Kahan p13 Example 3

Percentage Accurate: 100.0% → 100.0%

Time: 13.9s

Alternatives: 14

Speedup: 1.8×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

function tmp = code(t)
	t_1 = 2.0 - ((2.0 / t) / (1.0 + (1.0 / t)));
	tmp = 1.0 - (1.0 / (2.0 + (t_1 * t_1)));
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]}, N[(1.0 - N[(1.0 / N[(2.0 + N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

function tmp = code(t)
	t_1 = 2.0 - ((2.0 / t) / (1.0 + (1.0 / t)));
	tmp = 1.0 - (1.0 / (2.0 + (t_1 * t_1)));
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]}, N[(1.0 - N[(1.0 / N[(2.0 + N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Alternative 1: 100.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 2 + \frac{-2}{1 + t}\\ \mathsf{fma}\left(\mathsf{fma}\left(t\_1, t\_1, -2\right), \frac{1}{\mathsf{fma}\left(t\_1, \left(-2 + \frac{2}{1 + t}\right) \cdot \left(t\_1 \cdot t\_1\right), 4\right)}, 1\right) \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (let* ((t_1 (+ 2.0 (/ -2.0 (+ 1.0 t)))))
   (fma
    (fma t_1 t_1 -2.0)
    (/ 1.0 (fma t_1 (* (+ -2.0 (/ 2.0 (+ 1.0 t))) (* t_1 t_1)) 4.0))
    1.0)))

C

double code(double t) {
	double t_1 = 2.0 + (-2.0 / (1.0 + t));
	return fma(fma(t_1, t_1, -2.0), (1.0 / fma(t_1, ((-2.0 + (2.0 / (1.0 + t))) * (t_1 * t_1)), 4.0)), 1.0);
}

Julia

function code(t)
	t_1 = Float64(2.0 + Float64(-2.0 / Float64(1.0 + t)))
	return fma(fma(t_1, t_1, -2.0), Float64(1.0 / fma(t_1, Float64(Float64(-2.0 + Float64(2.0 / Float64(1.0 + t))) * Float64(t_1 * t_1)), 4.0)), 1.0)
end

Wolfram

code[t_] := Block[{t$95$1 = N[(2.0 + N[(-2.0 / N[(1.0 + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(N[(t$95$1 * t$95$1 + -2.0), $MachinePrecision] * N[(1.0 / N[(t$95$1 * N[(N[(-2.0 + N[(2.0 / N[(1.0 + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision] + 4.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]]

Derivation

1. Initial program 100.0%

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

4. Applied rewrites 100.0%

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

6. Applied rewrites 100.0%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-neg.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. distribute-rgt-neg-in N/A

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

   6. lift-+.f64 N/A

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

   7. +-commutative N/A

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

   8. distribute-neg-in N/A

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

   9. metadata-eval N/A

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

   10. sub-neg N/A

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

   11. lift--.f64 N/A

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

   12. lower-fma.f64 100.0

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

8. Applied rewrites 100.0%

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

   1. lift-fma.f64 N/A

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

   2. *-commutative N/A

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

   3. *-lft-identity N/A

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

   4. lower-fma.f64 N/A

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

10. Applied rewrites 100.0%

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

11. Final simplification 100.0%

    \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(2 + \frac{-2}{1 + t}, 2 + \frac{-2}{1 + t}, -2\right), \frac{1}{\mathsf{fma}\left(2 + \frac{-2}{1 + t}, \left(-2 + \frac{2}{1 + t}\right) \cdot \left(\left(2 + \frac{-2}{1 + t}\right) \cdot \left(2 + \frac{-2}{1 + t}\right)\right), 4\right)}, 1\right) \]
12. Add Preprocessing

Alternative 2: 100.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 2 + \frac{-2}{1 + t}\\ 1 + \frac{\mathsf{fma}\left(t\_1, t\_1, -2\right)}{\mathsf{fma}\left(t\_1, \left(-2 + \frac{2}{1 + t}\right) \cdot \left(t\_1 \cdot t\_1\right), 4\right)} \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (let* ((t_1 (+ 2.0 (/ -2.0 (+ 1.0 t)))))
   (+
    1.0
    (/
     (fma t_1 t_1 -2.0)
     (fma t_1 (* (+ -2.0 (/ 2.0 (+ 1.0 t))) (* t_1 t_1)) 4.0)))))

