xlohi (overflows)

Percentage Accurate: 3.1% → 20.6%

Time: 44.0s

Alternatives: 8

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: 20.6% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{log1p}\left(\frac{lo}{-hi}\right)\\ \sqrt[3]{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(x, 3 \cdot \left({t\_0}^{2} \cdot \left(\frac{1}{hi} + \frac{\frac{lo}{{hi}^{2}}}{1 - \frac{lo}{hi}}\right)\right), {t\_0}^{3}\right)\right)\right)} \end{array} \end{array} \]

FPCore

(FPCore (lo hi x)
 :precision binary64
 (let* ((t_0 (log1p (/ lo (- hi)))))
   (cbrt
    (expm1
     (log1p
      (fma
       x
       (*
        3.0
        (*
         (pow t_0 2.0)
         (+ (/ 1.0 hi) (/ (/ lo (pow hi 2.0)) (- 1.0 (/ lo hi))))))
       (pow t_0 3.0)))))))

C

double code(double lo, double hi, double x) {
	double t_0 = log1p((lo / -hi));
	return cbrt(expm1(log1p(fma(x, (3.0 * (pow(t_0, 2.0) * ((1.0 / hi) + ((lo / pow(hi, 2.0)) / (1.0 - (lo / hi)))))), pow(t_0, 3.0)))));
}

Julia

function code(lo, hi, x)
	t_0 = log1p(Float64(lo / Float64(-hi)))
	return cbrt(expm1(log1p(fma(x, Float64(3.0 * Float64((t_0 ^ 2.0) * Float64(Float64(1.0 / hi) + Float64(Float64(lo / (hi ^ 2.0)) / Float64(1.0 - Float64(lo / hi)))))), (t_0 ^ 3.0)))))
end

Wolfram

code[lo_, hi_, x_] := Block[{t$95$0 = N[Log[1 + N[(lo / (-hi)), $MachinePrecision]], $MachinePrecision]}, N[Power[N[(Exp[N[Log[1 + N[(x * N[(3.0 * N[(N[Power[t$95$0, 2.0], $MachinePrecision] * N[(N[(1.0 / hi), $MachinePrecision] + N[(N[(lo / N[Power[hi, 2.0], $MachinePrecision]), $MachinePrecision] / N[(1.0 - N[(lo / hi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[Power[t$95$0, 3.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]] - 1), $MachinePrecision], 1/3], $MachinePrecision]]

Derivation

1. Initial program 3.1%

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

3. Taylor expanded in lo around 0 18.8%

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

   1. +-commutative 18.8%

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

   2. mul-1-neg 18.8%

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

   3. unsub-neg 18.8%

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

   4. +-commutative 18.8%

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

   5. mul-1-neg 18.8%

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

   6. unsub-neg 18.8%

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

5. Simplified 18.8%

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

   1. add-exp-log 18.8%

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

   2. div-inv 18.8%

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

   3. pow-flip 18.8%

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

   4. metadata-eval 18.8%

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

7. Applied egg-rr 18.8%

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

   1. add-log-exp 18.8%

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

   2. *-commutative 18.8%

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

   3. rem-exp-log 18.8%

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

9. Applied egg-rr 18.8%

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

10. Taylor expanded in lo around 0 20.6%

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

    1. *-commutative 20.6%

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

12. Simplified 20.6%

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

    1. add-cbrt-cube 20.6%

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

    2. pow3 20.6%

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

14. Applied egg-rr 20.6%

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

    1. expm1-log1p-u 20.6%

       \[\leadsto \sqrt[3]{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\log \left(\mathsf{fma}\left(lo, \mathsf{fma}\left(x, {hi}^{-2}, -\frac{1}{hi}\right) \cdot e^{\frac{x}{hi}}, e^{\frac{x}{hi}}\right)\right)}^{3}\right)\right)}} \]

