xlohi (overflows)

Percentage Accurate: 3.1% → 24.3%

Time: 18.5s

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: 24.3% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{hi + x}{lo}\\ t_1 := \frac{hi}{\frac{lo \cdot lo}{hi - x}}\\ \frac{{t_1}^{3} + {t_0}^{3}}{\mathsf{fma}\left(\frac{hi}{lo} \cdot \frac{hi}{lo}, t_0 \cdot \frac{hi - x}{lo}, t_0 \cdot \left(t_0 - t_1\right)\right)} + 1 \end{array} \end{array} \]

FPCore

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

C

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

Julia

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

Wolfram

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

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

   2. pow3 19.0%

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

   3. sub-neg 19.0%

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

   4. add-sqr-sqrt 9.5%

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

   5. sqrt-unprod 14.5%

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

   6. sqr-neg 14.5%

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

   7. sqrt-unprod 9.4%

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

   8. add-sqr-sqrt 19.0%

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

6. Applied egg-rr 19.0%

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

   1. fma-udef 19.0%

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

   2. flip3-+ 19.0%

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

   3. rem-cube-cbrt 19.0%

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

   4. pow2 19.0%

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

   5. rem-cbrt-cube 19.0%

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

   6. rem-cbrt-cube 19.0%

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

8. Applied egg-rr 19.0%

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

   1. times-frac 0.0%

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

   2. unpow2 0.0%

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

   3. associate-/l* 21.2%

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

   4. unpow2 21.2%

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

   5. +-commutative 21.2%

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

   6. swap-sqr 21.2%

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

   7. fma-def 21.2%

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

10. Simplified 24.4%

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

    1. expm1-log1p-u 13.5%

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

    2. expm1-udef 13.5%

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

    3. sub-neg 13.5%

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

    4. add-sqr-sqrt 6.8%

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

    5. sqrt-unprod 10.5%

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

    6. sqr-neg 10.5%

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

    7. sqrt-unprod 6.7%

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

    8. add-sqr-sqrt 13.5%

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

12. Applied egg-rr 13.5%

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

    1. expm1-def 13.5%

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

    2. expm1-log1p 24.4%

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

    3. +-commutative 24.4%

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

14. Simplified 24.4%

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

15. Final simplification 24.4%

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

Alternative 2: 24.3% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{hi + x}{lo}\\ t_1 := \frac{hi}{\frac{lo \cdot lo}{hi - x}}\\ \frac{{t_1}^{3} + {t_0}^{3}}{\mathsf{fma}\left(\frac{hi}{lo} \cdot \frac{hi}{lo}, \frac{hi}{lo} \cdot \frac{hi - x}{lo}, t_0 \cdot \left(t_0 - t_1\right)\right)} + 1 \end{array} \end{array} \]

FPCore

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

C

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

Julia

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

Wolfram

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

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

   2. pow3 19.0%

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

   3. sub-neg 19.0%

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

   4. add-sqr-sqrt 9.5%

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

   5. sqrt-unprod 14.5%

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

   6. sqr-neg 14.5%

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

   7. sqrt-unprod 9.4%

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

   8. add-sqr-sqrt 19.0%

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

6. Applied egg-rr 19.0%

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

   1. fma-udef 19.0%

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

   2. flip3-+ 19.0%

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

   3. rem-cube-cbrt 19.0%

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

   4. pow2 19.0%

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

   5. rem-cbrt-cube 19.0%

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

   6. rem-cbrt-cube 19.0%

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

8. Applied egg-rr 19.0%

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

   1. times-frac 0.0%

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

   2. unpow2 0.0%

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

   3. associate-/l* 21.2%

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

   4. unpow2 21.2%

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

   5. +-commutative 21.2%

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

   6. swap-sqr 21.2%

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

   7. fma-def 21.2%

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

10. Simplified 24.4%

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

11. Taylor expanded in hi around inf 24.4%

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

12. Final simplification 24.4%

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

Alternative 3: 24.3% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{hi - x}{lo}\\ t_1 := \frac{hi + x}{lo}\\ \frac{{t_1}^{3}}{\mathsf{fma}\left(\frac{hi}{lo} \cdot \frac{hi}{lo}, t_0 \cdot t_0, t_1 \cdot \left(t_1 - \frac{hi}{\frac{lo \cdot lo}{hi - x}}\right)\right)} + 1 \end{array} \end{array} \]