C

double code(double t) {
	double t_1 = 2.0 + (-2.0 / (1.0 + t));
	return 1.0 + (fma(t_1, t_1, -2.0) / fma(t_1, ((-2.0 + (2.0 / (1.0 + t))) * (t_1 * t_1)), 4.0));
}

Julia

function code(t)
	t_1 = Float64(2.0 + Float64(-2.0 / Float64(1.0 + t)))
	return Float64(1.0 + Float64(fma(t_1, t_1, -2.0) / fma(t_1, Float64(Float64(-2.0 + Float64(2.0 / Float64(1.0 + t))) * Float64(t_1 * t_1)), 4.0)))
end

Wolfram

code[t_] := Block[{t$95$1 = N[(2.0 + N[(-2.0 / N[(1.0 + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(1.0 + N[(N[(t$95$1 * t$95$1 + -2.0), $MachinePrecision] / N[(t$95$1 * N[(N[(-2.0 + N[(2.0 / N[(1.0 + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision] + 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 100.0%

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

4. Applied rewrites 100.0%

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

6. Applied rewrites 100.0%

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

8. Applied rewrites 100.0%

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

9. Final simplification 100.0%

   \[\leadsto 1 + \frac{\mathsf{fma}\left(2 + \frac{-2}{1 + t}, 2 + \frac{-2}{1 + t}, -2\right)}{\mathsf{fma}\left(2 + \frac{-2}{1 + t}, \left(-2 + \frac{2}{1 + t}\right) \cdot \left(\left(2 + \frac{-2}{1 + t}\right) \cdot \left(2 + \frac{-2}{1 + t}\right)\right), 4\right)} \]
10. Add Preprocessing

Alternative 3: 99.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\frac{2}{t}}{1 + \frac{1}{t}} \leq 0.005:\\ \;\;\;\;0.8333333333333334 + \frac{\frac{0.037037037037037035 + \frac{0.04938271604938271}{t}}{t} + -0.2222222222222222}{t}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{2 + \left(t \cdot t\right) \cdot \mathsf{fma}\left(t, \mathsf{fma}\left(t, \mathsf{fma}\left(t, -16, 12\right), -8\right), 4\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (<= (/ (/ 2.0 t) (+ 1.0 (/ 1.0 t))) 0.005)
   (+
    0.8333333333333334
    (/
     (+
      (/ (+ 0.037037037037037035 (/ 0.04938271604938271 t)) t)
      -0.2222222222222222)
     t))
   (+
    1.0
    (/ -1.0 (+ 2.0 (* (* t t) (fma t (fma t (fma t -16.0 12.0) -8.0) 4.0)))))))

C

double code(double t) {
	double tmp;
	if (((2.0 / t) / (1.0 + (1.0 / t))) <= 0.005) {
		tmp = 0.8333333333333334 + ((((0.037037037037037035 + (0.04938271604938271 / t)) / t) + -0.2222222222222222) / t);
	} else {
		tmp = 1.0 + (-1.0 / (2.0 + ((t * t) * fma(t, fma(t, fma(t, -16.0, 12.0), -8.0), 4.0))));
	}
	return tmp;
}

Julia

function code(t)
	tmp = 0.0
	if (Float64(Float64(2.0 / t) / Float64(1.0 + Float64(1.0 / t))) <= 0.005)
		tmp = Float64(0.8333333333333334 + Float64(Float64(Float64(Float64(0.037037037037037035 + Float64(0.04938271604938271 / t)) / t) + -0.2222222222222222) / t));
	else
		tmp = Float64(1.0 + Float64(-1.0 / Float64(2.0 + Float64(Float64(t * t) * fma(t, fma(t, fma(t, -16.0, 12.0), -8.0), 4.0)))));
	end
	return tmp
end

Wolfram

code[t_] := If[LessEqual[N[(N[(2.0 / t), $MachinePrecision] / N[(1.0 + N[(1.0 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.005], N[(0.8333333333333334 + N[(N[(N[(N[(0.037037037037037035 + N[(0.04938271604938271 / t), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision] + -0.2222222222222222), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(-1.0 / N[(2.0 + N[(N[(t * t), $MachinePrecision] * N[(t * N[(t * N[(t * -16.0 + 12.0), $MachinePrecision] + -8.0), $MachinePrecision] + 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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))) < 0.0050000000000000001

   1. Initial program 100.0%

      \[1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]
   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 99.8%

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

   if 0.0050000000000000001 < (/.f64 (/.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) t)))

