xlohi (overflows)

Percentage Accurate: 3.1% → 26.0%

Time: 25.6s

Alternatives: 9

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: 26.0% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x - hi}{lo}\\ t_1 := {t\_0}^{2}\\ t_2 := t\_0 + -1\\ t_3 := \frac{\mathsf{fma}\left(hi, t\_2, x\right)}{lo}\\ \mathbf{if}\;hi \leq 1.482 \cdot 10^{+308}:\\ \;\;\;\;\frac{{\left(\frac{x + \left(hi \cdot t\_0 - hi\right)}{lo}\right)}^{3} + 1}{t\_1 + \left(1 - \frac{x + \left|hi \cdot t\_2\right|}{lo}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + {t\_3}^{3}}{\left(1 + t\_1\right) - t\_3}\\ \end{array} \end{array} \]

FPCore

(FPCore (lo hi x)
 :precision binary64
 (let* ((t_0 (/ (- x hi) lo))
        (t_1 (pow t_0 2.0))
        (t_2 (+ t_0 -1.0))
        (t_3 (/ (fma hi t_2 x) lo)))
   (if (<= hi 1.482e+308)
     (/
      (+ (pow (/ (+ x (- (* hi t_0) hi)) lo) 3.0) 1.0)
      (+ t_1 (- 1.0 (/ (+ x (fabs (* hi t_2))) lo))))
     (/ (+ 1.0 (pow t_3 3.0)) (- (+ 1.0 t_1) t_3)))))

C

double code(double lo, double hi, double x) {
	double t_0 = (x - hi) / lo;
	double t_1 = pow(t_0, 2.0);
	double t_2 = t_0 + -1.0;
	double t_3 = fma(hi, t_2, x) / lo;
	double tmp;
	if (hi <= 1.482e+308) {
		tmp = (pow(((x + ((hi * t_0) - hi)) / lo), 3.0) + 1.0) / (t_1 + (1.0 - ((x + fabs((hi * t_2))) / lo)));
	} else {
		tmp = (1.0 + pow(t_3, 3.0)) / ((1.0 + t_1) - t_3);
	}
	return tmp;
}

Julia

function code(lo, hi, x)
	t_0 = Float64(Float64(x - hi) / lo)
	t_1 = t_0 ^ 2.0
	t_2 = Float64(t_0 + -1.0)
	t_3 = Float64(fma(hi, t_2, x) / lo)
	tmp = 0.0
	if (hi <= 1.482e+308)
		tmp = Float64(Float64((Float64(Float64(x + Float64(Float64(hi * t_0) - hi)) / lo) ^ 3.0) + 1.0) / Float64(t_1 + Float64(1.0 - Float64(Float64(x + abs(Float64(hi * t_2))) / lo))));
	else
		tmp = Float64(Float64(1.0 + (t_3 ^ 3.0)) / Float64(Float64(1.0 + t_1) - t_3));
	end
	return tmp
end

Wolfram

code[lo_, hi_, x_] := Block[{t$95$0 = N[(N[(x - hi), $MachinePrecision] / lo), $MachinePrecision]}, Block[{t$95$1 = N[Power[t$95$0, 2.0], $MachinePrecision]}, Block[{t$95$2 = N[(t$95$0 + -1.0), $MachinePrecision]}, Block[{t$95$3 = N[(N[(hi * t$95$2 + x), $MachinePrecision] / lo), $MachinePrecision]}, If[LessEqual[hi, 1.482e+308], N[(N[(N[Power[N[(N[(x + N[(N[(hi * t$95$0), $MachinePrecision] - hi), $MachinePrecision]), $MachinePrecision] / lo), $MachinePrecision], 3.0], $MachinePrecision] + 1.0), $MachinePrecision] / N[(t$95$1 + N[(1.0 - N[(N[(x + N[Abs[N[(hi * t$95$2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / lo), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[Power[t$95$3, 3.0], $MachinePrecision]), $MachinePrecision] / N[(N[(1.0 + t$95$1), $MachinePrecision] - t$95$3), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 2 regimes