    2. expm1-undefine 20.6%

       \[\leadsto \sqrt[3]{\color{blue}{e^{\mathsf{log1p}\left({\log \left(\mathsf{fma}\left(lo, \mathsf{fma}\left(x, {hi}^{-2}, -\frac{1}{hi}\right) \cdot e^{\frac{x}{hi}}, e^{\frac{x}{hi}}\right)\right)}^{3}\right)} - 1}} \]

16. Applied egg-rr 20.6%

    \[\leadsto \sqrt[3]{\color{blue}{e^{\mathsf{log1p}\left({\log \left(\mathsf{fma}\left(lo \cdot \mathsf{fma}\left(x, {hi}^{-2}, \frac{-1}{hi}\right), e^{\frac{x}{hi}}, e^{\frac{x}{hi}}\right)\right)}^{3}\right)} - 1}} \]
17. Step-by-step derivation

    1. expm1-define 20.6%

       \[\leadsto \sqrt[3]{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\log \left(\mathsf{fma}\left(lo \cdot \mathsf{fma}\left(x, {hi}^{-2}, \frac{-1}{hi}\right), e^{\frac{x}{hi}}, e^{\frac{x}{hi}}\right)\right)}^{3}\right)\right)}} \]

    2. fma-define 20.6%

       \[\leadsto \sqrt[3]{\mathsf{expm1}\left(\mathsf{log1p}\left({\log \color{blue}{\left(\left(lo \cdot \mathsf{fma}\left(x, {hi}^{-2}, \frac{-1}{hi}\right)\right) \cdot e^{\frac{x}{hi}} + e^{\frac{x}{hi}}\right)}}^{3}\right)\right)} \]

    3. distribute-lft1-in 20.6%

       \[\leadsto \sqrt[3]{\mathsf{expm1}\left(\mathsf{log1p}\left({\log \color{blue}{\left(\left(lo \cdot \mathsf{fma}\left(x, {hi}^{-2}, \frac{-1}{hi}\right) + 1\right) \cdot e^{\frac{x}{hi}}\right)}}^{3}\right)\right)} \]

18. Simplified 20.6%

    \[\leadsto \sqrt[3]{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\log \left(\left(lo \cdot \mathsf{fma}\left(x, {hi}^{-2}, \frac{-1}{hi}\right) + 1\right) \cdot e^{\frac{x}{hi}}\right)}^{3}\right)\right)}} \]
19. Step-by-step derivation

    1. add-log-exp 20.6%

       \[\leadsto \sqrt[3]{\mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\log \left(e^{{\log \left(\left(lo \cdot \mathsf{fma}\left(x, {hi}^{-2}, \frac{-1}{hi}\right) + 1\right) \cdot e^{\frac{x}{hi}}\right)}^{3}}\right)}\right)\right)} \]

    2. *-commutative 20.6%

       \[\leadsto \sqrt[3]{\mathsf{expm1}\left(\mathsf{log1p}\left(\log \left(e^{{\log \color{blue}{\left(e^{\frac{x}{hi}} \cdot \left(lo \cdot \mathsf{fma}\left(x, {hi}^{-2}, \frac{-1}{hi}\right) + 1\right)\right)}}^{3}}\right)\right)\right)} \]

    3. log-prod 20.6%

       \[\leadsto \sqrt[3]{\mathsf{expm1}\left(\mathsf{log1p}\left(\log \left(e^{{\color{blue}{\left(\log \left(e^{\frac{x}{hi}}\right) + \log \left(lo \cdot \mathsf{fma}\left(x, {hi}^{-2}, \frac{-1}{hi}\right) + 1\right)\right)}}^{3}}\right)\right)\right)} \]

    4. add-log-exp 20.6%

       \[\leadsto \sqrt[3]{\mathsf{expm1}\left(\mathsf{log1p}\left(\log \left(e^{{\left(\color{blue}{\frac{x}{hi}} + \log \left(lo \cdot \mathsf{fma}\left(x, {hi}^{-2}, \frac{-1}{hi}\right) + 1\right)\right)}^{3}}\right)\right)\right)} \]

