Kahan p13 Example 2

Percentage Accurate: 100.0% → 100.0%

Time: 10.5s

Alternatives: 12

Speedup: 2.0×

Specification

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\\ t_2 := t\_1 \cdot t\_1\\ \frac{1 + t\_2}{2 + t\_2} \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (let* ((t_1 (- 2.0 (/ (/ 2.0 t) (+ 1.0 (/ 1.0 t))))) (t_2 (* t_1 t_1)))
   (/ (+ 1.0 t_2) (+ 2.0 t_2))))

C

double code(double t) {
	double t_1 = 2.0 - ((2.0 / t) / (1.0 + (1.0 / t)));
	double t_2 = t_1 * t_1;
	return (1.0 + t_2) / (2.0 + t_2);
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    t_1 = 2.0d0 - ((2.0d0 / t) / (1.0d0 + (1.0d0 / t)))
    t_2 = t_1 * t_1
    code = (1.0d0 + t_2) / (2.0d0 + t_2)
end function

Java

public static double code(double t) {
	double t_1 = 2.0 - ((2.0 / t) / (1.0 + (1.0 / t)));
	double t_2 = t_1 * t_1;
	return (1.0 + t_2) / (2.0 + t_2);
}

Python

def code(t):
	t_1 = 2.0 - ((2.0 / t) / (1.0 + (1.0 / t)))
	t_2 = t_1 * t_1
	return (1.0 + t_2) / (2.0 + t_2)

Julia

function code(t)
	t_1 = Float64(2.0 - Float64(Float64(2.0 / t) / Float64(1.0 + Float64(1.0 / t))))
	t_2 = Float64(t_1 * t_1)
	return Float64(Float64(1.0 + t_2) / Float64(2.0 + t_2))
end

MATLAB

function tmp = code(t)
	t_1 = 2.0 - ((2.0 / t) / (1.0 + (1.0 / t)));
	t_2 = t_1 * t_1;
	tmp = (1.0 + t_2) / (2.0 + t_2);
end

Wolfram

code[t_] := Block[{t$95$1 = N[(2.0 - N[(N[(2.0 / t), $MachinePrecision] / N[(1.0 + N[(1.0 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * t$95$1), $MachinePrecision]}, N[(N[(1.0 + t$95$2), $MachinePrecision] / N[(2.0 + t$95$2), $MachinePrecision]), $MachinePrecision]]]

Initial Program: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\\ t_2 := t\_1 \cdot t\_1\\ \frac{1 + t\_2}{2 + t\_2} \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (let* ((t_1 (- 2.0 (/ (/ 2.0 t) (+ 1.0 (/ 1.0 t))))) (t_2 (* t_1 t_1)))
   (/ (+ 1.0 t_2) (+ 2.0 t_2))))

C

double code(double t) {
	double t_1 = 2.0 - ((2.0 / t) / (1.0 + (1.0 / t)));
	double t_2 = t_1 * t_1;
	return (1.0 + t_2) / (2.0 + t_2);
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    t_1 = 2.0d0 - ((2.0d0 / t) / (1.0d0 + (1.0d0 / t)))
    t_2 = t_1 * t_1
    code = (1.0d0 + t_2) / (2.0d0 + t_2)
end function

Java

public static double code(double t) {
	double t_1 = 2.0 - ((2.0 / t) / (1.0 + (1.0 / t)));
	double t_2 = t_1 * t_1;
	return (1.0 + t_2) / (2.0 + t_2);
}

Python

def code(t):
	t_1 = 2.0 - ((2.0 / t) / (1.0 + (1.0 / t)))
	t_2 = t_1 * t_1
	return (1.0 + t_2) / (2.0 + t_2)

Julia

function code(t)
	t_1 = Float64(2.0 - Float64(Float64(2.0 / t) / Float64(1.0 + Float64(1.0 / t))))
	t_2 = Float64(t_1 * t_1)
	return Float64(Float64(1.0 + t_2) / Float64(2.0 + t_2))
end

MATLAB

function tmp = code(t)
	t_1 = 2.0 - ((2.0 / t) / (1.0 + (1.0 / t)));
	t_2 = t_1 * t_1;
	tmp = (1.0 + t_2) / (2.0 + t_2);
end

Wolfram

code[t_] := Block[{t$95$1 = N[(2.0 - N[(N[(2.0 / t), $MachinePrecision] / N[(1.0 + N[(1.0 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * t$95$1), $MachinePrecision]}, N[(N[(1.0 + t$95$2), $MachinePrecision] / N[(2.0 + t$95$2), $MachinePrecision]), $MachinePrecision]]]

Alternative 1: 100.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{2}{1 + t} - 2\\ \frac{\mathsf{fma}\left(t\_1, t\_1, 1\right)}{\mathsf{fma}\left(t\_1, t\_1, 2\right)} \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (let* ((t_1 (- (/ 2.0 (+ 1.0 t)) 2.0)))
   (/ (fma t_1 t_1 1.0) (fma t_1 t_1 2.0))))

C

double code(double t) {
	double t_1 = (2.0 / (1.0 + t)) - 2.0;
	return fma(t_1, t_1, 1.0) / fma(t_1, t_1, 2.0);
}

Julia

function code(t)
	t_1 = Float64(Float64(2.0 / Float64(1.0 + t)) - 2.0)
	return Float64(fma(t_1, t_1, 1.0) / fma(t_1, t_1, 2.0))
end

Wolfram

code[t_] := Block[{t$95$1 = N[(N[(2.0 / N[(1.0 + t), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision]}, N[(N[(t$95$1 * t$95$1 + 1.0), $MachinePrecision] / N[(t$95$1 * t$95$1 + 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 100.0%

   \[\frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-+.f64 N/A

      \[\leadsto \frac{\color{blue}{1 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]

   2. +-commutative N/A

      \[\leadsto \frac{\color{blue}{\left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) + 1}}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{\color{blue}{\left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} + 1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]

   4. sqr-abs-rev N/A

      \[\leadsto \frac{\color{blue}{\left|2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right| \cdot \left|2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right|} + 1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]

   5. lift--.f64 N/A

      \[\leadsto \frac{\left|\color{blue}{2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}}\right| \cdot \left|2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right| + 1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]

   6. fabs-sub N/A

      \[\leadsto \frac{\color{blue}{\left|\frac{\frac{2}{t}}{1 + \frac{1}{t}} - 2\right|} \cdot \left|2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right| + 1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]

   7. lift--.f64 N/A

      \[\leadsto \frac{\left|\frac{\frac{2}{t}}{1 + \frac{1}{t}} - 2\right| \cdot \left|\color{blue}{2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}}\right| + 1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]

   8. fabs-sub N/A

      \[\leadsto \frac{\left|\frac{\frac{2}{t}}{1 + \frac{1}{t}} - 2\right| \cdot \color{blue}{\left|\frac{\frac{2}{t}}{1 + \frac{1}{t}} - 2\right|} + 1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]

   9. sqr-abs N/A

      \[\leadsto \frac{\color{blue}{\left(\frac{\frac{2}{t}}{1 + \frac{1}{t}} - 2\right) \cdot \left(\frac{\frac{2}{t}}{1 + \frac{1}{t}} - 2\right)} + 1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]

   10. lower-fma.f64 N/A

       \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{\frac{2}{t}}{1 + \frac{1}{t}} - 2, \frac{\frac{2}{t}}{1 + \frac{1}{t}} - 2, 1\right)}}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]

4. Applied rewrites 100.0%

   \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{2}{1 + t} - 2, \frac{2}{1 + t} - 2, 1\right)}}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]
5. Step-by-step derivation

   1. lift-+.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{2}{1 + t} - 2, \frac{2}{1 + t} - 2, 1\right)}{\color{blue}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}} \]

   2. +-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{2}{1 + t} - 2, \frac{2}{1 + t} - 2, 1\right)}{\color{blue}{\left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) + 2}} \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{2}{1 + t} - 2, \frac{2}{1 + t} - 2, 1\right)}{\color{blue}{\left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} + 2} \]

   4. lower-fma.f64 100.0

      \[\leadsto \frac{\mathsf{fma}\left(\frac{2}{1 + t} - 2, \frac{2}{1 + t} - 2, 1\right)}{\color{blue}{\mathsf{fma}\left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}, 2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}, 2\right)}} \]