2. if hi < 1.482e308

   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* 18.7%

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

   5. Simplified 18.7%

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

      1. +-commutative 18.7%

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

      2. flip3-+ 18.5%

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

   7. Applied egg-rr 17.9%

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

   8. Taylor expanded in lo around inf 21.9%

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

      1. neg-mul-1 21.9%

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

   10. Simplified 21.9%

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

       1. add-sqr-sqrt 5.5%

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

       2. sqrt-unprod 3.0%

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

       3. pow2 3.0%

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

   12. Applied egg-rr 3.0%

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

       1. unpow2 3.0%

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

       2. rem-sqrt-square 29.7%

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

       3. sub-neg 29.7%

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

       4. +-commutative 29.7%

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

       5. neg-mul-1 29.7%

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

       6. *-commutative 29.7%

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

       7. distribute-rgt-out 29.7%

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

   14. Simplified 29.7%

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

   if 1.482e308 < hi

   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* 7.2%

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

   5. Simplified 7.2%

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

      1. +-commutative 7.2%

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

      2. flip3-+ 4.9%

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

   7. Applied egg-rr 4.9%

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

   8. Taylor expanded in lo around inf 6.5%

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

      1. neg-mul-1 6.5%

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

   10. Simplified 6.5%

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

       1. *-un-lft-identity 6.5%

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

   12. Applied egg-rr 20.8%

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

       1. *-lft-identity 20.8%

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

       2. +-commutative 20.8%

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

       3. associate-+r- 20.8%

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

       4. +-commutative 20.8%

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

   14. Simplified 20.8%

       \[\leadsto \color{blue}{\frac{1 + {\left(\frac{\mathsf{fma}\left(hi, -1 + \frac{x - hi}{lo}, x\right)}{lo}\right)}^{3}}{\left({\left(\frac{x - hi}{lo}\right)}^{2} + 1\right) - \frac{\mathsf{fma}\left(hi, -1 + \frac{x - hi}{lo}, x\right)}{lo}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 26.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;hi \leq 1.482 \cdot 10^{+308}:\\ \;\;\;\;\frac{{\left(\frac{x + \left(hi \cdot \frac{x - hi}{lo} - hi\right)}{lo}\right)}^{3} + 1}{{\left(\frac{x - hi}{lo}\right)}^{2} + \left(1 - \frac{x + \left|hi \cdot \left(\frac{x - hi}{lo} + -1\right)\right|}{lo}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + {\left(\frac{\mathsf{fma}\left(hi, \frac{x - hi}{lo} + -1, x\right)}{lo}\right)}^{3}}{\left(1 + {\left(\frac{x - hi}{lo}\right)}^{2}\right) - \frac{\mathsf{fma}\left(hi, \frac{x - hi}{lo} + -1, x\right)}{lo}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 25.2% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x - hi}{lo}\\ \mathbf{if}\;hi \leq 1.5 \cdot 10^{+308}:\\ \;\;\;\;\frac{{\left(\frac{x + \left(hi \cdot t\_0 - hi\right)}{lo}\right)}^{3} + 1}{{t\_0}^{2} + \left(1 - \frac{x + \left|hi \cdot \left(t\_0 + -1\right)\right|}{lo}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + hi \cdot \frac{-1 - \frac{hi}{lo}}{lo}\right) + x \cdot \left(\frac{1}{lo} + \frac{hi}{{lo}^{2}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (lo hi x)
 :precision binary64
 (let* ((t_0 (/ (- x hi) lo)))
   (if (<= hi 1.5e+308)
     (/
      (+ (pow (/ (+ x (- (* hi t_0) hi)) lo) 3.0) 1.0)
      (+ (pow t_0 2.0) (- 1.0 (/ (+ x (fabs (* hi (+ t_0 -1.0)))) lo))))
     (+
      (+ 1.0 (* hi (/ (- -1.0 (/ hi lo)) lo)))
      (* x (+ (/ 1.0 lo) (/ hi (pow lo 2.0))))))))

C

double code(double lo, double hi, double x) {
	double t_0 = (x - hi) / lo;
	double tmp;
	if (hi <= 1.5e+308) {
		tmp = (pow(((x + ((hi * t_0) - hi)) / lo), 3.0) + 1.0) / (pow(t_0, 2.0) + (1.0 - ((x + fabs((hi * (t_0 + -1.0)))) / lo)));
	} else {
		tmp = (1.0 + (hi * ((-1.0 - (hi / lo)) / lo))) + (x * ((1.0 / lo) + (hi / pow(lo, 2.0))));
	}
	return tmp;
}

Fortran

real(8) function code(lo, hi, x)
    real(8), intent (in) :: lo
    real(8), intent (in) :: hi
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x - hi) / lo
    if (hi <= 1.5d+308) then
        tmp = ((((x + ((hi * t_0) - hi)) / lo) ** 3.0d0) + 1.0d0) / ((t_0 ** 2.0d0) + (1.0d0 - ((x + abs((hi * (t_0 + (-1.0d0))))) / lo)))
    else
        tmp = (1.0d0 + (hi * (((-1.0d0) - (hi / lo)) / lo))) + (x * ((1.0d0 / lo) + (hi / (lo ** 2.0d0))))
    end if
    code = tmp
end function

Java

public static double code(double lo, double hi, double x) {
	double t_0 = (x - hi) / lo;
	double tmp;
	if (hi <= 1.5e+308) {
		tmp = (Math.pow(((x + ((hi * t_0) - hi)) / lo), 3.0) + 1.0) / (Math.pow(t_0, 2.0) + (1.0 - ((x + Math.abs((hi * (t_0 + -1.0)))) / lo)));
	} else {
		tmp = (1.0 + (hi * ((-1.0 - (hi / lo)) / lo))) + (x * ((1.0 / lo) + (hi / Math.pow(lo, 2.0))));
	}
	return tmp;
}

Python

def code(lo, hi, x):
	t_0 = (x - hi) / lo
	tmp = 0
	if hi <= 1.5e+308:
		tmp = (math.pow(((x + ((hi * t_0) - hi)) / lo), 3.0) + 1.0) / (math.pow(t_0, 2.0) + (1.0 - ((x + math.fabs((hi * (t_0 + -1.0)))) / lo)))
	else:
		tmp = (1.0 + (hi * ((-1.0 - (hi / lo)) / lo))) + (x * ((1.0 / lo) + (hi / math.pow(lo, 2.0))))
	return tmp

Julia

function code(lo, hi, x)
	t_0 = Float64(Float64(x - hi) / lo)
	tmp = 0.0
	if (hi <= 1.5e+308)
		tmp = Float64(Float64((Float64(Float64(x + Float64(Float64(hi * t_0) - hi)) / lo) ^ 3.0) + 1.0) / Float64((t_0 ^ 2.0) + Float64(1.0 - Float64(Float64(x + abs(Float64(hi * Float64(t_0 + -1.0)))) / lo))));
	else
		tmp = Float64(Float64(1.0 + Float64(hi * Float64(Float64(-1.0 - Float64(hi / lo)) / lo))) + Float64(x * Float64(Float64(1.0 / lo) + Float64(hi / (lo ^ 2.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(lo, hi, x)
	t_0 = (x - hi) / lo;
	tmp = 0.0;
	if (hi <= 1.5e+308)
		tmp = ((((x + ((hi * t_0) - hi)) / lo) ^ 3.0) + 1.0) / ((t_0 ^ 2.0) + (1.0 - ((x + abs((hi * (t_0 + -1.0)))) / lo)));
	else
		tmp = (1.0 + (hi * ((-1.0 - (hi / lo)) / lo))) + (x * ((1.0 / lo) + (hi / (lo ^ 2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[lo_, hi_, x_] := Block[{t$95$0 = N[(N[(x - hi), $MachinePrecision] / lo), $MachinePrecision]}, If[LessEqual[hi, 1.5e+308], N[(N[(N[Power[N[(N[(x + N[(N[(hi * t$95$0), $MachinePrecision] - hi), $MachinePrecision]), $MachinePrecision] / lo), $MachinePrecision], 3.0], $MachinePrecision] + 1.0), $MachinePrecision] / N[(N[Power[t$95$0, 2.0], $MachinePrecision] + N[(1.0 - N[(N[(x + N[Abs[N[(hi * N[(t$95$0 + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / lo), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[(hi * N[(N[(-1.0 - N[(hi / lo), $MachinePrecision]), $MachinePrecision] / lo), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x * N[(N[(1.0 / lo), $MachinePrecision] + N[(hi / N[Power[lo, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if hi < 1.5e308