    5. +-commutative 20.6%

       \[\leadsto \sqrt[3]{\mathsf{expm1}\left(\mathsf{log1p}\left(\log \left(e^{{\left(\frac{x}{hi} + \log \color{blue}{\left(1 + lo \cdot \mathsf{fma}\left(x, {hi}^{-2}, \frac{-1}{hi}\right)\right)}\right)}^{3}}\right)\right)\right)} \]

    6. log1p-define 20.6%

       \[\leadsto \sqrt[3]{\mathsf{expm1}\left(\mathsf{log1p}\left(\log \left(e^{{\left(\frac{x}{hi} + \color{blue}{\mathsf{log1p}\left(lo \cdot \mathsf{fma}\left(x, {hi}^{-2}, \frac{-1}{hi}\right)\right)}\right)}^{3}}\right)\right)\right)} \]

20. Applied egg-rr 20.6%

    \[\leadsto \sqrt[3]{\mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\log \left(e^{{\left(\frac{x}{hi} + \mathsf{log1p}\left(lo \cdot \mathsf{fma}\left(x, {hi}^{-2}, \frac{-1}{hi}\right)\right)\right)}^{3}}\right)}\right)\right)} \]

21. Taylor expanded in x around 0 20.6%

    \[\leadsto \sqrt[3]{\mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{x \cdot \left(2 \cdot \left({\log \left(1 + -1 \cdot \frac{lo}{hi}\right)}^{2} \cdot \left(\frac{1}{hi} + \frac{lo}{{hi}^{2} \cdot \left(1 + -1 \cdot \frac{lo}{hi}\right)}\right)\right) + {\log \left(1 + -1 \cdot \frac{lo}{hi}\right)}^{2} \cdot \left(\frac{1}{hi} + \frac{lo}{{hi}^{2} \cdot \left(1 + -1 \cdot \frac{lo}{hi}\right)}\right)\right) + {\log \left(1 + -1 \cdot \frac{lo}{hi}\right)}^{3}}\right)\right)} \]
22. Step-by-step derivation

    1. fma-define 20.6%

       \[\leadsto \sqrt[3]{\mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\mathsf{fma}\left(x, 2 \cdot \left({\log \left(1 + -1 \cdot \frac{lo}{hi}\right)}^{2} \cdot \left(\frac{1}{hi} + \frac{lo}{{hi}^{2} \cdot \left(1 + -1 \cdot \frac{lo}{hi}\right)}\right)\right) + {\log \left(1 + -1 \cdot \frac{lo}{hi}\right)}^{2} \cdot \left(\frac{1}{hi} + \frac{lo}{{hi}^{2} \cdot \left(1 + -1 \cdot \frac{lo}{hi}\right)}\right), {\log \left(1 + -1 \cdot \frac{lo}{hi}\right)}^{3}\right)}\right)\right)} \]

23. Simplified 20.6%

    \[\leadsto \sqrt[3]{\mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\mathsf{fma}\left(x, 3 \cdot \left({\left(\mathsf{log1p}\left(\frac{-lo}{hi}\right)\right)}^{2} \cdot \left(\frac{1}{hi} + \frac{\frac{lo}{{hi}^{2}}}{1 - \frac{lo}{hi}}\right)\right), {\left(\mathsf{log1p}\left(\frac{-lo}{hi}\right)\right)}^{3}\right)}\right)\right)} \]

24. Final simplification 20.6%

    \[\leadsto \sqrt[3]{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(x, 3 \cdot \left({\left(\mathsf{log1p}\left(\frac{lo}{-hi}\right)\right)}^{2} \cdot \left(\frac{1}{hi} + \frac{\frac{lo}{{hi}^{2}}}{1 - \frac{lo}{hi}}\right)\right), {\left(\mathsf{log1p}\left(\frac{lo}{-hi}\right)\right)}^{3}\right)\right)\right)} \]
25. Add Preprocessing