   1. Initial program 100.0%

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

   3. Taylor expanded in t around 0

      \[\leadsto 1 - \frac{1}{2 + \color{blue}{{t}^{2} \cdot \left(4 + t \cdot \left(t \cdot \left(12 + -16 \cdot t\right) - 8\right)\right)}} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto 1 - \frac{1}{2 + \color{blue}{{t}^{2} \cdot \left(4 + t \cdot \left(t \cdot \left(12 + -16 \cdot t\right) - 8\right)\right)}} \]

      2. unpow2 N/A

         \[\leadsto 1 - \frac{1}{2 + \color{blue}{\left(t \cdot t\right)} \cdot \left(4 + t \cdot \left(t \cdot \left(12 + -16 \cdot t\right) - 8\right)\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto 1 - \frac{1}{2 + \color{blue}{\left(t \cdot t\right)} \cdot \left(4 + t \cdot \left(t \cdot \left(12 + -16 \cdot t\right) - 8\right)\right)} \]

      4. +-commutative N/A

         \[\leadsto 1 - \frac{1}{2 + \left(t \cdot t\right) \cdot \color{blue}{\left(t \cdot \left(t \cdot \left(12 + -16 \cdot t\right) - 8\right) + 4\right)}} \]

      5. lower-fma.f64 N/A

         \[\leadsto 1 - \frac{1}{2 + \left(t \cdot t\right) \cdot \color{blue}{\mathsf{fma}\left(t, t \cdot \left(12 + -16 \cdot t\right) - 8, 4\right)}} \]

      6. sub-neg N/A

         \[\leadsto 1 - \frac{1}{2 + \left(t \cdot t\right) \cdot \mathsf{fma}\left(t, \color{blue}{t \cdot \left(12 + -16 \cdot t\right) + \left(\mathsf{neg}\left(8\right)\right)}, 4\right)} \]

      7. metadata-eval N/A

         \[\leadsto 1 - \frac{1}{2 + \left(t \cdot t\right) \cdot \mathsf{fma}\left(t, t \cdot \left(12 + -16 \cdot t\right) + \color{blue}{-8}, 4\right)} \]

      8. lower-fma.f64 N/A

         \[\leadsto 1 - \frac{1}{2 + \left(t \cdot t\right) \cdot \mathsf{fma}\left(t, \color{blue}{\mathsf{fma}\left(t, 12 + -16 \cdot t, -8\right)}, 4\right)} \]

      9. +-commutative N/A

         \[\leadsto 1 - \frac{1}{2 + \left(t \cdot t\right) \cdot \mathsf{fma}\left(t, \mathsf{fma}\left(t, \color{blue}{-16 \cdot t + 12}, -8\right), 4\right)} \]

      10. *-commutative N/A

          \[\leadsto 1 - \frac{1}{2 + \left(t \cdot t\right) \cdot \mathsf{fma}\left(t, \mathsf{fma}\left(t, \color{blue}{t \cdot -16} + 12, -8\right), 4\right)} \]

      11. lower-fma.f64 99.8

          \[\leadsto 1 - \frac{1}{2 + \left(t \cdot t\right) \cdot \mathsf{fma}\left(t, \mathsf{fma}\left(t, \color{blue}{\mathsf{fma}\left(t, -16, 12\right)}, -8\right), 4\right)} \]

   5. Applied rewrites 99.8%

      \[\leadsto 1 - \frac{1}{2 + \color{blue}{\left(t \cdot t\right) \cdot \mathsf{fma}\left(t, \mathsf{fma}\left(t, \mathsf{fma}\left(t, -16, 12\right), -8\right), 4\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{2}{t}}{1 + \frac{1}{t}} \leq 0.005:\\ \;\;\;\;0.8333333333333334 + \frac{\frac{0.037037037037037035 + \frac{0.04938271604938271}{t}}{t} + -0.2222222222222222}{t}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{2 + \left(t \cdot t\right) \cdot \mathsf{fma}\left(t, \mathsf{fma}\left(t, \mathsf{fma}\left(t, -16, 12\right), -8\right), 4\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\frac{2}{t}}{1 + \frac{1}{t}} \leq 0.005:\\ \;\;\;\;0.8333333333333334 + \frac{\frac{0.037037037037037035 + \frac{0.04938271604938271}{t}}{t} + -0.2222222222222222}{t}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{\mathsf{fma}\left(t \cdot t, \mathsf{fma}\left(t, \mathsf{fma}\left(t, 12, -8\right), 4\right), 2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (<= (/ (/ 2.0 t) (+ 1.0 (/ 1.0 t))) 0.005)
   (+
    0.8333333333333334
    (/
     (+
      (/ (+ 0.037037037037037035 (/ 0.04938271604938271 t)) t)
      -0.2222222222222222)
     t))
   (+ 1.0 (/ -1.0 (fma (* t t) (fma t (fma t 12.0 -8.0) 4.0) 2.0)))))