   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* 18.6%

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

   5. Simplified 18.6%

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

      1. +-commutative 18.6%

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

      2. flip3-+ 18.5%

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

   7. Applied egg-rr 17.9%

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

   8. Taylor expanded in lo around inf 21.9%

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

      1. neg-mul-1 21.9%

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

   10. Simplified 21.9%

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

       1. add-sqr-sqrt 6.0%

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

       2. sqrt-unprod 3.0%

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

       3. pow2 3.0%

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

   12. Applied egg-rr 3.0%

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

       1. unpow2 3.0%

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

       2. rem-sqrt-square 29.5%

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

       3. sub-neg 29.5%

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

       4. +-commutative 29.5%

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

       5. neg-mul-1 29.5%

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

       6. *-commutative 29.5%

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

       7. distribute-rgt-out 29.5%

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

   14. Simplified 29.5%

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

   if 1.5e308 < hi

   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* 6.7%

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

   5. Simplified 6.7%

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

      1. expm1-log1p-u 6.7%

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

      2. expm1-undefine 6.7%

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

      3. add-sqr-sqrt 4.4%

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

      4. sqrt-unprod 6.6%

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

      5. sqr-neg 6.6%

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

      6. sqrt-unprod 2.2%

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

      7. add-sqr-sqrt 4.3%

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

   7. Applied egg-rr 4.3%

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

      1. expm1-define 4.3%

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

      2. associate-+r- 4.3%

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

      3. div-sub 4.3%

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

      4. associate-*r/ 0.0%

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

      5. div-sub 0.0%

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

      6. associate-*r/ 4.3%

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

      7. associate-+r- 4.3%

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

      8. fma-neg 19.5%

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

   9. Simplified 19.5%

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

   10. Taylor expanded in hi around 0 19.5%

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

       1. sub-neg 19.5%

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

       2. mul-1-neg 19.5%

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

       3. +-commutative 19.5%

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

       4. sub-neg 19.5%

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

       5. div-sub 19.5%

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

       6. metadata-eval 19.5%

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

   12. Simplified 19.5%

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

   13. Taylor expanded in x around 0 19.5%

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

       1. associate-+r+ 19.5%

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

       2. mul-1-neg 19.5%

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

       3. associate-/l* 19.5%

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

       4. distribute-rgt-neg-in 19.5%

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

       5. mul-1-neg 19.5%

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

       6. associate-*r/ 19.5%

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

       7. distribute-lft-in 19.5%

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

       8. metadata-eval 19.5%

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

       9. mul-1-neg 19.5%

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

       10. unsub-neg 19.5%

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

   15. Simplified 19.5%

       \[\leadsto \color{blue}{\left(1 + hi \cdot \frac{-1 - \frac{hi}{lo}}{lo}\right) + x \cdot \left(\frac{1}{lo} + \frac{hi}{{lo}^{2}}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 25.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;hi \leq 1.5 \cdot 10^{+308}:\\ \;\;\;\;\frac{{\left(\frac{x + \left(hi \cdot \frac{x - hi}{lo} - hi\right)}{lo}\right)}^{3} + 1}{{\left(\frac{x - hi}{lo}\right)}^{2} + \left(1 - \frac{x + \left|hi \cdot \left(\frac{x - hi}{lo} + -1\right)\right|}{lo}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + hi \cdot \frac{-1 - \frac{hi}{lo}}{lo}\right) + x \cdot \left(\frac{1}{lo} + \frac{hi}{{lo}^{2}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 21.7% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x + \left(hi \cdot \frac{x - hi}{lo} - hi\right)}{lo}\\ \mathbf{if}\;hi \leq 1.5 \cdot 10^{+308}:\\ \;\;\;\;\frac{{t\_0}^{3} + 1}{{t\_0}^{2} + \left(1 - \frac{x + \left(x - hi\right) \cdot \frac{hi}{lo}}{lo}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + hi \cdot \frac{-1 - \frac{hi}{lo}}{lo}\right) + x \cdot \left(\frac{1}{lo} + \frac{hi}{{lo}^{2}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (lo hi x)
 :precision binary64
 (let* ((t_0 (/ (+ x (- (* hi (/ (- x hi) lo)) hi)) lo)))
   (if (<= hi 1.5e+308)
     (/
      (+ (pow t_0 3.0) 1.0)
      (+ (pow t_0 2.0) (- 1.0 (/ (+ x (* (- x hi) (/ hi lo))) lo))))
     (+
      (+ 1.0 (* hi (/ (- -1.0 (/ hi lo)) lo)))
      (* x (+ (/ 1.0 lo) (/ hi (pow lo 2.0))))))))

C

double code(double lo, double hi, double x) {
	double t_0 = (x + ((hi * ((x - hi) / lo)) - hi)) / lo;
	double tmp;
	if (hi <= 1.5e+308) {
		tmp = (pow(t_0, 3.0) + 1.0) / (pow(t_0, 2.0) + (1.0 - ((x + ((x - hi) * (hi / lo))) / lo)));
	} else {
		tmp = (1.0 + (hi * ((-1.0 - (hi / lo)) / lo))) + (x * ((1.0 / lo) + (hi / pow(lo, 2.0))));
	}
	return tmp;
}

