Kahan p13 Example 3

Percentage Accurate: 100.0% → 100.0%

Time: 11.7s

Alternatives: 13

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, 1.8× speedup

Math

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

FPCore

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

C

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

Julia

function code(t)
	t_1 = Float64(2.0 + Float64(2.0 / Float64(-1.0 - t)))
	return Float64(1.0 + Float64(-1.0 / fma(t_1, t_1, 2.0)))
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[(-1.0 / N[(t$95$1 * t$95$1 + 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
3. Step-by-step derivation

   1. +-commutative N/A

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

   2. accelerator-lowering-fma.f64 N/A

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

4. Applied egg-rr 100.0%

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

   1. accelerator-lowering-fma.f64 N/A

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

   2. --lowering--.f64 N/A

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

   3. /-lowering-/.f64 N/A

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

   4. rgt-mult-inverse N/A

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

   5. +-commutative N/A

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

   6. +-lowering-+.f64 N/A

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

   7. --lowering--.f64 N/A

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

   8. /-lowering-/.f64 N/A

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

   9. rgt-mult-inverse N/A

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

   10. +-commutative N/A

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

   11. +-lowering-+.f64 100.0

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

6. Applied egg-rr 100.0%

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

7. Final simplification 100.0%

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

Alternative 2: 99.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.4:\\ \;\;\;\;\left(1 - \frac{0.2222222222222222}{t}\right) - 0.16666666666666666\\ \mathbf{elif}\;t \leq 0.52:\\ \;\;\;\;1 + \frac{-1}{\mathsf{fma}\left(\mathsf{fma}\left(t, \mathsf{fma}\left(t, \mathsf{fma}\left(t, -16, 12\right), -8\right), 4\right), t \cdot t, 2\right)}\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334 - \frac{\mathsf{fma}\left(t, 0.2222222222222222, -0.037037037037037035\right)}{\mathsf{fma}\left(t, t, 0\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (<= t -1.4)
   (- (- 1.0 (/ 0.2222222222222222 t)) 0.16666666666666666)
   (if (<= t 0.52)
     (+
      1.0
      (/ -1.0 (fma (fma t (fma t (fma t -16.0 12.0) -8.0) 4.0) (* t t) 2.0)))
     (-
      0.8333333333333334
      (/ (fma t 0.2222222222222222 -0.037037037037037035) (fma t t 0.0))))))

C

double code(double t) {
	double tmp;
	if (t <= -1.4) {
		tmp = (1.0 - (0.2222222222222222 / t)) - 0.16666666666666666;
	} else if (t <= 0.52) {
		tmp = 1.0 + (-1.0 / fma(fma(t, fma(t, fma(t, -16.0, 12.0), -8.0), 4.0), (t * t), 2.0));
	} else {
		tmp = 0.8333333333333334 - (fma(t, 0.2222222222222222, -0.037037037037037035) / fma(t, t, 0.0));
	}
	return tmp;
}

Julia

function code(t)
	tmp = 0.0
	if (t <= -1.4)
		tmp = Float64(Float64(1.0 - Float64(0.2222222222222222 / t)) - 0.16666666666666666);
	elseif (t <= 0.52)
		tmp = Float64(1.0 + Float64(-1.0 / fma(fma(t, fma(t, fma(t, -16.0, 12.0), -8.0), 4.0), Float64(t * t), 2.0)));
	else
		tmp = Float64(0.8333333333333334 - Float64(fma(t, 0.2222222222222222, -0.037037037037037035) / fma(t, t, 0.0)));
	end
	return tmp
end

Wolfram

code[t_] := If[LessEqual[t, -1.4], N[(N[(1.0 - N[(0.2222222222222222 / t), $MachinePrecision]), $MachinePrecision] - 0.16666666666666666), $MachinePrecision], If[LessEqual[t, 0.52], N[(1.0 + N[(-1.0 / N[(N[(t * N[(t * N[(t * -16.0 + 12.0), $MachinePrecision] + -8.0), $MachinePrecision] + 4.0), $MachinePrecision] * N[(t * t), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.8333333333333334 - N[(N[(t * 0.2222222222222222 + -0.037037037037037035), $MachinePrecision] / N[(t * t + 0.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -1.3999999999999999

   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. +-lowering-+.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. /-lowering-/.f64 100.0

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

   5. Simplified 100.0%

      \[\leadsto 1 - \color{blue}{\left(\frac{0.2222222222222222}{t} + 0.16666666666666666\right)} \]
   6. Step-by-step derivation

      1. associate--r+ N/A

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

      2. --lowering--.f64 N/A

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

      3. --lowering--.f64 N/A

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

      4. /-lowering-/.f64 100.0

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

   7. Applied egg-rr 100.0%

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

   if -1.3999999999999999 < t < 0.52000000000000002

   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. *-lowering-*.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. *-lowering-*.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. accelerator-lowering-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. accelerator-lowering-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. accelerator-lowering-fma.f64 98.8

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

   5. Simplified 98.8%

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. *-lowering-*.f64 98.8

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

   7. Applied egg-rr 98.8%

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

   if 0.52000000000000002 < 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. Step-by-step derivation

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in t around inf

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

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

      6. unpow2 N/A

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

      7. associate-/r* N/A

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

      8. metadata-eval N/A

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

      9. associate-*r/ N/A

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

      10. sub-neg N/A

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

      11. associate-*r/ N/A

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

      12. metadata-eval N/A

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

      13. div-sub N/A

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

      14. --lowering--.f64 N/A

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

      15. /-lowering-/.f64 N/A

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

   7. Simplified 100.0%

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

   8. Taylor expanded in t around 0

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

      1. /-lowering-/.f64 N/A

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

      2. sub-neg N/A

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

      3. *-commutative N/A

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

      4. metadata-eval N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. remove-double-neg N/A

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

      7. mul-1-neg N/A

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

      8. +-lft-identity N/A

         \[\leadsto \frac{5}{6} - \frac{\mathsf{fma}\left(t, \frac{2}{9}, \frac{-1}{27}\right)}{\mathsf{neg}\left(\color{blue}{\left(0 + -1 \cdot {t}^{2}\right)}\right)} \]

