xlohi (overflows)

Percentage Accurate: 3.1% → 19.0%

Time: 10.9s

Alternatives: 11

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: 19.0% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ 1 + \mathsf{fma}\left(\frac{hi}{lo}, \frac{hi - x}{lo}, \log \left(1 + \mathsf{fma}\left(0.16666666666666666, {\left(\frac{hi + x}{lo}\right)}^{3}, \mathsf{fma}\left(0.5, \frac{hi}{lo} \cdot \frac{hi}{lo}, \mathsf{fma}\left(x, \frac{1}{lo} + \frac{hi}{lo \cdot lo}, \frac{hi}{lo}\right)\right)\right)\right)\right) \end{array} \]

FPCore

(FPCore (lo hi x)
 :precision binary64
 (+
  1.0
  (fma
   (/ hi lo)
   (/ (- hi x) lo)
   (log
    (+
     1.0
     (fma
      0.16666666666666666
      (pow (/ (+ hi x) lo) 3.0)
      (fma
       0.5
       (* (/ hi lo) (/ hi lo))
       (fma x (+ (/ 1.0 lo) (/ hi (* lo lo))) (/ hi lo)))))))))

C

double code(double lo, double hi, double x) {
	return 1.0 + fma((hi / lo), ((hi - x) / lo), log((1.0 + fma(0.16666666666666666, pow(((hi + x) / lo), 3.0), fma(0.5, ((hi / lo) * (hi / lo)), fma(x, ((1.0 / lo) + (hi / (lo * lo))), (hi / lo)))))));
}

Julia

function code(lo, hi, x)
	return Float64(1.0 + fma(Float64(hi / lo), Float64(Float64(hi - x) / lo), log(Float64(1.0 + fma(0.16666666666666666, (Float64(Float64(hi + x) / lo) ^ 3.0), fma(0.5, Float64(Float64(hi / lo) * Float64(hi / lo)), fma(x, Float64(Float64(1.0 / lo) + Float64(hi / Float64(lo * lo))), Float64(hi / lo))))))))
end

Wolfram

code[lo_, hi_, x_] := N[(1.0 + N[(N[(hi / lo), $MachinePrecision] * N[(N[(hi - x), $MachinePrecision] / lo), $MachinePrecision] + N[Log[N[(1.0 + N[(0.16666666666666666 * N[Power[N[(N[(hi + x), $MachinePrecision] / lo), $MachinePrecision], 3.0], $MachinePrecision] + N[(0.5 * N[(N[(hi / lo), $MachinePrecision] * N[(hi / lo), $MachinePrecision]), $MachinePrecision] + N[(x * N[(N[(1.0 / lo), $MachinePrecision] + N[(hi / N[(lo * lo), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(hi / lo), $MachinePrecision]), $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 19.0%

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

   10. distribute-lft-out-- 19.0%

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

   11. associate-*r/ 19.0%

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

   12. fma-neg 19.0%

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

4. Simplified 19.0%

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

   1. add-log-exp 19.0%

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

   2. sub-neg 19.0%

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

   3. add-sqr-sqrt 10.2%

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

   4. sqrt-unprod 14.6%

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

   5. sqr-neg 14.6%

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

   6. sqrt-unprod 8.8%

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

   7. add-sqr-sqrt 19.0%

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

6. Applied egg-rr 19.0%

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

7. Taylor expanded in lo around inf 0.0%

   \[\leadsto 1 + \mathsf{fma}\left(\frac{hi}{lo}, \frac{hi - x}{lo}, \log \color{blue}{\left(1 + \left(0.16666666666666666 \cdot \frac{{\left(hi + x\right)}^{3}}{{lo}^{3}} + \left(0.5 \cdot \frac{{\left(hi + x\right)}^{2}}{{lo}^{2}} + \left(\frac{hi}{lo} + \frac{x}{lo}\right)\right)\right)\right)}\right) \]
8. Step-by-step derivation

   1. fma-def 0.0%

      \[\leadsto 1 + \mathsf{fma}\left(\frac{hi}{lo}, \frac{hi - x}{lo}, \log \left(1 + \color{blue}{\mathsf{fma}\left(0.16666666666666666, \frac{{\left(hi + x\right)}^{3}}{{lo}^{3}}, 0.5 \cdot \frac{{\left(hi + x\right)}^{2}}{{lo}^{2}} + \left(\frac{hi}{lo} + \frac{x}{lo}\right)\right)}\right)\right) \]

   2. *-lft-identity 0.0%

      \[\leadsto 1 + \mathsf{fma}\left(\frac{hi}{lo}, \frac{hi - x}{lo}, \log \left(1 + \mathsf{fma}\left(0.16666666666666666, \frac{{\color{blue}{\left(1 \cdot \left(hi + x\right)\right)}}^{3}}{{lo}^{3}}, 0.5 \cdot \frac{{\left(hi + x\right)}^{2}}{{lo}^{2}} + \left(\frac{hi}{lo} + \frac{x}{lo}\right)\right)\right)\right) \]

   3. cube-div 0.0%

      \[\leadsto 1 + \mathsf{fma}\left(\frac{hi}{lo}, \frac{hi - x}{lo}, \log \left(1 + \mathsf{fma}\left(0.16666666666666666, \color{blue}{{\left(\frac{1 \cdot \left(hi + x\right)}{lo}\right)}^{3}}, 0.5 \cdot \frac{{\left(hi + x\right)}^{2}}{{lo}^{2}} + \left(\frac{hi}{lo} + \frac{x}{lo}\right)\right)\right)\right) \]

   4. *-lft-identity 0.0%

      \[\leadsto 1 + \mathsf{fma}\left(\frac{hi}{lo}, \frac{hi - x}{lo}, \log \left(1 + \mathsf{fma}\left(0.16666666666666666, {\left(\frac{\color{blue}{hi + x}}{lo}\right)}^{3}, 0.5 \cdot \frac{{\left(hi + x\right)}^{2}}{{lo}^{2}} + \left(\frac{hi}{lo} + \frac{x}{lo}\right)\right)\right)\right) \]

   5. +-commutative 0.0%

      \[\leadsto 1 + \mathsf{fma}\left(\frac{hi}{lo}, \frac{hi - x}{lo}, \log \left(1 + \mathsf{fma}\left(0.16666666666666666, {\left(\frac{\color{blue}{x + hi}}{lo}\right)}^{3}, 0.5 \cdot \frac{{\left(hi + x\right)}^{2}}{{lo}^{2}} + \left(\frac{hi}{lo} + \frac{x}{lo}\right)\right)\right)\right) \]

   6. fma-def 0.0%

      \[\leadsto 1 + \mathsf{fma}\left(\frac{hi}{lo}, \frac{hi - x}{lo}, \log \left(1 + \mathsf{fma}\left(0.16666666666666666, {\left(\frac{x + hi}{lo}\right)}^{3}, \color{blue}{\mathsf{fma}\left(0.5, \frac{{\left(hi + x\right)}^{2}}{{lo}^{2}}, \frac{hi}{lo} + \frac{x}{lo}\right)}\right)\right)\right) \]

   7. unpow2 0.0%

      \[\leadsto 1 + \mathsf{fma}\left(\frac{hi}{lo}, \frac{hi - x}{lo}, \log \left(1 + \mathsf{fma}\left(0.16666666666666666, {\left(\frac{x + hi}{lo}\right)}^{3}, \mathsf{fma}\left(0.5, \frac{\color{blue}{\left(hi + x\right) \cdot \left(hi + x\right)}}{{lo}^{2}}, \frac{hi}{lo} + \frac{x}{lo}\right)\right)\right)\right) \]

   8. unpow2 0.0%

      \[\leadsto 1 + \mathsf{fma}\left(\frac{hi}{lo}, \frac{hi - x}{lo}, \log \left(1 + \mathsf{fma}\left(0.16666666666666666, {\left(\frac{x + hi}{lo}\right)}^{3}, \mathsf{fma}\left(0.5, \frac{\left(hi + x\right) \cdot \left(hi + x\right)}{\color{blue}{lo \cdot lo}}, \frac{hi}{lo} + \frac{x}{lo}\right)\right)\right)\right) \]

   9. times-frac 19.1%

      \[\leadsto 1 + \mathsf{fma}\left(\frac{hi}{lo}, \frac{hi - x}{lo}, \log \left(1 + \mathsf{fma}\left(0.16666666666666666, {\left(\frac{x + hi}{lo}\right)}^{3}, \mathsf{fma}\left(0.5, \color{blue}{\frac{hi + x}{lo} \cdot \frac{hi + x}{lo}}, \frac{hi}{lo} + \frac{x}{lo}\right)\right)\right)\right) \]

   10. unpow2 19.1%

       \[\leadsto 1 + \mathsf{fma}\left(\frac{hi}{lo}, \frac{hi - x}{lo}, \log \left(1 + \mathsf{fma}\left(0.16666666666666666, {\left(\frac{x + hi}{lo}\right)}^{3}, \mathsf{fma}\left(0.5, \color{blue}{{\left(\frac{hi + x}{lo}\right)}^{2}}, \frac{hi}{lo} + \frac{x}{lo}\right)\right)\right)\right) \]

   11. +-commutative 19.1%

       \[\leadsto 1 + \mathsf{fma}\left(\frac{hi}{lo}, \frac{hi - x}{lo}, \log \left(1 + \mathsf{fma}\left(0.16666666666666666, {\left(\frac{x + hi}{lo}\right)}^{3}, \mathsf{fma}\left(0.5, {\left(\frac{\color{blue}{x + hi}}{lo}\right)}^{2}, \frac{hi}{lo} + \frac{x}{lo}\right)\right)\right)\right) \]

   12. *-lft-identity 19.1%

       \[\leadsto 1 + \mathsf{fma}\left(\frac{hi}{lo}, \frac{hi - x}{lo}, \log \left(1 + \mathsf{fma}\left(0.16666666666666666, {\left(\frac{x + hi}{lo}\right)}^{3}, \mathsf{fma}\left(0.5, {\left(\frac{x + hi}{lo}\right)}^{2}, \frac{\color{blue}{1 \cdot hi}}{lo} + \frac{x}{lo}\right)\right)\right)\right) \]

   13. associate-*l/ 19.1%

       \[\leadsto 1 + \mathsf{fma}\left(\frac{hi}{lo}, \frac{hi - x}{lo}, \log \left(1 + \mathsf{fma}\left(0.16666666666666666, {\left(\frac{x + hi}{lo}\right)}^{3}, \mathsf{fma}\left(0.5, {\left(\frac{x + hi}{lo}\right)}^{2}, \color{blue}{\frac{1}{lo} \cdot hi} + \frac{x}{lo}\right)\right)\right)\right) \]

   14. *-lft-identity 19.1%

       \[\leadsto 1 + \mathsf{fma}\left(\frac{hi}{lo}, \frac{hi - x}{lo}, \log \left(1 + \mathsf{fma}\left(0.16666666666666666, {\left(\frac{x + hi}{lo}\right)}^{3}, \mathsf{fma}\left(0.5, {\left(\frac{x + hi}{lo}\right)}^{2}, \frac{1}{lo} \cdot hi + \frac{\color{blue}{1 \cdot x}}{lo}\right)\right)\right)\right) \]

