xlohi (overflows)

Percentage Accurate: 3.1% → 42.6%

Time: 14.8s

Alternatives: 7

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: 42.6% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{lo \cdot lo}{hi}\\ t_1 := \frac{hi - x}{t_0}\\ t_2 := \frac{hi + x}{lo}\\ 1 + \frac{\mathsf{fma}\left(t_2, t_2 + \frac{x - hi}{t_0}, {t_1}^{2}\right) \cdot \left(t_2 + t_1\right)}{{\left(\frac{hi - x}{lo \cdot \frac{lo}{hi}}\right)}^{2} + \left({t_2}^{2} + \frac{hi}{lo} \cdot \left(t_2 \cdot \frac{x - hi}{lo}\right)\right)} \end{array} \end{array} \]

FPCore

(FPCore (lo hi x)
 :precision binary64
 (let* ((t_0 (/ (* lo lo) hi)) (t_1 (/ (- hi x) t_0)) (t_2 (/ (+ hi x) lo)))
   (+
    1.0
    (/
     (* (fma t_2 (+ t_2 (/ (- x hi) t_0)) (pow t_1 2.0)) (+ t_2 t_1))
     (+
      (pow (/ (- hi x) (* lo (/ lo hi))) 2.0)
      (+ (pow t_2 2.0) (* (/ hi lo) (* t_2 (/ (- x hi) lo)))))))))

C

double code(double lo, double hi, double x) {
	double t_0 = (lo * lo) / hi;
	double t_1 = (hi - x) / t_0;
	double t_2 = (hi + x) / lo;
	return 1.0 + ((fma(t_2, (t_2 + ((x - hi) / t_0)), pow(t_1, 2.0)) * (t_2 + t_1)) / (pow(((hi - x) / (lo * (lo / hi))), 2.0) + (pow(t_2, 2.0) + ((hi / lo) * (t_2 * ((x - hi) / lo))))));
}

Julia

function code(lo, hi, x)
	t_0 = Float64(Float64(lo * lo) / hi)
	t_1 = Float64(Float64(hi - x) / t_0)
	t_2 = Float64(Float64(hi + x) / lo)
	return Float64(1.0 + Float64(Float64(fma(t_2, Float64(t_2 + Float64(Float64(x - hi) / t_0)), (t_1 ^ 2.0)) * Float64(t_2 + t_1)) / Float64((Float64(Float64(hi - x) / Float64(lo * Float64(lo / hi))) ^ 2.0) + Float64((t_2 ^ 2.0) + Float64(Float64(hi / lo) * Float64(t_2 * Float64(Float64(x - hi) / lo)))))))
end

Wolfram

code[lo_, hi_, x_] := Block[{t$95$0 = N[(N[(lo * lo), $MachinePrecision] / hi), $MachinePrecision]}, Block[{t$95$1 = N[(N[(hi - x), $MachinePrecision] / t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(hi + x), $MachinePrecision] / lo), $MachinePrecision]}, N[(1.0 + N[(N[(N[(t$95$2 * N[(t$95$2 + N[(N[(x - hi), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision] + N[Power[t$95$1, 2.0], $MachinePrecision]), $MachinePrecision] * N[(t$95$2 + t$95$1), $MachinePrecision]), $MachinePrecision] / N[(N[Power[N[(N[(hi - x), $MachinePrecision] / N[(lo * N[(lo / hi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[(N[Power[t$95$2, 2.0], $MachinePrecision] + N[(N[(hi / lo), $MachinePrecision] * N[(t$95$2 * N[(N[(x - hi), $MachinePrecision] / lo), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Initial program 3.1%

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

2. Taylor expanded in lo around inf 0.0%

   \[\leadsto \color{blue}{\left(1 + \left(-1 \cdot \frac{x}{lo} + \frac{hi \cdot \left(-1 \cdot x - -1 \cdot hi\right)}{{lo}^{2}}\right)\right) - -1 \cdot \frac{hi}{lo}} \]
3. 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. distribute-lft-out-- 0.0%