      9. +-commutative N/A

         \[\leadsto \frac{5}{6} - \frac{\mathsf{fma}\left(t, \frac{2}{9}, \frac{-1}{27}\right)}{\mathsf{neg}\left(\color{blue}{\left(-1 \cdot {t}^{2} + 0\right)}\right)} \]

      10. distribute-neg-in N/A

          \[\leadsto \frac{5}{6} - \frac{\mathsf{fma}\left(t, \frac{2}{9}, \frac{-1}{27}\right)}{\color{blue}{\left(\mathsf{neg}\left(-1 \cdot {t}^{2}\right)\right) + \left(\mathsf{neg}\left(0\right)\right)}} \]

      11. mul-1-neg N/A

          \[\leadsto \frac{5}{6} - \frac{\mathsf{fma}\left(t, \frac{2}{9}, \frac{-1}{27}\right)}{\left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left({t}^{2}\right)\right)}\right)\right) + \left(\mathsf{neg}\left(0\right)\right)} \]

      12. remove-double-neg N/A

          \[\leadsto \frac{5}{6} - \frac{\mathsf{fma}\left(t, \frac{2}{9}, \frac{-1}{27}\right)}{\color{blue}{{t}^{2}} + \left(\mathsf{neg}\left(0\right)\right)} \]

      13. metadata-eval N/A

          \[\leadsto \frac{5}{6} - \frac{\mathsf{fma}\left(t, \frac{2}{9}, \frac{-1}{27}\right)}{{t}^{2} + \color{blue}{0}} \]

      14. unpow2 N/A

          \[\leadsto \frac{5}{6} - \frac{\mathsf{fma}\left(t, \frac{2}{9}, \frac{-1}{27}\right)}{\color{blue}{t \cdot t} + 0} \]

      15. accelerator-lowering-fma.f64 100.0

          \[\leadsto 0.8333333333333334 - \frac{\mathsf{fma}\left(t, 0.2222222222222222, -0.037037037037037035\right)}{\color{blue}{\mathsf{fma}\left(t, t, 0\right)}} \]

   10. Simplified 100.0%

       \[\leadsto 0.8333333333333334 - \color{blue}{\frac{\mathsf{fma}\left(t, 0.2222222222222222, -0.037037037037037035\right)}{\mathsf{fma}\left(t, t, 0\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.4:\\ \;\;\;\;\left(1 - \frac{0.2222222222222222}{t}\right) - 0.16666666666666666\\ \mathbf{elif}\;t \leq 0.52:\\ \;\;\;\;1 + \frac{-1}{\mathsf{fma}\left(\mathsf{fma}\left(t, \mathsf{fma}\left(t, \mathsf{fma}\left(t, -16, 12\right), -8\right), 4\right), t \cdot t, 2\right)}\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334 - \frac{\mathsf{fma}\left(t, 0.2222222222222222, -0.037037037037037035\right)}{\mathsf{fma}\left(t, t, 0\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.4% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -0.46:\\ \;\;\;\;1 + \frac{-1}{6 + \frac{-8}{t}}\\ \mathbf{elif}\;t \leq 0.58:\\ \;\;\;\;\mathsf{fma}\left(t \cdot t, \mathsf{fma}\left(t, t + -2, 1\right), 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334 - \frac{\mathsf{fma}\left(t, 0.2222222222222222, -0.037037037037037035\right)}{\mathsf{fma}\left(t, t, 0\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (<= t -0.46)
   (+ 1.0 (/ -1.0 (+ 6.0 (/ -8.0 t))))
   (if (<= t 0.58)
     (fma (* t t) (fma t (+ t -2.0) 1.0) 0.5)
     (-
      0.8333333333333334
      (/ (fma t 0.2222222222222222 -0.037037037037037035) (fma t t 0.0))))))

C

double code(double t) {
	double tmp;
	if (t <= -0.46) {
		tmp = 1.0 + (-1.0 / (6.0 + (-8.0 / t)));
	} else if (t <= 0.58) {
		tmp = fma((t * t), fma(t, (t + -2.0), 1.0), 0.5);
	} else {
		tmp = 0.8333333333333334 - (fma(t, 0.2222222222222222, -0.037037037037037035) / fma(t, t, 0.0));
	}
	return tmp;
}

Julia

function code(t)
	tmp = 0.0
	if (t <= -0.46)
		tmp = Float64(1.0 + Float64(-1.0 / Float64(6.0 + Float64(-8.0 / t))));
	elseif (t <= 0.58)
		tmp = fma(Float64(t * t), fma(t, Float64(t + -2.0), 1.0), 0.5);
	else
		tmp = Float64(0.8333333333333334 - Float64(fma(t, 0.2222222222222222, -0.037037037037037035) / fma(t, t, 0.0)));
	end
	return tmp
end

Wolfram

code[t_] := If[LessEqual[t, -0.46], N[(1.0 + N[(-1.0 / N[(6.0 + N[(-8.0 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 0.58], N[(N[(t * t), $MachinePrecision] * N[(t * N[(t + -2.0), $MachinePrecision] + 1.0), $MachinePrecision] + 0.5), $MachinePrecision], N[(0.8333333333333334 - N[(N[(t * 0.2222222222222222 + -0.037037037037037035), $MachinePrecision] / N[(t * t + 0.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -0.46000000000000002

   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

      \[\leadsto 1 - \frac{1}{\color{blue}{6 - 8 \cdot \frac{1}{t}}} \]
   4. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto 1 - \frac{1}{\color{blue}{6 + \left(\mathsf{neg}\left(8 \cdot \frac{1}{t}\right)\right)}} \]

      2. +-lowering-+.f64 N/A

         \[\leadsto 1 - \frac{1}{\color{blue}{6 + \left(\mathsf{neg}\left(8 \cdot \frac{1}{t}\right)\right)}} \]

      3. associate-*r/ N/A

         \[\leadsto 1 - \frac{1}{6 + \left(\mathsf{neg}\left(\color{blue}{\frac{8 \cdot 1}{t}}\right)\right)} \]

      4. metadata-eval N/A

         \[\leadsto 1 - \frac{1}{6 + \left(\mathsf{neg}\left(\frac{\color{blue}{8}}{t}\right)\right)} \]