   15. associate-*l/ 19.1%

       \[\leadsto 1 + \mathsf{fma}\left(\frac{hi}{lo}, \frac{hi - x}{lo}, \log \left(1 + \mathsf{fma}\left(0.16666666666666666, {\left(\frac{x + hi}{lo}\right)}^{3}, \mathsf{fma}\left(0.5, {\left(\frac{x + hi}{lo}\right)}^{2}, \frac{1}{lo} \cdot hi + \color{blue}{\frac{1}{lo} \cdot x}\right)\right)\right)\right) \]

   16. distribute-lft-in 19.1%

       \[\leadsto 1 + \mathsf{fma}\left(\frac{hi}{lo}, \frac{hi - x}{lo}, \log \left(1 + \mathsf{fma}\left(0.16666666666666666, {\left(\frac{x + hi}{lo}\right)}^{3}, \mathsf{fma}\left(0.5, {\left(\frac{x + hi}{lo}\right)}^{2}, \color{blue}{\frac{1}{lo} \cdot \left(hi + x\right)}\right)\right)\right)\right) \]

9. Simplified 19.1%

   \[\leadsto 1 + \mathsf{fma}\left(\frac{hi}{lo}, \frac{hi - x}{lo}, \log \color{blue}{\left(1 + \mathsf{fma}\left(0.16666666666666666, {\left(\frac{x + hi}{lo}\right)}^{3}, \mathsf{fma}\left(0.5, {\left(\frac{x + hi}{lo}\right)}^{2}, \frac{x + hi}{lo}\right)\right)\right)}\right) \]

10. Taylor expanded in x around 0 0.0%

    \[\leadsto 1 + \mathsf{fma}\left(\frac{hi}{lo}, \frac{hi - x}{lo}, \log \left(1 + \mathsf{fma}\left(0.16666666666666666, {\left(\frac{x + hi}{lo}\right)}^{3}, \color{blue}{0.5 \cdot \frac{{hi}^{2}}{{lo}^{2}} + \left(x \cdot \left(\frac{1}{lo} + \frac{hi}{{lo}^{2}}\right) + \frac{hi}{lo}\right)}\right)\right)\right) \]
11. Step-by-step derivation

    1. fma-def 0.0%

       \[\leadsto 1 + \mathsf{fma}\left(\frac{hi}{lo}, \frac{hi - x}{lo}, \log \left(1 + \mathsf{fma}\left(0.16666666666666666, {\left(\frac{x + hi}{lo}\right)}^{3}, \color{blue}{\mathsf{fma}\left(0.5, \frac{{hi}^{2}}{{lo}^{2}}, x \cdot \left(\frac{1}{lo} + \frac{hi}{{lo}^{2}}\right) + \frac{hi}{lo}\right)}\right)\right)\right) \]

    2. unpow2 0.0%

       \[\leadsto 1 + \mathsf{fma}\left(\frac{hi}{lo}, \frac{hi - x}{lo}, \log \left(1 + \mathsf{fma}\left(0.16666666666666666, {\left(\frac{x + hi}{lo}\right)}^{3}, \mathsf{fma}\left(0.5, \frac{\color{blue}{hi \cdot hi}}{{lo}^{2}}, x \cdot \left(\frac{1}{lo} + \frac{hi}{{lo}^{2}}\right) + \frac{hi}{lo}\right)\right)\right)\right) \]

    3. unpow2 0.0%

       \[\leadsto 1 + \mathsf{fma}\left(\frac{hi}{lo}, \frac{hi - x}{lo}, \log \left(1 + \mathsf{fma}\left(0.16666666666666666, {\left(\frac{x + hi}{lo}\right)}^{3}, \mathsf{fma}\left(0.5, \frac{hi \cdot hi}{\color{blue}{lo \cdot lo}}, x \cdot \left(\frac{1}{lo} + \frac{hi}{{lo}^{2}}\right) + \frac{hi}{lo}\right)\right)\right)\right) \]

    4. times-frac 19.1%

       \[\leadsto 1 + \mathsf{fma}\left(\frac{hi}{lo}, \frac{hi - x}{lo}, \log \left(1 + \mathsf{fma}\left(0.16666666666666666, {\left(\frac{x + hi}{lo}\right)}^{3}, \mathsf{fma}\left(0.5, \color{blue}{\frac{hi}{lo} \cdot \frac{hi}{lo}}, x \cdot \left(\frac{1}{lo} + \frac{hi}{{lo}^{2}}\right) + \frac{hi}{lo}\right)\right)\right)\right) \]

    5. fma-def 19.1%

       \[\leadsto 1 + \mathsf{fma}\left(\frac{hi}{lo}, \frac{hi - x}{lo}, \log \left(1 + \mathsf{fma}\left(0.16666666666666666, {\left(\frac{x + hi}{lo}\right)}^{3}, \mathsf{fma}\left(0.5, \frac{hi}{lo} \cdot \frac{hi}{lo}, \color{blue}{\mathsf{fma}\left(x, \frac{1}{lo} + \frac{hi}{{lo}^{2}}, \frac{hi}{lo}\right)}\right)\right)\right)\right) \]

    6. unpow2 19.1%

       \[\leadsto 1 + \mathsf{fma}\left(\frac{hi}{lo}, \frac{hi - x}{lo}, \log \left(1 + \mathsf{fma}\left(0.16666666666666666, {\left(\frac{x + hi}{lo}\right)}^{3}, \mathsf{fma}\left(0.5, \frac{hi}{lo} \cdot \frac{hi}{lo}, \mathsf{fma}\left(x, \frac{1}{lo} + \frac{hi}{\color{blue}{lo \cdot lo}}, \frac{hi}{lo}\right)\right)\right)\right)\right) \]

12. Simplified 19.1%

    \[\leadsto 1 + \mathsf{fma}\left(\frac{hi}{lo}, \frac{hi - x}{lo}, \log \left(1 + \mathsf{fma}\left(0.16666666666666666, {\left(\frac{x + hi}{lo}\right)}^{3}, \color{blue}{\mathsf{fma}\left(0.5, \frac{hi}{lo} \cdot \frac{hi}{lo}, \mathsf{fma}\left(x, \frac{1}{lo} + \frac{hi}{lo \cdot lo}, \frac{hi}{lo}\right)\right)}\right)\right)\right) \]

13. Final simplification 19.1%

    \[\leadsto 1 + \mathsf{fma}\left(\frac{hi}{lo}, \frac{hi - x}{lo}, \log \left(1 + \mathsf{fma}\left(0.16666666666666666, {\left(\frac{hi + x}{lo}\right)}^{3}, \mathsf{fma}\left(0.5, \frac{hi}{lo} \cdot \frac{hi}{lo}, \mathsf{fma}\left(x, \frac{1}{lo} + \frac{hi}{lo \cdot lo}, \frac{hi}{lo}\right)\right)\right)\right)\right) \]

Alternative 2: 19.0% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ 1 + \mathsf{fma}\left(\frac{hi}{lo}, \frac{hi - x}{lo}, \log \left(1 + \mathsf{fma}\left(0.16666666666666666, {\left(\frac{hi + x}{lo}\right)}^{3}, \mathsf{fma}\left(0.5, \frac{hi}{lo} \cdot \frac{hi}{lo}, hi \cdot \left(\frac{1}{lo} + \frac{x}{lo \cdot lo}\right)\right)\right)\right)\right) \end{array} \]

FPCore

(FPCore (lo hi x)
 :precision binary64
 (+
  1.0
  (fma
   (/ hi lo)
   (/ (- hi x) lo)
   (log
    (+
     1.0
     (fma
      0.16666666666666666
      (pow (/ (+ hi x) lo) 3.0)
      (fma
       0.5
       (* (/ hi lo) (/ hi lo))
       (* hi (+ (/ 1.0 lo) (/ x (* lo lo)))))))))))

C

double code(double lo, double hi, double x) {
	return 1.0 + fma((hi / lo), ((hi - x) / lo), log((1.0 + fma(0.16666666666666666, pow(((hi + x) / lo), 3.0), fma(0.5, ((hi / lo) * (hi / lo)), (hi * ((1.0 / lo) + (x / (lo * lo)))))))));
}

Julia

function code(lo, hi, x)
	return Float64(1.0 + fma(Float64(hi / lo), Float64(Float64(hi - x) / lo), log(Float64(1.0 + fma(0.16666666666666666, (Float64(Float64(hi + x) / lo) ^ 3.0), fma(0.5, Float64(Float64(hi / lo) * Float64(hi / lo)), Float64(hi * Float64(Float64(1.0 / lo) + Float64(x / Float64(lo * lo))))))))))
end

Wolfram

code[lo_, hi_, x_] := N[(1.0 + N[(N[(hi / lo), $MachinePrecision] * N[(N[(hi - x), $MachinePrecision] / lo), $MachinePrecision] + N[Log[N[(1.0 + N[(0.16666666666666666 * N[Power[N[(N[(hi + x), $MachinePrecision] / lo), $MachinePrecision], 3.0], $MachinePrecision] + N[(0.5 * N[(N[(hi / lo), $MachinePrecision] * N[(hi / lo), $MachinePrecision]), $MachinePrecision] + N[(hi * N[(N[(1.0 / lo), $MachinePrecision] + N[(x / N[(lo * lo), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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 19.0%

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

   10. distribute-lft-out-- 19.0%

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

   11. associate-*r/ 19.0%

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

   12. fma-neg 19.0%

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

4. Simplified 19.0%

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

   1. add-log-exp 19.0%

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

   2. sub-neg 19.0%

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

   3. add-sqr-sqrt 10.2%

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

   4. sqrt-unprod 14.6%

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

   5. sqr-neg 14.6%

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

   6. sqrt-unprod 8.8%

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

   7. add-sqr-sqrt 19.0%

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

6. Applied egg-rr 19.0%

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

7. Taylor expanded in lo around inf 0.0%

   \[\leadsto 1 + \mathsf{fma}\left(\frac{hi}{lo}, \frac{hi - x}{lo}, \log \color{blue}{\left(1 + \left(0.16666666666666666 \cdot \frac{{\left(hi + x\right)}^{3}}{{lo}^{3}} + \left(0.5 \cdot \frac{{\left(hi + x\right)}^{2}}{{lo}^{2}} + \left(\frac{hi}{lo} + \frac{x}{lo}\right)\right)\right)\right)}\right) \]
8. Step-by-step derivation

   1. fma-def 0.0%

      \[\leadsto 1 + \mathsf{fma}\left(\frac{hi}{lo}, \frac{hi - x}{lo}, \log \left(1 + \color{blue}{\mathsf{fma}\left(0.16666666666666666, \frac{{\left(hi + x\right)}^{3}}{{lo}^{3}}, 0.5 \cdot \frac{{\left(hi + x\right)}^{2}}{{lo}^{2}} + \left(\frac{hi}{lo} + \frac{x}{lo}\right)\right)}\right)\right) \]

   2. *-lft-identity 0.0%

      \[\leadsto 1 + \mathsf{fma}\left(\frac{hi}{lo}, \frac{hi - x}{lo}, \log \left(1 + \mathsf{fma}\left(0.16666666666666666, \frac{{\color{blue}{\left(1 \cdot \left(hi + x\right)\right)}}^{3}}{{lo}^{3}}, 0.5 \cdot \frac{{\left(hi + x\right)}^{2}}{{lo}^{2}} + \left(\frac{hi}{lo} + \frac{x}{lo}\right)\right)\right)\right) \]