Alternative 2: 20.6% accurate, 0.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[lo_, hi_, x_] := Block[{t$95$0 = N[Exp[N[(x / hi), $MachinePrecision]], $MachinePrecision]}, N[Log[N[(lo * N[Power[N[Power[N[(t$95$0 * N[(N[(x * N[Power[hi, -2.0], $MachinePrecision]), $MachinePrecision] + N[(-1.0 / hi), $MachinePrecision]), $MachinePrecision] + N[(t$95$0 / lo), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Initial program 3.1%

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

3. Taylor expanded in lo around 0 18.8%

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

   1. +-commutative 18.8%

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

   2. mul-1-neg 18.8%

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

   3. unsub-neg 18.8%

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

   4. +-commutative 18.8%

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

   5. mul-1-neg 18.8%

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

   6. unsub-neg 18.8%

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

5. Simplified 18.8%

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

   1. add-exp-log 18.8%

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

   2. div-inv 18.8%

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

   3. pow-flip 18.8%

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

   4. metadata-eval 18.8%

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

7. Applied egg-rr 18.8%

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

   1. add-log-exp 18.8%

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

   2. *-commutative 18.8%

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

   3. rem-exp-log 18.8%

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

9. Applied egg-rr 18.8%

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

10. Taylor expanded in lo around 0 20.6%

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

    1. *-commutative 20.6%

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

12. Simplified 20.6%

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

13. Taylor expanded in lo around inf 20.6%

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

    1. add-cube-cbrt 20.6%

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

    2. pow3 20.6%

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

    3. fma-define 20.6%

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

    4. div-inv 20.6%

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

    5. pow-flip 20.6%

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

    6. metadata-eval 20.6%

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

15. Applied egg-rr 20.6%

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

16. Final simplification 20.6%

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

Alternative 3: 20.6% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{log1p}\left(\frac{lo}{-hi}\right) + x \cdot \left(\frac{1}{hi} + \frac{\frac{lo}{{hi}^{2}}}{1 - \frac{lo}{hi}}\right) \end{array} \]

FPCore

(FPCore (lo hi x)
 :precision binary64
 (+
  (log1p (/ lo (- hi)))
  (* x (+ (/ 1.0 hi) (/ (/ lo (pow hi 2.0)) (- 1.0 (/ lo hi)))))))

C

double code(double lo, double hi, double x) {
	return log1p((lo / -hi)) + (x * ((1.0 / hi) + ((lo / pow(hi, 2.0)) / (1.0 - (lo / hi)))));
}

Java

public static double code(double lo, double hi, double x) {
	return Math.log1p((lo / -hi)) + (x * ((1.0 / hi) + ((lo / Math.pow(hi, 2.0)) / (1.0 - (lo / hi)))));
}

Python

def code(lo, hi, x):
	return math.log1p((lo / -hi)) + (x * ((1.0 / hi) + ((lo / math.pow(hi, 2.0)) / (1.0 - (lo / hi)))))

Julia

function code(lo, hi, x)
	return Float64(log1p(Float64(lo / Float64(-hi))) + Float64(x * Float64(Float64(1.0 / hi) + Float64(Float64(lo / (hi ^ 2.0)) / Float64(1.0 - Float64(lo / hi))))))
end

Wolfram

code[lo_, hi_, x_] := N[(N[Log[1 + N[(lo / (-hi)), $MachinePrecision]], $MachinePrecision] + N[(x * N[(N[(1.0 / hi), $MachinePrecision] + N[(N[(lo / N[Power[hi, 2.0], $MachinePrecision]), $MachinePrecision] / N[(1.0 - N[(lo / hi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 3.1%