      5. distribute-neg-frac N/A

         \[\leadsto 1 - \frac{1}{6 + \color{blue}{\frac{\mathsf{neg}\left(8\right)}{t}}} \]

      6. metadata-eval N/A

         \[\leadsto 1 - \frac{1}{6 + \frac{\color{blue}{-8}}{t}} \]

      7. /-lowering-/.f64 98.8

         \[\leadsto 1 - \frac{1}{6 + \color{blue}{\frac{-8}{t}}} \]

   5. Simplified 98.8%

      \[\leadsto 1 - \frac{1}{\color{blue}{6 + \frac{-8}{t}}} \]

   if -0.46000000000000002 < t < 0.57999999999999996

   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. accelerator-lowering-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 N/A

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

      5. +-commutative N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. sub-neg N/A

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

      8. metadata-eval N/A

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

      9. +-commutative N/A

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

      10. +-lowering-+.f64 99.3

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

   5. Simplified 99.3%

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

   if 0.57999999999999996 < 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. Step-by-step derivation

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in t around inf

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

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

      6. unpow2 N/A

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

      7. associate-/r* N/A

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

      8. metadata-eval N/A

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

      9. associate-*r/ N/A

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

      10. sub-neg N/A

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

      11. associate-*r/ N/A

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

      12. metadata-eval N/A

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

      13. div-sub N/A

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

      14. --lowering--.f64 N/A

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

      15. /-lowering-/.f64 N/A

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

   7. Simplified 100.0%

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

   8. Taylor expanded in t around 0

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

      1. /-lowering-/.f64 N/A

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

      2. sub-neg N/A

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

      3. *-commutative N/A

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

      4. metadata-eval N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. remove-double-neg N/A

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

      7. mul-1-neg N/A

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

      8. +-lft-identity N/A

         \[\leadsto \frac{5}{6} - \frac{\mathsf{fma}\left(t, \frac{2}{9}, \frac{-1}{27}\right)}{\mathsf{neg}\left(\color{blue}{\left(0 + -1 \cdot {t}^{2}\right)}\right)} \]

      9. +-commutative N/A

         \[\leadsto \frac{5}{6} - \frac{\mathsf{fma}\left(t, \frac{2}{9}, \frac{-1}{27}\right)}{\mathsf{neg}\left(\color{blue}{\left(-1 \cdot {t}^{2} + 0\right)}\right)} \]

      10. distribute-neg-in N/A

          \[\leadsto \frac{5}{6} - \frac{\mathsf{fma}\left(t, \frac{2}{9}, \frac{-1}{27}\right)}{\color{blue}{\left(\mathsf{neg}\left(-1 \cdot {t}^{2}\right)\right) + \left(\mathsf{neg}\left(0\right)\right)}} \]

      11. mul-1-neg N/A

          \[\leadsto \frac{5}{6} - \frac{\mathsf{fma}\left(t, \frac{2}{9}, \frac{-1}{27}\right)}{\left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left({t}^{2}\right)\right)}\right)\right) + \left(\mathsf{neg}\left(0\right)\right)} \]

      12. remove-double-neg N/A

          \[\leadsto \frac{5}{6} - \frac{\mathsf{fma}\left(t, \frac{2}{9}, \frac{-1}{27}\right)}{\color{blue}{{t}^{2}} + \left(\mathsf{neg}\left(0\right)\right)} \]

      13. metadata-eval N/A

          \[\leadsto \frac{5}{6} - \frac{\mathsf{fma}\left(t, \frac{2}{9}, \frac{-1}{27}\right)}{{t}^{2} + \color{blue}{0}} \]

      14. unpow2 N/A

          \[\leadsto \frac{5}{6} - \frac{\mathsf{fma}\left(t, \frac{2}{9}, \frac{-1}{27}\right)}{\color{blue}{t \cdot t} + 0} \]

      15. accelerator-lowering-fma.f64 100.0

          \[\leadsto 0.8333333333333334 - \frac{\mathsf{fma}\left(t, 0.2222222222222222, -0.037037037037037035\right)}{\color{blue}{\mathsf{fma}\left(t, t, 0\right)}} \]

   10. Simplified 100.0%

       \[\leadsto 0.8333333333333334 - \color{blue}{\frac{\mathsf{fma}\left(t, 0.2222222222222222, -0.037037037037037035\right)}{\mathsf{fma}\left(t, t, 0\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.46:\\ \;\;\;\;1 + \frac{-1}{6 + \frac{-8}{t}}\\ \mathbf{elif}\;t \leq 0.58:\\ \;\;\;\;\mathsf{fma}\left(t \cdot t, \mathsf{fma}\left(t, t + -2, 1\right), 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334 - \frac{\mathsf{fma}\left(t, 0.2222222222222222, -0.037037037037037035\right)}{\mathsf{fma}\left(t, t, 0\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.4% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -0.55:\\ \;\;\;\;\left(1 - \frac{0.2222222222222222}{t}\right) - 0.16666666666666666\\ \mathbf{elif}\;t \leq 0.58:\\ \;\;\;\;\mathsf{fma}\left(t \cdot t, \mathsf{fma}\left(t, t + -2, 1\right), 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334 - \frac{\mathsf{fma}\left(t, 0.2222222222222222, -0.037037037037037035\right)}{\mathsf{fma}\left(t, t, 0\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (<= t -0.55)
   (- (- 1.0 (/ 0.2222222222222222 t)) 0.16666666666666666)
   (if (<= t 0.58)
     (fma (* t t) (fma t (+ t -2.0) 1.0) 0.5)
     (-
      0.8333333333333334
      (/ (fma t 0.2222222222222222 -0.037037037037037035) (fma t t 0.0))))))

C

double code(double t) {
	double tmp;
	if (t <= -0.55) {
		tmp = (1.0 - (0.2222222222222222 / t)) - 0.16666666666666666;
	} else if (t <= 0.58) {
		tmp = fma((t * t), fma(t, (t + -2.0), 1.0), 0.5);
	} else {
		tmp = 0.8333333333333334 - (fma(t, 0.2222222222222222, -0.037037037037037035) / fma(t, t, 0.0));
	}
	return tmp;
}

