xlohi (overflows)

Percentage Accurate: 3.1% → 19.5%

Time: 13.2s

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: 19.5% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \sqrt[3]{{\left(1 + \frac{{\left(\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\sqrt[3]{\mathsf{fma}\left(hi, \frac{hi + x}{lo}, hi + x\right)}\right)}^{2}\right)\right)\right)}^{1.5}}{lo}\right)}^{3}} \end{array} \]

FPCore

(FPCore (lo hi x)
 :precision binary64
 (cbrt
  (pow
   (+
    1.0
    (/
     (pow
      (expm1 (log1p (pow (cbrt (fma hi (/ (+ hi x) lo) (+ hi x))) 2.0)))
      1.5)
     lo))
   3.0)))

C

double code(double lo, double hi, double x) {
	return cbrt(pow((1.0 + (pow(expm1(log1p(pow(cbrt(fma(hi, ((hi + x) / lo), (hi + x))), 2.0))), 1.5) / lo)), 3.0));
}

Julia

function code(lo, hi, x)
	return cbrt((Float64(1.0 + Float64((expm1(log1p((cbrt(fma(hi, Float64(Float64(hi + x) / lo), Float64(hi + x))) ^ 2.0))) ^ 1.5) / lo)) ^ 3.0))
end

Wolfram

code[lo_, hi_, x_] := N[Power[N[Power[N[(1.0 + N[(N[Power[N[(Exp[N[Log[1 + N[Power[N[Power[N[(hi * N[(N[(hi + x), $MachinePrecision] / lo), $MachinePrecision] + N[(hi + x), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision]] - 1), $MachinePrecision], 1.5], $MachinePrecision] / lo), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision], 1/3], $MachinePrecision]

Derivation

1. Initial program 3.1%

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

3. Taylor expanded in lo around -inf 3.1%

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

   1. mul-1-neg 3.1%

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

   2. associate--l+ 3.1%

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

   3. associate-/l* 14.7%

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

5. Simplified 14.7%

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

   1. add-cbrt-cube 14.7%

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

   2. pow3 14.7%

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

7. Applied egg-rr 18.8%

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

   1. rem-cube-cbrt 18.8%

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

   2. sqr-pow 9.2%

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

   3. pow-prod-down 19.6%

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

   4. pow2 19.6%

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

   5. metadata-eval 19.6%

      \[\leadsto \sqrt[3]{{\left(1 + \frac{{\left({\left(\sqrt[3]{x + \mathsf{fma}\left(hi, \frac{x + hi}{lo}, hi\right)}\right)}^{2}\right)}^{\color{blue}{1.5}}}{lo}\right)}^{3}} \]

9. Applied egg-rr 19.6%

   \[\leadsto \sqrt[3]{{\left(1 + \frac{\color{blue}{{\left({\left(\sqrt[3]{x + \mathsf{fma}\left(hi, \frac{x + hi}{lo}, hi\right)}\right)}^{2}\right)}^{1.5}}}{lo}\right)}^{3}} \]
10. Step-by-step derivation

    1. fma-define 14.2%

       \[\leadsto \sqrt[3]{{\left(1 + \frac{{\left({\left(\sqrt[3]{x + \color{blue}{\left(hi \cdot \frac{x + hi}{lo} + hi\right)}}\right)}^{2}\right)}^{1.5}}{lo}\right)}^{3}} \]

    2. associate-+l+ 14.2%

       \[\leadsto \sqrt[3]{{\left(1 + \frac{{\left({\left(\sqrt[3]{\color{blue}{\left(x + hi \cdot \frac{x + hi}{lo}\right) + hi}}\right)}^{2}\right)}^{1.5}}{lo}\right)}^{3}} \]

    3. +-commutative 14.2%

       \[\leadsto \sqrt[3]{{\left(1 + \frac{{\left({\left(\sqrt[3]{\color{blue}{\left(hi \cdot \frac{x + hi}{lo} + x\right)} + hi}\right)}^{2}\right)}^{1.5}}{lo}\right)}^{3}} \]

    4. associate-+r+ 14.2%

       \[\leadsto \sqrt[3]{{\left(1 + \frac{{\left({\left(\sqrt[3]{\color{blue}{hi \cdot \frac{x + hi}{lo} + \left(x + hi\right)}}\right)}^{2}\right)}^{1.5}}{lo}\right)}^{3}} \]