   3. cube-div 0.0%

      \[\leadsto 1 + \mathsf{fma}\left(\frac{hi}{lo}, \frac{hi - x}{lo}, \log \left(1 + \mathsf{fma}\left(0.16666666666666666, \color{blue}{{\left(\frac{1 \cdot \left(hi + x\right)}{lo}\right)}^{3}}, 0.5 \cdot \frac{{\left(hi + x\right)}^{2}}{{lo}^{2}} + \left(\frac{hi}{lo} + \frac{x}{lo}\right)\right)\right)\right) \]

   4. *-lft-identity 0.0%

      \[\leadsto 1 + \mathsf{fma}\left(\frac{hi}{lo}, \frac{hi - x}{lo}, \log \left(1 + \mathsf{fma}\left(0.16666666666666666, {\left(\frac{\color{blue}{hi + x}}{lo}\right)}^{3}, 0.5 \cdot \frac{{\left(hi + x\right)}^{2}}{{lo}^{2}} + \left(\frac{hi}{lo} + \frac{x}{lo}\right)\right)\right)\right) \]

   5. +-commutative 0.0%

      \[\leadsto 1 + \mathsf{fma}\left(\frac{hi}{lo}, \frac{hi - x}{lo}, \log \left(1 + \mathsf{fma}\left(0.16666666666666666, {\left(\frac{\color{blue}{x + hi}}{lo}\right)}^{3}, 0.5 \cdot \frac{{\left(hi + x\right)}^{2}}{{lo}^{2}} + \left(\frac{hi}{lo} + \frac{x}{lo}\right)\right)\right)\right) \]

   6. fma-def 0.0%

      \[\leadsto 1 + \mathsf{fma}\left(\frac{hi}{lo}, \frac{hi - x}{lo}, \log \left(1 + \mathsf{fma}\left(0.16666666666666666, {\left(\frac{x + hi}{lo}\right)}^{3}, \color{blue}{\mathsf{fma}\left(0.5, \frac{{\left(hi + x\right)}^{2}}{{lo}^{2}}, \frac{hi}{lo} + \frac{x}{lo}\right)}\right)\right)\right) \]

   7. unpow2 0.0%

      \[\leadsto 1 + \mathsf{fma}\left(\frac{hi}{lo}, \frac{hi - x}{lo}, \log \left(1 + \mathsf{fma}\left(0.16666666666666666, {\left(\frac{x + hi}{lo}\right)}^{3}, \mathsf{fma}\left(0.5, \frac{\color{blue}{\left(hi + x\right) \cdot \left(hi + x\right)}}{{lo}^{2}}, \frac{hi}{lo} + \frac{x}{lo}\right)\right)\right)\right) \]

   8. unpow2 0.0%

      \[\leadsto 1 + \mathsf{fma}\left(\frac{hi}{lo}, \frac{hi - x}{lo}, \log \left(1 + \mathsf{fma}\left(0.16666666666666666, {\left(\frac{x + hi}{lo}\right)}^{3}, \mathsf{fma}\left(0.5, \frac{\left(hi + x\right) \cdot \left(hi + x\right)}{\color{blue}{lo \cdot lo}}, \frac{hi}{lo} + \frac{x}{lo}\right)\right)\right)\right) \]

   9. times-frac 19.1%

      \[\leadsto 1 + \mathsf{fma}\left(\frac{hi}{lo}, \frac{hi - x}{lo}, \log \left(1 + \mathsf{fma}\left(0.16666666666666666, {\left(\frac{x + hi}{lo}\right)}^{3}, \mathsf{fma}\left(0.5, \color{blue}{\frac{hi + x}{lo} \cdot \frac{hi + x}{lo}}, \frac{hi}{lo} + \frac{x}{lo}\right)\right)\right)\right) \]

   10. unpow2 19.1%

       \[\leadsto 1 + \mathsf{fma}\left(\frac{hi}{lo}, \frac{hi - x}{lo}, \log \left(1 + \mathsf{fma}\left(0.16666666666666666, {\left(\frac{x + hi}{lo}\right)}^{3}, \mathsf{fma}\left(0.5, \color{blue}{{\left(\frac{hi + x}{lo}\right)}^{2}}, \frac{hi}{lo} + \frac{x}{lo}\right)\right)\right)\right) \]

   11. +-commutative 19.1%

       \[\leadsto 1 + \mathsf{fma}\left(\frac{hi}{lo}, \frac{hi - x}{lo}, \log \left(1 + \mathsf{fma}\left(0.16666666666666666, {\left(\frac{x + hi}{lo}\right)}^{3}, \mathsf{fma}\left(0.5, {\left(\frac{\color{blue}{x + hi}}{lo}\right)}^{2}, \frac{hi}{lo} + \frac{x}{lo}\right)\right)\right)\right) \]

   12. *-lft-identity 19.1%

       \[\leadsto 1 + \mathsf{fma}\left(\frac{hi}{lo}, \frac{hi - x}{lo}, \log \left(1 + \mathsf{fma}\left(0.16666666666666666, {\left(\frac{x + hi}{lo}\right)}^{3}, \mathsf{fma}\left(0.5, {\left(\frac{x + hi}{lo}\right)}^{2}, \frac{\color{blue}{1 \cdot hi}}{lo} + \frac{x}{lo}\right)\right)\right)\right) \]

   13. associate-*l/ 19.1%

       \[\leadsto 1 + \mathsf{fma}\left(\frac{hi}{lo}, \frac{hi - x}{lo}, \log \left(1 + \mathsf{fma}\left(0.16666666666666666, {\left(\frac{x + hi}{lo}\right)}^{3}, \mathsf{fma}\left(0.5, {\left(\frac{x + hi}{lo}\right)}^{2}, \color{blue}{\frac{1}{lo} \cdot hi} + \frac{x}{lo}\right)\right)\right)\right) \]

   14. *-lft-identity 19.1%

       \[\leadsto 1 + \mathsf{fma}\left(\frac{hi}{lo}, \frac{hi - x}{lo}, \log \left(1 + \mathsf{fma}\left(0.16666666666666666, {\left(\frac{x + hi}{lo}\right)}^{3}, \mathsf{fma}\left(0.5, {\left(\frac{x + hi}{lo}\right)}^{2}, \frac{1}{lo} \cdot hi + \frac{\color{blue}{1 \cdot x}}{lo}\right)\right)\right)\right) \]

   15. associate-*l/ 19.1%

       \[\leadsto 1 + \mathsf{fma}\left(\frac{hi}{lo}, \frac{hi - x}{lo}, \log \left(1 + \mathsf{fma}\left(0.16666666666666666, {\left(\frac{x + hi}{lo}\right)}^{3}, \mathsf{fma}\left(0.5, {\left(\frac{x + hi}{lo}\right)}^{2}, \frac{1}{lo} \cdot hi + \color{blue}{\frac{1}{lo} \cdot x}\right)\right)\right)\right) \]

   16. distribute-lft-in 19.1%

       \[\leadsto 1 + \mathsf{fma}\left(\frac{hi}{lo}, \frac{hi - x}{lo}, \log \left(1 + \mathsf{fma}\left(0.16666666666666666, {\left(\frac{x + hi}{lo}\right)}^{3}, \mathsf{fma}\left(0.5, {\left(\frac{x + hi}{lo}\right)}^{2}, \color{blue}{\frac{1}{lo} \cdot \left(hi + x\right)}\right)\right)\right)\right) \]

9. Simplified 19.1%

   \[\leadsto 1 + \mathsf{fma}\left(\frac{hi}{lo}, \frac{hi - x}{lo}, \log \color{blue}{\left(1 + \mathsf{fma}\left(0.16666666666666666, {\left(\frac{x + hi}{lo}\right)}^{3}, \mathsf{fma}\left(0.5, {\left(\frac{x + hi}{lo}\right)}^{2}, \frac{x + hi}{lo}\right)\right)\right)}\right) \]

10. Taylor expanded in hi around inf 0.0%

    \[\leadsto 1 + \mathsf{fma}\left(\frac{hi}{lo}, \frac{hi - x}{lo}, \log \left(1 + \mathsf{fma}\left(0.16666666666666666, {\left(\frac{x + hi}{lo}\right)}^{3}, \color{blue}{0.5 \cdot \frac{{hi}^{2}}{{lo}^{2}} + hi \cdot \left(\frac{1}{lo} + \frac{x}{{lo}^{2}}\right)}\right)\right)\right) \]
11. Step-by-step derivation

    1. fma-def 0.0%

       \[\leadsto 1 + \mathsf{fma}\left(\frac{hi}{lo}, \frac{hi - x}{lo}, \log \left(1 + \mathsf{fma}\left(0.16666666666666666, {\left(\frac{x + hi}{lo}\right)}^{3}, \color{blue}{\mathsf{fma}\left(0.5, \frac{{hi}^{2}}{{lo}^{2}}, hi \cdot \left(\frac{1}{lo} + \frac{x}{{lo}^{2}}\right)\right)}\right)\right)\right) \]

    2. unpow2 0.0%

       \[\leadsto 1 + \mathsf{fma}\left(\frac{hi}{lo}, \frac{hi - x}{lo}, \log \left(1 + \mathsf{fma}\left(0.16666666666666666, {\left(\frac{x + hi}{lo}\right)}^{3}, \mathsf{fma}\left(0.5, \frac{\color{blue}{hi \cdot hi}}{{lo}^{2}}, hi \cdot \left(\frac{1}{lo} + \frac{x}{{lo}^{2}}\right)\right)\right)\right)\right) \]

    3. unpow2 0.0%

       \[\leadsto 1 + \mathsf{fma}\left(\frac{hi}{lo}, \frac{hi - x}{lo}, \log \left(1 + \mathsf{fma}\left(0.16666666666666666, {\left(\frac{x + hi}{lo}\right)}^{3}, \mathsf{fma}\left(0.5, \frac{hi \cdot hi}{\color{blue}{lo \cdot lo}}, hi \cdot \left(\frac{1}{lo} + \frac{x}{{lo}^{2}}\right)\right)\right)\right)\right) \]

    4. times-frac 19.1%

       \[\leadsto 1 + \mathsf{fma}\left(\frac{hi}{lo}, \frac{hi - x}{lo}, \log \left(1 + \mathsf{fma}\left(0.16666666666666666, {\left(\frac{x + hi}{lo}\right)}^{3}, \mathsf{fma}\left(0.5, \color{blue}{\frac{hi}{lo} \cdot \frac{hi}{lo}}, hi \cdot \left(\frac{1}{lo} + \frac{x}{{lo}^{2}}\right)\right)\right)\right)\right) \]

    5. +-commutative 19.1%

       \[\leadsto 1 + \mathsf{fma}\left(\frac{hi}{lo}, \frac{hi - x}{lo}, \log \left(1 + \mathsf{fma}\left(0.16666666666666666, {\left(\frac{x + hi}{lo}\right)}^{3}, \mathsf{fma}\left(0.5, \frac{hi}{lo} \cdot \frac{hi}{lo}, hi \cdot \color{blue}{\left(\frac{x}{{lo}^{2}} + \frac{1}{lo}\right)}\right)\right)\right)\right) \]

    6. unpow2 19.1%

       \[\leadsto 1 + \mathsf{fma}\left(\frac{hi}{lo}, \frac{hi - x}{lo}, \log \left(1 + \mathsf{fma}\left(0.16666666666666666, {\left(\frac{x + hi}{lo}\right)}^{3}, \mathsf{fma}\left(0.5, \frac{hi}{lo} \cdot \frac{hi}{lo}, hi \cdot \left(\frac{x}{\color{blue}{lo \cdot lo}} + \frac{1}{lo}\right)\right)\right)\right)\right) \]