Julia

function code(t)
	tmp = 0.0
	if (t <= -0.55)
		tmp = Float64(Float64(1.0 - Float64(0.2222222222222222 / t)) - 0.16666666666666666);
	elseif (t <= 0.58)
		tmp = fma(Float64(t * t), fma(t, Float64(t + -2.0), 1.0), 0.5);
	else
		tmp = Float64(0.8333333333333334 - Float64(fma(t, 0.2222222222222222, -0.037037037037037035) / fma(t, t, 0.0)));
	end
	return tmp
end

Wolfram

code[t_] := If[LessEqual[t, -0.55], N[(N[(1.0 - N[(0.2222222222222222 / t), $MachinePrecision]), $MachinePrecision] - 0.16666666666666666), $MachinePrecision], If[LessEqual[t, 0.58], N[(N[(t * t), $MachinePrecision] * N[(t * N[(t + -2.0), $MachinePrecision] + 1.0), $MachinePrecision] + 0.5), $MachinePrecision], N[(0.8333333333333334 - N[(N[(t * 0.2222222222222222 + -0.037037037037037035), $MachinePrecision] / N[(t * t + 0.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -0.55000000000000004

   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. +-lowering-+.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. /-lowering-/.f64 98.8

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

   5. Simplified 98.8%

      \[\leadsto 1 - \color{blue}{\left(\frac{0.2222222222222222}{t} + 0.16666666666666666\right)} \]
   6. Step-by-step derivation

      1. associate--r+ N/A

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

      2. --lowering--.f64 N/A

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

      3. --lowering--.f64 N/A

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

      4. /-lowering-/.f64 98.8

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

   7. Applied egg-rr 98.8%

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

   if -0.55000000000000004 < t < 0.57999999999999996

   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. accelerator-lowering-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 N/A

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

      5. +-commutative N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. sub-neg N/A

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

      8. metadata-eval N/A

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

      9. +-commutative N/A

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

      10. +-lowering-+.f64 99.3

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

   5. Simplified 99.3%

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

   if 0.57999999999999996 < 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. Step-by-step derivation

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in t around inf

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

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

      6. unpow2 N/A

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

      7. associate-/r* N/A

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

      8. metadata-eval N/A

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

      9. associate-*r/ N/A

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

      10. sub-neg N/A

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

      11. associate-*r/ N/A

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

      12. metadata-eval N/A

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

      13. div-sub N/A

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

      14. --lowering--.f64 N/A

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

      15. /-lowering-/.f64 N/A

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

   7. Simplified 100.0%

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

   8. Taylor expanded in t around 0

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

      1. /-lowering-/.f64 N/A

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

      2. sub-neg N/A

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

      3. *-commutative N/A

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

      4. metadata-eval N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. remove-double-neg N/A

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

      7. mul-1-neg N/A

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

      8. +-lft-identity N/A

         \[\leadsto \frac{5}{6} - \frac{\mathsf{fma}\left(t, \frac{2}{9}, \frac{-1}{27}\right)}{\mathsf{neg}\left(\color{blue}{\left(0 + -1 \cdot {t}^{2}\right)}\right)} \]

      9. +-commutative N/A

         \[\leadsto \frac{5}{6} - \frac{\mathsf{fma}\left(t, \frac{2}{9}, \frac{-1}{27}\right)}{\mathsf{neg}\left(\color{blue}{\left(-1 \cdot {t}^{2} + 0\right)}\right)} \]

      10. distribute-neg-in N/A

          \[\leadsto \frac{5}{6} - \frac{\mathsf{fma}\left(t, \frac{2}{9}, \frac{-1}{27}\right)}{\color{blue}{\left(\mathsf{neg}\left(-1 \cdot {t}^{2}\right)\right) + \left(\mathsf{neg}\left(0\right)\right)}} \]

      11. mul-1-neg N/A

          \[\leadsto \frac{5}{6} - \frac{\mathsf{fma}\left(t, \frac{2}{9}, \frac{-1}{27}\right)}{\left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left({t}^{2}\right)\right)}\right)\right) + \left(\mathsf{neg}\left(0\right)\right)} \]

      12. remove-double-neg N/A

          \[\leadsto \frac{5}{6} - \frac{\mathsf{fma}\left(t, \frac{2}{9}, \frac{-1}{27}\right)}{\color{blue}{{t}^{2}} + \left(\mathsf{neg}\left(0\right)\right)} \]

      13. metadata-eval N/A

          \[\leadsto \frac{5}{6} - \frac{\mathsf{fma}\left(t, \frac{2}{9}, \frac{-1}{27}\right)}{{t}^{2} + \color{blue}{0}} \]

      14. unpow2 N/A

          \[\leadsto \frac{5}{6} - \frac{\mathsf{fma}\left(t, \frac{2}{9}, \frac{-1}{27}\right)}{\color{blue}{t \cdot t} + 0} \]

      15. accelerator-lowering-fma.f64 100.0

          \[\leadsto 0.8333333333333334 - \frac{\mathsf{fma}\left(t, 0.2222222222222222, -0.037037037037037035\right)}{\color{blue}{\mathsf{fma}\left(t, t, 0\right)}} \]

   10. Simplified 100.0%

       \[\leadsto 0.8333333333333334 - \color{blue}{\frac{\mathsf{fma}\left(t, 0.2222222222222222, -0.037037037037037035\right)}{\mathsf{fma}\left(t, t, 0\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.55:\\ \;\;\;\;\left(1 - \frac{0.2222222222222222}{t}\right) - 0.16666666666666666\\ \mathbf{elif}\;t \leq 0.58:\\ \;\;\;\;\mathsf{fma}\left(t \cdot t, \mathsf{fma}\left(t, t + -2, 1\right), 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334 - \frac{\mathsf{fma}\left(t, 0.2222222222222222, -0.037037037037037035\right)}{\mathsf{fma}\left(t, t, 0\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.4% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -0.55:\\ \;\;\;\;\left(1 - \frac{0.2222222222222222}{t}\right) - 0.16666666666666666\\ \mathbf{elif}\;t \leq 0.75:\\ \;\;\;\;\mathsf{fma}\left(t \cdot t, \mathsf{fma}\left(t, t + -2, 1\right), 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \left(\frac{0.2222222222222222}{t} + 0.16666666666666666\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (<= t -0.55)
   (- (- 1.0 (/ 0.2222222222222222 t)) 0.16666666666666666)
   (if (<= t 0.75)
     (fma (* t t) (fma t (+ t -2.0) 1.0) 0.5)
     (- 1.0 (+ (/ 0.2222222222222222 t) 0.16666666666666666)))))