    5. +-commutative 14.2%

       \[\leadsto \sqrt[3]{{\left(1 + \frac{{\left({\left(\sqrt[3]{hi \cdot \frac{x + hi}{lo} + \color{blue}{\left(hi + x\right)}}\right)}^{2}\right)}^{1.5}}{lo}\right)}^{3}} \]

    6. fma-undefine 19.6%

       \[\leadsto \sqrt[3]{{\left(1 + \frac{{\left({\left(\sqrt[3]{\color{blue}{\mathsf{fma}\left(hi, \frac{x + hi}{lo}, hi + x\right)}}\right)}^{2}\right)}^{1.5}}{lo}\right)}^{3}} \]

    7. +-commutative 19.6%

       \[\leadsto \sqrt[3]{{\left(1 + \frac{{\left({\left(\sqrt[3]{\mathsf{fma}\left(hi, \frac{\color{blue}{hi + x}}{lo}, hi + x\right)}\right)}^{2}\right)}^{1.5}}{lo}\right)}^{3}} \]

11. Simplified 19.6%

    \[\leadsto \sqrt[3]{{\left(1 + \frac{\color{blue}{{\left({\left(\sqrt[3]{\mathsf{fma}\left(hi, \frac{hi + x}{lo}, hi + x\right)}\right)}^{2}\right)}^{1.5}}}{lo}\right)}^{3}} \]
12. Step-by-step derivation

    1. expm1-log1p-u 19.6%

       \[\leadsto \sqrt[3]{{\left(1 + \frac{{\color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\sqrt[3]{\mathsf{fma}\left(hi, \frac{hi + x}{lo}, hi + x\right)}\right)}^{2}\right)\right)\right)}}^{1.5}}{lo}\right)}^{3}} \]

13. Applied egg-rr 19.6%

    \[\leadsto \sqrt[3]{{\left(1 + \frac{{\color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\sqrt[3]{\mathsf{fma}\left(hi, \frac{hi + x}{lo}, hi + x\right)}\right)}^{2}\right)\right)\right)}}^{1.5}}{lo}\right)}^{3}} \]
14. Add Preprocessing

Alternative 2: 19.5% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \sqrt[3]{{\left(1 + \frac{{\left({\left(\sqrt[3]{\mathsf{fma}\left(hi, \frac{hi + x}{lo}, hi + x\right)}\right)}^{2}\right)}^{1.5}}{lo}\right)}^{3}} \end{array} \]

FPCore

(FPCore (lo hi x)
 :precision binary64
 (cbrt
  (pow
   (+ 1.0 (/ (pow (pow (cbrt (fma hi (/ (+ hi x) lo) (+ hi x))) 2.0) 1.5) lo))
   3.0)))

C

double code(double lo, double hi, double x) {
	return cbrt(pow((1.0 + (pow(pow(cbrt(fma(hi, ((hi + x) / lo), (hi + x))), 2.0), 1.5) / lo)), 3.0));
}

Julia

function code(lo, hi, x)
	return cbrt((Float64(1.0 + Float64(((cbrt(fma(hi, Float64(Float64(hi + x) / lo), Float64(hi + x))) ^ 2.0) ^ 1.5) / lo)) ^ 3.0))
end

Wolfram

code[lo_, hi_, x_] := N[Power[N[Power[N[(1.0 + N[(N[Power[N[Power[N[Power[N[(hi * N[(N[(hi + x), $MachinePrecision] / lo), $MachinePrecision] + N[(hi + x), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision], 2.0], $MachinePrecision], 1.5], $MachinePrecision] / lo), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision], 1/3], $MachinePrecision]

Derivation

1. Initial program 3.1%

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

3. Taylor expanded in lo around -inf 3.1%

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

   1. mul-1-neg 3.1%

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

   2. associate--l+ 3.1%

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

   3. associate-/l* 14.7%

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

5. Simplified 14.7%

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

   1. add-cbrt-cube 14.7%

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

   2. pow3 14.7%

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

7. Applied egg-rr 18.8%

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

   1. rem-cube-cbrt 18.8%

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

   2. sqr-pow 9.2%

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

   3. pow-prod-down 19.6%

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

   4. pow2 19.6%

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

   5. metadata-eval 19.6%

      \[\leadsto \sqrt[3]{{\left(1 + \frac{{\left({\left(\sqrt[3]{x + \mathsf{fma}\left(hi, \frac{x + hi}{lo}, hi\right)}\right)}^{2}\right)}^{\color{blue}{1.5}}}{lo}\right)}^{3}} \]