6. Applied rewrites 100.0%

   \[\leadsto \frac{\mathsf{fma}\left(\frac{2}{1 + t} - 2, \frac{2}{1 + t} - 2, 1\right)}{\color{blue}{\mathsf{fma}\left(\frac{2}{1 + t} - 2, \frac{2}{1 + t} - 2, 2\right)}} \]
7. Add Preprocessing

Alternative 2: 99.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;2 - \frac{\frac{2}{t}}{1 + {t}^{-1}} \leq 0.0002:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{2}{1 + t} - 2, \left(\mathsf{fma}\left(-2, t, 2\right) \cdot t - 2\right) \cdot t, 1\right)}{\mathsf{fma}\left(\mathsf{fma}\left(12 \cdot t - 8, t, 4\right), t \cdot t, 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(0.8333333333333334 - \frac{0.2222222222222222}{t}\right) + \frac{\frac{0.037037037037037035 + \frac{0.04938271604938271}{t}}{t}}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (<= (- 2.0 (/ (/ 2.0 t) (+ 1.0 (pow t -1.0)))) 0.0002)
   (/
    (fma (- (/ 2.0 (+ 1.0 t)) 2.0) (* (- (* (fma -2.0 t 2.0) t) 2.0) t) 1.0)
    (fma (fma (- (* 12.0 t) 8.0) t 4.0) (* t t) 2.0))
   (+
    (- 0.8333333333333334 (/ 0.2222222222222222 t))
    (/ (/ (+ 0.037037037037037035 (/ 0.04938271604938271 t)) t) t))))

C

double code(double t) {
	double tmp;
	if ((2.0 - ((2.0 / t) / (1.0 + pow(t, -1.0)))) <= 0.0002) {
		tmp = fma(((2.0 / (1.0 + t)) - 2.0), (((fma(-2.0, t, 2.0) * t) - 2.0) * t), 1.0) / fma(fma(((12.0 * t) - 8.0), t, 4.0), (t * t), 2.0);
	} else {
		tmp = (0.8333333333333334 - (0.2222222222222222 / t)) + (((0.037037037037037035 + (0.04938271604938271 / t)) / t) / t);
	}
	return tmp;
}

Julia

function code(t)
	tmp = 0.0
	if (Float64(2.0 - Float64(Float64(2.0 / t) / Float64(1.0 + (t ^ -1.0)))) <= 0.0002)
		tmp = Float64(fma(Float64(Float64(2.0 / Float64(1.0 + t)) - 2.0), Float64(Float64(Float64(fma(-2.0, t, 2.0) * t) - 2.0) * t), 1.0) / fma(fma(Float64(Float64(12.0 * t) - 8.0), t, 4.0), Float64(t * t), 2.0));
	else
		tmp = Float64(Float64(0.8333333333333334 - Float64(0.2222222222222222 / t)) + Float64(Float64(Float64(0.037037037037037035 + Float64(0.04938271604938271 / t)) / t) / t));
	end
	return tmp
end

Wolfram

code[t_] := If[LessEqual[N[(2.0 - N[(N[(2.0 / t), $MachinePrecision] / N[(1.0 + N[Power[t, -1.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0002], N[(N[(N[(N[(2.0 / N[(1.0 + t), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision] * N[(N[(N[(N[(-2.0 * t + 2.0), $MachinePrecision] * t), $MachinePrecision] - 2.0), $MachinePrecision] * t), $MachinePrecision] + 1.0), $MachinePrecision] / N[(N[(N[(N[(12.0 * t), $MachinePrecision] - 8.0), $MachinePrecision] * t + 4.0), $MachinePrecision] * N[(t * t), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(0.8333333333333334 - N[(0.2222222222222222 / t), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(0.037037037037037035 + N[(0.04938271604938271 / t), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 #s(literal 2 binary64) (/.f64 (/.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) t)))) < 2.0000000000000001e-4

   1. Initial program 100.0%

      \[\frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{1 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]

      2. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) + 1}}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} + 1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]

      4. sqr-abs-rev N/A

         \[\leadsto \frac{\color{blue}{\left|2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right| \cdot \left|2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right|} + 1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]

      5. lift--.f64 N/A

         \[\leadsto \frac{\left|\color{blue}{2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}}\right| \cdot \left|2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right| + 1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]

      6. fabs-sub N/A

         \[\leadsto \frac{\color{blue}{\left|\frac{\frac{2}{t}}{1 + \frac{1}{t}} - 2\right|} \cdot \left|2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right| + 1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]

      7. lift--.f64 N/A

         \[\leadsto \frac{\left|\frac{\frac{2}{t}}{1 + \frac{1}{t}} - 2\right| \cdot \left|\color{blue}{2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}}\right| + 1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]

      8. fabs-sub N/A

         \[\leadsto \frac{\left|\frac{\frac{2}{t}}{1 + \frac{1}{t}} - 2\right| \cdot \color{blue}{\left|\frac{\frac{2}{t}}{1 + \frac{1}{t}} - 2\right|} + 1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]

      9. sqr-abs N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{\frac{2}{t}}{1 + \frac{1}{t}} - 2\right) \cdot \left(\frac{\frac{2}{t}}{1 + \frac{1}{t}} - 2\right)} + 1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]

      10. lower-fma.f64 N/A

          \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{\frac{2}{t}}{1 + \frac{1}{t}} - 2, \frac{\frac{2}{t}}{1 + \frac{1}{t}} - 2, 1\right)}}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]

   4. Applied rewrites 100.0%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{2}{1 + t} - 2, \frac{2}{1 + t} - 2, 1\right)}}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]

   5. Taylor expanded in t around 0

      \[\leadsto \frac{\mathsf{fma}\left(\frac{2}{1 + t} - 2, \frac{2}{1 + t} - 2, 1\right)}{\color{blue}{2 + {t}^{2} \cdot \left(4 + t \cdot \left(12 \cdot t - 8\right)\right)}} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{2}{1 + t} - 2, \frac{2}{1 + t} - 2, 1\right)}{\color{blue}{{t}^{2} \cdot \left(4 + t \cdot \left(12 \cdot t - 8\right)\right) + 2}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{2}{1 + t} - 2, \frac{2}{1 + t} - 2, 1\right)}{\color{blue}{\left(4 + t \cdot \left(12 \cdot t - 8\right)\right) \cdot {t}^{2}} + 2} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{2}{1 + t} - 2, \frac{2}{1 + t} - 2, 1\right)}{\color{blue}{\mathsf{fma}\left(4 + t \cdot \left(12 \cdot t - 8\right), {t}^{2}, 2\right)}} \]

      4. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{2}{1 + t} - 2, \frac{2}{1 + t} - 2, 1\right)}{\mathsf{fma}\left(\color{blue}{t \cdot \left(12 \cdot t - 8\right) + 4}, {t}^{2}, 2\right)} \]

      5. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{2}{1 + t} - 2, \frac{2}{1 + t} - 2, 1\right)}{\mathsf{fma}\left(\color{blue}{\left(12 \cdot t - 8\right) \cdot t} + 4, {t}^{2}, 2\right)} \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{2}{1 + t} - 2, \frac{2}{1 + t} - 2, 1\right)}{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(12 \cdot t - 8, t, 4\right)}, {t}^{2}, 2\right)} \]

      7. lower--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{2}{1 + t} - 2, \frac{2}{1 + t} - 2, 1\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{12 \cdot t - 8}, t, 4\right), {t}^{2}, 2\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{2}{1 + t} - 2, \frac{2}{1 + t} - 2, 1\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{12 \cdot t} - 8, t, 4\right), {t}^{2}, 2\right)} \]

      9. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{2}{1 + t} - 2, \frac{2}{1 + t} - 2, 1\right)}{\mathsf{fma}\left(\mathsf{fma}\left(12 \cdot t - 8, t, 4\right), \color{blue}{t \cdot t}, 2\right)} \]

      10. lower-*.f64 99.8

          \[\leadsto \frac{\mathsf{fma}\left(\frac{2}{1 + t} - 2, \frac{2}{1 + t} - 2, 1\right)}{\mathsf{fma}\left(\mathsf{fma}\left(12 \cdot t - 8, t, 4\right), \color{blue}{t \cdot t}, 2\right)} \]

   7. Applied rewrites 99.8%

      \[\leadsto \frac{\mathsf{fma}\left(\frac{2}{1 + t} - 2, \frac{2}{1 + t} - 2, 1\right)}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(12 \cdot t - 8, t, 4\right), t \cdot t, 2\right)}} \]