C

double code(double t) {
	double tmp;
	if (((2.0 / t) / (1.0 + (1.0 / t))) <= 0.005) {
		tmp = 0.8333333333333334 + ((((0.037037037037037035 + (0.04938271604938271 / t)) / t) + -0.2222222222222222) / t);
	} else {
		tmp = 1.0 + (-1.0 / fma((t * t), fma(t, fma(t, 12.0, -8.0), 4.0), 2.0));
	}
	return tmp;
}

Julia

function code(t)
	tmp = 0.0
	if (Float64(Float64(2.0 / t) / Float64(1.0 + Float64(1.0 / t))) <= 0.005)
		tmp = Float64(0.8333333333333334 + Float64(Float64(Float64(Float64(0.037037037037037035 + Float64(0.04938271604938271 / t)) / t) + -0.2222222222222222) / t));
	else
		tmp = Float64(1.0 + Float64(-1.0 / fma(Float64(t * t), fma(t, fma(t, 12.0, -8.0), 4.0), 2.0)));
	end
	return tmp
end

Wolfram

code[t_] := If[LessEqual[N[(N[(2.0 / t), $MachinePrecision] / N[(1.0 + N[(1.0 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.005], N[(0.8333333333333334 + N[(N[(N[(N[(0.037037037037037035 + N[(0.04938271604938271 / t), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision] + -0.2222222222222222), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(-1.0 / N[(N[(t * t), $MachinePrecision] * N[(t * N[(t * 12.0 + -8.0), $MachinePrecision] + 4.0), $MachinePrecision] + 2.0), $MachinePrecision]), $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))) < 0.0050000000000000001

   1. Initial program 100.0%

      \[1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]
   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 99.8%

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

   if 0.0050000000000000001 < (/.f64 (/.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) t)))

   1. Initial program 100.0%

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

   3. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. sub-neg N/A

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

      8. *-commutative N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 99.7

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

   5. Applied rewrites 99.7%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{2}{t}}{1 + \frac{1}{t}} \leq 0.005:\\ \;\;\;\;0.8333333333333334 + \frac{\frac{0.037037037037037035 + \frac{0.04938271604938271}{t}}{t} + -0.2222222222222222}{t}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{\mathsf{fma}\left(t \cdot t, \mathsf{fma}\left(t, \mathsf{fma}\left(t, 12, -8\right), 4\right), 2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\frac{2}{t}}{1 + \frac{1}{t}} \leq 0.005:\\ \;\;\;\;0.8333333333333334 + \frac{\mathsf{fma}\left(t, -0.2222222222222222, 0.037037037037037035\right)}{t \cdot t}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{\mathsf{fma}\left(t \cdot t, \mathsf{fma}\left(t, \mathsf{fma}\left(t, 12, -8\right), 4\right), 2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (<= (/ (/ 2.0 t) (+ 1.0 (/ 1.0 t))) 0.005)
   (+
    0.8333333333333334
    (/ (fma t -0.2222222222222222 0.037037037037037035) (* t t)))
   (+ 1.0 (/ -1.0 (fma (* t t) (fma t (fma t 12.0 -8.0) 4.0) 2.0)))))

C

double code(double t) {
	double tmp;
	if (((2.0 / t) / (1.0 + (1.0 / t))) <= 0.005) {
		tmp = 0.8333333333333334 + (fma(t, -0.2222222222222222, 0.037037037037037035) / (t * t));
	} else {
		tmp = 1.0 + (-1.0 / fma((t * t), fma(t, fma(t, 12.0, -8.0), 4.0), 2.0));
	}
	return tmp;
}

Julia

function code(t)
	tmp = 0.0
	if (Float64(Float64(2.0 / t) / Float64(1.0 + Float64(1.0 / t))) <= 0.005)
		tmp = Float64(0.8333333333333334 + Float64(fma(t, -0.2222222222222222, 0.037037037037037035) / Float64(t * t)));
	else
		tmp = Float64(1.0 + Float64(-1.0 / fma(Float64(t * t), fma(t, fma(t, 12.0, -8.0), 4.0), 2.0)));
	end
	return tmp
end