12. Simplified 19.1%

    \[\leadsto 1 + \mathsf{fma}\left(\frac{hi}{lo}, \frac{hi - x}{lo}, \log \left(1 + \mathsf{fma}\left(0.16666666666666666, {\left(\frac{x + hi}{lo}\right)}^{3}, \color{blue}{\mathsf{fma}\left(0.5, \frac{hi}{lo} \cdot \frac{hi}{lo}, hi \cdot \left(\frac{x}{lo \cdot lo} + \frac{1}{lo}\right)\right)}\right)\right)\right) \]

13. Final simplification 19.1%

    \[\leadsto 1 + \mathsf{fma}\left(\frac{hi}{lo}, \frac{hi - x}{lo}, \log \left(1 + \mathsf{fma}\left(0.16666666666666666, {\left(\frac{hi + x}{lo}\right)}^{3}, \mathsf{fma}\left(0.5, \frac{hi}{lo} \cdot \frac{hi}{lo}, hi \cdot \left(\frac{1}{lo} + \frac{x}{lo \cdot lo}\right)\right)\right)\right)\right) \]

Alternative 3: 19.0% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ 1 + \mathsf{fma}\left(\frac{hi}{lo}, \frac{hi - x}{lo}, \log \left(1 + \mathsf{fma}\left(0.16666666666666666, {\left(\frac{hi + x}{lo}\right)}^{3}, \mathsf{fma}\left(0.5, \frac{hi}{lo} \cdot \frac{hi}{lo}, \frac{hi}{lo}\right)\right)\right)\right) \end{array} \]

FPCore

(FPCore (lo hi x)
 :precision binary64
 (+
  1.0
  (fma
   (/ hi lo)
   (/ (- hi x) lo)
   (log
    (+
     1.0
     (fma
      0.16666666666666666
      (pow (/ (+ hi x) lo) 3.0)
      (fma 0.5 (* (/ hi lo) (/ hi lo)) (/ hi lo))))))))

C

double code(double lo, double hi, double x) {
	return 1.0 + fma((hi / lo), ((hi - x) / lo), log((1.0 + fma(0.16666666666666666, pow(((hi + x) / lo), 3.0), fma(0.5, ((hi / lo) * (hi / lo)), (hi / lo))))));
}

Julia

function code(lo, hi, x)
	return Float64(1.0 + fma(Float64(hi / lo), Float64(Float64(hi - x) / lo), log(Float64(1.0 + fma(0.16666666666666666, (Float64(Float64(hi + x) / lo) ^ 3.0), fma(0.5, Float64(Float64(hi / lo) * Float64(hi / lo)), Float64(hi / lo)))))))
end

Wolfram

code[lo_, hi_, x_] := N[(1.0 + N[(N[(hi / lo), $MachinePrecision] * N[(N[(hi - x), $MachinePrecision] / lo), $MachinePrecision] + N[Log[N[(1.0 + N[(0.16666666666666666 * N[Power[N[(N[(hi + x), $MachinePrecision] / lo), $MachinePrecision], 3.0], $MachinePrecision] + N[(0.5 * N[(N[(hi / lo), $MachinePrecision] * N[(hi / lo), $MachinePrecision]), $MachinePrecision] + N[(hi / 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 19.0%

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

   10. distribute-lft-out-- 19.0%

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

   11. associate-*r/ 19.0%

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

   12. fma-neg 19.0%

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

4. Simplified 19.0%

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

   1. add-log-exp 19.0%

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

   2. sub-neg 19.0%

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

   3. add-sqr-sqrt 10.2%

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

   4. sqrt-unprod 14.6%

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

   5. sqr-neg 14.6%

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

   6. sqrt-unprod 8.8%

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

   7. add-sqr-sqrt 19.0%

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

6. Applied egg-rr 19.0%

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

7. Taylor expanded in lo around inf 0.0%

   \[\leadsto 1 + \mathsf{fma}\left(\frac{hi}{lo}, \frac{hi - x}{lo}, \log \color{blue}{\left(1 + \left(0.16666666666666666 \cdot \frac{{\left(hi + x\right)}^{3}}{{lo}^{3}} + \left(0.5 \cdot \frac{{\left(hi + x\right)}^{2}}{{lo}^{2}} + \left(\frac{hi}{lo} + \frac{x}{lo}\right)\right)\right)\right)}\right) \]
8. Step-by-step derivation

   1. fma-def 0.0%

      \[\leadsto 1 + \mathsf{fma}\left(\frac{hi}{lo}, \frac{hi - x}{lo}, \log \left(1 + \color{blue}{\mathsf{fma}\left(0.16666666666666666, \frac{{\left(hi + x\right)}^{3}}{{lo}^{3}}, 0.5 \cdot \frac{{\left(hi + x\right)}^{2}}{{lo}^{2}} + \left(\frac{hi}{lo} + \frac{x}{lo}\right)\right)}\right)\right) \]

   2. *-lft-identity 0.0%

      \[\leadsto 1 + \mathsf{fma}\left(\frac{hi}{lo}, \frac{hi - x}{lo}, \log \left(1 + \mathsf{fma}\left(0.16666666666666666, \frac{{\color{blue}{\left(1 \cdot \left(hi + x\right)\right)}}^{3}}{{lo}^{3}}, 0.5 \cdot \frac{{\left(hi + x\right)}^{2}}{{lo}^{2}} + \left(\frac{hi}{lo} + \frac{x}{lo}\right)\right)\right)\right) \]

   3. cube-div 0.0%

      \[\leadsto 1 + \mathsf{fma}\left(\frac{hi}{lo}, \frac{hi - x}{lo}, \log \left(1 + \mathsf{fma}\left(0.16666666666666666, \color{blue}{{\left(\frac{1 \cdot \left(hi + x\right)}{lo}\right)}^{3}}, 0.5 \cdot \frac{{\left(hi + x\right)}^{2}}{{lo}^{2}} + \left(\frac{hi}{lo} + \frac{x}{lo}\right)\right)\right)\right) \]

   4. *-lft-identity 0.0%

      \[\leadsto 1 + \mathsf{fma}\left(\frac{hi}{lo}, \frac{hi - x}{lo}, \log \left(1 + \mathsf{fma}\left(0.16666666666666666, {\left(\frac{\color{blue}{hi + x}}{lo}\right)}^{3}, 0.5 \cdot \frac{{\left(hi + x\right)}^{2}}{{lo}^{2}} + \left(\frac{hi}{lo} + \frac{x}{lo}\right)\right)\right)\right) \]

   5. +-commutative 0.0%

      \[\leadsto 1 + \mathsf{fma}\left(\frac{hi}{lo}, \frac{hi - x}{lo}, \log \left(1 + \mathsf{fma}\left(0.16666666666666666, {\left(\frac{\color{blue}{x + hi}}{lo}\right)}^{3}, 0.5 \cdot \frac{{\left(hi + x\right)}^{2}}{{lo}^{2}} + \left(\frac{hi}{lo} + \frac{x}{lo}\right)\right)\right)\right) \]

   6. fma-def 0.0%

      \[\leadsto 1 + \mathsf{fma}\left(\frac{hi}{lo}, \frac{hi - x}{lo}, \log \left(1 + \mathsf{fma}\left(0.16666666666666666, {\left(\frac{x + hi}{lo}\right)}^{3}, \color{blue}{\mathsf{fma}\left(0.5, \frac{{\left(hi + x\right)}^{2}}{{lo}^{2}}, \frac{hi}{lo} + \frac{x}{lo}\right)}\right)\right)\right) \]

   7. unpow2 0.0%

      \[\leadsto 1 + \mathsf{fma}\left(\frac{hi}{lo}, \frac{hi - x}{lo}, \log \left(1 + \mathsf{fma}\left(0.16666666666666666, {\left(\frac{x + hi}{lo}\right)}^{3}, \mathsf{fma}\left(0.5, \frac{\color{blue}{\left(hi + x\right) \cdot \left(hi + x\right)}}{{lo}^{2}}, \frac{hi}{lo} + \frac{x}{lo}\right)\right)\right)\right) \]

   8. unpow2 0.0%

      \[\leadsto 1 + \mathsf{fma}\left(\frac{hi}{lo}, \frac{hi - x}{lo}, \log \left(1 + \mathsf{fma}\left(0.16666666666666666, {\left(\frac{x + hi}{lo}\right)}^{3}, \mathsf{fma}\left(0.5, \frac{\left(hi + x\right) \cdot \left(hi + x\right)}{\color{blue}{lo \cdot lo}}, \frac{hi}{lo} + \frac{x}{lo}\right)\right)\right)\right) \]

   9. times-frac 19.1%

      \[\leadsto 1 + \mathsf{fma}\left(\frac{hi}{lo}, \frac{hi - x}{lo}, \log \left(1 + \mathsf{fma}\left(0.16666666666666666, {\left(\frac{x + hi}{lo}\right)}^{3}, \mathsf{fma}\left(0.5, \color{blue}{\frac{hi + x}{lo} \cdot \frac{hi + x}{lo}}, \frac{hi}{lo} + \frac{x}{lo}\right)\right)\right)\right) \]

   10. unpow2 19.1%

       \[\leadsto 1 + \mathsf{fma}\left(\frac{hi}{lo}, \frac{hi - x}{lo}, \log \left(1 + \mathsf{fma}\left(0.16666666666666666, {\left(\frac{x + hi}{lo}\right)}^{3}, \mathsf{fma}\left(0.5, \color{blue}{{\left(\frac{hi + x}{lo}\right)}^{2}}, \frac{hi}{lo} + \frac{x}{lo}\right)\right)\right)\right) \]

   11. +-commutative 19.1%

       \[\leadsto 1 + \mathsf{fma}\left(\frac{hi}{lo}, \frac{hi - x}{lo}, \log \left(1 + \mathsf{fma}\left(0.16666666666666666, {\left(\frac{x + hi}{lo}\right)}^{3}, \mathsf{fma}\left(0.5, {\left(\frac{\color{blue}{x + hi}}{lo}\right)}^{2}, \frac{hi}{lo} + \frac{x}{lo}\right)\right)\right)\right) \]

   12. *-lft-identity 19.1%

       \[\leadsto 1 + \mathsf{fma}\left(\frac{hi}{lo}, \frac{hi - x}{lo}, \log \left(1 + \mathsf{fma}\left(0.16666666666666666, {\left(\frac{x + hi}{lo}\right)}^{3}, \mathsf{fma}\left(0.5, {\left(\frac{x + hi}{lo}\right)}^{2}, \frac{\color{blue}{1 \cdot hi}}{lo} + \frac{x}{lo}\right)\right)\right)\right) \]

   13. associate-*l/ 19.1%

       \[\leadsto 1 + \mathsf{fma}\left(\frac{hi}{lo}, \frac{hi - x}{lo}, \log \left(1 + \mathsf{fma}\left(0.16666666666666666, {\left(\frac{x + hi}{lo}\right)}^{3}, \mathsf{fma}\left(0.5, {\left(\frac{x + hi}{lo}\right)}^{2}, \color{blue}{\frac{1}{lo} \cdot hi} + \frac{x}{lo}\right)\right)\right)\right) \]