C

double code(double t) {
	double tmp;
	if (t <= -0.55) {
		tmp = (1.0 - (0.2222222222222222 / t)) - 0.16666666666666666;
	} else if (t <= 0.75) {
		tmp = fma((t * t), fma(t, (t + -2.0), 1.0), 0.5);
	} else {
		tmp = 1.0 - ((0.2222222222222222 / t) + 0.16666666666666666);
	}
	return tmp;
}

Julia

function code(t)
	tmp = 0.0
	if (t <= -0.55)
		tmp = Float64(Float64(1.0 - Float64(0.2222222222222222 / t)) - 0.16666666666666666);
	elseif (t <= 0.75)
		tmp = fma(Float64(t * t), fma(t, Float64(t + -2.0), 1.0), 0.5);
	else
		tmp = Float64(1.0 - Float64(Float64(0.2222222222222222 / t) + 0.16666666666666666));
	end
	return tmp
end

Wolfram

code[t_] := If[LessEqual[t, -0.55], N[(N[(1.0 - N[(0.2222222222222222 / t), $MachinePrecision]), $MachinePrecision] - 0.16666666666666666), $MachinePrecision], If[LessEqual[t, 0.75], N[(N[(t * t), $MachinePrecision] * N[(t * N[(t + -2.0), $MachinePrecision] + 1.0), $MachinePrecision] + 0.5), $MachinePrecision], N[(1.0 - N[(N[(0.2222222222222222 / t), $MachinePrecision] + 0.16666666666666666), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -0.55000000000000004

   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. +-lowering-+.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. /-lowering-/.f64 98.8

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

   5. Simplified 98.8%

      \[\leadsto 1 - \color{blue}{\left(\frac{0.2222222222222222}{t} + 0.16666666666666666\right)} \]
   6. Step-by-step derivation

      1. associate--r+ N/A

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

      2. --lowering--.f64 N/A

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

      3. --lowering--.f64 N/A

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

      4. /-lowering-/.f64 98.8

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

   7. Applied egg-rr 98.8%

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

   if -0.55000000000000004 < t < 0.75

   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. accelerator-lowering-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 N/A

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

      5. +-commutative N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. sub-neg N/A

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

      8. metadata-eval N/A

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

      9. +-commutative N/A

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

      10. +-lowering-+.f64 99.3

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

   5. Simplified 99.3%

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

   if 0.75 < 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 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. +-lowering-+.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. /-lowering-/.f64 99.8

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

   5. Simplified 99.8%

      \[\leadsto 1 - \color{blue}{\left(\frac{0.2222222222222222}{t} + 0.16666666666666666\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.55:\\ \;\;\;\;\left(1 - \frac{0.2222222222222222}{t}\right) - 0.16666666666666666\\ \mathbf{elif}\;t \leq 0.75:\\ \;\;\;\;\mathsf{fma}\left(t \cdot t, \mathsf{fma}\left(t, t + -2, 1\right), 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \left(\frac{0.2222222222222222}{t} + 0.16666666666666666\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.3% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -0.58:\\ \;\;\;\;\left(1 - \frac{0.2222222222222222}{t}\right) - 0.16666666666666666\\ \mathbf{elif}\;t \leq 0.58:\\ \;\;\;\;\mathsf{fma}\left(t, t \cdot \mathsf{fma}\left(-2, t, 1\right), 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \left(\frac{0.2222222222222222}{t} + 0.16666666666666666\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (<= t -0.58)
   (- (- 1.0 (/ 0.2222222222222222 t)) 0.16666666666666666)
   (if (<= t 0.58)
     (fma t (* t (fma -2.0 t 1.0)) 0.5)
     (- 1.0 (+ (/ 0.2222222222222222 t) 0.16666666666666666)))))

C

double code(double t) {
	double tmp;
	if (t <= -0.58) {
		tmp = (1.0 - (0.2222222222222222 / t)) - 0.16666666666666666;
	} else if (t <= 0.58) {
		tmp = fma(t, (t * fma(-2.0, t, 1.0)), 0.5);
	} else {
		tmp = 1.0 - ((0.2222222222222222 / t) + 0.16666666666666666);
	}
	return tmp;
}

Julia

function code(t)
	tmp = 0.0
	if (t <= -0.58)
		tmp = Float64(Float64(1.0 - Float64(0.2222222222222222 / t)) - 0.16666666666666666);
	elseif (t <= 0.58)
		tmp = fma(t, Float64(t * fma(-2.0, t, 1.0)), 0.5);
	else
		tmp = Float64(1.0 - Float64(Float64(0.2222222222222222 / t) + 0.16666666666666666));
	end
	return tmp
end

Wolfram

code[t_] := If[LessEqual[t, -0.58], N[(N[(1.0 - N[(0.2222222222222222 / t), $MachinePrecision]), $MachinePrecision] - 0.16666666666666666), $MachinePrecision], If[LessEqual[t, 0.58], N[(t * N[(t * N[(-2.0 * t + 1.0), $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision], N[(1.0 - N[(N[(0.2222222222222222 / t), $MachinePrecision] + 0.16666666666666666), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -0.57999999999999996

   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. +-lowering-+.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. /-lowering-/.f64 98.8

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

   5. Simplified 98.8%

      \[\leadsto 1 - \color{blue}{\left(\frac{0.2222222222222222}{t} + 0.16666666666666666\right)} \]
   6. Step-by-step derivation

      1. associate--r+ N/A

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

      2. --lowering--.f64 N/A

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

      3. --lowering--.f64 N/A

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

      4. /-lowering-/.f64 98.8

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

   7. Applied egg-rr 98.8%

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

   if -0.57999999999999996 < t < 0.57999999999999996

   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. accelerator-lowering-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