Fortran

real(8) function code(lo, hi, x)
    real(8), intent (in) :: lo
    real(8), intent (in) :: hi
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x + ((hi * ((x - hi) / lo)) - hi)) / lo
    if (hi <= 1.5d+308) then
        tmp = ((t_0 ** 3.0d0) + 1.0d0) / ((t_0 ** 2.0d0) + (1.0d0 - ((x + ((x - hi) * (hi / lo))) / lo)))
    else
        tmp = (1.0d0 + (hi * (((-1.0d0) - (hi / lo)) / lo))) + (x * ((1.0d0 / lo) + (hi / (lo ** 2.0d0))))
    end if
    code = tmp
end function

Java

public static double code(double lo, double hi, double x) {
	double t_0 = (x + ((hi * ((x - hi) / lo)) - hi)) / lo;
	double tmp;
	if (hi <= 1.5e+308) {
		tmp = (Math.pow(t_0, 3.0) + 1.0) / (Math.pow(t_0, 2.0) + (1.0 - ((x + ((x - hi) * (hi / lo))) / lo)));
	} else {
		tmp = (1.0 + (hi * ((-1.0 - (hi / lo)) / lo))) + (x * ((1.0 / lo) + (hi / Math.pow(lo, 2.0))));
	}
	return tmp;
}

Python

def code(lo, hi, x):
	t_0 = (x + ((hi * ((x - hi) / lo)) - hi)) / lo
	tmp = 0
	if hi <= 1.5e+308:
		tmp = (math.pow(t_0, 3.0) + 1.0) / (math.pow(t_0, 2.0) + (1.0 - ((x + ((x - hi) * (hi / lo))) / lo)))
	else:
		tmp = (1.0 + (hi * ((-1.0 - (hi / lo)) / lo))) + (x * ((1.0 / lo) + (hi / math.pow(lo, 2.0))))
	return tmp

Julia

function code(lo, hi, x)
	t_0 = Float64(Float64(x + Float64(Float64(hi * Float64(Float64(x - hi) / lo)) - hi)) / lo)
	tmp = 0.0
	if (hi <= 1.5e+308)
		tmp = Float64(Float64((t_0 ^ 3.0) + 1.0) / Float64((t_0 ^ 2.0) + Float64(1.0 - Float64(Float64(x + Float64(Float64(x - hi) * Float64(hi / lo))) / lo))));
	else
		tmp = Float64(Float64(1.0 + Float64(hi * Float64(Float64(-1.0 - Float64(hi / lo)) / lo))) + Float64(x * Float64(Float64(1.0 / lo) + Float64(hi / (lo ^ 2.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(lo, hi, x)
	t_0 = (x + ((hi * ((x - hi) / lo)) - hi)) / lo;
	tmp = 0.0;
	if (hi <= 1.5e+308)
		tmp = ((t_0 ^ 3.0) + 1.0) / ((t_0 ^ 2.0) + (1.0 - ((x + ((x - hi) * (hi / lo))) / lo)));
	else
		tmp = (1.0 + (hi * ((-1.0 - (hi / lo)) / lo))) + (x * ((1.0 / lo) + (hi / (lo ^ 2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[lo_, hi_, x_] := Block[{t$95$0 = N[(N[(x + N[(N[(hi * N[(N[(x - hi), $MachinePrecision] / lo), $MachinePrecision]), $MachinePrecision] - hi), $MachinePrecision]), $MachinePrecision] / lo), $MachinePrecision]}, If[LessEqual[hi, 1.5e+308], N[(N[(N[Power[t$95$0, 3.0], $MachinePrecision] + 1.0), $MachinePrecision] / N[(N[Power[t$95$0, 2.0], $MachinePrecision] + N[(1.0 - N[(N[(x + N[(N[(x - hi), $MachinePrecision] * N[(hi / lo), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / lo), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[(hi * N[(N[(-1.0 - N[(hi / lo), $MachinePrecision]), $MachinePrecision] / lo), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x * N[(N[(1.0 / lo), $MachinePrecision] + N[(hi / N[Power[lo, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if hi < 1.5e308