   14. *-lft-identity 19.1%

       \[\leadsto 1 + \mathsf{fma}\left(\frac{hi}{lo}, \frac{hi - x}{lo}, \log \left(1 + \mathsf{fma}\left(0.16666666666666666, {\left(\frac{x + hi}{lo}\right)}^{3}, \mathsf{fma}\left(0.5, {\left(\frac{x + hi}{lo}\right)}^{2}, \frac{1}{lo} \cdot hi + \frac{\color{blue}{1 \cdot x}}{lo}\right)\right)\right)\right) \]

   15. associate-*l/ 19.1%

       \[\leadsto 1 + \mathsf{fma}\left(\frac{hi}{lo}, \frac{hi - x}{lo}, \log \left(1 + \mathsf{fma}\left(0.16666666666666666, {\left(\frac{x + hi}{lo}\right)}^{3}, \mathsf{fma}\left(0.5, {\left(\frac{x + hi}{lo}\right)}^{2}, \frac{1}{lo} \cdot hi + \color{blue}{\frac{1}{lo} \cdot x}\right)\right)\right)\right) \]

   16. distribute-lft-in 19.1%

       \[\leadsto 1 + \mathsf{fma}\left(\frac{hi}{lo}, \frac{hi - x}{lo}, \log \left(1 + \mathsf{fma}\left(0.16666666666666666, {\left(\frac{x + hi}{lo}\right)}^{3}, \mathsf{fma}\left(0.5, {\left(\frac{x + hi}{lo}\right)}^{2}, \color{blue}{\frac{1}{lo} \cdot \left(hi + x\right)}\right)\right)\right)\right) \]

9. Simplified 19.1%

   \[\leadsto 1 + \mathsf{fma}\left(\frac{hi}{lo}, \frac{hi - x}{lo}, \log \color{blue}{\left(1 + \mathsf{fma}\left(0.16666666666666666, {\left(\frac{x + hi}{lo}\right)}^{3}, \mathsf{fma}\left(0.5, {\left(\frac{x + hi}{lo}\right)}^{2}, \frac{x + hi}{lo}\right)\right)\right)}\right) \]

10. Taylor expanded in x around 0 0.0%

    \[\leadsto 1 + \mathsf{fma}\left(\frac{hi}{lo}, \frac{hi - x}{lo}, \log \left(1 + \mathsf{fma}\left(0.16666666666666666, {\left(\frac{x + hi}{lo}\right)}^{3}, \color{blue}{0.5 \cdot \frac{{hi}^{2}}{{lo}^{2}} + \frac{hi}{lo}}\right)\right)\right) \]
11. Step-by-step derivation

    1. fma-def 0.0%

       \[\leadsto 1 + \mathsf{fma}\left(\frac{hi}{lo}, \frac{hi - x}{lo}, \log \left(1 + \mathsf{fma}\left(0.16666666666666666, {\left(\frac{x + hi}{lo}\right)}^{3}, \color{blue}{\mathsf{fma}\left(0.5, \frac{{hi}^{2}}{{lo}^{2}}, \frac{hi}{lo}\right)}\right)\right)\right) \]

    2. unpow2 0.0%

       \[\leadsto 1 + \mathsf{fma}\left(\frac{hi}{lo}, \frac{hi - x}{lo}, \log \left(1 + \mathsf{fma}\left(0.16666666666666666, {\left(\frac{x + hi}{lo}\right)}^{3}, \mathsf{fma}\left(0.5, \frac{\color{blue}{hi \cdot hi}}{{lo}^{2}}, \frac{hi}{lo}\right)\right)\right)\right) \]

    3. unpow2 0.0%

       \[\leadsto 1 + \mathsf{fma}\left(\frac{hi}{lo}, \frac{hi - x}{lo}, \log \left(1 + \mathsf{fma}\left(0.16666666666666666, {\left(\frac{x + hi}{lo}\right)}^{3}, \mathsf{fma}\left(0.5, \frac{hi \cdot hi}{\color{blue}{lo \cdot lo}}, \frac{hi}{lo}\right)\right)\right)\right) \]

    4. times-frac 19.1%

       \[\leadsto 1 + \mathsf{fma}\left(\frac{hi}{lo}, \frac{hi - x}{lo}, \log \left(1 + \mathsf{fma}\left(0.16666666666666666, {\left(\frac{x + hi}{lo}\right)}^{3}, \mathsf{fma}\left(0.5, \color{blue}{\frac{hi}{lo} \cdot \frac{hi}{lo}}, \frac{hi}{lo}\right)\right)\right)\right) \]

12. Simplified 19.1%

    \[\leadsto 1 + \mathsf{fma}\left(\frac{hi}{lo}, \frac{hi - x}{lo}, \log \left(1 + \mathsf{fma}\left(0.16666666666666666, {\left(\frac{x + hi}{lo}\right)}^{3}, \color{blue}{\mathsf{fma}\left(0.5, \frac{hi}{lo} \cdot \frac{hi}{lo}, \frac{hi}{lo}\right)}\right)\right)\right) \]

13. Final simplification 19.1%

    \[\leadsto 1 + \mathsf{fma}\left(\frac{hi}{lo}, \frac{hi - x}{lo}, \log \left(1 + \mathsf{fma}\left(0.16666666666666666, {\left(\frac{hi + x}{lo}\right)}^{3}, \mathsf{fma}\left(0.5, \frac{hi}{lo} \cdot \frac{hi}{lo}, \frac{hi}{lo}\right)\right)\right)\right) \]

Alternative 4: 19.0% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ 1 + \mathsf{fma}\left(\frac{hi}{lo}, \frac{hi - x}{lo}, \log \left(1 + \mathsf{fma}\left(0.16666666666666666, {\left(\frac{hi}{lo}\right)}^{3}, \mathsf{fma}\left(0.5, \frac{hi}{lo} \cdot \frac{hi}{lo}, \frac{hi}{lo}\right)\right)\right)\right) \end{array} \]

FPCore

(FPCore (lo hi x)
 :precision binary64
 (+
  1.0
  (fma
   (/ hi lo)
   (/ (- hi x) lo)
   (log
    (+
     1.0
     (fma
      0.16666666666666666
      (pow (/ hi lo) 3.0)
      (fma 0.5 (* (/ hi lo) (/ hi lo)) (/ hi lo))))))))

C

double code(double lo, double hi, double x) {
	return 1.0 + fma((hi / lo), ((hi - x) / lo), log((1.0 + fma(0.16666666666666666, pow((hi / lo), 3.0), fma(0.5, ((hi / lo) * (hi / lo)), (hi / lo))))));
}

Julia

function code(lo, hi, x)
	return Float64(1.0 + fma(Float64(hi / lo), Float64(Float64(hi - x) / lo), log(Float64(1.0 + fma(0.16666666666666666, (Float64(hi / lo) ^ 3.0), fma(0.5, Float64(Float64(hi / lo) * Float64(hi / lo)), Float64(hi / lo)))))))
end

Wolfram

code[lo_, hi_, x_] := N[(1.0 + N[(N[(hi / lo), $MachinePrecision] * N[(N[(hi - x), $MachinePrecision] / lo), $MachinePrecision] + N[Log[N[(1.0 + N[(0.16666666666666666 * N[Power[N[(hi / lo), $MachinePrecision], 3.0], $MachinePrecision] + N[(0.5 * N[(N[(hi / lo), $MachinePrecision] * N[(hi / lo), $MachinePrecision]), $MachinePrecision] + N[(hi / 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 19.0%

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

   10. distribute-lft-out-- 19.0%

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

   11. associate-*r/ 19.0%

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

   12. fma-neg 19.0%

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

4. Simplified 19.0%

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

   1. add-log-exp 19.0%

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

   2. sub-neg 19.0%

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

   3. add-sqr-sqrt 10.2%

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

   4. sqrt-unprod 14.6%

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

   5. sqr-neg 14.6%

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

   6. sqrt-unprod 8.8%

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

   7. add-sqr-sqrt 19.0%

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

6. Applied egg-rr 19.0%

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

7. Taylor expanded in lo around inf 0.0%

   \[\leadsto 1 + \mathsf{fma}\left(\frac{hi}{lo}, \frac{hi - x}{lo}, \log \color{blue}{\left(1 + \left(0.16666666666666666 \cdot \frac{{\left(hi + x\right)}^{3}}{{lo}^{3}} + \left(0.5 \cdot \frac{{\left(hi + x\right)}^{2}}{{lo}^{2}} + \left(\frac{hi}{lo} + \frac{x}{lo}\right)\right)\right)\right)}\right) \]
8. Step-by-step derivation

   1. fma-def 0.0%

      \[\leadsto 1 + \mathsf{fma}\left(\frac{hi}{lo}, \frac{hi - x}{lo}, \log \left(1 + \color{blue}{\mathsf{fma}\left(0.16666666666666666, \frac{{\left(hi + x\right)}^{3}}{{lo}^{3}}, 0.5 \cdot \frac{{\left(hi + x\right)}^{2}}{{lo}^{2}} + \left(\frac{hi}{lo} + \frac{x}{lo}\right)\right)}\right)\right) \]

   2. *-lft-identity 0.0%

      \[\leadsto 1 + \mathsf{fma}\left(\frac{hi}{lo}, \frac{hi - x}{lo}, \log \left(1 + \mathsf{fma}\left(0.16666666666666666, \frac{{\color{blue}{\left(1 \cdot \left(hi + x\right)\right)}}^{3}}{{lo}^{3}}, 0.5 \cdot \frac{{\left(hi + x\right)}^{2}}{{lo}^{2}} + \left(\frac{hi}{lo} + \frac{x}{lo}\right)\right)\right)\right) \]

   3. cube-div 0.0%

      \[\leadsto 1 + \mathsf{fma}\left(\frac{hi}{lo}, \frac{hi - x}{lo}, \log \left(1 + \mathsf{fma}\left(0.16666666666666666, \color{blue}{{\left(\frac{1 \cdot \left(hi + x\right)}{lo}\right)}^{3}}, 0.5 \cdot \frac{{\left(hi + x\right)}^{2}}{{lo}^{2}} + \left(\frac{hi}{lo} + \frac{x}{lo}\right)\right)\right)\right) \]

   4. *-lft-identity 0.0%

      \[\leadsto 1 + \mathsf{fma}\left(\frac{hi}{lo}, \frac{hi - x}{lo}, \log \left(1 + \mathsf{fma}\left(0.16666666666666666, {\left(\frac{\color{blue}{hi + x}}{lo}\right)}^{3}, 0.5 \cdot \frac{{\left(hi + x\right)}^{2}}{{lo}^{2}} + \left(\frac{hi}{lo} + \frac{x}{lo}\right)\right)\right)\right) \]

   5. +-commutative 0.0%

      \[\leadsto 1 + \mathsf{fma}\left(\frac{hi}{lo}, \frac{hi - x}{lo}, \log \left(1 + \mathsf{fma}\left(0.16666666666666666, {\left(\frac{\color{blue}{x + hi}}{lo}\right)}^{3}, 0.5 \cdot \frac{{\left(hi + x\right)}^{2}}{{lo}^{2}} + \left(\frac{hi}{lo} + \frac{x}{lo}\right)\right)\right)\right) \]

   6. fma-def 0.0%

      \[\leadsto 1 + \mathsf{fma}\left(\frac{hi}{lo}, \frac{hi - x}{lo}, \log \left(1 + \mathsf{fma}\left(0.16666666666666666, {\left(\frac{x + hi}{lo}\right)}^{3}, \color{blue}{\mathsf{fma}\left(0.5, \frac{{\left(hi + x\right)}^{2}}{{lo}^{2}}, \frac{hi}{lo} + \frac{x}{lo}\right)}\right)\right)\right) \]