Wolfram

code[t_] := If[LessEqual[N[(N[(2.0 / t), $MachinePrecision] / N[(1.0 + N[(1.0 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.005], N[(0.8333333333333334 + N[(N[(t * -0.2222222222222222 + 0.037037037037037035), $MachinePrecision] / N[(t * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(-1.0 / N[(N[(t * t), $MachinePrecision] * N[(t * N[(t * 12.0 + -8.0), $MachinePrecision] + 4.0), $MachinePrecision] + 2.0), $MachinePrecision]), $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))) < 0.0050000000000000001

   1. Initial program 100.0%

      \[1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]
   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. associate--l+ N/A

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

      2. unpow2 N/A

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

      3. associate-/r* N/A

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

      4. metadata-eval N/A

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

      5. associate-*r/ N/A

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

      6. associate-*r/ N/A

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

      7. metadata-eval N/A

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

      8. div-sub N/A

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

      9. lower-+.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. sub-neg N/A

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

      12. lower-+.f64 N/A

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

      13. associate-*r/ N/A

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

      14. metadata-eval N/A

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

      15. lower-/.f64 N/A

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

      16. metadata-eval 99.6

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

   5. Applied rewrites 99.6%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 99.6%

      \[\leadsto 0.8333333333333334 + \frac{\mathsf{fma}\left(t, -0.2222222222222222, 0.037037037037037035\right)}{\color{blue}{t \cdot t}} \]

   if 0.0050000000000000001 < (/.f64 (/.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) t)))

   1. Initial program 100.0%

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

   3. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. sub-neg N/A

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

      8. *-commutative N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 99.7

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

   5. Applied rewrites 99.7%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{2}{t}}{1 + \frac{1}{t}} \leq 0.005:\\ \;\;\;\;0.8333333333333334 + \frac{\mathsf{fma}\left(t, -0.2222222222222222, 0.037037037037037035\right)}{t \cdot t}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{\mathsf{fma}\left(t \cdot t, \mathsf{fma}\left(t, \mathsf{fma}\left(t, 12, -8\right), 4\right), 2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\frac{2}{t}}{1 + \frac{1}{t}} \leq 0.005:\\ \;\;\;\;0.8333333333333334 + \frac{\mathsf{fma}\left(t, -0.2222222222222222, 0.037037037037037035\right)}{t \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t, \mathsf{fma}\left(t \cdot t, -2 + t, t\right), 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (<= (/ (/ 2.0 t) (+ 1.0 (/ 1.0 t))) 0.005)
   (+
    0.8333333333333334
    (/ (fma t -0.2222222222222222 0.037037037037037035) (* t t)))
   (fma t (fma (* t t) (+ -2.0 t) t) 0.5)))

C

double code(double t) {
	double tmp;
	if (((2.0 / t) / (1.0 + (1.0 / t))) <= 0.005) {
		tmp = 0.8333333333333334 + (fma(t, -0.2222222222222222, 0.037037037037037035) / (t * t));
	} else {
		tmp = fma(t, fma((t * t), (-2.0 + t), t), 0.5);
	}
	return tmp;
}

Julia

function code(t)
	tmp = 0.0
	if (Float64(Float64(2.0 / t) / Float64(1.0 + Float64(1.0 / t))) <= 0.005)
		tmp = Float64(0.8333333333333334 + Float64(fma(t, -0.2222222222222222, 0.037037037037037035) / Float64(t * t)));
	else
		tmp = fma(t, fma(Float64(t * t), Float64(-2.0 + t), t), 0.5);
	end
	return tmp
end

Wolfram

code[t_] := If[LessEqual[N[(N[(2.0 / t), $MachinePrecision] / N[(1.0 + N[(1.0 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.005], N[(0.8333333333333334 + N[(N[(t * -0.2222222222222222 + 0.037037037037037035), $MachinePrecision] / N[(t * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t * N[(N[(t * t), $MachinePrecision] * N[(-2.0 + t), $MachinePrecision] + 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))) < 0.0050000000000000001

   1. Initial program 100.0%

      \[1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]
   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. associate--l+ N/A