   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* 18.6%

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

   5. Simplified 18.6%

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

      1. +-commutative 18.6%

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

      2. flip3-+ 18.5%

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

   7. Applied egg-rr 17.9%

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

   8. Taylor expanded in lo around 0 3.0%

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

      1. associate-*r/ 23.5%

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

      2. div-sub 23.5%

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

      3. sub-neg 23.5%

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

      4. mul-1-neg 23.5%

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

      5. distribute-lft-in 23.5%

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

      6. associate-/l* 13.4%

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

      7. *-commutative 13.4%

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

      8. associate-*r/ 23.5%

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

      9. associate-*r* 23.5%

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

      10. *-commutative 23.5%

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

      11. neg-mul-1 23.5%

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

      12. distribute-rgt-in 23.5%

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

      13. sub-neg 23.5%

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

   10. Simplified 23.5%

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

   if 1.5e308 < hi

   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* 6.7%

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

   5. Simplified 6.7%

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

      1. expm1-log1p-u 6.7%

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

      2. expm1-undefine 6.7%

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

      3. add-sqr-sqrt 4.4%

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

      4. sqrt-unprod 6.6%

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

      5. sqr-neg 6.6%

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

      6. sqrt-unprod 2.2%

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

      7. add-sqr-sqrt 4.3%

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

   7. Applied egg-rr 4.3%

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

      1. expm1-define 4.3%

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

      2. associate-+r- 4.3%

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

      3. div-sub 4.3%

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

      4. associate-*r/ 0.0%

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

      5. div-sub 0.0%

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

      6. associate-*r/ 4.3%

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

      7. associate-+r- 4.3%

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

      8. fma-neg 19.5%

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

   9. Simplified 19.5%

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

   10. Taylor expanded in hi around 0 19.5%

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

       1. sub-neg 19.5%

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

       2. mul-1-neg 19.5%

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

       3. +-commutative 19.5%

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

       4. sub-neg 19.5%

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

       5. div-sub 19.5%

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

       6. metadata-eval 19.5%

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

   12. Simplified 19.5%

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

   13. Taylor expanded in x around 0 19.5%

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

       1. associate-+r+ 19.5%

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

       2. mul-1-neg 19.5%

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

       3. associate-/l* 19.5%

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

       4. distribute-rgt-neg-in 19.5%

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

       5. mul-1-neg 19.5%

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

       6. associate-*r/ 19.5%

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

       7. distribute-lft-in 19.5%

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

       8. metadata-eval 19.5%

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

       9. mul-1-neg 19.5%

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

       10. unsub-neg 19.5%

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

   15. Simplified 19.5%

       \[\leadsto \color{blue}{\left(1 + hi \cdot \frac{-1 - \frac{hi}{lo}}{lo}\right) + x \cdot \left(\frac{1}{lo} + \frac{hi}{{lo}^{2}}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 22.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;hi \leq 1.5 \cdot 10^{+308}:\\ \;\;\;\;\frac{{\left(\frac{x + \left(hi \cdot \frac{x - hi}{lo} - hi\right)}{lo}\right)}^{3} + 1}{{\left(\frac{x + \left(hi \cdot \frac{x - hi}{lo} - hi\right)}{lo}\right)}^{2} + \left(1 - \frac{x + \left(x - hi\right) \cdot \frac{hi}{lo}}{lo}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + hi \cdot \frac{-1 - \frac{hi}{lo}}{lo}\right) + x \cdot \left(\frac{1}{lo} + \frac{hi}{{lo}^{2}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 21.6% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x - hi}{lo}\\ \mathbf{if}\;hi \leq 1.5 \cdot 10^{+308}:\\ \;\;\;\;\frac{{\left(\frac{x + \left(hi \cdot t\_0 - hi\right)}{lo}\right)}^{3} + 1}{{t\_0}^{2} + \left(1 - \frac{x + x \cdot \frac{hi}{lo}}{lo}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + hi \cdot \frac{-1 - \frac{hi}{lo}}{lo}\right) + x \cdot \left(\frac{1}{lo} + \frac{hi}{{lo}^{2}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (lo hi x)
 :precision binary64
 (let* ((t_0 (/ (- x hi) lo)))
   (if (<= hi 1.5e+308)
     (/
      (+ (pow (/ (+ x (- (* hi t_0) hi)) lo) 3.0) 1.0)
      (+ (pow t_0 2.0) (- 1.0 (/ (+ x (* x (/ hi lo))) lo))))
     (+
      (+ 1.0 (* hi (/ (- -1.0 (/ hi lo)) lo)))
      (* x (+ (/ 1.0 lo) (/ hi (pow lo 2.0))))))))

C

double code(double lo, double hi, double x) {
	double t_0 = (x - hi) / lo;
	double tmp;
	if (hi <= 1.5e+308) {
		tmp = (pow(((x + ((hi * t_0) - hi)) / lo), 3.0) + 1.0) / (pow(t_0, 2.0) + (1.0 - ((x + (x * (hi / lo))) / lo)));
	} else {
		tmp = (1.0 + (hi * ((-1.0 - (hi / lo)) / lo))) + (x * ((1.0 / lo) + (hi / pow(lo, 2.0))));
	}
	return tmp;
}

Fortran

real(8) function code(lo, hi, x)
    real(8), intent (in) :: lo
    real(8), intent (in) :: hi
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x - hi) / lo
    if (hi <= 1.5d+308) then
        tmp = ((((x + ((hi * t_0) - hi)) / lo) ** 3.0d0) + 1.0d0) / ((t_0 ** 2.0d0) + (1.0d0 - ((x + (x * (hi / lo))) / lo)))
    else
        tmp = (1.0d0 + (hi * (((-1.0d0) - (hi / lo)) / lo))) + (x * ((1.0d0 / lo) + (hi / (lo ** 2.0d0))))
    end if
    code = tmp
end function

Java

public static double code(double lo, double hi, double x) {
	double t_0 = (x - hi) / lo;
	double tmp;
	if (hi <= 1.5e+308) {
		tmp = (Math.pow(((x + ((hi * t_0) - hi)) / lo), 3.0) + 1.0) / (Math.pow(t_0, 2.0) + (1.0 - ((x + (x * (hi / lo))) / lo)));
	} else {
		tmp = (1.0 + (hi * ((-1.0 - (hi / lo)) / lo))) + (x * ((1.0 / lo) + (hi / Math.pow(lo, 2.0))));
	}
	return tmp;
}

Python

def code(lo, hi, x):
	t_0 = (x - hi) / lo
	tmp = 0
	if hi <= 1.5e+308:
		tmp = (math.pow(((x + ((hi * t_0) - hi)) / lo), 3.0) + 1.0) / (math.pow(t_0, 2.0) + (1.0 - ((x + (x * (hi / lo))) / lo)))
	else:
		tmp = (1.0 + (hi * ((-1.0 - (hi / lo)) / lo))) + (x * ((1.0 / lo) + (hi / math.pow(lo, 2.0))))
	return tmp

