xlohi (overflows)

Percentage Accurate: 3.1% → 97.9%

Time: 17.2s

Alternatives: 10

Speedup: 7.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 3.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.9% accurate, 0.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

   \[\leadsto \color{blue}{\left(1 + \left(-1 \cdot \frac{x}{lo} + \frac{hi \cdot \left(-1 \cdot x - -1 \cdot hi\right)}{{lo}^{2}}\right)\right) - -1 \cdot \frac{hi}{lo}} \]
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 18.9%

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

   1. +-commutative 18.9%

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

   2. flip3-+ 18.9%

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

   3. pow3 18.9%

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

   4. metadata-eval 18.9%

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

   5. +-commutative 18.9%

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

   6. pow3 18.9%

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

   7. *-commutative 18.9%

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

7. Applied egg-rr 18.9%

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

   1. *-commutative 18.9%

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

   2. *-commutative 18.9%

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

   3. *-commutative 18.9%

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

9. Simplified 18.9%

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

10. Taylor expanded in hi around 0 32.2%

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

    1. distribute-lft-out 32.2%

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

    2. mul-1-neg 32.2%

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

    3. fma-define 32.2%

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

    4. sub-neg 32.2%

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

    5. distribute-neg-frac 32.2%

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

    6. metadata-eval 32.2%

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

12. Simplified 32.2%

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

13. Taylor expanded in x around inf 97.8%

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

    1. mul-1-neg 97.8%

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

    2. associate-/l* 97.8%

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

    3. distribute-lft-neg-in 97.8%

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

    4. +-commutative 97.8%

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

15. Simplified 97.8%

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

16. Final simplification 97.8%

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

Alternative 2: 97.9% accurate, 0.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

   \[\leadsto \color{blue}{\left(1 + \left(-1 \cdot \frac{x}{lo} + \frac{hi \cdot \left(-1 \cdot x - -1 \cdot hi\right)}{{lo}^{2}}\right)\right) - -1 \cdot \frac{hi}{lo}} \]
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 18.9%

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

   1. +-commutative 18.9%

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

   2. flip3-+ 18.9%

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

   3. pow3 18.9%

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

   4. metadata-eval 18.9%

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

   5. +-commutative 18.9%

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

   6. pow3 18.9%

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

   7. *-commutative 18.9%

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

7. Applied egg-rr 18.9%

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

   1. *-commutative 18.9%

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

   2. *-commutative 18.9%

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

   3. *-commutative 18.9%

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

9. Simplified 18.9%

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

10. Taylor expanded in hi around 0 32.2%

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

    1. distribute-lft-out 32.2%

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

    2. mul-1-neg 32.2%

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

    3. fma-define 32.2%

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

    4. sub-neg 32.2%

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

    5. distribute-neg-frac 32.2%

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

    6. metadata-eval 32.2%

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

12. Simplified 32.2%

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

13. Taylor expanded in hi around 0 97.8%

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

    1. mul-1-neg 32.1%

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

    2. distribute-frac-neg2 32.1%

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

15. Simplified 97.8%

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

16. Final simplification 97.8%

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

Alternative 3: 32.0% accurate, 0.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[lo_, hi_, x_] := Block[{t$95$0 = N[(1.0 + N[(hi / lo), $MachinePrecision]), $MachinePrecision]}, N[(N[(1.0 + N[Power[N[(t$95$0 * N[(N[(hi - x), $MachinePrecision] / lo), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision] / N[(N[(1.0 + N[(t$95$0 * N[(N[(x - hi), $MachinePrecision] / lo), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[Power[N[(N[(hi * N[(N[(1.0 / lo), $MachinePrecision] - N[(x / N[Power[lo, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x / lo), $MachinePrecision]), $MachinePrecision], 2.0], $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 18.9%