   7. unpow2 0.0%

      \[\leadsto 1 + \mathsf{fma}\left(\frac{hi}{lo}, \frac{hi - x}{lo}, \log \left(1 + \mathsf{fma}\left(0.16666666666666666, {\left(\frac{x + hi}{lo}\right)}^{3}, \mathsf{fma}\left(0.5, \frac{\color{blue}{\left(hi + x\right) \cdot \left(hi + x\right)}}{{lo}^{2}}, \frac{hi}{lo} + \frac{x}{lo}\right)\right)\right)\right) \]

   8. unpow2 0.0%

      \[\leadsto 1 + \mathsf{fma}\left(\frac{hi}{lo}, \frac{hi - x}{lo}, \log \left(1 + \mathsf{fma}\left(0.16666666666666666, {\left(\frac{x + hi}{lo}\right)}^{3}, \mathsf{fma}\left(0.5, \frac{\left(hi + x\right) \cdot \left(hi + x\right)}{\color{blue}{lo \cdot lo}}, \frac{hi}{lo} + \frac{x}{lo}\right)\right)\right)\right) \]

   9. times-frac 19.1%

      \[\leadsto 1 + \mathsf{fma}\left(\frac{hi}{lo}, \frac{hi - x}{lo}, \log \left(1 + \mathsf{fma}\left(0.16666666666666666, {\left(\frac{x + hi}{lo}\right)}^{3}, \mathsf{fma}\left(0.5, \color{blue}{\frac{hi + x}{lo} \cdot \frac{hi + x}{lo}}, \frac{hi}{lo} + \frac{x}{lo}\right)\right)\right)\right) \]

   10. unpow2 19.1%

       \[\leadsto 1 + \mathsf{fma}\left(\frac{hi}{lo}, \frac{hi - x}{lo}, \log \left(1 + \mathsf{fma}\left(0.16666666666666666, {\left(\frac{x + hi}{lo}\right)}^{3}, \mathsf{fma}\left(0.5, \color{blue}{{\left(\frac{hi + x}{lo}\right)}^{2}}, \frac{hi}{lo} + \frac{x}{lo}\right)\right)\right)\right) \]

   11. +-commutative 19.1%

       \[\leadsto 1 + \mathsf{fma}\left(\frac{hi}{lo}, \frac{hi - x}{lo}, \log \left(1 + \mathsf{fma}\left(0.16666666666666666, {\left(\frac{x + hi}{lo}\right)}^{3}, \mathsf{fma}\left(0.5, {\left(\frac{\color{blue}{x + hi}}{lo}\right)}^{2}, \frac{hi}{lo} + \frac{x}{lo}\right)\right)\right)\right) \]

   12. *-lft-identity 19.1%

       \[\leadsto 1 + \mathsf{fma}\left(\frac{hi}{lo}, \frac{hi - x}{lo}, \log \left(1 + \mathsf{fma}\left(0.16666666666666666, {\left(\frac{x + hi}{lo}\right)}^{3}, \mathsf{fma}\left(0.5, {\left(\frac{x + hi}{lo}\right)}^{2}, \frac{\color{blue}{1 \cdot hi}}{lo} + \frac{x}{lo}\right)\right)\right)\right) \]

   13. associate-*l/ 19.1%

       \[\leadsto 1 + \mathsf{fma}\left(\frac{hi}{lo}, \frac{hi - x}{lo}, \log \left(1 + \mathsf{fma}\left(0.16666666666666666, {\left(\frac{x + hi}{lo}\right)}^{3}, \mathsf{fma}\left(0.5, {\left(\frac{x + hi}{lo}\right)}^{2}, \color{blue}{\frac{1}{lo} \cdot hi} + \frac{x}{lo}\right)\right)\right)\right) \]

   14. *-lft-identity 19.1%

       \[\leadsto 1 + \mathsf{fma}\left(\frac{hi}{lo}, \frac{hi - x}{lo}, \log \left(1 + \mathsf{fma}\left(0.16666666666666666, {\left(\frac{x + hi}{lo}\right)}^{3}, \mathsf{fma}\left(0.5, {\left(\frac{x + hi}{lo}\right)}^{2}, \frac{1}{lo} \cdot hi + \frac{\color{blue}{1 \cdot x}}{lo}\right)\right)\right)\right) \]

   15. associate-*l/ 19.1%

       \[\leadsto 1 + \mathsf{fma}\left(\frac{hi}{lo}, \frac{hi - x}{lo}, \log \left(1 + \mathsf{fma}\left(0.16666666666666666, {\left(\frac{x + hi}{lo}\right)}^{3}, \mathsf{fma}\left(0.5, {\left(\frac{x + hi}{lo}\right)}^{2}, \frac{1}{lo} \cdot hi + \color{blue}{\frac{1}{lo} \cdot x}\right)\right)\right)\right) \]

   16. distribute-lft-in 19.1%

       \[\leadsto 1 + \mathsf{fma}\left(\frac{hi}{lo}, \frac{hi - x}{lo}, \log \left(1 + \mathsf{fma}\left(0.16666666666666666, {\left(\frac{x + hi}{lo}\right)}^{3}, \mathsf{fma}\left(0.5, {\left(\frac{x + hi}{lo}\right)}^{2}, \color{blue}{\frac{1}{lo} \cdot \left(hi + x\right)}\right)\right)\right)\right) \]

9. Simplified 19.1%

   \[\leadsto 1 + \mathsf{fma}\left(\frac{hi}{lo}, \frac{hi - x}{lo}, \log \color{blue}{\left(1 + \mathsf{fma}\left(0.16666666666666666, {\left(\frac{x + hi}{lo}\right)}^{3}, \mathsf{fma}\left(0.5, {\left(\frac{x + hi}{lo}\right)}^{2}, \frac{x + hi}{lo}\right)\right)\right)}\right) \]

10. Taylor expanded in x around 0 0.0%

    \[\leadsto 1 + \mathsf{fma}\left(\frac{hi}{lo}, \frac{hi - x}{lo}, \log \color{blue}{\left(1 + \left(0.16666666666666666 \cdot \frac{{hi}^{3}}{{lo}^{3}} + \left(0.5 \cdot \frac{{hi}^{2}}{{lo}^{2}} + \frac{hi}{lo}\right)\right)\right)}\right) \]
11. Step-by-step derivation

    1. fma-def 0.0%

       \[\leadsto 1 + \mathsf{fma}\left(\frac{hi}{lo}, \frac{hi - x}{lo}, \log \left(1 + \color{blue}{\mathsf{fma}\left(0.16666666666666666, \frac{{hi}^{3}}{{lo}^{3}}, 0.5 \cdot \frac{{hi}^{2}}{{lo}^{2}} + \frac{hi}{lo}\right)}\right)\right) \]

    2. cube-div 0.0%

       \[\leadsto 1 + \mathsf{fma}\left(\frac{hi}{lo}, \frac{hi - x}{lo}, \log \left(1 + \mathsf{fma}\left(0.16666666666666666, \color{blue}{{\left(\frac{hi}{lo}\right)}^{3}}, 0.5 \cdot \frac{{hi}^{2}}{{lo}^{2}} + \frac{hi}{lo}\right)\right)\right) \]

    3. fma-def 0.0%

       \[\leadsto 1 + \mathsf{fma}\left(\frac{hi}{lo}, \frac{hi - x}{lo}, \log \left(1 + \mathsf{fma}\left(0.16666666666666666, {\left(\frac{hi}{lo}\right)}^{3}, \color{blue}{\mathsf{fma}\left(0.5, \frac{{hi}^{2}}{{lo}^{2}}, \frac{hi}{lo}\right)}\right)\right)\right) \]

    4. unpow2 0.0%

       \[\leadsto 1 + \mathsf{fma}\left(\frac{hi}{lo}, \frac{hi - x}{lo}, \log \left(1 + \mathsf{fma}\left(0.16666666666666666, {\left(\frac{hi}{lo}\right)}^{3}, \mathsf{fma}\left(0.5, \frac{\color{blue}{hi \cdot hi}}{{lo}^{2}}, \frac{hi}{lo}\right)\right)\right)\right) \]

    5. unpow2 0.0%

       \[\leadsto 1 + \mathsf{fma}\left(\frac{hi}{lo}, \frac{hi - x}{lo}, \log \left(1 + \mathsf{fma}\left(0.16666666666666666, {\left(\frac{hi}{lo}\right)}^{3}, \mathsf{fma}\left(0.5, \frac{hi \cdot hi}{\color{blue}{lo \cdot lo}}, \frac{hi}{lo}\right)\right)\right)\right) \]

    6. times-frac 19.1%

       \[\leadsto 1 + \mathsf{fma}\left(\frac{hi}{lo}, \frac{hi - x}{lo}, \log \left(1 + \mathsf{fma}\left(0.16666666666666666, {\left(\frac{hi}{lo}\right)}^{3}, \mathsf{fma}\left(0.5, \color{blue}{\frac{hi}{lo} \cdot \frac{hi}{lo}}, \frac{hi}{lo}\right)\right)\right)\right) \]

12. Simplified 19.1%

    \[\leadsto 1 + \mathsf{fma}\left(\frac{hi}{lo}, \frac{hi - x}{lo}, \log \color{blue}{\left(1 + \mathsf{fma}\left(0.16666666666666666, {\left(\frac{hi}{lo}\right)}^{3}, \mathsf{fma}\left(0.5, \frac{hi}{lo} \cdot \frac{hi}{lo}, \frac{hi}{lo}\right)\right)\right)}\right) \]

13. Final simplification 19.1%

    \[\leadsto 1 + \mathsf{fma}\left(\frac{hi}{lo}, \frac{hi - x}{lo}, \log \left(1 + \mathsf{fma}\left(0.16666666666666666, {\left(\frac{hi}{lo}\right)}^{3}, \mathsf{fma}\left(0.5, \frac{hi}{lo} \cdot \frac{hi}{lo}, \frac{hi}{lo}\right)\right)\right)\right) \]

Alternative 5: 19.0% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{hi}{lo} \cdot \frac{hi}{lo}\\ 1 + \left(t_0 + \mathsf{log1p}\left(\mathsf{fma}\left(0.16666666666666666, {\left(\frac{hi}{lo}\right)}^{3}, \mathsf{fma}\left(0.5, t_0, \frac{hi}{lo}\right)\right)\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (lo hi x)
 :precision binary64
 (let* ((t_0 (* (/ hi lo) (/ hi lo))))
   (+
    1.0
    (+
     t_0
     (log1p
      (fma
       0.16666666666666666
       (pow (/ hi lo) 3.0)
       (fma 0.5 t_0 (/ hi lo))))))))

C

double code(double lo, double hi, double x) {
	double t_0 = (hi / lo) * (hi / lo);
	return 1.0 + (t_0 + log1p(fma(0.16666666666666666, pow((hi / lo), 3.0), fma(0.5, t_0, (hi / lo)))));
}

Julia

function code(lo, hi, x)
	t_0 = Float64(Float64(hi / lo) * Float64(hi / lo))
	return Float64(1.0 + Float64(t_0 + log1p(fma(0.16666666666666666, (Float64(hi / lo) ^ 3.0), fma(0.5, t_0, Float64(hi / lo))))))
end