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

      7. *-lowering-*.f64 N/A

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

      8. +-commutative N/A

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

      9. accelerator-lowering-fma.f64 99.1

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

   5. Simplified 99.1%

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

   if 0.57999999999999996 < 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 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. +-lowering-+.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. /-lowering-/.f64 99.8

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

   5. Simplified 99.8%

      \[\leadsto 1 - \color{blue}{\left(\frac{0.2222222222222222}{t} + 0.16666666666666666\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 7: 99.2% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -0.78:\\ \;\;\;\;\left(1 - \frac{0.2222222222222222}{t}\right) - 0.16666666666666666\\ \mathbf{elif}\;t \leq 0.58:\\ \;\;\;\;1 + \left(t \cdot t - 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \left(\frac{0.2222222222222222}{t} + 0.16666666666666666\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (<= t -0.78)
   (- (- 1.0 (/ 0.2222222222222222 t)) 0.16666666666666666)
   (if (<= t 0.58)
     (+ 1.0 (- (* t t) 0.5))
     (- 1.0 (+ (/ 0.2222222222222222 t) 0.16666666666666666)))))

C

double code(double t) {
	double tmp;
	if (t <= -0.78) {
		tmp = (1.0 - (0.2222222222222222 / t)) - 0.16666666666666666;
	} else if (t <= 0.58) {
		tmp = 1.0 + ((t * t) - 0.5);
	} else {
		tmp = 1.0 - ((0.2222222222222222 / t) + 0.16666666666666666);
	}
	return tmp;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-0.78d0)) then
        tmp = (1.0d0 - (0.2222222222222222d0 / t)) - 0.16666666666666666d0
    else if (t <= 0.58d0) then
        tmp = 1.0d0 + ((t * t) - 0.5d0)
    else
        tmp = 1.0d0 - ((0.2222222222222222d0 / t) + 0.16666666666666666d0)
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double tmp;
	if (t <= -0.78) {
		tmp = (1.0 - (0.2222222222222222 / t)) - 0.16666666666666666;
	} else if (t <= 0.58) {
		tmp = 1.0 + ((t * t) - 0.5);
	} else {
		tmp = 1.0 - ((0.2222222222222222 / t) + 0.16666666666666666);
	}
	return tmp;
}

Python

def code(t):
	tmp = 0
	if t <= -0.78:
		tmp = (1.0 - (0.2222222222222222 / t)) - 0.16666666666666666
	elif t <= 0.58:
		tmp = 1.0 + ((t * t) - 0.5)
	else:
		tmp = 1.0 - ((0.2222222222222222 / t) + 0.16666666666666666)
	return tmp

Julia

function code(t)
	tmp = 0.0
	if (t <= -0.78)
		tmp = Float64(Float64(1.0 - Float64(0.2222222222222222 / t)) - 0.16666666666666666);
	elseif (t <= 0.58)
		tmp = Float64(1.0 + Float64(Float64(t * t) - 0.5));
	else
		tmp = Float64(1.0 - Float64(Float64(0.2222222222222222 / t) + 0.16666666666666666));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	tmp = 0.0;
	if (t <= -0.78)
		tmp = (1.0 - (0.2222222222222222 / t)) - 0.16666666666666666;
	elseif (t <= 0.58)
		tmp = 1.0 + ((t * t) - 0.5);
	else
		tmp = 1.0 - ((0.2222222222222222 / t) + 0.16666666666666666);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := If[LessEqual[t, -0.78], N[(N[(1.0 - N[(0.2222222222222222 / t), $MachinePrecision]), $MachinePrecision] - 0.16666666666666666), $MachinePrecision], If[LessEqual[t, 0.58], N[(1.0 + N[(N[(t * t), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(N[(0.2222222222222222 / t), $MachinePrecision] + 0.16666666666666666), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -0.78000000000000003

   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. +-lowering-+.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. /-lowering-/.f64 100.0

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

   5. Simplified 100.0%

      \[\leadsto 1 - \color{blue}{\left(\frac{0.2222222222222222}{t} + 0.16666666666666666\right)} \]
   6. Step-by-step derivation

      1. associate--r+ N/A

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

      2. --lowering--.f64 N/A

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

      3. --lowering--.f64 N/A

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

      4. /-lowering-/.f64 100.0

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

   7. Applied egg-rr 100.0%

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

   if -0.78000000000000003 < t < 0.57999999999999996

   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

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. --lowering--.f64 N/A

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 98.2

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

   5. Simplified 98.2%

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

   if 0.57999999999999996 < 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 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. +-lowering-+.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. /-lowering-/.f64 99.8

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

   5. Simplified 99.8%

      \[\leadsto 1 - \color{blue}{\left(\frac{0.2222222222222222}{t} + 0.16666666666666666\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.78:\\ \;\;\;\;\left(1 - \frac{0.2222222222222222}{t}\right) - 0.16666666666666666\\ \mathbf{elif}\;t \leq 0.58:\\ \;\;\;\;1 + \left(t \cdot t - 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \left(\frac{0.2222222222222222}{t} + 0.16666666666666666\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 99.2% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -0.78:\\ \;\;\;\;0.8333333333333334 + \frac{-0.2222222222222222}{t}\\ \mathbf{elif}\;t \leq 0.58:\\ \;\;\;\;1 + \left(t \cdot t - 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \left(\frac{0.2222222222222222}{t} + 0.16666666666666666\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (<= t -0.78)
   (+ 0.8333333333333334 (/ -0.2222222222222222 t))
   (if (<= t 0.58)
     (+ 1.0 (- (* t t) 0.5))
     (- 1.0 (+ (/ 0.2222222222222222 t) 0.16666666666666666)))))