9. Applied egg-rr 19.6%

   \[\leadsto \sqrt[3]{{\left(1 + \frac{\color{blue}{{\left({\left(\sqrt[3]{x + \mathsf{fma}\left(hi, \frac{x + hi}{lo}, hi\right)}\right)}^{2}\right)}^{1.5}}}{lo}\right)}^{3}} \]
10. Step-by-step derivation

    1. fma-define 14.2%

       \[\leadsto \sqrt[3]{{\left(1 + \frac{{\left({\left(\sqrt[3]{x + \color{blue}{\left(hi \cdot \frac{x + hi}{lo} + hi\right)}}\right)}^{2}\right)}^{1.5}}{lo}\right)}^{3}} \]

    2. associate-+l+ 14.2%

       \[\leadsto \sqrt[3]{{\left(1 + \frac{{\left({\left(\sqrt[3]{\color{blue}{\left(x + hi \cdot \frac{x + hi}{lo}\right) + hi}}\right)}^{2}\right)}^{1.5}}{lo}\right)}^{3}} \]

    3. +-commutative 14.2%

       \[\leadsto \sqrt[3]{{\left(1 + \frac{{\left({\left(\sqrt[3]{\color{blue}{\left(hi \cdot \frac{x + hi}{lo} + x\right)} + hi}\right)}^{2}\right)}^{1.5}}{lo}\right)}^{3}} \]

    4. associate-+r+ 14.2%

       \[\leadsto \sqrt[3]{{\left(1 + \frac{{\left({\left(\sqrt[3]{\color{blue}{hi \cdot \frac{x + hi}{lo} + \left(x + hi\right)}}\right)}^{2}\right)}^{1.5}}{lo}\right)}^{3}} \]

    5. +-commutative 14.2%

       \[\leadsto \sqrt[3]{{\left(1 + \frac{{\left({\left(\sqrt[3]{hi \cdot \frac{x + hi}{lo} + \color{blue}{\left(hi + x\right)}}\right)}^{2}\right)}^{1.5}}{lo}\right)}^{3}} \]

    6. fma-undefine 19.6%

       \[\leadsto \sqrt[3]{{\left(1 + \frac{{\left({\left(\sqrt[3]{\color{blue}{\mathsf{fma}\left(hi, \frac{x + hi}{lo}, hi + x\right)}}\right)}^{2}\right)}^{1.5}}{lo}\right)}^{3}} \]

    7. +-commutative 19.6%

       \[\leadsto \sqrt[3]{{\left(1 + \frac{{\left({\left(\sqrt[3]{\mathsf{fma}\left(hi, \frac{\color{blue}{hi + x}}{lo}, hi + x\right)}\right)}^{2}\right)}^{1.5}}{lo}\right)}^{3}} \]

11. Simplified 19.6%

    \[\leadsto \sqrt[3]{{\left(1 + \frac{\color{blue}{{\left({\left(\sqrt[3]{\mathsf{fma}\left(hi, \frac{hi + x}{lo}, hi + x\right)}\right)}^{2}\right)}^{1.5}}}{lo}\right)}^{3}} \]
12. Add Preprocessing

Alternative 3: 19.5% accurate, 0.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 3.1%