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

      2. unpow2 N/A

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

      3. associate-/r* N/A

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

      4. metadata-eval N/A

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

      5. associate-*r/ N/A

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

      6. associate-*r/ N/A

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

      7. metadata-eval N/A

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

      8. div-sub N/A

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

      9. lower-+.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. sub-neg N/A

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

      12. lower-+.f64 N/A

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

      13. associate-*r/ N/A

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

      14. metadata-eval N/A

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

      15. lower-/.f64 N/A

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

      16. metadata-eval 99.6

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

   5. Applied rewrites 99.6%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 99.6%

      \[\leadsto 0.8333333333333334 + \frac{\mathsf{fma}\left(t, -0.2222222222222222, 0.037037037037037035\right)}{\color{blue}{t \cdot t}} \]

   if 0.0050000000000000001 < (/.f64 (/.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) t)))

   1. Initial program 100.0%

      \[1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]
   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. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. distribute-lft-in N/A

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

      7. associate-*r* N/A

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

      8. unpow2 N/A

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

      9. *-rgt-identity N/A

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

      10. lower-fma.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. sub-neg N/A

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

      14. metadata-eval N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 99.7

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

   5. Applied rewrites 99.7%

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

Alternative 7: 99.2% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\frac{2}{t}}{1 + \frac{1}{t}} \leq 0.005:\\ \;\;\;\;1 - \left(\frac{0.2222222222222222}{t} + 0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t, \mathsf{fma}\left(t \cdot t, -2 + t, t\right), 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (<= (/ (/ 2.0 t) (+ 1.0 (/ 1.0 t))) 0.005)
   (- 1.0 (+ (/ 0.2222222222222222 t) 0.16666666666666666))
   (fma t (fma (* t t) (+ -2.0 t) t) 0.5)))

C

double code(double t) {
	double tmp;
	if (((2.0 / t) / (1.0 + (1.0 / t))) <= 0.005) {
		tmp = 1.0 - ((0.2222222222222222 / t) + 0.16666666666666666);
	} else {
		tmp = fma(t, fma((t * t), (-2.0 + t), t), 0.5);
	}
	return tmp;
}

Julia

function code(t)
	tmp = 0.0
	if (Float64(Float64(2.0 / t) / Float64(1.0 + Float64(1.0 / t))) <= 0.005)
		tmp = Float64(1.0 - Float64(Float64(0.2222222222222222 / t) + 0.16666666666666666));
	else
		tmp = fma(t, fma(Float64(t * t), Float64(-2.0 + t), t), 0.5);
	end
	return tmp
end

Wolfram

code[t_] := If[LessEqual[N[(N[(2.0 / t), $MachinePrecision] / N[(1.0 + N[(1.0 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.005], N[(1.0 - N[(N[(0.2222222222222222 / t), $MachinePrecision] + 0.16666666666666666), $MachinePrecision]), $MachinePrecision], N[(t * N[(N[(t * t), $MachinePrecision] * N[(-2.0 + t), $MachinePrecision] + 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))) < 0.0050000000000000001

   1. Initial program 100.0%

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

   3. Taylor expanded in t around inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

      5. lower-/.f64 99.3

         \[\leadsto 1 - \left(\color{blue}{\frac{0.2222222222222222}{t}} + 0.16666666666666666\right) \]

   5. Applied rewrites 99.3%

      \[\leadsto 1 - \color{blue}{\left(\frac{0.2222222222222222}{t} + 0.16666666666666666\right)} \]

   if 0.0050000000000000001 < (/.f64 (/.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) t)))

   1. Initial program 100.0%

      \[1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]
   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. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. distribute-lft-in N/A

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

      7. associate-*r* N/A

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

      8. unpow2 N/A

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

      9. *-rgt-identity N/A

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

      10. lower-fma.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. sub-neg N/A

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

      14. metadata-eval N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 99.7

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

   5. Applied rewrites 99.7%

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

Alternative 8: 99.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\frac{2}{t}}{1 + \frac{1}{t}} \leq 0.005:\\ \;\;\;\;1 - \left(\frac{0.2222222222222222}{t} + 0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t, \mathsf{fma}\left(-2, t \cdot t, t\right), 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (<= (/ (/ 2.0 t) (+ 1.0 (/ 1.0 t))) 0.005)
   (- 1.0 (+ (/ 0.2222222222222222 t) 0.16666666666666666))
   (fma t (fma -2.0 (* t t) t) 0.5)))

C

double code(double t) {
	double tmp;
	if (((2.0 / t) / (1.0 + (1.0 / t))) <= 0.005) {
		tmp = 1.0 - ((0.2222222222222222 / t) + 0.16666666666666666);
	} else {
		tmp = fma(t, fma(-2.0, (t * t), t), 0.5);
	}
	return tmp;
}