C

double code(double t) {
	double tmp;
	if (t <= -0.78) {
		tmp = 0.8333333333333334 + (-0.2222222222222222 / t);
	} else if (t <= 0.58) {
		tmp = 1.0 + ((t * t) - 0.5);
	} else {
		tmp = 1.0 - ((0.2222222222222222 / t) + 0.16666666666666666);
	}
	return tmp;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-0.78d0)) then
        tmp = 0.8333333333333334d0 + ((-0.2222222222222222d0) / t)
    else if (t <= 0.58d0) then
        tmp = 1.0d0 + ((t * t) - 0.5d0)
    else
        tmp = 1.0d0 - ((0.2222222222222222d0 / t) + 0.16666666666666666d0)
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double tmp;
	if (t <= -0.78) {
		tmp = 0.8333333333333334 + (-0.2222222222222222 / t);
	} else if (t <= 0.58) {
		tmp = 1.0 + ((t * t) - 0.5);
	} else {
		tmp = 1.0 - ((0.2222222222222222 / t) + 0.16666666666666666);
	}
	return tmp;
}

Python

def code(t):
	tmp = 0
	if t <= -0.78:
		tmp = 0.8333333333333334 + (-0.2222222222222222 / t)
	elif t <= 0.58:
		tmp = 1.0 + ((t * t) - 0.5)
	else:
		tmp = 1.0 - ((0.2222222222222222 / t) + 0.16666666666666666)
	return tmp

Julia

function code(t)
	tmp = 0.0
	if (t <= -0.78)
		tmp = Float64(0.8333333333333334 + Float64(-0.2222222222222222 / t));
	elseif (t <= 0.58)
		tmp = Float64(1.0 + Float64(Float64(t * t) - 0.5));
	else
		tmp = Float64(1.0 - Float64(Float64(0.2222222222222222 / t) + 0.16666666666666666));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	tmp = 0.0;
	if (t <= -0.78)
		tmp = 0.8333333333333334 + (-0.2222222222222222 / t);
	elseif (t <= 0.58)
		tmp = 1.0 + ((t * t) - 0.5);
	else
		tmp = 1.0 - ((0.2222222222222222 / t) + 0.16666666666666666);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := If[LessEqual[t, -0.78], N[(0.8333333333333334 + N[(-0.2222222222222222 / t), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 0.58], N[(1.0 + N[(N[(t * t), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(N[(0.2222222222222222 / t), $MachinePrecision] + 0.16666666666666666), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -0.78000000000000003

   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. +-lowering-+.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. /-lowering-/.f64 N/A

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

      7. metadata-eval 100.0

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

   5. Simplified 100.0%

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

   if -0.78000000000000003 < t < 0.57999999999999996

   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

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. --lowering--.f64 N/A

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 98.2

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

   5. Simplified 98.2%

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

   if 0.57999999999999996 < 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 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. +-lowering-+.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. /-lowering-/.f64 99.8

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

   5. Simplified 99.8%

      \[\leadsto 1 - \color{blue}{\left(\frac{0.2222222222222222}{t} + 0.16666666666666666\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.78:\\ \;\;\;\;0.8333333333333334 + \frac{-0.2222222222222222}{t}\\ \mathbf{elif}\;t \leq 0.58:\\ \;\;\;\;1 + \left(t \cdot t - 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \left(\frac{0.2222222222222222}{t} + 0.16666666666666666\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 99.2% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 0.8333333333333334 + \frac{-0.2222222222222222}{t}\\ \mathbf{if}\;t \leq -0.78:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 0.58:\\ \;\;\;\;1 + \left(t \cdot t - 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (let* ((t_1 (+ 0.8333333333333334 (/ -0.2222222222222222 t))))
   (if (<= t -0.78) t_1 (if (<= t 0.58) (+ 1.0 (- (* t t) 0.5)) t_1))))

C

double code(double t) {
	double t_1 = 0.8333333333333334 + (-0.2222222222222222 / t);
	double tmp;
	if (t <= -0.78) {
		tmp = t_1;
	} else if (t <= 0.58) {
		tmp = 1.0 + ((t * t) - 0.5);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = 0.8333333333333334d0 + ((-0.2222222222222222d0) / t)
    if (t <= (-0.78d0)) then
        tmp = t_1
    else if (t <= 0.58d0) then
        tmp = 1.0d0 + ((t * t) - 0.5d0)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double t_1 = 0.8333333333333334 + (-0.2222222222222222 / t);
	double tmp;
	if (t <= -0.78) {
		tmp = t_1;
	} else if (t <= 0.58) {
		tmp = 1.0 + ((t * t) - 0.5);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(t):
	t_1 = 0.8333333333333334 + (-0.2222222222222222 / t)
	tmp = 0
	if t <= -0.78:
		tmp = t_1
	elif t <= 0.58:
		tmp = 1.0 + ((t * t) - 0.5)
	else:
		tmp = t_1
	return tmp

Julia

function code(t)
	t_1 = Float64(0.8333333333333334 + Float64(-0.2222222222222222 / t))
	tmp = 0.0
	if (t <= -0.78)
		tmp = t_1;
	elseif (t <= 0.58)
		tmp = Float64(1.0 + Float64(Float64(t * t) - 0.5));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	t_1 = 0.8333333333333334 + (-0.2222222222222222 / t);
	tmp = 0.0;
	if (t <= -0.78)
		tmp = t_1;
	elseif (t <= 0.58)
		tmp = 1.0 + ((t * t) - 0.5);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := Block[{t$95$1 = N[(0.8333333333333334 + N[(-0.2222222222222222 / t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -0.78], t$95$1, If[LessEqual[t, 0.58], N[(1.0 + N[(N[(t * t), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -0.78000000000000003 or 0.57999999999999996 < 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 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. +-lowering-+.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. /-lowering-/.f64 N/A

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

      7. metadata-eval 99.9

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

   5. Simplified 99.9%

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

   if -0.78000000000000003 < t < 0.57999999999999996

   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

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. --lowering--.f64 N/A

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 98.2

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

   5. Simplified 98.2%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.78:\\ \;\;\;\;0.8333333333333334 + \frac{-0.2222222222222222}{t}\\ \mathbf{elif}\;t \leq 0.58:\\ \;\;\;\;1 + \left(t \cdot t - 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334 + \frac{-0.2222222222222222}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 98.7% accurate, 4.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -0.92:\\ \;\;\;\;0.8333333333333334\\ \mathbf{elif}\;t \leq 0.58:\\ \;\;\;\;1 + \left(t \cdot t - 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (<= t -0.92)
   0.8333333333333334
   (if (<= t 0.58) (+ 1.0 (- (* t t) 0.5)) 0.8333333333333334)))