Julia

function code(lo, hi, x)
	t_0 = Float64(Float64(x - hi) / lo)
	tmp = 0.0
	if (hi <= 1.5e+308)
		tmp = Float64(Float64((Float64(Float64(x + Float64(Float64(hi * t_0) - hi)) / lo) ^ 3.0) + 1.0) / Float64((t_0 ^ 2.0) + Float64(1.0 - Float64(Float64(x + Float64(x * Float64(hi / lo))) / lo))));
	else
		tmp = Float64(Float64(1.0 + Float64(hi * Float64(Float64(-1.0 - Float64(hi / lo)) / lo))) + Float64(x * Float64(Float64(1.0 / lo) + Float64(hi / (lo ^ 2.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(lo, hi, x)
	t_0 = (x - hi) / lo;
	tmp = 0.0;
	if (hi <= 1.5e+308)
		tmp = ((((x + ((hi * t_0) - hi)) / lo) ^ 3.0) + 1.0) / ((t_0 ^ 2.0) + (1.0 - ((x + (x * (hi / lo))) / lo)));
	else
		tmp = (1.0 + (hi * ((-1.0 - (hi / lo)) / lo))) + (x * ((1.0 / lo) + (hi / (lo ^ 2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[lo_, hi_, x_] := Block[{t$95$0 = N[(N[(x - hi), $MachinePrecision] / lo), $MachinePrecision]}, If[LessEqual[hi, 1.5e+308], N[(N[(N[Power[N[(N[(x + N[(N[(hi * t$95$0), $MachinePrecision] - hi), $MachinePrecision]), $MachinePrecision] / lo), $MachinePrecision], 3.0], $MachinePrecision] + 1.0), $MachinePrecision] / N[(N[Power[t$95$0, 2.0], $MachinePrecision] + N[(1.0 - N[(N[(x + N[(x * N[(hi / lo), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / lo), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[(hi * N[(N[(-1.0 - N[(hi / lo), $MachinePrecision]), $MachinePrecision] / lo), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x * N[(N[(1.0 / lo), $MachinePrecision] + N[(hi / N[Power[lo, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if hi < 1.5e308

   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* 18.6%

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

   5. Simplified 18.6%

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

      1. +-commutative 18.6%

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

      2. flip3-+ 18.5%

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

   7. Applied egg-rr 17.9%

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

   8. Taylor expanded in lo around inf 21.9%

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

      1. neg-mul-1 21.9%

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

   10. Simplified 21.9%

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

   11. Taylor expanded in x around inf 13.3%

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

       1. *-commutative 13.3%

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

       2. associate-*r/ 23.5%

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

   13. Simplified 23.5%

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

   if 1.5e308 < hi

   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* 6.7%

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

   5. Simplified 6.7%

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

      1. expm1-log1p-u 6.7%

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

      2. expm1-undefine 6.7%

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

      3. add-sqr-sqrt 4.4%

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

      4. sqrt-unprod 6.6%

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

      5. sqr-neg 6.6%

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

      6. sqrt-unprod 2.2%

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

      7. add-sqr-sqrt 4.3%

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

   7. Applied egg-rr 4.3%

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

      1. expm1-define 4.3%

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

      2. associate-+r- 4.3%

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

      3. div-sub 4.3%

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

      4. associate-*r/ 0.0%

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

      5. div-sub 0.0%

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

      6. associate-*r/ 4.3%

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

      7. associate-+r- 4.3%

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

      8. fma-neg 19.5%

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

   9. Simplified 19.5%

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

   10. Taylor expanded in hi around 0 19.5%

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

       1. sub-neg 19.5%

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

       2. mul-1-neg 19.5%

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

       3. +-commutative 19.5%

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

       4. sub-neg 19.5%

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

       5. div-sub 19.5%

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

       6. metadata-eval 19.5%

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

   12. Simplified 19.5%

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

   13. Taylor expanded in x around 0 19.5%

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

       1. associate-+r+ 19.5%

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

       2. mul-1-neg 19.5%

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

       3. associate-/l* 19.5%

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

       4. distribute-rgt-neg-in 19.5%

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

       5. mul-1-neg 19.5%

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

       6. associate-*r/ 19.5%

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

       7. distribute-lft-in 19.5%

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

       8. metadata-eval 19.5%

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

       9. mul-1-neg 19.5%

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

       10. unsub-neg 19.5%

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

   15. Simplified 19.5%

       \[\leadsto \color{blue}{\left(1 + hi \cdot \frac{-1 - \frac{hi}{lo}}{lo}\right) + x \cdot \left(\frac{1}{lo} + \frac{hi}{{lo}^{2}}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 22.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;hi \leq 1.5 \cdot 10^{+308}:\\ \;\;\;\;\frac{{\left(\frac{x + \left(hi \cdot \frac{x - hi}{lo} - hi\right)}{lo}\right)}^{3} + 1}{{\left(\frac{x - hi}{lo}\right)}^{2} + \left(1 - \frac{x + x \cdot \frac{hi}{lo}}{lo}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + hi \cdot \frac{-1 - \frac{hi}{lo}}{lo}\right) + x \cdot \left(\frac{1}{lo} + \frac{hi}{{lo}^{2}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 20.6% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{hi - x}{lo}\\ \mathbf{if}\;hi \leq 1.5 \cdot 10^{+308}:\\ \;\;\;\;\frac{{\left(\frac{x + \left(hi \cdot \frac{x - hi}{lo} - hi\right)}{lo}\right)}^{3} + 1}{t\_0 \cdot t\_0 + \left(1 + \frac{\left(hi + hi \cdot t\_0\right) - x}{lo}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + hi \cdot \frac{-1 - \frac{hi}{lo}}{lo}\right) + x \cdot \left(\frac{1}{lo} + \frac{hi}{{lo}^{2}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (lo hi x)
 :precision binary64
 (let* ((t_0 (/ (- hi x) lo)))
   (if (<= hi 1.5e+308)
     (/
      (+ (pow (/ (+ x (- (* hi (/ (- x hi) lo)) hi)) lo) 3.0) 1.0)
      (+ (* t_0 t_0) (+ 1.0 (/ (- (+ hi (* hi t_0)) x) lo))))
     (+
      (+ 1.0 (* hi (/ (- -1.0 (/ hi lo)) lo)))
      (* x (+ (/ 1.0 lo) (/ hi (pow lo 2.0))))))))