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

   5. div-sub 0.0%

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

   6. mul-1-neg 0.0%

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

   7. sub-neg 0.0%

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

   8. unpow2 0.0%

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

   9. times-frac 18.9%

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

   10. distribute-lft-out-- 18.9%

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

   11. associate-*r/ 18.9%

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

   12. fma-neg 18.9%

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

4. Simplified 18.9%

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

   1. clear-num 18.9%

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

   2. inv-pow 18.9%

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

   3. sub-neg 18.9%

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

   4. add-sqr-sqrt 9.5%

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

   5. sqrt-unprod 13.4%

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

   6. sqr-neg 13.4%

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

   7. sqrt-unprod 9.3%

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

   8. add-sqr-sqrt 18.9%

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

6. Applied egg-rr 18.9%

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

   1. fma-udef 18.9%

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

   2. unpow-1 18.9%

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

   3. clear-num 18.9%

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

   4. flip3-+ 18.9%

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

   5. pow2 18.9%

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

   6. clear-num 18.9%

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

   7. unpow-1 18.9%

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

   8. clear-num 18.9%

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

   9. unpow-1 18.9%

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

   10. pow-prod-up 18.9%

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

   11. metadata-eval 18.9%

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

8. Applied egg-rr 18.9%

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

   1. times-frac 0.0%

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

   2. *-commutative 0.0%

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

   3. associate-/l* 21.1%

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

   4. associate-*r/ 19.4%

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

   5. +-commutative 19.4%

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

   6. times-frac 0.0%

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

   7. *-commutative 0.0%

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

   8. associate-/l* 39.3%

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

   9. associate-*r/ 39.4%

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

10. Simplified 39.4%

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

    1. sum-cubes 39.4%

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

    2. +-commutative 39.4%

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

    3. +-commutative 39.4%

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

    4. +-commutative 39.4%

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

    5. +-commutative 39.4%

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

12. Applied egg-rr 39.4%

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

14. Simplified 41.1%

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

15. Final simplification 41.1%

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

Alternative 2: 42.6% accurate, 0.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 3.1%

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

2. Taylor expanded in lo around inf 0.0%

   \[\leadsto \color{blue}{\left(1 + \left(-1 \cdot \frac{x}{lo} + \frac{hi \cdot \left(-1 \cdot x - -1 \cdot hi\right)}{{lo}^{2}}\right)\right) - -1 \cdot \frac{hi}{lo}} \]
3. 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. distribute-lft-out-- 0.0%

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

   5. div-sub 0.0%

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

   6. mul-1-neg 0.0%

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

   7. sub-neg 0.0%

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

   8. unpow2 0.0%

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

   9. times-frac 18.9%

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

   10. distribute-lft-out-- 18.9%

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

   11. associate-*r/ 18.9%

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

   12. fma-neg 18.9%

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

4. Simplified 18.9%

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

   1. clear-num 18.9%

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

   2. inv-pow 18.9%

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

   3. sub-neg 18.9%

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

   4. add-sqr-sqrt 9.5%

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

   5. sqrt-unprod 13.4%

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

   6. sqr-neg 13.4%

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

   7. sqrt-unprod 9.3%

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

   8. add-sqr-sqrt 18.9%

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

6. Applied egg-rr 18.9%

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

   1. fma-udef 18.9%

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

   2. unpow-1 18.9%

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

   3. clear-num 18.9%

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

   4. flip3-+ 18.9%

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

   5. pow2 18.9%

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

   6. clear-num 18.9%

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

   7. unpow-1 18.9%

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

   8. clear-num 18.9%

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

   9. unpow-1 18.9%

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

   10. pow-prod-up 18.9%

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

   11. metadata-eval 18.9%

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

8. Applied egg-rr 18.9%

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

   1. times-frac 0.0%

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

   2. *-commutative 0.0%

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

   3. associate-/l* 21.1%

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

   4. associate-*r/ 19.4%

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

   5. +-commutative 19.4%

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

   6. times-frac 0.0%

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

   7. *-commutative 0.0%

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

   8. associate-/l* 39.3%

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

   9. associate-*r/ 39.4%

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

10. Simplified 39.4%

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

11. Taylor expanded in lo around inf 0.0%

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

    1. cube-div 41.1%

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

13. Simplified 41.1%

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

14. Final simplification 41.1%

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

Alternative 3: 41.0% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{hi + x}{lo}\\ t_1 := lo \cdot \frac{lo}{hi}\\ t_2 := \frac{x - hi}{t_1}\\ 1 + \frac{\left(t_2 \cdot t_2 + \left(t_0 \cdot t_0 + t_0 \cdot t_2\right)\right) \cdot \left(t_0 - t_2\right)}{{\left(\frac{hi - x}{t_1}\right)}^{2} + \left({t_0}^{2} + \frac{hi}{lo} \cdot \left(t_0 \cdot \frac{x - hi}{lo}\right)\right)} \end{array} \end{array} \]