   8. Taylor expanded in t around 0

      \[\leadsto \frac{\mathsf{fma}\left(\frac{2}{1 + t} - 2, \color{blue}{t \cdot \left(t \cdot \left(2 + -2 \cdot t\right) - 2\right)}, 1\right)}{\mathsf{fma}\left(\mathsf{fma}\left(12 \cdot t - 8, t, 4\right), t \cdot t, 2\right)} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{2}{1 + t} - 2, \color{blue}{\left(t \cdot \left(2 + -2 \cdot t\right) - 2\right) \cdot t}, 1\right)}{\mathsf{fma}\left(\mathsf{fma}\left(12 \cdot t - 8, t, 4\right), t \cdot t, 2\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{2}{1 + t} - 2, \color{blue}{\left(t \cdot \left(2 + -2 \cdot t\right) - 2\right) \cdot t}, 1\right)}{\mathsf{fma}\left(\mathsf{fma}\left(12 \cdot t - 8, t, 4\right), t \cdot t, 2\right)} \]

      3. lower--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{2}{1 + t} - 2, \color{blue}{\left(t \cdot \left(2 + -2 \cdot t\right) - 2\right)} \cdot t, 1\right)}{\mathsf{fma}\left(\mathsf{fma}\left(12 \cdot t - 8, t, 4\right), t \cdot t, 2\right)} \]

      4. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{2}{1 + t} - 2, \left(\color{blue}{\left(2 + -2 \cdot t\right) \cdot t} - 2\right) \cdot t, 1\right)}{\mathsf{fma}\left(\mathsf{fma}\left(12 \cdot t - 8, t, 4\right), t \cdot t, 2\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{2}{1 + t} - 2, \left(\color{blue}{\left(2 + -2 \cdot t\right) \cdot t} - 2\right) \cdot t, 1\right)}{\mathsf{fma}\left(\mathsf{fma}\left(12 \cdot t - 8, t, 4\right), t \cdot t, 2\right)} \]

      6. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{2}{1 + t} - 2, \left(\color{blue}{\left(-2 \cdot t + 2\right)} \cdot t - 2\right) \cdot t, 1\right)}{\mathsf{fma}\left(\mathsf{fma}\left(12 \cdot t - 8, t, 4\right), t \cdot t, 2\right)} \]

      7. lower-fma.f64 99.8

         \[\leadsto \frac{\mathsf{fma}\left(\frac{2}{1 + t} - 2, \left(\color{blue}{\mathsf{fma}\left(-2, t, 2\right)} \cdot t - 2\right) \cdot t, 1\right)}{\mathsf{fma}\left(\mathsf{fma}\left(12 \cdot t - 8, t, 4\right), t \cdot t, 2\right)} \]

   10. Applied rewrites 99.8%

       \[\leadsto \frac{\mathsf{fma}\left(\frac{2}{1 + t} - 2, \color{blue}{\left(\mathsf{fma}\left(-2, t, 2\right) \cdot t - 2\right) \cdot t}, 1\right)}{\mathsf{fma}\left(\mathsf{fma}\left(12 \cdot t - 8, t, 4\right), t \cdot t, 2\right)} \]

   if 2.0000000000000001e-4 < (-.f64 #s(literal 2 binary64) (/.f64 (/.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) t))))

   1. Initial program 100.0%

      \[\frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\left(\frac{5}{6} + \left(\frac{\frac{1}{27}}{{t}^{2}} + \frac{4}{81} \cdot \frac{1}{{t}^{3}}\right)\right) - \frac{2}{9} \cdot \frac{1}{t}} \]

   5. Applied rewrites 99.1%

      \[\leadsto \color{blue}{0.8333333333333334 - \frac{0.2222222222222222 - \frac{\frac{0.04938271604938271}{t} + 0.037037037037037035}{t}}{t}} \]
   6. Step-by-step derivation

   7. Applied rewrites 99.1%

      \[\leadsto \left(0.8333333333333334 - \frac{0.2222222222222222}{t}\right) + \color{blue}{\frac{\frac{0.037037037037037035 + \frac{0.04938271604938271}{t}}{t}}{t}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;2 - \frac{\frac{2}{t}}{1 + {t}^{-1}} \leq 0.0002:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{2}{1 + t} - 2, \left(\mathsf{fma}\left(-2, t, 2\right) \cdot t - 2\right) \cdot t, 1\right)}{\mathsf{fma}\left(\mathsf{fma}\left(12 \cdot t - 8, t, 4\right), t \cdot t, 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(0.8333333333333334 - \frac{0.2222222222222222}{t}\right) + \frac{\frac{0.037037037037037035 + \frac{0.04938271604938271}{t}}{t}}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\mathsf{fma}\left(-2, t, 2\right) \cdot t - 2\right) \cdot t\\ \mathbf{if}\;\frac{\frac{2}{t}}{1 + {t}^{-1}} \leq 1:\\ \;\;\;\;\left(0.8333333333333334 - \frac{0.2222222222222222}{t}\right) + \frac{\frac{0.037037037037037035 + \frac{0.04938271604938271}{t}}{t}}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_1, t\_1, 1\right)}{\mathsf{fma}\left(\mathsf{fma}\left(12 \cdot t - 8, t, 4\right), t \cdot t, 2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (let* ((t_1 (* (- (* (fma -2.0 t 2.0) t) 2.0) t)))
   (if (<= (/ (/ 2.0 t) (+ 1.0 (pow t -1.0))) 1.0)
     (+
      (- 0.8333333333333334 (/ 0.2222222222222222 t))
      (/ (/ (+ 0.037037037037037035 (/ 0.04938271604938271 t)) t) t))
     (/ (fma t_1 t_1 1.0) (fma (fma (- (* 12.0 t) 8.0) t 4.0) (* t t) 2.0)))))

C

double code(double t) {
	double t_1 = ((fma(-2.0, t, 2.0) * t) - 2.0) * t;
	double tmp;
	if (((2.0 / t) / (1.0 + pow(t, -1.0))) <= 1.0) {
		tmp = (0.8333333333333334 - (0.2222222222222222 / t)) + (((0.037037037037037035 + (0.04938271604938271 / t)) / t) / t);
	} else {
		tmp = fma(t_1, t_1, 1.0) / fma(fma(((12.0 * t) - 8.0), t, 4.0), (t * t), 2.0);
	}
	return tmp;
}

Julia

function code(t)
	t_1 = Float64(Float64(Float64(fma(-2.0, t, 2.0) * t) - 2.0) * t)
	tmp = 0.0
	if (Float64(Float64(2.0 / t) / Float64(1.0 + (t ^ -1.0))) <= 1.0)
		tmp = Float64(Float64(0.8333333333333334 - Float64(0.2222222222222222 / t)) + Float64(Float64(Float64(0.037037037037037035 + Float64(0.04938271604938271 / t)) / t) / t));
	else
		tmp = Float64(fma(t_1, t_1, 1.0) / fma(fma(Float64(Float64(12.0 * t) - 8.0), t, 4.0), Float64(t * t), 2.0));
	end
	return tmp
end

Wolfram

code[t_] := Block[{t$95$1 = N[(N[(N[(N[(-2.0 * t + 2.0), $MachinePrecision] * t), $MachinePrecision] - 2.0), $MachinePrecision] * t), $MachinePrecision]}, If[LessEqual[N[(N[(2.0 / t), $MachinePrecision] / N[(1.0 + N[Power[t, -1.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1.0], N[(N[(0.8333333333333334 - N[(0.2222222222222222 / t), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(0.037037037037037035 + N[(0.04938271604938271 / t), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$1 * t$95$1 + 1.0), $MachinePrecision] / N[(N[(N[(N[(12.0 * t), $MachinePrecision] - 8.0), $MachinePrecision] * t + 4.0), $MachinePrecision] * N[(t * t), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (/.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) t))) < 1

   1. Initial program 100.0%

      \[\frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\left(\frac{5}{6} + \left(\frac{\frac{1}{27}}{{t}^{2}} + \frac{4}{81} \cdot \frac{1}{{t}^{3}}\right)\right) - \frac{2}{9} \cdot \frac{1}{t}} \]