C

double code(double t) {
	double tmp;
	if (t <= -0.92) {
		tmp = 0.8333333333333334;
	} else if (t <= 0.58) {
		tmp = 1.0 + ((t * t) - 0.5);
	} else {
		tmp = 0.8333333333333334;
	}
	return tmp;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-0.92d0)) then
        tmp = 0.8333333333333334d0
    else if (t <= 0.58d0) then
        tmp = 1.0d0 + ((t * t) - 0.5d0)
    else
        tmp = 0.8333333333333334d0
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double tmp;
	if (t <= -0.92) {
		tmp = 0.8333333333333334;
	} else if (t <= 0.58) {
		tmp = 1.0 + ((t * t) - 0.5);
	} else {
		tmp = 0.8333333333333334;
	}
	return tmp;
}

Python

def code(t):
	tmp = 0
	if t <= -0.92:
		tmp = 0.8333333333333334
	elif t <= 0.58:
		tmp = 1.0 + ((t * t) - 0.5)
	else:
		tmp = 0.8333333333333334
	return tmp

Julia

function code(t)
	tmp = 0.0
	if (t <= -0.92)
		tmp = 0.8333333333333334;
	elseif (t <= 0.58)
		tmp = Float64(1.0 + Float64(Float64(t * t) - 0.5));
	else
		tmp = 0.8333333333333334;
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	tmp = 0.0;
	if (t <= -0.92)
		tmp = 0.8333333333333334;
	elseif (t <= 0.58)
		tmp = 1.0 + ((t * t) - 0.5);
	else
		tmp = 0.8333333333333334;
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := If[LessEqual[t, -0.92], 0.8333333333333334, If[LessEqual[t, 0.58], N[(1.0 + N[(N[(t * t), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision], 0.8333333333333334]]

Derivation

1. Split input into 2 regimes

2. if t < -0.92000000000000004 or 0.57999999999999996 < 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 inf

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

   5. Simplified 99.2%

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

   if -0.92000000000000004 < t < 0.57999999999999996

   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

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. --lowering--.f64 N/A

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 98.2

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

   5. Simplified 98.2%

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

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.92:\\ \;\;\;\;0.8333333333333334\\ \mathbf{elif}\;t \leq 0.58:\\ \;\;\;\;1 + \left(t \cdot t - 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 98.7% accurate, 5.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -0.92:\\ \;\;\;\;0.8333333333333334\\ \mathbf{elif}\;t \leq 0.58:\\ \;\;\;\;\mathsf{fma}\left(t, t, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (<= t -0.92)
   0.8333333333333334
   (if (<= t 0.58) (fma t t 0.5) 0.8333333333333334)))

C

double code(double t) {
	double tmp;
	if (t <= -0.92) {
		tmp = 0.8333333333333334;
	} else if (t <= 0.58) {
		tmp = fma(t, t, 0.5);
	} else {
		tmp = 0.8333333333333334;
	}
	return tmp;
}

Julia

function code(t)
	tmp = 0.0
	if (t <= -0.92)
		tmp = 0.8333333333333334;
	elseif (t <= 0.58)
		tmp = fma(t, t, 0.5);
	else
		tmp = 0.8333333333333334;
	end
	return tmp
end

Wolfram

code[t_] := If[LessEqual[t, -0.92], 0.8333333333333334, If[LessEqual[t, 0.58], N[(t * t + 0.5), $MachinePrecision], 0.8333333333333334]]

Derivation

1. Split input into 2 regimes

2. if t < -0.92000000000000004 or 0.57999999999999996 < 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 inf

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

   5. Simplified 99.2%

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

   if -0.92000000000000004 < t < 0.57999999999999996

   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. accelerator-lowering-fma.f64 98.2

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

   5. Simplified 98.2%

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

Alternative 12: 98.6% accurate, 7.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -0.33:\\ \;\;\;\;0.8333333333333334\\ \mathbf{elif}\;t \leq 1:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (<= t -0.33) 0.8333333333333334 (if (<= t 1.0) 0.5 0.8333333333333334)))

C

double code(double t) {
	double tmp;
	if (t <= -0.33) {
		tmp = 0.8333333333333334;
	} else if (t <= 1.0) {
		tmp = 0.5;
	} else {
		tmp = 0.8333333333333334;
	}
	return tmp;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-0.33d0)) then
        tmp = 0.8333333333333334d0
    else if (t <= 1.0d0) then
        tmp = 0.5d0
    else
        tmp = 0.8333333333333334d0
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double tmp;
	if (t <= -0.33) {
		tmp = 0.8333333333333334;
	} else if (t <= 1.0) {
		tmp = 0.5;
	} else {
		tmp = 0.8333333333333334;
	}
	return tmp;
}

Python

def code(t):
	tmp = 0
	if t <= -0.33:
		tmp = 0.8333333333333334
	elif t <= 1.0:
		tmp = 0.5
	else:
		tmp = 0.8333333333333334
	return tmp

Julia

function code(t)
	tmp = 0.0
	if (t <= -0.33)
		tmp = 0.8333333333333334;
	elseif (t <= 1.0)
		tmp = 0.5;
	else
		tmp = 0.8333333333333334;
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	tmp = 0.0;
	if (t <= -0.33)
		tmp = 0.8333333333333334;
	elseif (t <= 1.0)
		tmp = 0.5;
	else
		tmp = 0.8333333333333334;
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := If[LessEqual[t, -0.33], 0.8333333333333334, If[LessEqual[t, 1.0], 0.5, 0.8333333333333334]]

Derivation

1. Split input into 2 regimes

2. if t < -0.330000000000000016 or 1 < 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 inf

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

   5. Simplified 98.6%

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

   if -0.330000000000000016 < 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 0

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

   5. Simplified 98.6%

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

Alternative 13: 59.4% 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. Simplified 58.8%

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

Reproduce

herbie shell --seed 2024197 
(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)))))))))