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

   1. +-commutative 18.9%

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

   2. flip3-+ 18.9%

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

   3. pow3 18.9%

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

   4. metadata-eval 18.9%

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

   5. +-commutative 18.9%

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

   6. pow3 18.9%

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

   7. *-commutative 18.9%

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

7. Applied egg-rr 18.9%

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

   1. *-commutative 18.9%

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

   2. *-commutative 18.9%

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

   3. *-commutative 18.9%

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

9. Simplified 18.9%

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

10. Taylor expanded in hi around 0 32.2%

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

    1. distribute-lft-out 32.2%

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

    2. mul-1-neg 32.2%

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

    3. fma-define 32.2%

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

    4. sub-neg 32.2%

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

    5. distribute-neg-frac 32.2%

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

    6. metadata-eval 32.2%

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

12. Simplified 32.2%

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

13. Taylor expanded in hi around 0 32.2%

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

14. Final simplification 32.2%

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

Alternative 4: 32.0% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x - hi}{lo}\\ t_1 := 1 + \frac{hi}{lo}\\ \frac{1 + {\left(t\_1 \cdot \frac{hi - x}{lo}\right)}^{3}}{\left(1 + t\_1 \cdot t\_0\right) + {t\_0}^{2}} \end{array} \end{array} \]

FPCore

(FPCore (lo hi x)
 :precision binary64
 (let* ((t_0 (/ (- x hi) lo)) (t_1 (+ 1.0 (/ hi lo))))
   (/
    (+ 1.0 (pow (* t_1 (/ (- hi x) lo)) 3.0))
    (+ (+ 1.0 (* t_1 t_0)) (pow t_0 2.0)))))

C

double code(double lo, double hi, double x) {
	double t_0 = (x - hi) / lo;
	double t_1 = 1.0 + (hi / lo);
	return (1.0 + pow((t_1 * ((hi - x) / lo)), 3.0)) / ((1.0 + (t_1 * t_0)) + pow(t_0, 2.0));
}

Fortran

real(8) function code(lo, hi, x)
    real(8), intent (in) :: lo
    real(8), intent (in) :: hi
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    t_0 = (x - hi) / lo
    t_1 = 1.0d0 + (hi / lo)
    code = (1.0d0 + ((t_1 * ((hi - x) / lo)) ** 3.0d0)) / ((1.0d0 + (t_1 * t_0)) + (t_0 ** 2.0d0))
end function

Java

public static double code(double lo, double hi, double x) {
	double t_0 = (x - hi) / lo;
	double t_1 = 1.0 + (hi / lo);
	return (1.0 + Math.pow((t_1 * ((hi - x) / lo)), 3.0)) / ((1.0 + (t_1 * t_0)) + Math.pow(t_0, 2.0));
}

Python

def code(lo, hi, x):
	t_0 = (x - hi) / lo
	t_1 = 1.0 + (hi / lo)
	return (1.0 + math.pow((t_1 * ((hi - x) / lo)), 3.0)) / ((1.0 + (t_1 * t_0)) + math.pow(t_0, 2.0))

Julia

function code(lo, hi, x)
	t_0 = Float64(Float64(x - hi) / lo)
	t_1 = Float64(1.0 + Float64(hi / lo))
	return Float64(Float64(1.0 + (Float64(t_1 * Float64(Float64(hi - x) / lo)) ^ 3.0)) / Float64(Float64(1.0 + Float64(t_1 * t_0)) + (t_0 ^ 2.0)))
end

MATLAB

function tmp = code(lo, hi, x)
	t_0 = (x - hi) / lo;
	t_1 = 1.0 + (hi / lo);
	tmp = (1.0 + ((t_1 * ((hi - x) / lo)) ^ 3.0)) / ((1.0 + (t_1 * t_0)) + (t_0 ^ 2.0));
end

Wolfram

code[lo_, hi_, x_] := Block[{t$95$0 = N[(N[(x - hi), $MachinePrecision] / lo), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 + N[(hi / lo), $MachinePrecision]), $MachinePrecision]}, N[(N[(1.0 + N[Power[N[(t$95$1 * N[(N[(hi - x), $MachinePrecision] / lo), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision] / N[(N[(1.0 + N[(t$95$1 * t$95$0), $MachinePrecision]), $MachinePrecision] + N[Power[t$95$0, 2.0], $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 18.9%

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

   1. +-commutative 18.9%

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

   2. flip3-+ 18.9%

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

   3. pow3 18.9%

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

   4. metadata-eval 18.9%

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

   5. +-commutative 18.9%

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

   6. pow3 18.9%

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

   7. *-commutative 18.9%

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

7. Applied egg-rr 18.9%

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

   1. *-commutative 18.9%

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

   2. *-commutative 18.9%

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

   3. *-commutative 18.9%

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

9. Simplified 18.9%

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

10. Taylor expanded in lo around inf 0.0%

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

    1. unpow2 0.0%

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

    2. unpow2 0.0%

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

    3. times-frac 32.2%

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

    4. unpow2 32.2%

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

12. Simplified 32.2%

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

13. Final simplification 32.2%

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

Alternative 5: 32.0% accurate, 0.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[lo_, hi_, x_] := Block[{t$95$0 = N[(N[(hi - x), $MachinePrecision] / lo), $MachinePrecision]}, N[(N[(1.0 + N[Power[N[(N[(1.0 + N[(hi / lo), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision] / N[(N[Power[N[(x / (-lo)), $MachinePrecision], 2.0], $MachinePrecision] + N[(1.0 - t$95$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 18.9%