   5. Applied rewrites 99.1%

      \[\leadsto \color{blue}{0.8333333333333334 - \frac{0.2222222222222222 - \frac{\frac{0.04938271604938271}{t} + 0.037037037037037035}{t}}{t}} \]
   6. Step-by-step derivation

   7. Applied rewrites 99.1%

      \[\leadsto \left(0.8333333333333334 - \frac{0.2222222222222222}{t}\right) + \color{blue}{\frac{\frac{0.037037037037037035 + \frac{0.04938271604938271}{t}}{t}}{t}} \]

   if 1 < (/.f64 (/.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) t)))

   1. Initial program 100.0%

      \[\frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{1 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]

      2. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) + 1}}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} + 1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]

      4. sqr-abs-rev N/A

         \[\leadsto \frac{\color{blue}{\left|2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right| \cdot \left|2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right|} + 1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]

      5. lift--.f64 N/A

         \[\leadsto \frac{\left|\color{blue}{2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}}\right| \cdot \left|2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right| + 1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]

      6. fabs-sub N/A

         \[\leadsto \frac{\color{blue}{\left|\frac{\frac{2}{t}}{1 + \frac{1}{t}} - 2\right|} \cdot \left|2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right| + 1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]

      7. lift--.f64 N/A

         \[\leadsto \frac{\left|\frac{\frac{2}{t}}{1 + \frac{1}{t}} - 2\right| \cdot \left|\color{blue}{2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}}\right| + 1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]

      8. fabs-sub N/A

         \[\leadsto \frac{\left|\frac{\frac{2}{t}}{1 + \frac{1}{t}} - 2\right| \cdot \color{blue}{\left|\frac{\frac{2}{t}}{1 + \frac{1}{t}} - 2\right|} + 1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]

      9. sqr-abs N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{\frac{2}{t}}{1 + \frac{1}{t}} - 2\right) \cdot \left(\frac{\frac{2}{t}}{1 + \frac{1}{t}} - 2\right)} + 1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]

      10. lower-fma.f64 N/A

          \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{\frac{2}{t}}{1 + \frac{1}{t}} - 2, \frac{\frac{2}{t}}{1 + \frac{1}{t}} - 2, 1\right)}}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]

   4. Applied rewrites 100.0%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{2}{1 + t} - 2, \frac{2}{1 + t} - 2, 1\right)}}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]

   5. Taylor expanded in t around 0

      \[\leadsto \frac{\mathsf{fma}\left(\frac{2}{1 + t} - 2, \frac{2}{1 + t} - 2, 1\right)}{\color{blue}{2 + {t}^{2} \cdot \left(4 + t \cdot \left(12 \cdot t - 8\right)\right)}} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{2}{1 + t} - 2, \frac{2}{1 + t} - 2, 1\right)}{\color{blue}{{t}^{2} \cdot \left(4 + t \cdot \left(12 \cdot t - 8\right)\right) + 2}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{2}{1 + t} - 2, \frac{2}{1 + t} - 2, 1\right)}{\color{blue}{\left(4 + t \cdot \left(12 \cdot t - 8\right)\right) \cdot {t}^{2}} + 2} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{2}{1 + t} - 2, \frac{2}{1 + t} - 2, 1\right)}{\color{blue}{\mathsf{fma}\left(4 + t \cdot \left(12 \cdot t - 8\right), {t}^{2}, 2\right)}} \]

      4. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{2}{1 + t} - 2, \frac{2}{1 + t} - 2, 1\right)}{\mathsf{fma}\left(\color{blue}{t \cdot \left(12 \cdot t - 8\right) + 4}, {t}^{2}, 2\right)} \]

      5. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{2}{1 + t} - 2, \frac{2}{1 + t} - 2, 1\right)}{\mathsf{fma}\left(\color{blue}{\left(12 \cdot t - 8\right) \cdot t} + 4, {t}^{2}, 2\right)} \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{2}{1 + t} - 2, \frac{2}{1 + t} - 2, 1\right)}{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(12 \cdot t - 8, t, 4\right)}, {t}^{2}, 2\right)} \]

      7. lower--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{2}{1 + t} - 2, \frac{2}{1 + t} - 2, 1\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{12 \cdot t - 8}, t, 4\right), {t}^{2}, 2\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{2}{1 + t} - 2, \frac{2}{1 + t} - 2, 1\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{12 \cdot t} - 8, t, 4\right), {t}^{2}, 2\right)} \]

      9. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{2}{1 + t} - 2, \frac{2}{1 + t} - 2, 1\right)}{\mathsf{fma}\left(\mathsf{fma}\left(12 \cdot t - 8, t, 4\right), \color{blue}{t \cdot t}, 2\right)} \]

      10. lower-*.f64 99.8

          \[\leadsto \frac{\mathsf{fma}\left(\frac{2}{1 + t} - 2, \frac{2}{1 + t} - 2, 1\right)}{\mathsf{fma}\left(\mathsf{fma}\left(12 \cdot t - 8, t, 4\right), \color{blue}{t \cdot t}, 2\right)} \]

   7. Applied rewrites 99.8%

      \[\leadsto \frac{\mathsf{fma}\left(\frac{2}{1 + t} - 2, \frac{2}{1 + t} - 2, 1\right)}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(12 \cdot t - 8, t, 4\right), t \cdot t, 2\right)}} \]

   8. Taylor expanded in t around 0

      \[\leadsto \frac{\mathsf{fma}\left(\frac{2}{1 + t} - 2, \color{blue}{t \cdot \left(t \cdot \left(2 + -2 \cdot t\right) - 2\right)}, 1\right)}{\mathsf{fma}\left(\mathsf{fma}\left(12 \cdot t - 8, t, 4\right), t \cdot t, 2\right)} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{2}{1 + t} - 2, \color{blue}{\left(t \cdot \left(2 + -2 \cdot t\right) - 2\right) \cdot t}, 1\right)}{\mathsf{fma}\left(\mathsf{fma}\left(12 \cdot t - 8, t, 4\right), t \cdot t, 2\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{2}{1 + t} - 2, \color{blue}{\left(t \cdot \left(2 + -2 \cdot t\right) - 2\right) \cdot t}, 1\right)}{\mathsf{fma}\left(\mathsf{fma}\left(12 \cdot t - 8, t, 4\right), t \cdot t, 2\right)} \]

      3. lower--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{2}{1 + t} - 2, \color{blue}{\left(t \cdot \left(2 + -2 \cdot t\right) - 2\right)} \cdot t, 1\right)}{\mathsf{fma}\left(\mathsf{fma}\left(12 \cdot t - 8, t, 4\right), t \cdot t, 2\right)} \]

      4. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{2}{1 + t} - 2, \left(\color{blue}{\left(2 + -2 \cdot t\right) \cdot t} - 2\right) \cdot t, 1\right)}{\mathsf{fma}\left(\mathsf{fma}\left(12 \cdot t - 8, t, 4\right), t \cdot t, 2\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{2}{1 + t} - 2, \left(\color{blue}{\left(2 + -2 \cdot t\right) \cdot t} - 2\right) \cdot t, 1\right)}{\mathsf{fma}\left(\mathsf{fma}\left(12 \cdot t - 8, t, 4\right), t \cdot t, 2\right)} \]

      6. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{2}{1 + t} - 2, \left(\color{blue}{\left(-2 \cdot t + 2\right)} \cdot t - 2\right) \cdot t, 1\right)}{\mathsf{fma}\left(\mathsf{fma}\left(12 \cdot t - 8, t, 4\right), t \cdot t, 2\right)} \]

      7. lower-fma.f64 99.8

         \[\leadsto \frac{\mathsf{fma}\left(\frac{2}{1 + t} - 2, \left(\color{blue}{\mathsf{fma}\left(-2, t, 2\right)} \cdot t - 2\right) \cdot t, 1\right)}{\mathsf{fma}\left(\mathsf{fma}\left(12 \cdot t - 8, t, 4\right), t \cdot t, 2\right)} \]

   10. Applied rewrites 99.8%

       \[\leadsto \frac{\mathsf{fma}\left(\frac{2}{1 + t} - 2, \color{blue}{\left(\mathsf{fma}\left(-2, t, 2\right) \cdot t - 2\right) \cdot t}, 1\right)}{\mathsf{fma}\left(\mathsf{fma}\left(12 \cdot t - 8, t, 4\right), t \cdot t, 2\right)} \]