C

double code(double lo, double hi, double x) {
	double t_0 = (hi - x) / lo;
	double tmp;
	if (hi <= 1.5e+308) {
		tmp = (pow(((x + ((hi * ((x - hi) / lo)) - hi)) / lo), 3.0) + 1.0) / ((t_0 * t_0) + (1.0 + (((hi + (hi * t_0)) - x) / lo)));
	} else {
		tmp = (1.0 + (hi * ((-1.0 - (hi / lo)) / lo))) + (x * ((1.0 / lo) + (hi / pow(lo, 2.0))));
	}
	return tmp;
}

Fortran

real(8) function code(lo, hi, x)
    real(8), intent (in) :: lo
    real(8), intent (in) :: hi
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (hi - x) / lo
    if (hi <= 1.5d+308) then
        tmp = ((((x + ((hi * ((x - hi) / lo)) - hi)) / lo) ** 3.0d0) + 1.0d0) / ((t_0 * t_0) + (1.0d0 + (((hi + (hi * t_0)) - x) / lo)))
    else
        tmp = (1.0d0 + (hi * (((-1.0d0) - (hi / lo)) / lo))) + (x * ((1.0d0 / lo) + (hi / (lo ** 2.0d0))))
    end if
    code = tmp
end function

Java

public static double code(double lo, double hi, double x) {
	double t_0 = (hi - x) / lo;
	double tmp;
	if (hi <= 1.5e+308) {
		tmp = (Math.pow(((x + ((hi * ((x - hi) / lo)) - hi)) / lo), 3.0) + 1.0) / ((t_0 * t_0) + (1.0 + (((hi + (hi * t_0)) - x) / lo)));
	} else {
		tmp = (1.0 + (hi * ((-1.0 - (hi / lo)) / lo))) + (x * ((1.0 / lo) + (hi / Math.pow(lo, 2.0))));
	}
	return tmp;
}

Python

def code(lo, hi, x):
	t_0 = (hi - x) / lo
	tmp = 0
	if hi <= 1.5e+308:
		tmp = (math.pow(((x + ((hi * ((x - hi) / lo)) - hi)) / lo), 3.0) + 1.0) / ((t_0 * t_0) + (1.0 + (((hi + (hi * t_0)) - x) / lo)))
	else:
		tmp = (1.0 + (hi * ((-1.0 - (hi / lo)) / lo))) + (x * ((1.0 / lo) + (hi / math.pow(lo, 2.0))))
	return tmp

Julia

function code(lo, hi, x)
	t_0 = Float64(Float64(hi - x) / lo)
	tmp = 0.0
	if (hi <= 1.5e+308)
		tmp = Float64(Float64((Float64(Float64(x + Float64(Float64(hi * Float64(Float64(x - hi) / lo)) - hi)) / lo) ^ 3.0) + 1.0) / Float64(Float64(t_0 * t_0) + Float64(1.0 + Float64(Float64(Float64(hi + Float64(hi * t_0)) - x) / lo))));
	else
		tmp = Float64(Float64(1.0 + Float64(hi * Float64(Float64(-1.0 - Float64(hi / lo)) / lo))) + Float64(x * Float64(Float64(1.0 / lo) + Float64(hi / (lo ^ 2.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(lo, hi, x)
	t_0 = (hi - x) / lo;
	tmp = 0.0;
	if (hi <= 1.5e+308)
		tmp = ((((x + ((hi * ((x - hi) / lo)) - hi)) / lo) ^ 3.0) + 1.0) / ((t_0 * t_0) + (1.0 + (((hi + (hi * t_0)) - x) / lo)));
	else
		tmp = (1.0 + (hi * ((-1.0 - (hi / lo)) / lo))) + (x * ((1.0 / lo) + (hi / (lo ^ 2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[lo_, hi_, x_] := Block[{t$95$0 = N[(N[(hi - x), $MachinePrecision] / lo), $MachinePrecision]}, If[LessEqual[hi, 1.5e+308], N[(N[(N[Power[N[(N[(x + N[(N[(hi * N[(N[(x - hi), $MachinePrecision] / lo), $MachinePrecision]), $MachinePrecision] - hi), $MachinePrecision]), $MachinePrecision] / lo), $MachinePrecision], 3.0], $MachinePrecision] + 1.0), $MachinePrecision] / N[(N[(t$95$0 * t$95$0), $MachinePrecision] + N[(1.0 + N[(N[(N[(hi + N[(hi * t$95$0), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / lo), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[(hi * N[(N[(-1.0 - N[(hi / lo), $MachinePrecision]), $MachinePrecision] / lo), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x * N[(N[(1.0 / lo), $MachinePrecision] + N[(hi / N[Power[lo, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if hi < 1.5e308