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

   1. +-commutative 18.9%

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

   2. flip3-+ 18.9%

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

   3. pow3 18.9%

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

   4. metadata-eval 18.9%

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

   5. +-commutative 18.9%

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

   6. pow3 18.9%

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

   7. *-commutative 18.9%

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

7. Applied egg-rr 18.9%

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

   1. *-commutative 18.9%

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

   2. *-commutative 18.9%

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

   3. *-commutative 18.9%

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

9. Simplified 18.9%

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

10. Taylor expanded in lo around inf 29.6%

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

    1. mul-1-neg 29.6%

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

    2. distribute-frac-neg 29.6%

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

    3. sub-neg 29.6%

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

    4. distribute-neg-in 29.6%

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

    5. neg-mul-1 29.6%

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

    6. remove-double-neg 29.6%

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

    7. +-commutative 29.6%

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

    8. neg-mul-1 29.6%

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

    9. sub-neg 29.6%

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

12. Simplified 29.6%

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

13. Taylor expanded in hi around 0 32.1%

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

    1. mul-1-neg 32.1%

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

    2. distribute-frac-neg2 32.1%

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

15. Simplified 32.1%

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

16. Final simplification 32.1%

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

Alternative 6: 29.5% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + \frac{hi}{lo}\\ t_1 := \frac{hi - x}{lo}\\ t_2 := t\_0 \cdot t\_1\\ \frac{1 + {t\_2}^{3}}{\left(1 - t\_1\right) + t\_1 \cdot \left(t\_0 \cdot t\_2\right)} \end{array} \end{array} \]

FPCore

(FPCore (lo hi x)
 :precision binary64
 (let* ((t_0 (+ 1.0 (/ hi lo))) (t_1 (/ (- hi x) lo)) (t_2 (* t_0 t_1)))
   (/ (+ 1.0 (pow t_2 3.0)) (+ (- 1.0 t_1) (* t_1 (* t_0 t_2))))))

C

double code(double lo, double hi, double x) {
	double t_0 = 1.0 + (hi / lo);
	double t_1 = (hi - x) / lo;
	double t_2 = t_0 * t_1;
	return (1.0 + pow(t_2, 3.0)) / ((1.0 - t_1) + (t_1 * (t_0 * t_2)));
}

Fortran

real(8) function code(lo, hi, x)
    real(8), intent (in) :: lo
    real(8), intent (in) :: hi
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    t_0 = 1.0d0 + (hi / lo)
    t_1 = (hi - x) / lo
    t_2 = t_0 * t_1
    code = (1.0d0 + (t_2 ** 3.0d0)) / ((1.0d0 - t_1) + (t_1 * (t_0 * t_2)))
end function

Java

public static double code(double lo, double hi, double x) {
	double t_0 = 1.0 + (hi / lo);
	double t_1 = (hi - x) / lo;
	double t_2 = t_0 * t_1;
	return (1.0 + Math.pow(t_2, 3.0)) / ((1.0 - t_1) + (t_1 * (t_0 * t_2)));
}

Python

def code(lo, hi, x):
	t_0 = 1.0 + (hi / lo)
	t_1 = (hi - x) / lo
	t_2 = t_0 * t_1
	return (1.0 + math.pow(t_2, 3.0)) / ((1.0 - t_1) + (t_1 * (t_0 * t_2)))

Julia

function code(lo, hi, x)
	t_0 = Float64(1.0 + Float64(hi / lo))
	t_1 = Float64(Float64(hi - x) / lo)
	t_2 = Float64(t_0 * t_1)
	return Float64(Float64(1.0 + (t_2 ^ 3.0)) / Float64(Float64(1.0 - t_1) + Float64(t_1 * Float64(t_0 * t_2))))
end