   11. Taylor expanded in t around 0

       \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{t \cdot \left(t \cdot \left(2 + -2 \cdot t\right) - 2\right)}, \left(\mathsf{fma}\left(-2, t, 2\right) \cdot t - 2\right) \cdot t, 1\right)}{\mathsf{fma}\left(\mathsf{fma}\left(12 \cdot t - 8, t, 4\right), t \cdot t, 2\right)} \]
   12. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\left(t \cdot \left(2 + -2 \cdot t\right) - 2\right) \cdot t}, \left(\mathsf{fma}\left(-2, t, 2\right) \cdot t - 2\right) \cdot t, 1\right)}{\mathsf{fma}\left(\mathsf{fma}\left(12 \cdot t - 8, t, 4\right), t \cdot t, 2\right)} \]

       2. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\left(t \cdot \left(2 + -2 \cdot t\right) - 2\right) \cdot t}, \left(\mathsf{fma}\left(-2, t, 2\right) \cdot t - 2\right) \cdot t, 1\right)}{\mathsf{fma}\left(\mathsf{fma}\left(12 \cdot t - 8, t, 4\right), t \cdot t, 2\right)} \]

       3. lower--.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\left(t \cdot \left(2 + -2 \cdot t\right) - 2\right)} \cdot t, \left(\mathsf{fma}\left(-2, t, 2\right) \cdot t - 2\right) \cdot t, 1\right)}{\mathsf{fma}\left(\mathsf{fma}\left(12 \cdot t - 8, t, 4\right), t \cdot t, 2\right)} \]

       4. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\left(\color{blue}{\left(2 + -2 \cdot t\right) \cdot t} - 2\right) \cdot t, \left(\mathsf{fma}\left(-2, t, 2\right) \cdot t - 2\right) \cdot t, 1\right)}{\mathsf{fma}\left(\mathsf{fma}\left(12 \cdot t - 8, t, 4\right), t \cdot t, 2\right)} \]

       5. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\left(\color{blue}{\left(2 + -2 \cdot t\right) \cdot t} - 2\right) \cdot t, \left(\mathsf{fma}\left(-2, t, 2\right) \cdot t - 2\right) \cdot t, 1\right)}{\mathsf{fma}\left(\mathsf{fma}\left(12 \cdot t - 8, t, 4\right), t \cdot t, 2\right)} \]

       6. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\left(\color{blue}{\left(-2 \cdot t + 2\right)} \cdot t - 2\right) \cdot t, \left(\mathsf{fma}\left(-2, t, 2\right) \cdot t - 2\right) \cdot t, 1\right)}{\mathsf{fma}\left(\mathsf{fma}\left(12 \cdot t - 8, t, 4\right), t \cdot t, 2\right)} \]

       7. lower-fma.f64 99.9

          \[\leadsto \frac{\mathsf{fma}\left(\left(\color{blue}{\mathsf{fma}\left(-2, t, 2\right)} \cdot t - 2\right) \cdot t, \left(\mathsf{fma}\left(-2, t, 2\right) \cdot t - 2\right) \cdot t, 1\right)}{\mathsf{fma}\left(\mathsf{fma}\left(12 \cdot t - 8, t, 4\right), t \cdot t, 2\right)} \]

   13. Applied rewrites 99.9%

       \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\left(\mathsf{fma}\left(-2, t, 2\right) \cdot t - 2\right) \cdot t}, \left(\mathsf{fma}\left(-2, t, 2\right) \cdot t - 2\right) \cdot t, 1\right)}{\mathsf{fma}\left(\mathsf{fma}\left(12 \cdot t - 8, t, 4\right), t \cdot t, 2\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{2}{t}}{1 + {t}^{-1}} \leq 1:\\ \;\;\;\;\left(0.8333333333333334 - \frac{0.2222222222222222}{t}\right) + \frac{\frac{0.037037037037037035 + \frac{0.04938271604938271}{t}}{t}}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(-2, t, 2\right) \cdot t - 2\right) \cdot t, \left(\mathsf{fma}\left(-2, t, 2\right) \cdot t - 2\right) \cdot t, 1\right)}{\mathsf{fma}\left(\mathsf{fma}\left(12 \cdot t - 8, t, 4\right), t \cdot t, 2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;2 - \frac{\frac{2}{t}}{1 + {t}^{-1}} \leq 0.0002:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(t - 2, t, 1\right), t \cdot t, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.8333333333333334 - \frac{0.2222222222222222}{t}\right) + \frac{\frac{0.037037037037037035 + \frac{0.04938271604938271}{t}}{t}}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (<= (- 2.0 (/ (/ 2.0 t) (+ 1.0 (pow t -1.0)))) 0.0002)
   (fma (fma (- t 2.0) t 1.0) (* t t) 0.5)
   (+
    (- 0.8333333333333334 (/ 0.2222222222222222 t))
    (/ (/ (+ 0.037037037037037035 (/ 0.04938271604938271 t)) t) t))))

C

double code(double t) {
	double tmp;
	if ((2.0 - ((2.0 / t) / (1.0 + pow(t, -1.0)))) <= 0.0002) {
		tmp = fma(fma((t - 2.0), t, 1.0), (t * t), 0.5);
	} else {
		tmp = (0.8333333333333334 - (0.2222222222222222 / t)) + (((0.037037037037037035 + (0.04938271604938271 / t)) / t) / t);
	}
	return tmp;
}

Julia

function code(t)
	tmp = 0.0
	if (Float64(2.0 - Float64(Float64(2.0 / t) / Float64(1.0 + (t ^ -1.0)))) <= 0.0002)
		tmp = fma(fma(Float64(t - 2.0), t, 1.0), Float64(t * t), 0.5);
	else
		tmp = Float64(Float64(0.8333333333333334 - Float64(0.2222222222222222 / t)) + Float64(Float64(Float64(0.037037037037037035 + Float64(0.04938271604938271 / t)) / t) / t));
	end
	return tmp
end

Wolfram

code[t_] := If[LessEqual[N[(2.0 - N[(N[(2.0 / t), $MachinePrecision] / N[(1.0 + N[Power[t, -1.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0002], N[(N[(N[(t - 2.0), $MachinePrecision] * t + 1.0), $MachinePrecision] * N[(t * t), $MachinePrecision] + 0.5), $MachinePrecision], N[(N[(0.8333333333333334 - N[(0.2222222222222222 / t), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(0.037037037037037035 + N[(0.04938271604938271 / t), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 #s(literal 2 binary64) (/.f64 (/.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) t)))) < 2.0000000000000001e-4

   1. Initial program 100.0%

      \[\frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(t - 2, t, 1\right)}, {t}^{2}, \frac{1}{2}\right) \]

      7. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{t - 2}, t, 1\right), {t}^{2}, \frac{1}{2}\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(t - 2, t, 1\right), \color{blue}{t \cdot t}, \frac{1}{2}\right) \]

      9. lower-*.f64 99.8

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(t - 2, t, 1\right), \color{blue}{t \cdot t}, 0.5\right) \]

   5. Applied rewrites 99.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(t - 2, t, 1\right), t \cdot t, 0.5\right)} \]

   if 2.0000000000000001e-4 < (-.f64 #s(literal 2 binary64) (/.f64 (/.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) t))))

   1. Initial program 100.0%

      \[\frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\left(\frac{5}{6} + \left(\frac{\frac{1}{27}}{{t}^{2}} + \frac{4}{81} \cdot \frac{1}{{t}^{3}}\right)\right) - \frac{2}{9} \cdot \frac{1}{t}} \]

   5. Applied rewrites 99.1%

      \[\leadsto \color{blue}{0.8333333333333334 - \frac{0.2222222222222222 - \frac{\frac{0.04938271604938271}{t} + 0.037037037037037035}{t}}{t}} \]
   6. Step-by-step derivation

   7. Applied rewrites 99.1%

      \[\leadsto \left(0.8333333333333334 - \frac{0.2222222222222222}{t}\right) + \color{blue}{\frac{\frac{0.037037037037037035 + \frac{0.04938271604938271}{t}}{t}}{t}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;2 - \frac{\frac{2}{t}}{1 + {t}^{-1}} \leq 0.0002:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(t - 2, t, 1\right), t \cdot t, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.8333333333333334 - \frac{0.2222222222222222}{t}\right) + \frac{\frac{0.037037037037037035 + \frac{0.04938271604938271}{t}}{t}}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\frac{2}{t}}{1 + {t}^{-1}} \leq 1:\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222 - \frac{\frac{0.04938271604938271}{t} + 0.037037037037037035}{t}}{t}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(t - 2, t, 1\right), t \cdot t, 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (<= (/ (/ 2.0 t) (+ 1.0 (pow t -1.0))) 1.0)
   (-
    0.8333333333333334
    (/
     (-
      0.2222222222222222
      (/ (+ (/ 0.04938271604938271 t) 0.037037037037037035) t))
     t))
   (fma (fma (- t 2.0) t 1.0) (* t t) 0.5)))