FPCore

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

C

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

Julia

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

Wolfram

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

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

   2. pow3 19.0%

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

   3. sub-neg 19.0%

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

   4. add-sqr-sqrt 9.5%

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

   5. sqrt-unprod 14.5%

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

   6. sqr-neg 14.5%

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

   7. sqrt-unprod 9.4%

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

   8. add-sqr-sqrt 19.0%

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

6. Applied egg-rr 19.0%

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

   1. fma-udef 19.0%

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

   2. flip3-+ 19.0%

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

   3. rem-cube-cbrt 19.0%

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

   4. pow2 19.0%

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

   5. rem-cbrt-cube 19.0%

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

   6. rem-cbrt-cube 19.0%

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

8. Applied egg-rr 19.0%

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

   1. times-frac 0.0%

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

   2. unpow2 0.0%

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

   3. associate-/l* 21.2%

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

   4. unpow2 21.2%

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

   5. +-commutative 21.2%

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

   6. swap-sqr 21.2%

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

   7. fma-def 21.2%

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

10. Simplified 24.4%

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

11. Taylor expanded in lo around inf 0.0%

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

    1. cube-div 24.4%

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

13. Simplified 24.4%

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

14. Final simplification 24.4%

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

Alternative 4: 19.2% accurate, 0.0× speedup

Math

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

FPCore

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

C

double code(double lo, double hi, double x) {
	return fabs(pow((lo / hi), 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 = abs(((lo / hi) ** 2.0d0))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 3.1%

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

2. Taylor expanded in hi around inf 0.0%

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

   1. +-commutative 0.0%

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

   2. associate--l+ 0.0%

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

   3. *-commutative 0.0%

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

   4. unpow2 0.0%

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

   5. times-frac 8.9%

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

   6. div-sub 8.9%

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

4. Simplified 8.9%

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

   1. add-sqr-sqrt 8.1%

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

   2. sqrt-unprod 18.1%

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

   3. pow2 18.1%

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

   4. fma-def 18.1%

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

6. Applied egg-rr 18.1%

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

   1. unpow2 18.1%

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

   2. rem-sqrt-square 18.1%

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

   3. fma-udef 18.1%

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

   4. *-rgt-identity 18.1%

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

   5. distribute-lft-out 18.1%

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

   6. distribute-rgt-out 18.1%

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

   7. *-lft-identity 18.1%

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

   8. distribute-lft1-in 18.1%

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

   9. distribute-rgt1-in 18.1%

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

   10. *-lft-identity 18.1%

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

   11. distribute-rgt-out 18.1%

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

8. Simplified 18.1%

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

9. Taylor expanded in lo around inf 0.0%

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

    1. mul-1-neg 0.0%

       \[\leadsto \left|\color{blue}{-\frac{{lo}^{2}}{{hi}^{2}}}\right| \]

    2. unpow2 0.0%

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

    3. unpow2 0.0%

       \[\leadsto \left|-\frac{lo \cdot lo}{\color{blue}{hi \cdot hi}}\right| \]

    4. times-frac 19.1%

       \[\leadsto \left|-\color{blue}{\frac{lo}{hi} \cdot \frac{lo}{hi}}\right| \]

    5. unpow2 19.1%

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

11. Simplified 19.1%

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

12. Final simplification 19.1%

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

Alternative 5: 19.2% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \left|\frac{lo}{hi} \cdot \frac{x - lo}{hi}\right| \end{array} \]

FPCore

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

C

double code(double lo, double hi, double x) {
	return fabs(((lo / hi) * ((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 = abs(((lo / hi) * ((x - lo) / hi)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 3.1%