Julia

function code(t)
	tmp = 0.0
	if (Float64(Float64(2.0 / t) / Float64(1.0 + Float64(1.0 / t))) <= 0.005)
		tmp = Float64(1.0 - Float64(Float64(0.2222222222222222 / t) + 0.16666666666666666));
	else
		tmp = fma(t, fma(-2.0, Float64(t * t), t), 0.5);
	end
	return tmp
end

Wolfram

code[t_] := If[LessEqual[N[(N[(2.0 / t), $MachinePrecision] / N[(1.0 + N[(1.0 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.005], N[(1.0 - N[(N[(0.2222222222222222 / t), $MachinePrecision] + 0.16666666666666666), $MachinePrecision]), $MachinePrecision], N[(t * N[(-2.0 * N[(t * t), $MachinePrecision] + 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))) < 0.0050000000000000001

   1. Initial program 100.0%

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

   3. Taylor expanded in t around inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

      5. lower-/.f64 99.3

         \[\leadsto 1 - \left(\color{blue}{\frac{0.2222222222222222}{t}} + 0.16666666666666666\right) \]

   5. Applied rewrites 99.3%

      \[\leadsto 1 - \color{blue}{\left(\frac{0.2222222222222222}{t} + 0.16666666666666666\right)} \]

   if 0.0050000000000000001 < (/.f64 (/.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) t)))

   1. Initial program 100.0%

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

   3. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. distribute-lft1-in N/A

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

      8. associate-*l* N/A

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

      9. unpow2 N/A

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

      10. lower-fma.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 99.7

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

   5. Applied rewrites 99.7%

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

Alternative 9: 99.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\frac{2}{t}}{1 + \frac{1}{t}} \leq 0.005:\\ \;\;\;\;0.8333333333333334 + \frac{-0.2222222222222222}{t}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t, \mathsf{fma}\left(-2, t \cdot t, t\right), 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (<= (/ (/ 2.0 t) (+ 1.0 (/ 1.0 t))) 0.005)
   (+ 0.8333333333333334 (/ -0.2222222222222222 t))
   (fma t (fma -2.0 (* t t) t) 0.5)))

C

double code(double t) {
	double tmp;
	if (((2.0 / t) / (1.0 + (1.0 / t))) <= 0.005) {
		tmp = 0.8333333333333334 + (-0.2222222222222222 / t);
	} else {
		tmp = fma(t, fma(-2.0, (t * t), t), 0.5);
	}
	return tmp;
}

Julia

function code(t)
	tmp = 0.0
	if (Float64(Float64(2.0 / t) / Float64(1.0 + Float64(1.0 / t))) <= 0.005)
		tmp = Float64(0.8333333333333334 + Float64(-0.2222222222222222 / t));
	else
		tmp = fma(t, fma(-2.0, Float64(t * t), t), 0.5);
	end
	return tmp
end

Wolfram

code[t_] := If[LessEqual[N[(N[(2.0 / t), $MachinePrecision] / N[(1.0 + N[(1.0 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.005], N[(0.8333333333333334 + N[(-0.2222222222222222 / t), $MachinePrecision]), $MachinePrecision], N[(t * N[(-2.0 * N[(t * t), $MachinePrecision] + 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))) < 0.0050000000000000001

   1. Initial program 100.0%

      \[1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]
   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. sub-neg N/A

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

      2. lower-+.f64 N/A

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

         \[\leadsto \frac{5}{6} + \left(\mathsf{neg}\left(\frac{\color{blue}{\frac{2}{9}}}{t}\right)\right) \]

      5. distribute-neg-frac N/A

         \[\leadsto \frac{5}{6} + \color{blue}{\frac{\mathsf{neg}\left(\frac{2}{9}\right)}{t}} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{5}{6} + \color{blue}{\frac{\mathsf{neg}\left(\frac{2}{9}\right)}{t}} \]

      7. metadata-eval 99.3

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

   5. Applied rewrites 99.3%

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

   if 0.0050000000000000001 < (/.f64 (/.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) t)))

   1. Initial program 100.0%

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

   3. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. distribute-lft1-in N/A

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

      8. associate-*l* N/A

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

      9. unpow2 N/A

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

      10. lower-fma.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 99.7

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

   5. Applied rewrites 99.7%

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

Alternative 10: 99.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\frac{2}{t}}{1 + \frac{1}{t}} \leq 0.005:\\ \;\;\;\;0.8333333333333334 + \frac{-0.2222222222222222}{t}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t, t, 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (<= (/ (/ 2.0 t) (+ 1.0 (/ 1.0 t))) 0.005)
   (+ 0.8333333333333334 (/ -0.2222222222222222 t))
   (fma t t 0.5)))

