neg 18.9%

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

   3. metadata-eval 18.9%

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

   4. +-commutative 18.9%

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

   5. metadata-eval 18.9%

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

7. Applied egg-rr 18.9%

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

   1. distribute-neg-in 18.9%

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

   2. metadata-eval 18.9%

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

   3. unsub-neg 18.9%

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

   4. associate-*r/ 18.9%

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

   5. *-commutative 18.9%

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

   6. *-lft-identity 18.9%

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

   7. times-frac 18.9%

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

   8. rem-square-sqrt 18.9%

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

   9. associate-*r/ 18.9%

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

   10. /-rgt-identity 18.9%

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

   11. rem-square-sqrt 18.9%

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

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 32.8%

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

    1. unpow2 32.8%

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

    2. clear-num 32.8%

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

    3. add-sqr-sqrt 0.0%

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

    4. fabs-sqr 0.0%

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

    5. add-sqr-sqrt 9.7%

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

    6. un-div-inv 9.7%

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

    7. add-sqr-sqrt 0.0%

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

    8. fabs-sqr 0.0%

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

    9. add-sqr-sqrt 32.8%

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

12. Applied egg-rr 32.8%

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

13. Final simplification 32.8%

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

Alternative 3: 32.0% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

function tmp = code(lo, hi, x)
	t_0 = (hi - x) * (((hi / lo) + 1.0) / lo);
	tmp = (-1.0 - (t_0 ^ 3.0)) / (-1.0 + (t_0 + (((hi - x) / lo) * ((x - hi) / lo))));
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[(-1.0 - N[Power[t$95$0, 3.0], $MachinePrecision]), $MachinePrecision] / N[(-1.0 + N[(t$95$0 + N[(N[(N[(hi - x), $MachinePrecision] / lo), $MachinePrecision] * N[(N[(x - hi), $MachinePrecision] / lo), $MachinePrecision]), $MachinePrecision]), $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%

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

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. frac-2neg 18.9%

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

   3. metadata-eval 18.9%

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

   4. +-commutative 18.9%

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

   5. metadata-eval 18.9%

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

7. Applied egg-rr 18.9%

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

   1. distribute-neg-in 18.9%

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

   2. metadata-eval 18.9%

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

   3. unsub-neg 18.9%

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

   4. associate-*r/ 18.9%

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

   5. *-commutative 18.9%

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

   6. *-lft-identity 18.9%

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

   7. times-frac 18.9%

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

   8. rem-square-sqrt 18.9%

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

   9. associate-*r/ 18.9%

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

   10. /-rgt-identity 18.9%

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

   11. rem-square-sqrt 18.9%

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

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 32.8%

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

    1. unpow2 32.8%

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

12. Applied egg-rr 32.8%

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

13. Final simplification 32.8%

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

Alternative 4: 18.9% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

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. add-sqr-sqrt 0.0%

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

   2. sqrt-unprod 19.1%

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

   3. pow2 19.1%

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

7. Applied egg-rr 19.1%

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

   1. unpow2 19.1%

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

   2. rem-sqrt-square 19.1%

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

9. Simplified 19.1%

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

10. Final simplification 19.1%

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

Alternative 5: 18.9% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

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. add-sqr-sqrt 0.0%

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

   2. sqrt-unprod 19.1%

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

   3. pow2 19.1%

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

7. Applied egg-rr 19.1%

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

   1. unpow2 19.1%

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

   2. rem-sqrt-square 19.1%

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

9. Simplified 19.1%

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

    1. div-inv 19.1%

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

11. Applied egg-rr 19.1%

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

12. Taylor expanded in hi around inf 19.1%

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

13. Final simplification 19.1%

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

Alternative 6: 18.9% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

5. Simplified 18.9%

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

6. Final simplification 18.9%

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

Alternative 7: 18.9% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

5. Simplified 18.9%

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

6. Taylor expanded in x around 0 18.9%

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

   1. associate-/l* 18.9%

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

8. Simplified 18.9%

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

9. Final simplification 18.9%

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

Alternative 8: 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(lo / Float64(-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. Add Preprocessing

3. Taylor expanded in hi around inf 18.8%

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

4. Taylor expanded in x around 0 18.8%

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

   1. associate-*r/ 18.8%

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

   2. neg-mul-1 18.8%

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

6. Simplified 18.8%

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

7. Final simplification 18.8%

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

Alternative 9: 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 2024095 
(FPCore (lo hi x)
  :name "xlohi (overflows)"
  :precision binary64
  :pre (and (< lo -1e+308) (> hi 1e+308))
  (/ (- x lo) (- hi lo)))