C

double code(double t) {
	double tmp;
	if (((2.0 / t) / (1.0 + pow(t, -1.0))) <= 1.0) {
		tmp = 0.8333333333333334 - ((0.2222222222222222 - (((0.04938271604938271 / t) + 0.037037037037037035) / t)) / t);
	} else {
		tmp = fma(fma((t - 2.0), t, 1.0), (t * t), 0.5);
	}
	return tmp;
}

Julia

function code(t)
	tmp = 0.0
	if (Float64(Float64(2.0 / t) / Float64(1.0 + (t ^ -1.0))) <= 1.0)
		tmp = Float64(0.8333333333333334 - Float64(Float64(0.2222222222222222 - Float64(Float64(Float64(0.04938271604938271 / t) + 0.037037037037037035) / t)) / t));
	else
		tmp = fma(fma(Float64(t - 2.0), t, 1.0), Float64(t * t), 0.5);
	end
	return tmp
end

Wolfram

code[t_] := If[LessEqual[N[(N[(2.0 / t), $MachinePrecision] / N[(1.0 + N[Power[t, -1.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1.0], N[(0.8333333333333334 - N[(N[(0.2222222222222222 - N[(N[(N[(0.04938271604938271 / t), $MachinePrecision] + 0.037037037037037035), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], N[(N[(N[(t - 2.0), $MachinePrecision] * t + 1.0), $MachinePrecision] * N[(t * t), $MachinePrecision] + 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (/.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) t))) < 1

   1. Initial program 100.0%

      \[\frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\left(\frac{5}{6} + \left(\frac{\frac{1}{27}}{{t}^{2}} + \frac{4}{81} \cdot \frac{1}{{t}^{3}}\right)\right) - \frac{2}{9} \cdot \frac{1}{t}} \]

   5. Applied rewrites 99.1%

      \[\leadsto \color{blue}{0.8333333333333334 - \frac{0.2222222222222222 - \frac{\frac{0.04938271604938271}{t} + 0.037037037037037035}{t}}{t}} \]

   if 1 < (/.f64 (/.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) t)))

   1. Initial program 100.0%

      \[\frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(t - 2, t, 1\right)}, {t}^{2}, \frac{1}{2}\right) \]

      7. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{t - 2}, t, 1\right), {t}^{2}, \frac{1}{2}\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(t - 2, t, 1\right), \color{blue}{t \cdot t}, \frac{1}{2}\right) \]

      9. lower-*.f64 99.8

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(t - 2, t, 1\right), \color{blue}{t \cdot t}, 0.5\right) \]

   5. Applied rewrites 99.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(t - 2, t, 1\right), t \cdot t, 0.5\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{2}{t}}{1 + {t}^{-1}} \leq 1:\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222 - \frac{\frac{0.04938271604938271}{t} + 0.037037037037037035}{t}}{t}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(t - 2, t, 1\right), t \cdot t, 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\frac{2}{t}}{1 + {t}^{-1}} \leq 1:\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222 - \frac{0.037037037037037035}{t}}{t}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(t - 2, t, 1\right), t \cdot t, 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (<= (/ (/ 2.0 t) (+ 1.0 (pow t -1.0))) 1.0)
   (-
    0.8333333333333334
    (/ (- 0.2222222222222222 (/ 0.037037037037037035 t)) t))
   (fma (fma (- t 2.0) t 1.0) (* t t) 0.5)))

C

double code(double t) {
	double tmp;
	if (((2.0 / t) / (1.0 + pow(t, -1.0))) <= 1.0) {
		tmp = 0.8333333333333334 - ((0.2222222222222222 - (0.037037037037037035 / t)) / t);
	} else {
		tmp = fma(fma((t - 2.0), t, 1.0), (t * t), 0.5);
	}
	return tmp;
}

Julia

function code(t)
	tmp = 0.0
	if (Float64(Float64(2.0 / t) / Float64(1.0 + (t ^ -1.0))) <= 1.0)
		tmp = Float64(0.8333333333333334 - Float64(Float64(0.2222222222222222 - Float64(0.037037037037037035 / t)) / t));
	else
		tmp = fma(fma(Float64(t - 2.0), t, 1.0), Float64(t * t), 0.5);
	end
	return tmp
end

Wolfram

code[t_] := If[LessEqual[N[(N[(2.0 / t), $MachinePrecision] / N[(1.0 + N[Power[t, -1.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1.0], N[(0.8333333333333334 - N[(N[(0.2222222222222222 - N[(0.037037037037037035 / t), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], N[(N[(N[(t - 2.0), $MachinePrecision] * t + 1.0), $MachinePrecision] * N[(t * t), $MachinePrecision] + 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (/.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) t))) < 1

   1. Initial program 100.0%

      \[\frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\left(\frac{5}{6} + \frac{\frac{1}{27}}{{t}^{2}}\right) - \frac{2}{9} \cdot \frac{1}{t}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{\frac{1}{27}}{{t}^{2}} + \frac{5}{6}\right)} - \frac{2}{9} \cdot \frac{1}{t} \]

      2. associate--l+ N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{27}}{{t}^{2}} + \left(\frac{5}{6} - \frac{2}{9} \cdot \frac{1}{t}\right)} \]

      3. +-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{5}{6} - \frac{2}{9} \cdot \frac{1}{t}\right) + \frac{\frac{1}{27}}{{t}^{2}}} \]

      4. associate--r- N/A

         \[\leadsto \color{blue}{\frac{5}{6} - \left(\frac{2}{9} \cdot \frac{1}{t} - \frac{\frac{1}{27}}{{t}^{2}}\right)} \]

      5. associate-*r/ N/A

         \[\leadsto \frac{5}{6} - \left(\color{blue}{\frac{\frac{2}{9} \cdot 1}{t}} - \frac{\frac{1}{27}}{{t}^{2}}\right) \]

      6. metadata-eval N/A

         \[\leadsto \frac{5}{6} - \left(\frac{\color{blue}{\frac{2}{9}}}{t} - \frac{\frac{1}{27}}{{t}^{2}}\right) \]

      7. unpow2 N/A

         \[\leadsto \frac{5}{6} - \left(\frac{\frac{2}{9}}{t} - \frac{\frac{1}{27}}{\color{blue}{t \cdot t}}\right) \]

      8. associate-/r* N/A

         \[\leadsto \frac{5}{6} - \left(\frac{\frac{2}{9}}{t} - \color{blue}{\frac{\frac{\frac{1}{27}}{t}}{t}}\right) \]

      9. metadata-eval N/A

         \[\leadsto \frac{5}{6} - \left(\frac{\frac{2}{9}}{t} - \frac{\frac{\color{blue}{\frac{1}{27} \cdot 1}}{t}}{t}\right) \]

      10. associate-*r/ N/A

          \[\leadsto \frac{5}{6} - \left(\frac{\frac{2}{9}}{t} - \frac{\color{blue}{\frac{1}{27} \cdot \frac{1}{t}}}{t}\right) \]

      11. div-sub N/A

          \[\leadsto \frac{5}{6} - \color{blue}{\frac{\frac{2}{9} - \frac{1}{27} \cdot \frac{1}{t}}{t}} \]

      12. lower--.f64 N/A

          \[\leadsto \color{blue}{\frac{5}{6} - \frac{\frac{2}{9} - \frac{1}{27} \cdot \frac{1}{t}}{t}} \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{5}{6} - \color{blue}{\frac{\frac{2}{9} - \frac{1}{27} \cdot \frac{1}{t}}{t}} \]

      14. lower--.f64 N/A

          \[\leadsto \frac{5}{6} - \frac{\color{blue}{\frac{2}{9} - \frac{1}{27} \cdot \frac{1}{t}}}{t} \]