   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* 18.6%

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

   5. Simplified 18.6%

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

      1. +-commutative 18.6%

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

      2. flip3-+ 18.5%

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

   7. Applied egg-rr 17.9%

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

   8. Taylor expanded in lo around inf 21.9%

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

      1. neg-mul-1 21.9%

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

   10. Simplified 21.9%

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

       1. sub-neg 21.9%

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

       2. pow2 21.9%

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

   12. Applied egg-rr 21.9%

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

   if 1.5e308 < hi

   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* 6.7%

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

   5. Simplified 6.7%

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

      1. expm1-log1p-u 6.7%

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

      2. expm1-undefine 6.7%

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

      3. add-sqr-sqrt 4.4%

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

      4. sqrt-unprod 6.6%

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

      5. sqr-neg 6.6%

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

      6. sqrt-unprod 2.2%

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

      7. add-sqr-sqrt 4.3%

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

   7. Applied egg-rr 4.3%

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

      1. expm1-define 4.3%

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

      2. associate-+r- 4.3%

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

      3. div-sub 4.3%

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

      4. associate-*r/ 0.0%

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

      5. div-sub 0.0%

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

      6. associate-*r/ 4.3%

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

      7. associate-+r- 4.3%

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

      8. fma-neg 19.5%

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

   9. Simplified 19.5%

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

   10. Taylor expanded in hi around 0 19.5%

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

       1. sub-neg 19.5%

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

       2. mul-1-neg 19.5%

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

       3. +-commutative 19.5%

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

       4. sub-neg 19.5%

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

       5. div-sub 19.5%

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

       6. metadata-eval 19.5%

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

   12. Simplified 19.5%

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

   13. Taylor expanded in x around 0 19.5%

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

       1. associate-+r+ 19.5%

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

       2. mul-1-neg 19.5%

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

       3. associate-/l* 19.5%

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

       4. distribute-rgt-neg-in 19.5%

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

       5. mul-1-neg 19.5%

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

       6. associate-*r/ 19.5%

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

       7. distribute-lft-in 19.5%

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

       8. metadata-eval 19.5%

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

       9. mul-1-neg 19.5%

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

       10. unsub-neg 19.5%

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

   15. Simplified 19.5%

       \[\leadsto \color{blue}{\left(1 + hi \cdot \frac{-1 - \frac{hi}{lo}}{lo}\right) + x \cdot \left(\frac{1}{lo} + \frac{hi}{{lo}^{2}}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 21.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;hi \leq 1.5 \cdot 10^{+308}:\\ \;\;\;\;\frac{{\left(\frac{x + \left(hi \cdot \frac{x - hi}{lo} - hi\right)}{lo}\right)}^{3} + 1}{\frac{hi - x}{lo} \cdot \frac{hi - x}{lo} + \left(1 + \frac{\left(hi + hi \cdot \frac{hi - x}{lo}\right) - x}{lo}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + hi \cdot \frac{-1 - \frac{hi}{lo}}{lo}\right) + x \cdot \left(\frac{1}{lo} + \frac{hi}{{lo}^{2}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 18.9% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[lo_, hi_, x_] := N[(1.0 - N[(N[(N[(hi * N[(N[(N[(hi / lo), $MachinePrecision] - N[(x / lo), $MachinePrecision]), $MachinePrecision] - -1.0), $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* 14.3%

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

5. Simplified 14.3%

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

   1. expm1-log1p-u 14.3%

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

   2. expm1-undefine 14.3%

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

   3. add-sqr-sqrt 4.8%

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

   4. sqrt-unprod 13.8%

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

   5. sqr-neg 13.8%

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

   6. sqrt-unprod 9.0%

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

   7. add-sqr-sqrt 13.0%

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

7. Applied egg-rr 13.0%

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

   1. expm1-define 13.0%

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

   2. associate-+r- 12.9%

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

   3. div-sub 12.9%

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

   4. associate-*r/ 0.0%

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

   5. div-sub 0.0%

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

   6. associate-*r/ 12.9%

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

   7. associate-+r- 13.0%

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

   8. fma-neg 19.1%

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

9. Simplified 19.1%

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

10. Taylor expanded in lo around -inf 0.7%

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

    1. mul-1-neg 0.7%

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

    2. distribute-lft-out 0.7%

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

    3. associate-/l* 13.2%

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

12. Simplified 13.2%

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

13. Taylor expanded in hi around 0 19.1%

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

14. Final simplification 19.1%

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

Alternative 7: 18.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ 1 + hi \cdot \frac{-1 - \frac{hi}{lo}}{lo} \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[lo_, hi_, x_] := N[(1.0 + N[(hi * N[(N[(-1.0 - N[(hi / lo), $MachinePrecision]), $MachinePrecision] / lo), $MachinePrecision]), $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* 14.3%

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

5. Simplified 14.3%

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

   1. expm1-log1p-u 14.3%

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

   2. expm1-undefine 14.3%

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

   3. add-sqr-sqrt 4.8%

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

   4. sqrt-unprod 13.8%

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

   5. sqr-neg 13.8%

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

   6. sqrt-unprod 9.0%

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

   7. add-sqr-sqrt 13.0%

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

7. Applied egg-rr 13.0%

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

   1. expm1-define 13.0%

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

   2. associate-+r- 12.9%

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

   3. div-sub 12.9%

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

   4. associate-*r/ 0.0%

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

   5. div-sub 0.0%

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

   6. associate-*r/ 12.9%

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

   7. associate-+r- 13.0%

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

   8. fma-neg 19.1%

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

9. Simplified 19.1%

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

10. Taylor expanded in hi around 0 19.1%

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

    1. sub-neg 19.1%

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

    2. mul-1-neg 19.1%

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

    3. +-commutative 19.1%

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

    4. sub-neg 19.1%

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

    5. div-sub 19.1%

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

    6. metadata-eval 19.1%

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

12. Simplified 19.1%

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

13. Taylor expanded in x around 0 19.1%

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

    1. mul-1-neg 19.1%

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

    2. associate-/l* 19.1%

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

    3. distribute-rgt-neg-in 19.1%

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

    4. mul-1-neg 19.1%

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

    5. associate-*r/ 19.1%

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

    6. distribute-lft-in 19.1%

       \[\leadsto 1 + hi \cdot \frac{\color{blue}{-1 \cdot 1 + -1 \cdot \frac{hi}{lo}}}{lo} \]

    7. metadata-eval 19.1%

       \[\leadsto 1 + hi \cdot \frac{\color{blue}{-1} + -1 \cdot \frac{hi}{lo}}{lo} \]

    8. mul-1-neg 19.1%

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

    9. unsub-neg 19.1%

       \[\leadsto 1 + hi \cdot \frac{\color{blue}{-1 - \frac{hi}{lo}}}{lo} \]

15. Simplified 19.1%

    \[\leadsto \color{blue}{1 + hi \cdot \frac{-1 - \frac{hi}{lo}}{lo}} \]
16. Add Preprocessing

Alternative 8: 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 18.8%

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

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 9: 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 2024110 
(FPCore (lo hi x)
  :name "xlohi (overflows)"
  :precision binary64
  :pre (and (< lo -1e+308) (> hi 1e+308))
  (/ (- x lo) (- hi lo)))