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

3. Taylor expanded in lo around -inf 3.1%

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

   1. mul-1-neg 3.1%

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

   2. associate--l+ 3.1%

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

   3. associate-/l* 14.7%

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

5. Simplified 14.7%

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

   1. add-cube-cbrt 14.7%

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

   2. pow3 14.7%

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

   3. fmm-def 18.8%

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

   4. add-sqr-sqrt 0.0%

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

   5. sqrt-unprod 3.1%

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

   6. sqr-neg 3.1%

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

   7. sqrt-prod 4.1%

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

   8. add-sqr-sqrt 4.1%

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

   9. sub-neg 4.1%

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

   10. add-sqr-sqrt 0.0%

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

   11. sqrt-unprod 0.7%

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

   12. sqr-neg 0.7%

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

   13. sqrt-prod 19.0%

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

   14. add-sqr-sqrt 19.0%

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

7. Applied egg-rr 19.0%

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

   1. +-commutative 19.0%

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

   2. rem-cube-cbrt 19.0%

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

   3. distribute-neg-frac2 19.0%

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

   4. add-sqr-sqrt 19.0%

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

   5. sqrt-unprod 18.7%

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

   6. sqr-neg 18.7%

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

   7. sqrt-unprod 0.0%

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

   8. add-sqr-sqrt 18.8%

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

   9. div-inv 18.8%

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

   10. fma-define 18.8%

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

9. Applied egg-rr 18.8%

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

    1. add-sqr-sqrt 9.2%

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

    2. sqrt-unprod 0.7%

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

    3. pow2 0.7%

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

11. Applied egg-rr 0.7%

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

    1. unpow2 0.7%

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

    2. rem-sqrt-square 19.6%

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

    3. +-commutative 19.6%

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

13. Simplified 19.6%

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

Alternative 4: 19.5% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ {\left(\frac{hi}{lo}\right)}^{2} \end{array} \]

FPCore

(FPCore (lo hi x) :precision binary64 (pow (/ hi lo) 2.0))

C

double code(double lo, double hi, double x) {
	return pow((hi / lo), 2.0);
}

Fortran

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

Java

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

Python

def code(lo, hi, x):
	return math.pow((hi / lo), 2.0)

Julia

function code(lo, hi, x)
	return Float64(hi / lo) ^ 2.0
end

MATLAB

function tmp = code(lo, hi, x)
	tmp = (hi / lo) ^ 2.0;
end

Wolfram

code[lo_, hi_, x_] := N[Power[N[(hi / lo), $MachinePrecision], 2.0], $MachinePrecision]

Derivation

1. Initial program 3.1%

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

3. Taylor expanded in lo around -inf 3.1%

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

   1. mul-1-neg 3.1%

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

   2. associate--l+ 3.1%

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

   3. associate-/l* 14.7%

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

5. Simplified 14.7%

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

   1. add-cbrt-cube 14.7%

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

   2. pow3 14.7%

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

7. Applied egg-rr 18.8%

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

   1. rem-cube-cbrt 18.8%

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

   2. sqr-pow 9.2%

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

   3. pow-prod-down 19.6%

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

   4. pow2 19.6%

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

   5. metadata-eval 19.6%

      \[\leadsto \sqrt[3]{{\left(1 + \frac{{\left({\left(\sqrt[3]{x + \mathsf{fma}\left(hi, \frac{x + hi}{lo}, hi\right)}\right)}^{2}\right)}^{\color{blue}{1.5}}}{lo}\right)}^{3}} \]

9. Applied egg-rr 19.6%

   \[\leadsto \sqrt[3]{{\left(1 + \frac{\color{blue}{{\left({\left(\sqrt[3]{x + \mathsf{fma}\left(hi, \frac{x + hi}{lo}, hi\right)}\right)}^{2}\right)}^{1.5}}}{lo}\right)}^{3}} \]
10. Step-by-step derivation

    1. fma-define 14.2%

       \[\leadsto \sqrt[3]{{\left(1 + \frac{{\left({\left(\sqrt[3]{x + \color{blue}{\left(hi \cdot \frac{x + hi}{lo} + hi\right)}}\right)}^{2}\right)}^{1.5}}{lo}\right)}^{3}} \]

    2. associate-+l+ 14.2%

       \[\leadsto \sqrt[3]{{\left(1 + \frac{{\left({\left(\sqrt[3]{\color{blue}{\left(x + hi \cdot \frac{x + hi}{lo}\right) + hi}}\right)}^{2}\right)}^{1.5}}{lo}\right)}^{3}} \]

    3. +-commutative 14.2%

       \[\leadsto \sqrt[3]{{\left(1 + \frac{{\left({\left(\sqrt[3]{\color{blue}{\left(hi \cdot \frac{x + hi}{lo} + x\right)} + hi}\right)}^{2}\right)}^{1.5}}{lo}\right)}^{3}} \]