FPCore

(FPCore (lo hi x)
 :precision binary64
 (let* ((t_0 (/ (+ hi x) lo)) (t_1 (* lo (/ lo hi))) (t_2 (/ (- x hi) t_1)))
   (+
    1.0
    (/
     (* (+ (* t_2 t_2) (+ (* t_0 t_0) (* t_0 t_2))) (- t_0 t_2))
     (+
      (pow (/ (- hi x) t_1) 2.0)
      (+ (pow t_0 2.0) (* (/ hi lo) (* t_0 (/ (- x hi) lo)))))))))

C

double code(double lo, double hi, double x) {
	double t_0 = (hi + x) / lo;
	double t_1 = lo * (lo / hi);
	double t_2 = (x - hi) / t_1;
	return 1.0 + ((((t_2 * t_2) + ((t_0 * t_0) + (t_0 * t_2))) * (t_0 - t_2)) / (pow(((hi - x) / t_1), 2.0) + (pow(t_0, 2.0) + ((hi / lo) * (t_0 * ((x - hi) / lo))))));
}

Fortran

real(8) function code(lo, hi, x)
    real(8), intent (in) :: lo
    real(8), intent (in) :: hi
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    t_0 = (hi + x) / lo
    t_1 = lo * (lo / hi)
    t_2 = (x - hi) / t_1
    code = 1.0d0 + ((((t_2 * t_2) + ((t_0 * t_0) + (t_0 * t_2))) * (t_0 - t_2)) / ((((hi - x) / t_1) ** 2.0d0) + ((t_0 ** 2.0d0) + ((hi / lo) * (t_0 * ((x - hi) / lo))))))
end function

Java

public static double code(double lo, double hi, double x) {
	double t_0 = (hi + x) / lo;
	double t_1 = lo * (lo / hi);
	double t_2 = (x - hi) / t_1;
	return 1.0 + ((((t_2 * t_2) + ((t_0 * t_0) + (t_0 * t_2))) * (t_0 - t_2)) / (Math.pow(((hi - x) / t_1), 2.0) + (Math.pow(t_0, 2.0) + ((hi / lo) * (t_0 * ((x - hi) / lo))))));
}

Python

def code(lo, hi, x):
	t_0 = (hi + x) / lo
	t_1 = lo * (lo / hi)
	t_2 = (x - hi) / t_1
	return 1.0 + ((((t_2 * t_2) + ((t_0 * t_0) + (t_0 * t_2))) * (t_0 - t_2)) / (math.pow(((hi - x) / t_1), 2.0) + (math.pow(t_0, 2.0) + ((hi / lo) * (t_0 * ((x - hi) / lo))))))

Julia

function code(lo, hi, x)
	t_0 = Float64(Float64(hi + x) / lo)
	t_1 = Float64(lo * Float64(lo / hi))
	t_2 = Float64(Float64(x - hi) / t_1)
	return Float64(1.0 + Float64(Float64(Float64(Float64(t_2 * t_2) + Float64(Float64(t_0 * t_0) + Float64(t_0 * t_2))) * Float64(t_0 - t_2)) / Float64((Float64(Float64(hi - x) / t_1) ^ 2.0) + Float64((t_0 ^ 2.0) + Float64(Float64(hi / lo) * Float64(t_0 * Float64(Float64(x - hi) / lo)))))))
end

MATLAB

function tmp = code(lo, hi, x)
	t_0 = (hi + x) / lo;
	t_1 = lo * (lo / hi);
	t_2 = (x - hi) / t_1;
	tmp = 1.0 + ((((t_2 * t_2) + ((t_0 * t_0) + (t_0 * t_2))) * (t_0 - t_2)) / ((((hi - x) / t_1) ^ 2.0) + ((t_0 ^ 2.0) + ((hi / lo) * (t_0 * ((x - hi) / lo))))));
end

Wolfram

code[lo_, hi_, x_] := Block[{t$95$0 = N[(N[(hi + x), $MachinePrecision] / lo), $MachinePrecision]}, Block[{t$95$1 = N[(lo * N[(lo / hi), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x - hi), $MachinePrecision] / t$95$1), $MachinePrecision]}, N[(1.0 + N[(N[(N[(N[(t$95$2 * t$95$2), $MachinePrecision] + N[(N[(t$95$0 * t$95$0), $MachinePrecision] + N[(t$95$0 * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$0 - t$95$2), $MachinePrecision]), $MachinePrecision] / N[(N[Power[N[(N[(hi - x), $MachinePrecision] / t$95$1), $MachinePrecision], 2.0], $MachinePrecision] + N[(N[Power[t$95$0, 2.0], $MachinePrecision] + N[(N[(hi / lo), $MachinePrecision] * N[(t$95$0 * N[(N[(x - hi), $MachinePrecision] / lo), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Initial program 3.1%