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

2. Taylor expanded in hi around inf 0.0%

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

   1. +-commutative 0.0%

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

   2. associate--l+ 0.0%

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

   3. *-commutative 0.0%

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

   4. unpow2 0.0%

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

   5. times-frac 8.9%

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

   6. div-sub 8.9%

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

4. Simplified 8.9%

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

   1. add-sqr-sqrt 8.1%

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

   2. sqrt-unprod 18.1%

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

   3. pow2 18.1%

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

   4. fma-def 18.1%

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

6. Applied egg-rr 18.1%

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

   1. unpow2 18.1%

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

   2. rem-sqrt-square 18.1%

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

   3. fma-udef 18.1%

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

   4. *-rgt-identity 18.1%

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

   5. distribute-lft-out 18.1%

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

   6. distribute-rgt-out 18.1%

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

   7. *-lft-identity 18.1%

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

   8. distribute-lft1-in 18.1%

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

   9. distribute-rgt1-in 18.1%

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

   10. *-lft-identity 18.1%

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

   11. distribute-rgt-out 18.1%

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

8. Simplified 18.1%

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

9. Taylor expanded in lo around inf 19.1%

   \[\leadsto \left|\frac{x - lo}{hi} \cdot \color{blue}{\frac{lo}{hi}}\right| \]

10. Final simplification 19.1%

    \[\leadsto \left|\frac{lo}{hi} \cdot \frac{x - lo}{hi}\right| \]

Alternative 6: 18.9% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   2. pow3 19.0%

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

   3. sub-neg 19.0%

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

   4. add-sqr-sqrt 9.5%

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

   5. sqrt-unprod 14.5%

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

   6. sqr-neg 14.5%

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

   7. sqrt-unprod 9.4%

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

   8. add-sqr-sqrt 19.0%

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

6. Applied egg-rr 19.0%

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

   1. fma-udef 19.0%

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

   2. rem-cbrt-cube 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%

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

Alternative 7: 18.9% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   2. pow3 19.0%

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

   3. sub-neg 19.0%

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

   4. add-sqr-sqrt 9.5%

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

   5. sqrt-unprod 14.5%

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

   6. sqr-neg 14.5%

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

   7. sqrt-unprod 9.4%

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

   8. add-sqr-sqrt 19.0%

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

6. Applied egg-rr 19.0%

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

   1. fma-udef 19.0%

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

   2. rem-cbrt-cube 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. Taylor expanded in hi around inf 19.0%

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

10. Final simplification 19.0%

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

Alternative 8: 18.9% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   2. sub-neg 19.0%

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

   3. add-sqr-sqrt 9.5%

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

   4. sqrt-unprod 15.1%

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

   5. sqr-neg 15.1%

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

   6. sqrt-unprod 9.4%

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

   7. add-sqr-sqrt 19.0%

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

   8. sub-neg 19.0%

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

   9. add-sqr-sqrt 9.5%

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

   10. sqrt-unprod 14.5%

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

   11. sqr-neg 14.5%

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

   12. sqrt-unprod 9.4%

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

   13. add-sqr-sqrt 19.0%

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

6. Applied egg-rr 19.0%

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

   1. distribute-lft1-in 19.0%

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

   2. +-commutative 19.0%

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

   3. +-commutative 19.0%

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

8. Simplified 19.0%

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

9. Final simplification 19.0%

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

Alternative 9: 18.9% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   2. pow3 19.0%

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

   3. sub-neg 19.0%

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

   4. add-sqr-sqrt 9.5%

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

   5. sqrt-unprod 14.5%

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

   6. sqr-neg 14.5%

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

   7. sqrt-unprod 9.4%

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

   8. add-sqr-sqrt 19.0%

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

6. Applied egg-rr 19.0%

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

   1. fma-udef 19.0%

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

   2. rem-cbrt-cube 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. Taylor expanded in x around 0 0.0%

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

    1. *-rgt-identity 0.0%

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

    2. unpow2 0.0%

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

    3. unpow2 0.0%

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

    4. times-frac 19.0%

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

    5. distribute-lft-in 19.0%

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

11. Simplified 19.0%

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

12. Final simplification 19.0%

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

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 2023299 
(FPCore (lo hi x)
  :name "xlohi (overflows)"
  :precision binary64
  :pre (and (< lo -1e+308) (> hi 1e+308))
  (/ (- x lo) (- hi lo)))