Wolfram

code[lo_, hi_, x_] := Block[{t$95$0 = N[(N[(hi / lo), $MachinePrecision] * N[(hi / lo), $MachinePrecision]), $MachinePrecision]}, N[(1.0 + N[(t$95$0 + N[Log[1 + N[(0.16666666666666666 * N[Power[N[(hi / lo), $MachinePrecision], 3.0], $MachinePrecision] + N[(0.5 * t$95$0 + N[(hi / lo), $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 19.0%

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

   10. distribute-lft-out-- 19.0%

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

   11. associate-*r/ 19.0%

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

   12. fma-neg 19.0%

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

4. Simplified 19.0%

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

   1. add-log-exp 19.0%

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

   2. sub-neg 19.0%

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

   3. add-sqr-sqrt 10.2%

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

   4. sqrt-unprod 14.6%

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

   5. sqr-neg 14.6%

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

   6. sqrt-unprod 8.8%

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

   7. add-sqr-sqrt 19.0%

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

6. Applied egg-rr 19.0%

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

7. Taylor expanded in lo around inf 0.0%

   \[\leadsto 1 + \mathsf{fma}\left(\frac{hi}{lo}, \frac{hi - x}{lo}, \log \color{blue}{\left(1 + \left(0.16666666666666666 \cdot \frac{{\left(hi + x\right)}^{3}}{{lo}^{3}} + \left(0.5 \cdot \frac{{\left(hi + x\right)}^{2}}{{lo}^{2}} + \left(\frac{hi}{lo} + \frac{x}{lo}\right)\right)\right)\right)}\right) \]
8. Step-by-step derivation

   1. fma-def 0.0%

      \[\leadsto 1 + \mathsf{fma}\left(\frac{hi}{lo}, \frac{hi - x}{lo}, \log \left(1 + \color{blue}{\mathsf{fma}\left(0.16666666666666666, \frac{{\left(hi + x\right)}^{3}}{{lo}^{3}}, 0.5 \cdot \frac{{\left(hi + x\right)}^{2}}{{lo}^{2}} + \left(\frac{hi}{lo} + \frac{x}{lo}\right)\right)}\right)\right) \]

   2. *-lft-identity 0.0%

      \[\leadsto 1 + \mathsf{fma}\left(\frac{hi}{lo}, \frac{hi - x}{lo}, \log \left(1 + \mathsf{fma}\left(0.16666666666666666, \frac{{\color{blue}{\left(1 \cdot \left(hi + x\right)\right)}}^{3}}{{lo}^{3}}, 0.5 \cdot \frac{{\left(hi + x\right)}^{2}}{{lo}^{2}} + \left(\frac{hi}{lo} + \frac{x}{lo}\right)\right)\right)\right) \]

   3. cube-div 0.0%

      \[\leadsto 1 + \mathsf{fma}\left(\frac{hi}{lo}, \frac{hi - x}{lo}, \log \left(1 + \mathsf{fma}\left(0.16666666666666666, \color{blue}{{\left(\frac{1 \cdot \left(hi + x\right)}{lo}\right)}^{3}}, 0.5 \cdot \frac{{\left(hi + x\right)}^{2}}{{lo}^{2}} + \left(\frac{hi}{lo} + \frac{x}{lo}\right)\right)\right)\right) \]

   4. *-lft-identity 0.0%

      \[\leadsto 1 + \mathsf{fma}\left(\frac{hi}{lo}, \frac{hi - x}{lo}, \log \left(1 + \mathsf{fma}\left(0.16666666666666666, {\left(\frac{\color{blue}{hi + x}}{lo}\right)}^{3}, 0.5 \cdot \frac{{\left(hi + x\right)}^{2}}{{lo}^{2}} + \left(\frac{hi}{lo} + \frac{x}{lo}\right)\right)\right)\right) \]

   5. +-commutative 0.0%

      \[\leadsto 1 + \mathsf{fma}\left(\frac{hi}{lo}, \frac{hi - x}{lo}, \log \left(1 + \mathsf{fma}\left(0.16666666666666666, {\left(\frac{\color{blue}{x + hi}}{lo}\right)}^{3}, 0.5 \cdot \frac{{\left(hi + x\right)}^{2}}{{lo}^{2}} + \left(\frac{hi}{lo} + \frac{x}{lo}\right)\right)\right)\right) \]

   6. fma-def 0.0%

      \[\leadsto 1 + \mathsf{fma}\left(\frac{hi}{lo}, \frac{hi - x}{lo}, \log \left(1 + \mathsf{fma}\left(0.16666666666666666, {\left(\frac{x + hi}{lo}\right)}^{3}, \color{blue}{\mathsf{fma}\left(0.5, \frac{{\left(hi + x\right)}^{2}}{{lo}^{2}}, \frac{hi}{lo} + \frac{x}{lo}\right)}\right)\right)\right) \]

   7. unpow2 0.0%

      \[\leadsto 1 + \mathsf{fma}\left(\frac{hi}{lo}, \frac{hi - x}{lo}, \log \left(1 + \mathsf{fma}\left(0.16666666666666666, {\left(\frac{x + hi}{lo}\right)}^{3}, \mathsf{fma}\left(0.5, \frac{\color{blue}{\left(hi + x\right) \cdot \left(hi + x\right)}}{{lo}^{2}}, \frac{hi}{lo} + \frac{x}{lo}\right)\right)\right)\right) \]

   8. unpow2 0.0%

      \[\leadsto 1 + \mathsf{fma}\left(\frac{hi}{lo}, \frac{hi - x}{lo}, \log \left(1 + \mathsf{fma}\left(0.16666666666666666, {\left(\frac{x + hi}{lo}\right)}^{3}, \mathsf{fma}\left(0.5, \frac{\left(hi + x\right) \cdot \left(hi + x\right)}{\color{blue}{lo \cdot lo}}, \frac{hi}{lo} + \frac{x}{lo}\right)\right)\right)\right) \]

   9. times-frac 19.1%

      \[\leadsto 1 + \mathsf{fma}\left(\frac{hi}{lo}, \frac{hi - x}{lo}, \log \left(1 + \mathsf{fma}\left(0.16666666666666666, {\left(\frac{x + hi}{lo}\right)}^{3}, \mathsf{fma}\left(0.5, \color{blue}{\frac{hi + x}{lo} \cdot \frac{hi + x}{lo}}, \frac{hi}{lo} + \frac{x}{lo}\right)\right)\right)\right) \]

   10. unpow2 19.1%

       \[\leadsto 1 + \mathsf{fma}\left(\frac{hi}{lo}, \frac{hi - x}{lo}, \log \left(1 + \mathsf{fma}\left(0.16666666666666666, {\left(\frac{x + hi}{lo}\right)}^{3}, \mathsf{fma}\left(0.5, \color{blue}{{\left(\frac{hi + x}{lo}\right)}^{2}}, \frac{hi}{lo} + \frac{x}{lo}\right)\right)\right)\right) \]

   11. +-commutative 19.1%

       \[\leadsto 1 + \mathsf{fma}\left(\frac{hi}{lo}, \frac{hi - x}{lo}, \log \left(1 + \mathsf{fma}\left(0.16666666666666666, {\left(\frac{x + hi}{lo}\right)}^{3}, \mathsf{fma}\left(0.5, {\left(\frac{\color{blue}{x + hi}}{lo}\right)}^{2}, \frac{hi}{lo} + \frac{x}{lo}\right)\right)\right)\right) \]

   12. *-lft-identity 19.1%

       \[\leadsto 1 + \mathsf{fma}\left(\frac{hi}{lo}, \frac{hi - x}{lo}, \log \left(1 + \mathsf{fma}\left(0.16666666666666666, {\left(\frac{x + hi}{lo}\right)}^{3}, \mathsf{fma}\left(0.5, {\left(\frac{x + hi}{lo}\right)}^{2}, \frac{\color{blue}{1 \cdot hi}}{lo} + \frac{x}{lo}\right)\right)\right)\right) \]

   13. associate-*l/ 19.1%

       \[\leadsto 1 + \mathsf{fma}\left(\frac{hi}{lo}, \frac{hi - x}{lo}, \log \left(1 + \mathsf{fma}\left(0.16666666666666666, {\left(\frac{x + hi}{lo}\right)}^{3}, \mathsf{fma}\left(0.5, {\left(\frac{x + hi}{lo}\right)}^{2}, \color{blue}{\frac{1}{lo} \cdot hi} + \frac{x}{lo}\right)\right)\right)\right) \]

   14. *-lft-identity 19.1%

       \[\leadsto 1 + \mathsf{fma}\left(\frac{hi}{lo}, \frac{hi - x}{lo}, \log \left(1 + \mathsf{fma}\left(0.16666666666666666, {\left(\frac{x + hi}{lo}\right)}^{3}, \mathsf{fma}\left(0.5, {\left(\frac{x + hi}{lo}\right)}^{2}, \frac{1}{lo} \cdot hi + \frac{\color{blue}{1 \cdot x}}{lo}\right)\right)\right)\right) \]

   15. associate-*l/ 19.1%

       \[\leadsto 1 + \mathsf{fma}\left(\frac{hi}{lo}, \frac{hi - x}{lo}, \log \left(1 + \mathsf{fma}\left(0.16666666666666666, {\left(\frac{x + hi}{lo}\right)}^{3}, \mathsf{fma}\left(0.5, {\left(\frac{x + hi}{lo}\right)}^{2}, \frac{1}{lo} \cdot hi + \color{blue}{\frac{1}{lo} \cdot x}\right)\right)\right)\right) \]

   16. distribute-lft-in 19.1%

       \[\leadsto 1 + \mathsf{fma}\left(\frac{hi}{lo}, \frac{hi - x}{lo}, \log \left(1 + \mathsf{fma}\left(0.16666666666666666, {\left(\frac{x + hi}{lo}\right)}^{3}, \mathsf{fma}\left(0.5, {\left(\frac{x + hi}{lo}\right)}^{2}, \color{blue}{\frac{1}{lo} \cdot \left(hi + x\right)}\right)\right)\right)\right) \]

9. Simplified 19.1%

   \[\leadsto 1 + \mathsf{fma}\left(\frac{hi}{lo}, \frac{hi - x}{lo}, \log \color{blue}{\left(1 + \mathsf{fma}\left(0.16666666666666666, {\left(\frac{x + hi}{lo}\right)}^{3}, \mathsf{fma}\left(0.5, {\left(\frac{x + hi}{lo}\right)}^{2}, \frac{x + hi}{lo}\right)\right)\right)}\right) \]

10. Taylor expanded in x around 0 0.0%

    \[\leadsto 1 + \color{blue}{\left(\log \left(1 + \left(0.16666666666666666 \cdot \frac{{hi}^{3}}{{lo}^{3}} + \left(0.5 \cdot \frac{{hi}^{2}}{{lo}^{2}} + \frac{hi}{lo}\right)\right)\right) + \frac{{hi}^{2}}{{lo}^{2}}\right)} \]
11. Step-by-step derivation

    1. +-commutative 0.0%

       \[\leadsto 1 + \color{blue}{\left(\frac{{hi}^{2}}{{lo}^{2}} + \log \left(1 + \left(0.16666666666666666 \cdot \frac{{hi}^{3}}{{lo}^{3}} + \left(0.5 \cdot \frac{{hi}^{2}}{{lo}^{2}} + \frac{hi}{lo}\right)\right)\right)\right)} \]