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

2. Taylor expanded in lo around inf 0.0%

   \[\leadsto \color{blue}{\left(1 + \left(-1 \cdot \frac{x}{lo} + \frac{hi \cdot \left(-1 \cdot x - -1 \cdot hi\right)}{{lo}^{2}}\right)\right) - -1 \cdot \frac{hi}{lo}} \]
3. 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. distribute-lft-out-- 0.0%

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

   5. div-sub 0.0%

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

   6. mul-1-neg 0.0%

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

   7. sub-neg 0.0%

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

   8. unpow2 0.0%

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

   9. times-frac 18.9%

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

   10. distribute-lft-out-- 18.9%

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

   11. associate-*r/ 18.9%

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

   12. fma-neg 18.9%

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

4. Simplified 18.9%

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

   1. clear-num 18.9%

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

   2. inv-pow 18.9%

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

   3. sub-neg 18.9%

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

   4. add-sqr-sqrt 9.5%

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

   5. sqrt-unprod 13.4%

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

   6. sqr-neg 13.4%

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

   7. sqrt-unprod 9.3%

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

   8. add-sqr-sqrt 18.9%

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

6. Applied egg-rr 18.9%

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

   1. fma-udef 18.9%

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

   2. unpow-1 18.9%

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

   3. clear-num 18.9%

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

   4. flip3-+ 18.9%

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

   5. pow2 18.9%

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

   6. clear-num 18.9%

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

   7. unpow-1 18.9%

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

   8. clear-num 18.9%

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

   9. unpow-1 18.9%

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

   10. pow-prod-up 18.9%

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

   11. metadata-eval 18.9%

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

8. Applied egg-rr 18.9%

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

   1. times-frac 0.0%

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

   2. *-commutative 0.0%

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

   3. associate-/l* 21.1%

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

   4. associate-*r/ 19.4%

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

   5. +-commutative 19.4%

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

   6. times-frac 0.0%

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

   7. *-commutative 0.0%

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

   8. associate-/l* 39.3%

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

   9. associate-*r/ 39.4%

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

10. Simplified 39.4%

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

    1. sum-cubes 39.4%

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

    2. +-commutative 39.4%

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

    3. +-commutative 39.4%

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

    4. +-commutative 39.4%

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

    5. +-commutative 39.4%

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

12. Applied egg-rr 39.4%

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

13. Final simplification 39.4%

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

Alternative 4: 22.7% accurate, 0.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 3.1%

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

2. Taylor expanded in lo around inf 0.0%

   \[\leadsto \color{blue}{\left(1 + \left(-1 \cdot \frac{x}{lo} + \frac{hi \cdot \left(-1 \cdot x - -1 \cdot hi\right)}{{lo}^{2}}\right)\right) - -1 \cdot \frac{hi}{lo}} \]
3. 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. distribute-lft-out-- 0.0%

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

   5. div-sub 0.0%

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

   6. mul-1-neg 0.0%

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

   7. sub-neg 0.0%

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

   8. unpow2 0.0%

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

   9. times-frac 18.9%

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

   10. distribute-lft-out-- 18.9%

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

   11. associate-*r/ 18.9%

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

   12. fma-neg 18.9%

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

4. Simplified 18.9%

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

5. Taylor expanded in lo around 0 0.0%

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

   1. +-commutative 0.0%

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

   2. associate--l+ 0.0%

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

   3. unpow2 0.0%

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

   4. times-frac 18.9%

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

   5. div-sub 18.9%

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

7. Simplified 18.9%

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

   1. flip-+ 18.9%

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

   2. div-sub 18.9%

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

   3. pow2 18.9%

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

   4. associate-*r/ 38.3%

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

   5. sub-div 95.3%

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

   6. pow2 95.3%

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

   7. associate-*r/ 76.0%

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

   8. sub-div 18.8%

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

9. Applied egg-rr 18.8%

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

    1. div-sub 18.8%

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

    2. associate-/r/ 18.8%

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

    3. times-frac 0.0%

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

    4. *-commutative 0.0%

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

    5. associate-/l* 22.7%

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

    6. associate-*r/ 22.4%

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

    7. associate--r- 22.1%

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

    8. *-commutative 22.1%

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

11. Simplified 22.1%

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

12. Final simplification 22.1%

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

Alternative 5: 22.8% accurate, 0.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 3.1%