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

3. Taylor expanded in lo around 0 18.8%

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

   1. +-commutative 18.8%

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

   2. mul-1-neg 18.8%

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

   3. unsub-neg 18.8%

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

   4. +-commutative 18.8%

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

   5. mul-1-neg 18.8%

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

   6. unsub-neg 18.8%

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

5. Simplified 18.8%

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

   1. add-exp-log 18.8%

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

   2. div-inv 18.8%

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

   3. pow-flip 18.8%

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

   4. metadata-eval 18.8%

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

7. Applied egg-rr 18.8%

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

   1. add-log-exp 18.8%

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

   2. *-commutative 18.8%

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

   3. rem-exp-log 18.8%

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

9. Applied egg-rr 18.8%

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

10. Taylor expanded in lo around 0 20.6%

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

    1. *-commutative 20.6%

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

12. Simplified 20.6%

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

    1. add-cbrt-cube 20.6%

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

    2. pow3 20.6%

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

14. Applied egg-rr 20.6%

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

    1. expm1-log1p-u 20.6%

       \[\leadsto \sqrt[3]{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\log \left(\mathsf{fma}\left(lo, \mathsf{fma}\left(x, {hi}^{-2}, -\frac{1}{hi}\right) \cdot e^{\frac{x}{hi}}, e^{\frac{x}{hi}}\right)\right)}^{3}\right)\right)}} \]

    2. expm1-undefine 20.6%

       \[\leadsto \sqrt[3]{\color{blue}{e^{\mathsf{log1p}\left({\log \left(\mathsf{fma}\left(lo, \mathsf{fma}\left(x, {hi}^{-2}, -\frac{1}{hi}\right) \cdot e^{\frac{x}{hi}}, e^{\frac{x}{hi}}\right)\right)}^{3}\right)} - 1}} \]

16. Applied egg-rr 20.6%

    \[\leadsto \sqrt[3]{\color{blue}{e^{\mathsf{log1p}\left({\log \left(\mathsf{fma}\left(lo \cdot \mathsf{fma}\left(x, {hi}^{-2}, \frac{-1}{hi}\right), e^{\frac{x}{hi}}, e^{\frac{x}{hi}}\right)\right)}^{3}\right)} - 1}} \]
17. Step-by-step derivation

    1. expm1-define 20.6%

       \[\leadsto \sqrt[3]{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\log \left(\mathsf{fma}\left(lo \cdot \mathsf{fma}\left(x, {hi}^{-2}, \frac{-1}{hi}\right), e^{\frac{x}{hi}}, e^{\frac{x}{hi}}\right)\right)}^{3}\right)\right)}} \]

    2. fma-define 20.6%

       \[\leadsto \sqrt[3]{\mathsf{expm1}\left(\mathsf{log1p}\left({\log \color{blue}{\left(\left(lo \cdot \mathsf{fma}\left(x, {hi}^{-2}, \frac{-1}{hi}\right)\right) \cdot e^{\frac{x}{hi}} + e^{\frac{x}{hi}}\right)}}^{3}\right)\right)} \]

    3. distribute-lft1-in 20.6%

       \[\leadsto \sqrt[3]{\mathsf{expm1}\left(\mathsf{log1p}\left({\log \color{blue}{\left(\left(lo \cdot \mathsf{fma}\left(x, {hi}^{-2}, \frac{-1}{hi}\right) + 1\right) \cdot e^{\frac{x}{hi}}\right)}}^{3}\right)\right)} \]

18. Simplified 20.6%

    \[\leadsto \sqrt[3]{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\log \left(\left(lo \cdot \mathsf{fma}\left(x, {hi}^{-2}, \frac{-1}{hi}\right) + 1\right) \cdot e^{\frac{x}{hi}}\right)}^{3}\right)\right)}} \]
19. Step-by-step derivation

    1. add-log-exp 20.6%

       \[\leadsto \sqrt[3]{\mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\log \left(e^{{\log \left(\left(lo \cdot \mathsf{fma}\left(x, {hi}^{-2}, \frac{-1}{hi}\right) + 1\right) \cdot e^{\frac{x}{hi}}\right)}^{3}}\right)}\right)\right)} \]