      15. associate-*r/ N/A

          \[\leadsto \frac{5}{6} - \frac{\frac{2}{9} - \color{blue}{\frac{\frac{1}{27} \cdot 1}{t}}}{t} \]

      16. metadata-eval N/A

          \[\leadsto \frac{5}{6} - \frac{\frac{2}{9} - \frac{\color{blue}{\frac{1}{27}}}{t}}{t} \]

      17. lower-/.f64 98.9

          \[\leadsto 0.8333333333333334 - \frac{0.2222222222222222 - \color{blue}{\frac{0.037037037037037035}{t}}}{t} \]

   5. Applied rewrites 98.9%

      \[\leadsto \color{blue}{0.8333333333333334 - \frac{0.2222222222222222 - \frac{0.037037037037037035}{t}}{t}} \]

   if 1 < (/.f64 (/.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) t)))

   1. Initial program 100.0%

      \[\frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(t - 2, t, 1\right)}, {t}^{2}, \frac{1}{2}\right) \]

      7. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{t - 2}, t, 1\right), {t}^{2}, \frac{1}{2}\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(t - 2, t, 1\right), \color{blue}{t \cdot t}, \frac{1}{2}\right) \]

      9. lower-*.f64 99.8

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(t - 2, t, 1\right), \color{blue}{t \cdot t}, 0.5\right) \]

   5. Applied rewrites 99.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(t - 2, t, 1\right), t \cdot t, 0.5\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{2}{t}}{1 + {t}^{-1}} \leq 1:\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222 - \frac{0.037037037037037035}{t}}{t}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(t - 2, t, 1\right), t \cdot t, 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 99.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\frac{2}{t}}{1 + {t}^{-1}} \leq 1:\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222}{t}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(t - 2, t, 1\right), t \cdot t, 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (<= (/ (/ 2.0 t) (+ 1.0 (pow t -1.0))) 1.0)
   (- 0.8333333333333334 (/ 0.2222222222222222 t))
   (fma (fma (- t 2.0) t 1.0) (* t t) 0.5)))

C

double code(double t) {
	double tmp;
	if (((2.0 / t) / (1.0 + pow(t, -1.0))) <= 1.0) {
		tmp = 0.8333333333333334 - (0.2222222222222222 / t);
	} else {
		tmp = fma(fma((t - 2.0), t, 1.0), (t * t), 0.5);
	}
	return tmp;
}

Julia

function code(t)
	tmp = 0.0
	if (Float64(Float64(2.0 / t) / Float64(1.0 + (t ^ -1.0))) <= 1.0)
		tmp = Float64(0.8333333333333334 - Float64(0.2222222222222222 / t));
	else
		tmp = fma(fma(Float64(t - 2.0), t, 1.0), Float64(t * t), 0.5);
	end
	return tmp
end

Wolfram

code[t_] := If[LessEqual[N[(N[(2.0 / t), $MachinePrecision] / N[(1.0 + N[Power[t, -1.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1.0], N[(0.8333333333333334 - N[(0.2222222222222222 / t), $MachinePrecision]), $MachinePrecision], N[(N[(N[(t - 2.0), $MachinePrecision] * t + 1.0), $MachinePrecision] * N[(t * t), $MachinePrecision] + 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (/.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) t))) < 1

   1. Initial program 100.0%

      \[\frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\frac{5}{6} - \frac{2}{9} \cdot \frac{1}{t}} \]
   4. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \color{blue}{\frac{5}{6} - \frac{2}{9} \cdot \frac{1}{t}} \]

      2. associate-*r/ N/A

         \[\leadsto \frac{5}{6} - \color{blue}{\frac{\frac{2}{9} \cdot 1}{t}} \]

      3. metadata-eval N/A

         \[\leadsto \frac{5}{6} - \frac{\color{blue}{\frac{2}{9}}}{t} \]

      4. lower-/.f64 98.6

         \[\leadsto 0.8333333333333334 - \color{blue}{\frac{0.2222222222222222}{t}} \]

   5. Applied rewrites 98.6%

      \[\leadsto \color{blue}{0.8333333333333334 - \frac{0.2222222222222222}{t}} \]

   if 1 < (/.f64 (/.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) t)))

   1. Initial program 100.0%

      \[\frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(t - 2, t, 1\right)}, {t}^{2}, \frac{1}{2}\right) \]

      7. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{t - 2}, t, 1\right), {t}^{2}, \frac{1}{2}\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(t - 2, t, 1\right), \color{blue}{t \cdot t}, \frac{1}{2}\right) \]

      9. lower-*.f64 99.8

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(t - 2, t, 1\right), \color{blue}{t \cdot t}, 0.5\right) \]

   5. Applied rewrites 99.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(t - 2, t, 1\right), t \cdot t, 0.5\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{2}{t}}{1 + {t}^{-1}} \leq 1:\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222}{t}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(t - 2, t, 1\right), t \cdot t, 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 99.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\frac{2}{t}}{1 + {t}^{-1}} \leq 1:\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222}{t}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-2, t, 1\right), t \cdot t, 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (<= (/ (/ 2.0 t) (+ 1.0 (pow t -1.0))) 1.0)
   (- 0.8333333333333334 (/ 0.2222222222222222 t))
   (fma (fma -2.0 t 1.0) (* t t) 0.5)))

C

double code(double t) {
	double tmp;
	if (((2.0 / t) / (1.0 + pow(t, -1.0))) <= 1.0) {
		tmp = 0.8333333333333334 - (0.2222222222222222 / t);
	} else {
		tmp = fma(fma(-2.0, t, 1.0), (t * t), 0.5);
	}
	return tmp;
}

Julia

function code(t)
	tmp = 0.0
	if (Float64(Float64(2.0 / t) / Float64(1.0 + (t ^ -1.0))) <= 1.0)
		tmp = Float64(0.8333333333333334 - Float64(0.2222222222222222 / t));
	else
		tmp = fma(fma(-2.0, t, 1.0), Float64(t * t), 0.5);
	end
	return tmp
end

Wolfram

code[t_] := If[LessEqual[N[(N[(2.0 / t), $MachinePrecision] / N[(1.0 + N[Power[t, -1.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1.0], N[(0.8333333333333334 - N[(0.2222222222222222 / t), $MachinePrecision]), $MachinePrecision], N[(N[(-2.0 * t + 1.0), $MachinePrecision] * N[(t * t), $MachinePrecision] + 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (/.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) t))) < 1

   1. Initial program 100.0%

      \[\frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\frac{5}{6} - \frac{2}{9} \cdot \frac{1}{t}} \]
   4. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \color{blue}{\frac{5}{6} - \frac{2}{9} \cdot \frac{1}{t}} \]

      2. associate-*r/ N/A

         \[\leadsto \frac{5}{6} - \color{blue}{\frac{\frac{2}{9} \cdot 1}{t}} \]

      3. metadata-eval N/A

         \[\leadsto \frac{5}{6} - \frac{\color{blue}{\frac{2}{9}}}{t} \]

      4. lower-/.f64 98.6

         \[\leadsto 0.8333333333333334 - \color{blue}{\frac{0.2222222222222222}{t}} \]

   5. Applied rewrites 98.6%

      \[\leadsto \color{blue}{0.8333333333333334 - \frac{0.2222222222222222}{t}} \]

   if 1 < (/.f64 (/.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) t)))

   1. Initial program 100.0%

      \[\frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(t - 2, t, 1\right)}, {t}^{2}, \frac{1}{2}\right) \]

      7. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{t - 2}, t, 1\right), {t}^{2}, \frac{1}{2}\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(t - 2, t, 1\right), \color{blue}{t \cdot t}, \frac{1}{2}\right) \]

      9. lower-*.f64 99.8

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(t - 2, t, 1\right), \color{blue}{t \cdot t}, 0.5\right) \]

   5. Applied rewrites 99.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(t - 2, t, 1\right), t \cdot t, 0.5\right)} \]

   6. Taylor expanded in t around 0

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-2, t, 1\right), t \cdot t, \frac{1}{2}\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 99.7%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-2, t, 1\right), t \cdot t, 0.5\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{2}{t}}{1 + {t}^{-1}} \leq 1:\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222}{t}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-2, t, 1\right), t \cdot t, 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 99.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\frac{2}{t}}{1 + {t}^{-1}} \leq 1:\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222}{t}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t, t, 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (<= (/ (/ 2.0 t) (+ 1.0 (pow t -1.0))) 1.0)
   (- 0.8333333333333334 (/ 0.2222222222222222 t))
   (fma t t 0.5)))