MATLAB

function tmp = code(lo, hi, x)
	t_0 = 1.0 + (hi / lo);
	t_1 = (hi - x) / lo;
	t_2 = t_0 * t_1;
	tmp = (1.0 + (t_2 ^ 3.0)) / ((1.0 - t_1) + (t_1 * (t_0 * t_2)));
end

Wolfram

code[lo_, hi_, x_] := Block[{t$95$0 = N[(1.0 + N[(hi / lo), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(hi - x), $MachinePrecision] / lo), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$0 * t$95$1), $MachinePrecision]}, N[(N[(1.0 + N[Power[t$95$2, 3.0], $MachinePrecision]), $MachinePrecision] / N[(N[(1.0 - t$95$1), $MachinePrecision] + N[(t$95$1 * N[(t$95$0 * t$95$2), $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%

   \[\leadsto \color{blue}{\left(1 + \left(-1 \cdot \frac{x}{lo} + \frac{hi \cdot \left(-1 \cdot x - -1 \cdot hi\right)}{{lo}^{2}}\right)\right) - -1 \cdot \frac{hi}{lo}} \]
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 18.9%

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

   1. +-commutative 18.9%

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

   2. flip3-+ 18.9%

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

   3. pow3 18.9%

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

   4. metadata-eval 18.9%

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

   5. +-commutative 18.9%

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

   6. pow3 18.9%

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

   7. *-commutative 18.9%

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

7. Applied egg-rr 18.9%

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

   1. *-commutative 18.9%

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

   2. *-commutative 18.9%

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

   3. *-commutative 18.9%

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

9. Simplified 18.9%

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

10. Taylor expanded in lo around inf 29.6%

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

    1. mul-1-neg 29.6%

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

    2. distribute-frac-neg 29.6%

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

    3. sub-neg 29.6%

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

    4. distribute-neg-in 29.6%

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

    5. neg-mul-1 29.6%

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

    6. remove-double-neg 29.6%

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

    7. +-commutative 29.6%

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

    8. neg-mul-1 29.6%

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

    9. sub-neg 29.6%

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

12. Simplified 29.6%

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

    1. unpow2 29.6%

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

    2. *-commutative 29.6%

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

    3. associate-*r* 29.6%

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

    4. *-commutative 29.6%

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

14. Applied egg-rr 29.6%

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

15. Final simplification 29.6%

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

Alternative 7: 21.1% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

   \[\leadsto \color{blue}{\left(1 + \left(-1 \cdot \frac{x}{lo} + \frac{hi \cdot \left(-1 \cdot x - -1 \cdot hi\right)}{{lo}^{2}}\right)\right) - -1 \cdot \frac{hi}{lo}} \]
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 18.9%

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

   1. flip3-- 18.9%

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

   2. associate-*r/ 18.9%

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

   3. metadata-eval 18.9%

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

   4. metadata-eval 18.9%

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

   5. distribute-rgt-out 18.9%

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

   6. +-commutative 18.9%

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

7. Applied egg-rr 18.9%

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

8. Taylor expanded in lo around inf 21.1%

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

   1. associate-*r/ 21.1%

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

   2. neg-mul-1 21.1%

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

10. Simplified 21.1%

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

11. Final simplification 21.1%

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

Alternative 8: 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 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 18.9%

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

   1. add-cbrt-cube 18.9%

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

   2. pow3 18.9%

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

   3. +-commutative 18.9%

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

   4. *-commutative 18.9%

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

   5. fma-define 18.9%

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

7. Applied egg-rr 18.9%

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

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 19.3%

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

   4. unpow2 19.3%

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

10. Simplified 19.3%

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

    1. unpow2 19.3%

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

12. Applied egg-rr 19.3%

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

13. Final simplification 19.3%

    \[\leadsto \frac{hi}{lo} \cdot \frac{hi}{lo} \]
14. Add Preprocessing

Alternative 9: 18.8% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 3.1%

   \[\frac{x - lo}{hi - lo} \]
2. 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. neg-mul-1 18.8%

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

   2. distribute-neg-frac2 18.8%

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

6. Simplified 18.8%

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

7. Final simplification 18.8%

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

Alternative 10: 18.7% accurate, 7.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[lo_, hi_, x_] := 1.0

Derivation

1. Initial program 3.1%

   \[\frac{x - lo}{hi - lo} \]
2. 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 2024039 
(FPCore (lo hi x)
  :name "xlohi (overflows)"
  :precision binary64
  :pre (and (< lo -1e+308) (> hi 1e+308))
  (/ (- x lo) (- hi lo)))