    2. *-commutative 20.6%

       \[\leadsto \sqrt[3]{\mathsf{expm1}\left(\mathsf{log1p}\left(\log \left(e^{{\log \color{blue}{\left(e^{\frac{x}{hi}} \cdot \left(lo \cdot \mathsf{fma}\left(x, {hi}^{-2}, \frac{-1}{hi}\right) + 1\right)\right)}}^{3}}\right)\right)\right)} \]

    3. log-prod 20.6%

       \[\leadsto \sqrt[3]{\mathsf{expm1}\left(\mathsf{log1p}\left(\log \left(e^{{\color{blue}{\left(\log \left(e^{\frac{x}{hi}}\right) + \log \left(lo \cdot \mathsf{fma}\left(x, {hi}^{-2}, \frac{-1}{hi}\right) + 1\right)\right)}}^{3}}\right)\right)\right)} \]

    4. add-log-exp 20.6%

       \[\leadsto \sqrt[3]{\mathsf{expm1}\left(\mathsf{log1p}\left(\log \left(e^{{\left(\color{blue}{\frac{x}{hi}} + \log \left(lo \cdot \mathsf{fma}\left(x, {hi}^{-2}, \frac{-1}{hi}\right) + 1\right)\right)}^{3}}\right)\right)\right)} \]

    5. +-commutative 20.6%

       \[\leadsto \sqrt[3]{\mathsf{expm1}\left(\mathsf{log1p}\left(\log \left(e^{{\left(\frac{x}{hi} + \log \color{blue}{\left(1 + lo \cdot \mathsf{fma}\left(x, {hi}^{-2}, \frac{-1}{hi}\right)\right)}\right)}^{3}}\right)\right)\right)} \]

    6. log1p-define 20.6%

       \[\leadsto \sqrt[3]{\mathsf{expm1}\left(\mathsf{log1p}\left(\log \left(e^{{\left(\frac{x}{hi} + \color{blue}{\mathsf{log1p}\left(lo \cdot \mathsf{fma}\left(x, {hi}^{-2}, \frac{-1}{hi}\right)\right)}\right)}^{3}}\right)\right)\right)} \]

20. Applied egg-rr 20.6%

    \[\leadsto \sqrt[3]{\mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\log \left(e^{{\left(\frac{x}{hi} + \mathsf{log1p}\left(lo \cdot \mathsf{fma}\left(x, {hi}^{-2}, \frac{-1}{hi}\right)\right)\right)}^{3}}\right)}\right)\right)} \]

21. Taylor expanded in x around 0 20.6%

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

    1. log1p-define 20.6%

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

    2. associate-*r/ 20.6%

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

    3. mul-1-neg 20.6%

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

    4. associate-/r* 20.6%

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

    5. mul-1-neg 20.6%

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

    6. unsub-neg 20.6%

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

23. Simplified 20.6%

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

24. Final simplification 20.6%

    \[\leadsto \mathsf{log1p}\left(\frac{lo}{-hi}\right) + x \cdot \left(\frac{1}{hi} + \frac{\frac{lo}{{hi}^{2}}}{1 - \frac{lo}{hi}}\right) \]
25. Add Preprocessing

Alternative 4: 20.6% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 3.1%

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

3. Taylor expanded in lo around 0 18.8%

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

   1. +-commutative 18.8%

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

   2. mul-1-neg 18.8%

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

   3. unsub-neg 18.8%

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

   4. +-commutative 18.8%

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

   5. mul-1-neg 18.8%

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

   6. unsub-neg 18.8%

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

5. Simplified 18.8%

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

   1. add-exp-log 18.8%

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

   2. div-inv 18.8%

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

   3. pow-flip 18.8%

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

   4. metadata-eval 18.8%

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

7. Applied egg-rr 18.8%

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

   1. add-log-exp 18.8%

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

   2. *-commutative 18.8%

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

   3. rem-exp-log 18.8%

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

9. Applied egg-rr 18.8%

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

10. Taylor expanded in hi around inf 20.6%

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

    1. associate--l+ 20.6%

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

    2. div-sub 20.6%

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

12. Simplified 20.6%

    \[\leadsto \log \color{blue}{\left(1 + \frac{x - lo}{hi}\right)} \]
13. Add Preprocessing