    4. associate-+r+ 14.2%

       \[\leadsto \sqrt[3]{{\left(1 + \frac{{\left({\left(\sqrt[3]{\color{blue}{hi \cdot \frac{x + hi}{lo} + \left(x + hi\right)}}\right)}^{2}\right)}^{1.5}}{lo}\right)}^{3}} \]

    5. +-commutative 14.2%

       \[\leadsto \sqrt[3]{{\left(1 + \frac{{\left({\left(\sqrt[3]{hi \cdot \frac{x + hi}{lo} + \color{blue}{\left(hi + x\right)}}\right)}^{2}\right)}^{1.5}}{lo}\right)}^{3}} \]

    6. fma-undefine 19.6%

       \[\leadsto \sqrt[3]{{\left(1 + \frac{{\left({\left(\sqrt[3]{\color{blue}{\mathsf{fma}\left(hi, \frac{x + hi}{lo}, hi + x\right)}}\right)}^{2}\right)}^{1.5}}{lo}\right)}^{3}} \]

    7. +-commutative 19.6%

       \[\leadsto \sqrt[3]{{\left(1 + \frac{{\left({\left(\sqrt[3]{\mathsf{fma}\left(hi, \frac{\color{blue}{hi + x}}{lo}, hi + x\right)}\right)}^{2}\right)}^{1.5}}{lo}\right)}^{3}} \]

11. Simplified 19.6%

    \[\leadsto \sqrt[3]{{\left(1 + \frac{\color{blue}{{\left({\left(\sqrt[3]{\mathsf{fma}\left(hi, \frac{hi + x}{lo}, hi + x\right)}\right)}^{2}\right)}^{1.5}}}{lo}\right)}^{3}} \]

12. Taylor expanded in hi around inf 0.0%

    \[\leadsto \color{blue}{\frac{{hi}^{2}}{{lo}^{2}}} \]
13. Step-by-step derivation

    1. unpow2 0.0%

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

    2. unpow2 0.0%

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

    3. times-frac 19.2%

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

    4. unpow2 19.2%

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

14. Simplified 19.2%

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

Alternative 5: 18.9% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 3.1%

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

3. Taylor expanded in lo around -inf 3.1%

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

   1. mul-1-neg 3.1%

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

   2. associate--l+ 3.1%

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

   3. associate-/l* 14.7%

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

5. Simplified 14.7%

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

   1. unsub-neg 14.7%

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

   2. fmm-def 18.8%

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

   3. add-sqr-sqrt 0.0%

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

   4. sqrt-unprod 3.1%

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

   5. sqr-neg 3.1%

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

   6. sqrt-prod 4.1%

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

   7. add-sqr-sqrt 4.1%

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

   8. sub-neg 4.1%

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

   9. add-sqr-sqrt 0.0%

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

   10. sqrt-unprod 0.7%

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

   11. sqr-neg 0.7%

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

   12. sqrt-prod 19.0%

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

   13. add-sqr-sqrt 19.0%

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

7. Applied egg-rr 19.0%

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

8. Taylor expanded in hi around 0 19.0%

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

9. Final simplification 19.0%

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

Alternative 6: 18.9% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 3.1%

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

3. Taylor expanded in lo around -inf 3.1%

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

   1. mul-1-neg 3.1%

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

   2. associate--l+ 3.1%

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

   3. associate-/l* 14.7%

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

5. Simplified 14.7%

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

6. Taylor expanded in hi around 0 18.8%

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

   1. sub-neg 18.8%

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

   2. +-commutative 18.8%

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

   3. mul-1-neg 18.8%

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

   4. sub-neg 18.8%

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

   5. div-sub 18.8%

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

   6. metadata-eval 18.8%

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

8. Simplified 18.8%

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

9. Final simplification 18.8%

   \[\leadsto 1 - \frac{x + hi \cdot \left(\frac{x - hi}{lo} + -1\right)}{lo} \]
10. 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 hi around inf 18.7%

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

4. Taylor expanded in x around 0 18.8%

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

   1. neg-mul-1 18.8%

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

   2. distribute-neg-frac 18.8%

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

6. Simplified 18.8%

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

7. Final simplification 18.8%

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

Alternative 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 2024149 
(FPCore (lo hi x)
  :name "xlohi (overflows)"
  :precision binary64
  :pre (and (< lo -1e+308) (> hi 1e+308))
  (/ (- x lo) (- hi lo)))