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

2. Taylor expanded in lo around inf 0.0%

   \[\leadsto \color{blue}{\left(1 + \left(-1 \cdot \frac{x}{lo} + \frac{hi \cdot \left(-1 \cdot x - -1 \cdot hi\right)}{{lo}^{2}}\right)\right) - -1 \cdot \frac{hi}{lo}} \]
3. 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. distribute-lft-out-- 0.0%

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

   5. div-sub 0.0%

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

   6. mul-1-neg 0.0%

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

   7. sub-neg 0.0%

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

   8. unpow2 0.0%

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

   9. times-frac 18.9%

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

   10. distribute-lft-out-- 18.9%

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

   11. associate-*r/ 18.9%

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

   12. fma-neg 18.9%

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

4. Simplified 18.9%

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

5. Taylor expanded in lo around 0 0.0%

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

   1. +-commutative 0.0%

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

   2. associate--l+ 0.0%

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

   3. unpow2 0.0%

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

   4. times-frac 18.9%

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

   5. div-sub 18.9%

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

7. Simplified 18.9%

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

   1. flip-+ 18.9%

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

   2. frac-2neg 18.9%

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

   3. pow2 18.9%

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

   4. pow2 18.9%

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

   5. associate-*r/ 19.1%

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

   6. sub-div 18.8%

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

9. Applied egg-rr 18.8%

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

    1. times-frac 0.0%

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

    2. *-commutative 0.0%

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

    3. associate-/l* 22.8%

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

    4. associate-*r/ 22.5%

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

    5. distribute-neg-frac 22.5%

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

    6. associate--r- 22.2%

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

    7. *-commutative 22.2%

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

11. Simplified 22.2%

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

12. Final simplification 22.2%

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

Alternative 6: 19.5% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 3.1%

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

2. Taylor expanded in lo around inf 0.0%

   \[\leadsto \color{blue}{\left(1 + \left(-1 \cdot \frac{x}{lo} + \frac{hi \cdot \left(-1 \cdot x - -1 \cdot hi\right)}{{lo}^{2}}\right)\right) - -1 \cdot \frac{hi}{lo}} \]
3. 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. distribute-lft-out-- 0.0%

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

   5. div-sub 0.0%

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

   6. mul-1-neg 0.0%

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

   7. sub-neg 0.0%

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

   8. unpow2 0.0%

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

   9. times-frac 18.9%

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

   10. distribute-lft-out-- 18.9%

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

   11. associate-*r/ 18.9%

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

   12. fma-neg 18.9%

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

4. Simplified 18.9%

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

   1. expm1-log1p-u 18.9%

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

   2. expm1-udef 18.9%

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

6. Applied egg-rr 18.8%

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

   1. expm1-def 18.8%

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

   2. expm1-log1p 18.8%

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

   3. fma-def 18.8%

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

   4. distribute-lft1-in 18.9%

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

   5. +-commutative 18.9%

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

   6. +-commutative 18.9%

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

   7. fma-def 18.9%

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

8. Simplified 18.9%

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

9. Taylor expanded in hi around -inf 0.0%

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

    1. +-commutative 0.0%

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

    2. mul-1-neg 0.0%

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

    3. unsub-neg 0.0%

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

    4. unpow2 0.0%

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

    5. unpow2 0.0%

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

    6. times-frac 9.9%

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

    7. unpow2 9.9%

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

    8. unpow2 9.9%

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

    9. associate-*r/ 9.9%

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

    10. neg-mul-1 9.9%

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

11. Simplified 9.9%

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

12. Taylor expanded in lo around 0 0.0%

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

    1. associate-*r* 0.0%

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

    2. neg-mul-1 0.0%

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

    3. cancel-sign-sub 0.0%

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

    4. unpow2 0.0%

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

    5. distribute-lft-in 0.0%

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

    6. unpow2 0.0%

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

    7. times-frac 19.4%

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

14. Simplified 19.4%

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

15. Final simplification 19.4%

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

Alternative 7: 18.7% accurate, 7.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[lo_, hi_, x_] := 1.0

Derivation

1. Initial program 3.1%

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

2. Taylor expanded in lo around inf 18.7%

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

3. Final simplification 18.7%

   \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023277 
(FPCore (lo hi x)
  :name "xlohi (overflows)"
  :precision binary64
  :pre (and (< lo -1e+308) (> hi 1e+308))
  (/ (- x lo) (- hi lo)))