C

double code(double t) {
	double tmp;
	if (((2.0 / t) / (1.0 + (1.0 / t))) <= 0.005) {
		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 + Float64(1.0 / t))) <= 0.005)
		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[(1.0 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.005], 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))) < 0.0050000000000000001

   1. Initial program 100.0%

      \[1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]
   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. sub-neg N/A

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

      2. lower-+.f64 N/A

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

         \[\leadsto \frac{5}{6} + \left(\mathsf{neg}\left(\frac{\color{blue}{\frac{2}{9}}}{t}\right)\right) \]

      5. distribute-neg-frac N/A

         \[\leadsto \frac{5}{6} + \color{blue}{\frac{\mathsf{neg}\left(\frac{2}{9}\right)}{t}} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{5}{6} + \color{blue}{\frac{\mathsf{neg}\left(\frac{2}{9}\right)}{t}} \]

      7. metadata-eval 99.3

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

   5. Applied rewrites 99.3%

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

   if 0.0050000000000000001 < (/.f64 (/.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) t)))

   1. Initial program 100.0%

      \[1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]
   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 99.6

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

   5. Applied rewrites 99.6%

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

Alternative 11: 100.0% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[t_] := N[(1.0 + N[(1.0 / N[(N[(-2.0 + N[(2.0 / N[(1.0 + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(2.0 + N[(-2.0 / N[(1.0 + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

4. Applied rewrites 100.0%

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

6. Applied rewrites 100.0%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-neg.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. distribute-rgt-neg-in N/A

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

   6. lift-+.f64 N/A

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

   7. +-commutative N/A

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

   8. distribute-neg-in N/A

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

   9. metadata-eval N/A

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

   10. sub-neg N/A

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

   11. lift--.f64 N/A

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

   12. lower-fma.f64 100.0

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

8. Applied rewrites 100.0%

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

10. Applied rewrites 100.0%

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

11. Final simplification 100.0%

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

Alternative 12: 98.5% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\frac{2}{t}}{1 + \frac{1}{t}} \leq 0.005:\\ \;\;\;\;0.8333333333333334\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t, t, 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (<= (/ (/ 2.0 t) (+ 1.0 (/ 1.0 t))) 0.005)
   0.8333333333333334
   (fma t t 0.5)))

C

double code(double t) {
	double tmp;
	if (((2.0 / t) / (1.0 + (1.0 / t))) <= 0.005) {
		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 + Float64(1.0 / t))) <= 0.005)
		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[(1.0 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.005], 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))) < 0.0050000000000000001

   1. Initial program 100.0%

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

   3. Taylor expanded in t around inf

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

   5. Applied rewrites 98.2%

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

   if 0.0050000000000000001 < (/.f64 (/.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) t)))

   1. Initial program 100.0%

      \[1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]
   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 99.6

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

   5. Applied rewrites 99.6%

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

Alternative 13: 98.4% accurate, 2.3× speedup

Math

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

FPCore

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

C

double code(double t) {
	double tmp;
	if (((2.0 / t) / (1.0 + (1.0 / t))) <= 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 + (1.0d0 / t))) <= 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 + (1.0 / t))) <= 1.0) {
		tmp = 0.8333333333333334;
	} else {
		tmp = 0.5;
	}
	return tmp;
}

Python

def code(t):
	tmp = 0
	if ((2.0 / t) / (1.0 + (1.0 / t))) <= 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 + Float64(1.0 / t))) <= 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 + (1.0 / t))) <= 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[(1.0 / t), $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%

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

   3. Taylor expanded in t around inf

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

   5. Applied rewrites 98.2%

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

   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%

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

   3. Taylor expanded in t around 0

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

   5. Applied rewrites 99.5%

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

Alternative 14: 59.0% accurate, 101.0× 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%

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

3. Taylor expanded in t around 0

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

5. Applied rewrites 57.7%

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

Reproduce

herbie shell --seed 2024238 
(FPCore (t)
  :name "Kahan p13 Example 3"
  :precision binary64
  (- 1.0 (/ 1.0 (+ 2.0 (* (- 2.0 (/ (/ 2.0 t) (+ 1.0 (/ 1.0 t)))) (- 2.0 (/ (/ 2.0 t) (+ 1.0 (/ 1.0 t)))))))))