C

double code(double t) {
	double tmp;
	if (((2.0 / t) / (1.0 + pow(t, -1.0))) <= 1.0) {
		tmp = 0.8333333333333334 - (0.2222222222222222 / t);
	} else {
		tmp = fma(t, t, 0.5);
	}
	return tmp;
}

Julia

function code(t)
	tmp = 0.0
	if (Float64(Float64(2.0 / t) / Float64(1.0 + (t ^ -1.0))) <= 1.0)
		tmp = Float64(0.8333333333333334 - Float64(0.2222222222222222 / t));
	else
		tmp = fma(t, t, 0.5);
	end
	return tmp
end

Wolfram

code[t_] := If[LessEqual[N[(N[(2.0 / t), $MachinePrecision] / N[(1.0 + N[Power[t, -1.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1.0], N[(0.8333333333333334 - N[(0.2222222222222222 / t), $MachinePrecision]), $MachinePrecision], N[(t * t + 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (/.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) t))) < 1

   1. Initial program 100.0%

      \[\frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\frac{5}{6} - \frac{2}{9} \cdot \frac{1}{t}} \]
   4. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \color{blue}{\frac{5}{6} - \frac{2}{9} \cdot \frac{1}{t}} \]

      2. associate-*r/ N/A

         \[\leadsto \frac{5}{6} - \color{blue}{\frac{\frac{2}{9} \cdot 1}{t}} \]

      3. metadata-eval N/A

         \[\leadsto \frac{5}{6} - \frac{\color{blue}{\frac{2}{9}}}{t} \]

      4. lower-/.f64 98.6

         \[\leadsto 0.8333333333333334 - \color{blue}{\frac{0.2222222222222222}{t}} \]

   5. Applied rewrites 98.6%

      \[\leadsto \color{blue}{0.8333333333333334 - \frac{0.2222222222222222}{t}} \]

   if 1 < (/.f64 (/.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) t)))

   1. Initial program 100.0%

      \[\frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\frac{1}{2} + {t}^{2}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{{t}^{2} + \frac{1}{2}} \]

      2. unpow2 N/A

         \[\leadsto \color{blue}{t \cdot t} + \frac{1}{2} \]

      3. lower-fma.f64 99.3

         \[\leadsto \color{blue}{\mathsf{fma}\left(t, t, 0.5\right)} \]

   5. Applied rewrites 99.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(t, t, 0.5\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{2}{t}}{1 + {t}^{-1}} \leq 1:\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222}{t}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t, t, 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 98.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\frac{2}{t}}{1 + {t}^{-1}} \leq 1:\\ \;\;\;\;0.8333333333333334\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t, t, 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (<= (/ (/ 2.0 t) (+ 1.0 (pow t -1.0))) 1.0)
   0.8333333333333334
   (fma t t 0.5)))

C

double code(double t) {
	double tmp;
	if (((2.0 / t) / (1.0 + pow(t, -1.0))) <= 1.0) {
		tmp = 0.8333333333333334;
	} else {
		tmp = fma(t, t, 0.5);
	}
	return tmp;
}

Julia

function code(t)
	tmp = 0.0
	if (Float64(Float64(2.0 / t) / Float64(1.0 + (t ^ -1.0))) <= 1.0)
		tmp = 0.8333333333333334;
	else
		tmp = fma(t, t, 0.5);
	end
	return tmp
end

Wolfram

code[t_] := If[LessEqual[N[(N[(2.0 / t), $MachinePrecision] / N[(1.0 + N[Power[t, -1.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1.0], 0.8333333333333334, N[(t * t + 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (/.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) t))) < 1

   1. Initial program 100.0%

      \[\frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\frac{5}{6}} \]
   4. Step-by-step derivation

   5. Applied rewrites 97.5%

      \[\leadsto \color{blue}{0.8333333333333334} \]

   if 1 < (/.f64 (/.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) t)))

   1. Initial program 100.0%

      \[\frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\frac{1}{2} + {t}^{2}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{{t}^{2} + \frac{1}{2}} \]

      2. unpow2 N/A

         \[\leadsto \color{blue}{t \cdot t} + \frac{1}{2} \]

      3. lower-fma.f64 99.3

         \[\leadsto \color{blue}{\mathsf{fma}\left(t, t, 0.5\right)} \]

   5. Applied rewrites 99.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(t, t, 0.5\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{2}{t}}{1 + {t}^{-1}} \leq 1:\\ \;\;\;\;0.8333333333333334\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t, t, 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 98.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\frac{2}{t}}{1 + {t}^{-1}} \leq 1:\\ \;\;\;\;0.8333333333333334\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (<= (/ (/ 2.0 t) (+ 1.0 (pow t -1.0))) 1.0) 0.8333333333333334 0.5))

C

double code(double t) {
	double tmp;
	if (((2.0 / t) / (1.0 + pow(t, -1.0))) <= 1.0) {
		tmp = 0.8333333333333334;
	} else {
		tmp = 0.5;
	}
	return tmp;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: tmp
    if (((2.0d0 / t) / (1.0d0 + (t ** (-1.0d0)))) <= 1.0d0) then
        tmp = 0.8333333333333334d0
    else
        tmp = 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double tmp;
	if (((2.0 / t) / (1.0 + Math.pow(t, -1.0))) <= 1.0) {
		tmp = 0.8333333333333334;
	} else {
		tmp = 0.5;
	}
	return tmp;
}

Python

def code(t):
	tmp = 0
	if ((2.0 / t) / (1.0 + math.pow(t, -1.0))) <= 1.0:
		tmp = 0.8333333333333334
	else:
		tmp = 0.5
	return tmp

Julia

function code(t)
	tmp = 0.0
	if (Float64(Float64(2.0 / t) / Float64(1.0 + (t ^ -1.0))) <= 1.0)
		tmp = 0.8333333333333334;
	else
		tmp = 0.5;
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	tmp = 0.0;
	if (((2.0 / t) / (1.0 + (t ^ -1.0))) <= 1.0)
		tmp = 0.8333333333333334;
	else
		tmp = 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := If[LessEqual[N[(N[(2.0 / t), $MachinePrecision] / N[(1.0 + N[Power[t, -1.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1.0], 0.8333333333333334, 0.5]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (/.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) t))) < 1

   1. Initial program 100.0%

      \[\frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\frac{5}{6}} \]
   4. Step-by-step derivation

   5. Applied rewrites 97.5%

      \[\leadsto \color{blue}{0.8333333333333334} \]

   if 1 < (/.f64 (/.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) t)))

   1. Initial program 100.0%

      \[\frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\frac{1}{2}} \]
   4. Step-by-step derivation

   5. Applied rewrites 98.8%

      \[\leadsto \color{blue}{0.5} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{2}{t}}{1 + {t}^{-1}} \leq 1:\\ \;\;\;\;0.8333333333333334\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 58.8% accurate, 184.0× speedup

Math

\[\begin{array}{l} \\ 0.5 \end{array} \]

FPCore

(FPCore (t) :precision binary64 0.5)

C

double code(double t) {
	return 0.5;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    code = 0.5d0
end function

Java

public static double code(double t) {
	return 0.5;
}

Python

def code(t):
	return 0.5

Julia

function code(t)
	return 0.5
end

MATLAB

function tmp = code(t)
	tmp = 0.5;
end

Wolfram

code[t_] := 0.5

Derivation

1. Initial program 100.0%

   \[\frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]
2. Add Preprocessing

3. Taylor expanded in t around 0

   \[\leadsto \color{blue}{\frac{1}{2}} \]
4. Step-by-step derivation

5. Applied rewrites 54.0%

   \[\leadsto \color{blue}{0.5} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024329 
(FPCore (t)
  :name "Kahan p13 Example 2"
  :precision binary64
  (/ (+ 1.0 (* (- 2.0 (/ (/ 2.0 t) (+ 1.0 (/ 1.0 t)))) (- 2.0 (/ (/ 2.0 t) (+ 1.0 (/ 1.0 t)))))) (+ 2.0 (* (- 2.0 (/ (/ 2.0 t) (+ 1.0 (/ 1.0 t)))) (- 2.0 (/ (/ 2.0 t) (+ 1.0 (/ 1.0 t))))))))