Alternative 5: 20.6% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \mathsf{log1p}\left(\frac{lo}{-hi}\right) \end{array} \]

FPCore

(FPCore (lo hi x) :precision binary64 (log1p (/ lo (- hi))))

C

double code(double lo, double hi, double x) {
	return log1p((lo / -hi));
}

Java

public static double code(double lo, double hi, double x) {
	return Math.log1p((lo / -hi));
}

Python

def code(lo, hi, x):
	return math.log1p((lo / -hi))

Julia

function code(lo, hi, x)
	return log1p(Float64(lo / Float64(-hi)))
end

Wolfram

code[lo_, hi_, x_] := N[Log[1 + N[(lo / (-hi)), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 3.1%

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

3. Taylor expanded in lo around 0 18.8%

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

   1. +-commutative 18.8%

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

   2. mul-1-neg 18.8%

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

   3. unsub-neg 18.8%

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

   4. +-commutative 18.8%

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

   5. mul-1-neg 18.8%

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

   6. unsub-neg 18.8%

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

5. Simplified 18.8%

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

   1. add-exp-log 18.8%

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

   2. div-inv 18.8%

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

   3. pow-flip 18.8%

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

   4. metadata-eval 18.8%

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

7. Applied egg-rr 18.8%

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

   1. add-log-exp 18.8%

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

   2. *-commutative 18.8%

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

   3. rem-exp-log 18.8%

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

9. Applied egg-rr 18.8%

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

10. Taylor expanded in lo around 0 20.6%

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

    1. *-commutative 20.6%

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

12. Simplified 20.6%

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

13. Taylor expanded in x around 0 20.6%

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

    1. log1p-define 20.6%

       \[\leadsto \color{blue}{\mathsf{log1p}\left(-1 \cdot \frac{lo}{hi}\right)} \]

    2. mul-1-neg 20.6%

       \[\leadsto \mathsf{log1p}\left(\color{blue}{-\frac{lo}{hi}}\right) \]

    3. distribute-neg-frac2 20.6%

       \[\leadsto \mathsf{log1p}\left(\color{blue}{\frac{lo}{-hi}}\right) \]

15. Simplified 20.6%

    \[\leadsto \color{blue}{\mathsf{log1p}\left(\frac{lo}{-hi}\right)} \]
16. Add Preprocessing

Alternative 6: 18.8% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 3.1%

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

3. Taylor expanded in hi around inf 18.8%

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

Alternative 7: 18.8% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 3.1%

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

3. Taylor expanded in lo around 0 18.8%

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

   1. +-commutative 18.8%

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

   2. mul-1-neg 18.8%

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

   3. unsub-neg 18.8%

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

   4. +-commutative 18.8%

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

   5. mul-1-neg 18.8%

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

   6. unsub-neg 18.8%

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

5. Simplified 18.8%

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

6. Taylor expanded in x around 0 18.8%

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

   1. neg-mul-1 18.8%

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

   2. distribute-neg-frac2 18.8%

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

8. Simplified 18.8%

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

Alternative 8: 18.7% accurate, 7.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[lo_, hi_, x_] := 1.0

Derivation

1. Initial program 3.1%

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

3. Taylor expanded in lo around inf 18.7%

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

Reproduce

herbie shell --seed 2024105 
(FPCore (lo hi x)
  :name "xlohi (overflows)"
  :precision binary64
  :pre (and (< lo -1e+308) (> hi 1e+308))
  (/ (- x lo) (- hi lo)))