    2. unpow2 0.0%

       \[\leadsto 1 + \left(\frac{\color{blue}{hi \cdot hi}}{{lo}^{2}} + \log \left(1 + \left(0.16666666666666666 \cdot \frac{{hi}^{3}}{{lo}^{3}} + \left(0.5 \cdot \frac{{hi}^{2}}{{lo}^{2}} + \frac{hi}{lo}\right)\right)\right)\right) \]

    3. unpow2 0.0%

       \[\leadsto 1 + \left(\frac{hi \cdot hi}{\color{blue}{lo \cdot lo}} + \log \left(1 + \left(0.16666666666666666 \cdot \frac{{hi}^{3}}{{lo}^{3}} + \left(0.5 \cdot \frac{{hi}^{2}}{{lo}^{2}} + \frac{hi}{lo}\right)\right)\right)\right) \]

    4. times-frac 0.0%

       \[\leadsto 1 + \left(\color{blue}{\frac{hi}{lo} \cdot \frac{hi}{lo}} + \log \left(1 + \left(0.16666666666666666 \cdot \frac{{hi}^{3}}{{lo}^{3}} + \left(0.5 \cdot \frac{{hi}^{2}}{{lo}^{2}} + \frac{hi}{lo}\right)\right)\right)\right) \]

    5. log1p-def 0.0%

       \[\leadsto 1 + \left(\frac{hi}{lo} \cdot \frac{hi}{lo} + \color{blue}{\mathsf{log1p}\left(0.16666666666666666 \cdot \frac{{hi}^{3}}{{lo}^{3}} + \left(0.5 \cdot \frac{{hi}^{2}}{{lo}^{2}} + \frac{hi}{lo}\right)\right)}\right) \]

    6. fma-def 0.0%

       \[\leadsto 1 + \left(\frac{hi}{lo} \cdot \frac{hi}{lo} + \mathsf{log1p}\left(\color{blue}{\mathsf{fma}\left(0.16666666666666666, \frac{{hi}^{3}}{{lo}^{3}}, 0.5 \cdot \frac{{hi}^{2}}{{lo}^{2}} + \frac{hi}{lo}\right)}\right)\right) \]

    7. cube-div 0.0%

       \[\leadsto 1 + \left(\frac{hi}{lo} \cdot \frac{hi}{lo} + \mathsf{log1p}\left(\mathsf{fma}\left(0.16666666666666666, \color{blue}{{\left(\frac{hi}{lo}\right)}^{3}}, 0.5 \cdot \frac{{hi}^{2}}{{lo}^{2}} + \frac{hi}{lo}\right)\right)\right) \]

    8. fma-def 0.0%

       \[\leadsto 1 + \left(\frac{hi}{lo} \cdot \frac{hi}{lo} + \mathsf{log1p}\left(\mathsf{fma}\left(0.16666666666666666, {\left(\frac{hi}{lo}\right)}^{3}, \color{blue}{\mathsf{fma}\left(0.5, \frac{{hi}^{2}}{{lo}^{2}}, \frac{hi}{lo}\right)}\right)\right)\right) \]

    9. unpow2 0.0%

       \[\leadsto 1 + \left(\frac{hi}{lo} \cdot \frac{hi}{lo} + \mathsf{log1p}\left(\mathsf{fma}\left(0.16666666666666666, {\left(\frac{hi}{lo}\right)}^{3}, \mathsf{fma}\left(0.5, \frac{\color{blue}{hi \cdot hi}}{{lo}^{2}}, \frac{hi}{lo}\right)\right)\right)\right) \]

    10. unpow2 0.0%

        \[\leadsto 1 + \left(\frac{hi}{lo} \cdot \frac{hi}{lo} + \mathsf{log1p}\left(\mathsf{fma}\left(0.16666666666666666, {\left(\frac{hi}{lo}\right)}^{3}, \mathsf{fma}\left(0.5, \frac{hi \cdot hi}{\color{blue}{lo \cdot lo}}, \frac{hi}{lo}\right)\right)\right)\right) \]

    11. times-frac 19.1%

        \[\leadsto 1 + \left(\frac{hi}{lo} \cdot \frac{hi}{lo} + \mathsf{log1p}\left(\mathsf{fma}\left(0.16666666666666666, {\left(\frac{hi}{lo}\right)}^{3}, \mathsf{fma}\left(0.5, \color{blue}{\frac{hi}{lo} \cdot \frac{hi}{lo}}, \frac{hi}{lo}\right)\right)\right)\right) \]

12. Simplified 19.1%

    \[\leadsto 1 + \color{blue}{\left(\frac{hi}{lo} \cdot \frac{hi}{lo} + \mathsf{log1p}\left(\mathsf{fma}\left(0.16666666666666666, {\left(\frac{hi}{lo}\right)}^{3}, \mathsf{fma}\left(0.5, \frac{hi}{lo} \cdot \frac{hi}{lo}, \frac{hi}{lo}\right)\right)\right)\right)} \]

13. Final simplification 19.1%

    \[\leadsto 1 + \left(\frac{hi}{lo} \cdot \frac{hi}{lo} + \mathsf{log1p}\left(\mathsf{fma}\left(0.16666666666666666, {\left(\frac{hi}{lo}\right)}^{3}, \mathsf{fma}\left(0.5, \frac{hi}{lo} \cdot \frac{hi}{lo}, \frac{hi}{lo}\right)\right)\right)\right) \]

Alternative 6: 18.9% accurate, 0.0× speedup

Math

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

FPCore

(FPCore (lo hi x)
 :precision binary64
 (let* ((t_0 (exp (/ hi lo))))
   (+ 1.0 (fma (/ hi lo) (/ (- hi x) lo) (log (+ t_0 (/ (* x t_0) lo)))))))

C

double code(double lo, double hi, double x) {
	double t_0 = exp((hi / lo));
	return 1.0 + fma((hi / lo), ((hi - x) / lo), log((t_0 + ((x * t_0) / lo))));
}

Julia

function code(lo, hi, x)
	t_0 = exp(Float64(hi / lo))
	return Float64(1.0 + fma(Float64(hi / lo), Float64(Float64(hi - x) / lo), log(Float64(t_0 + Float64(Float64(x * t_0) / lo)))))
end

Wolfram

code[lo_, hi_, x_] := Block[{t$95$0 = N[Exp[N[(hi / lo), $MachinePrecision]], $MachinePrecision]}, N[(1.0 + N[(N[(hi / lo), $MachinePrecision] * N[(N[(hi - x), $MachinePrecision] / lo), $MachinePrecision] + N[Log[N[(t$95$0 + N[(N[(x * t$95$0), $MachinePrecision] / lo), $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 19.0%

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

   10. distribute-lft-out-- 19.0%

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

   11. associate-*r/ 19.0%

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

   12. fma-neg 19.0%

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

4. Simplified 19.0%

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

   1. add-log-exp 19.0%

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

   2. sub-neg 19.0%

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

   3. add-sqr-sqrt 10.2%

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

   4. sqrt-unprod 14.6%

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

   5. sqr-neg 14.6%

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

   6. sqrt-unprod 8.8%

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

   7. add-sqr-sqrt 19.0%

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

6. Applied egg-rr 19.0%

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

7. Taylor expanded in x around 0 19.0%

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

8. Final simplification 19.0%

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

Alternative 7: 18.9% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[lo_, hi_, x_] := N[(1.0 + N[(N[(N[(hi / lo), $MachinePrecision] * N[(N[(hi - x), $MachinePrecision] / lo), $MachinePrecision]), $MachinePrecision] + N[(N[(hi + 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 19.0%

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

   10. distribute-lft-out-- 19.0%

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

   11. associate-*r/ 19.0%

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

   12. fma-neg 19.0%

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

4. Simplified 19.0%

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

   1. add-log-exp 19.0%

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

   2. sub-neg 19.0%

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

   3. add-sqr-sqrt 10.2%

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

   4. sqrt-unprod 14.6%

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

   5. sqr-neg 14.6%

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

   6. sqrt-unprod 8.8%

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

   7. add-sqr-sqrt 19.0%

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

6. Applied egg-rr 19.0%

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

   1. fma-udef 19.0%

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

   2. add-log-exp 19.0%

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

8. Applied egg-rr 19.0%

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

9. Final simplification 19.0%

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

Alternative 8: 18.9% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[lo_, hi_, x_] := N[(1.0 + N[(N[(N[(hi / lo), $MachinePrecision] * N[(hi / lo), $MachinePrecision]), $MachinePrecision] + N[(N[(hi + 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 19.0%

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

   10. distribute-lft-out-- 19.0%

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

   11. associate-*r/ 19.0%

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

   12. fma-neg 19.0%

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

4. Simplified 19.0%

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

5. Taylor expanded in hi around inf 19.0%

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

   1. add-cube-cbrt 19.0%

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

   2. *-un-lft-identity 19.0%

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

   3. times-frac 19.0%

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

   4. pow2 19.0%

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

   5. sub-neg 19.0%

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

   6. add-sqr-sqrt 10.2%

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

   7. sqrt-unprod 14.6%

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

   8. sqr-neg 14.6%

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

   9. sqrt-unprod 8.8%

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

   10. add-sqr-sqrt 19.0%

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

   11. sub-neg 19.0%

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

   12. add-sqr-sqrt 10.2%

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

   13. sqrt-unprod 14.6%

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

   14. sqr-neg 14.6%

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

   15. sqrt-unprod 8.8%

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

   16. add-sqr-sqrt 19.0%

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

7. Applied egg-rr 19.0%

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

   1. fma-udef 19.0%

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

   2. frac-times 19.0%

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

   3. unpow2 19.0%

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

   4. add-cube-cbrt 19.0%

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

   5. *-un-lft-identity 19.0%

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

9. Applied egg-rr 19.0%

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

10. Final simplification 19.0%

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

Alternative 9: 18.9% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[lo_, hi_, x_] := N[(1.0 + N[(N[(hi / lo), $MachinePrecision] * N[(1.0 + N[(hi / lo), $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 19.0%

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

   10. distribute-lft-out-- 19.0%

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

   11. associate-*r/ 19.0%

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

   12. fma-neg 19.0%

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

4. Simplified 19.0%

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

   1. div-inv 19.0%

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

   2. sub-neg 19.0%

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

   3. add-sqr-sqrt 10.2%

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

   4. sqrt-unprod 15.2%

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

   5. sqr-neg 15.2%

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

   6. sqrt-unprod 8.8%

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

   7. add-sqr-sqrt 19.0%

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

6. Applied egg-rr 19.0%

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

7. Taylor expanded in x around 0 0.0%

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

   1. *-lft-identity 0.0%

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

   2. unpow2 0.0%

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

   3. unpow2 0.0%

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

   4. times-frac 19.0%

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

   5. distribute-rgt-in 19.0%

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

9. Simplified 19.0%

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

10. Final simplification 19.0%

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

Alternative 10: 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. Taylor expanded in hi around inf 18.8%

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

3. Taylor expanded in x around 0 18.8%

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

   1. neg-mul-1 18.8%

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

   2. distribute-neg-frac 18.8%

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

5. Simplified 18.8%

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

6. Final simplification 18.8%

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

Alternative 11: 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 2023272 
(FPCore (lo hi x)
  :name "xlohi (overflows)"
  :precision binary64
  :pre (and (< lo -1e+308) (> hi 1e+308))
  (/ (- x lo) (- hi lo)))