Kahan p13 Example 1

Percentage Accurate: 100.0% → 99.8%

Time: 11.1s

Alternatives: 11

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t \cdot 4}{\mathsf{fma}\left(t, 2 + t, 1\right)}\\ \mathbf{if}\;\frac{2 \cdot t}{1 + t} \leq 1.99:\\ \;\;\;\;\frac{\mathsf{fma}\left(t, t\_1, 1\right)}{\mathsf{fma}\left(t, t\_1, 2\right)}\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334 + \frac{\frac{0.037037037037037035 + \frac{0.04938271604938271}{t}}{t} + -0.2222222222222222}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (let* ((t_1 (/ (* t 4.0) (fma t (+ 2.0 t) 1.0))))
   (if (<= (/ (* 2.0 t) (+ 1.0 t)) 1.99)
     (/ (fma t t_1 1.0) (fma t t_1 2.0))
     (+
      0.8333333333333334
      (/
       (+
        (/ (+ 0.037037037037037035 (/ 0.04938271604938271 t)) t)
        -0.2222222222222222)
       t)))))

C

double code(double t) {
	double t_1 = (t * 4.0) / fma(t, (2.0 + t), 1.0);
	double tmp;
	if (((2.0 * t) / (1.0 + t)) <= 1.99) {
		tmp = fma(t, t_1, 1.0) / fma(t, t_1, 2.0);
	} else {
		tmp = 0.8333333333333334 + ((((0.037037037037037035 + (0.04938271604938271 / t)) / t) + -0.2222222222222222) / t);
	}
	return tmp;
}

Julia

function code(t)
	t_1 = Float64(Float64(t * 4.0) / fma(t, Float64(2.0 + t), 1.0))
	tmp = 0.0
	if (Float64(Float64(2.0 * t) / Float64(1.0 + t)) <= 1.99)
		tmp = Float64(fma(t, t_1, 1.0) / fma(t, t_1, 2.0));
	else
		tmp = Float64(0.8333333333333334 + Float64(Float64(Float64(Float64(0.037037037037037035 + Float64(0.04938271604938271 / t)) / t) + -0.2222222222222222) / t));
	end
	return tmp
end

Wolfram

code[t_] := Block[{t$95$1 = N[(N[(t * 4.0), $MachinePrecision] / N[(t * N[(2.0 + t), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(2.0 * t), $MachinePrecision] / N[(1.0 + t), $MachinePrecision]), $MachinePrecision], 1.99], N[(N[(t * t$95$1 + 1.0), $MachinePrecision] / N[(t * t$95$1 + 2.0), $MachinePrecision]), $MachinePrecision], N[(0.8333333333333334 + N[(N[(N[(N[(0.037037037037037035 + N[(0.04938271604938271 / t), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision] + -0.2222222222222222), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) t)) < 1.98999999999999999

   1. Initial program 100.0%

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

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

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

   4. Applied egg-rr 100.0%

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

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

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

      10. +-commutative N/A

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

      11. associate-/l* N/A

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

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

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

      3. rgt-mult-inverse N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. *-rgt-identity N/A

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

      7. distribute-lft-in N/A

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

      8. +-commutative N/A

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

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

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

      10. distribute-lft-in N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. rgt-mult-inverse N/A

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

      14. metadata-eval N/A

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

      15. *-rgt-identity N/A

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

      16. +-commutative N/A

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

      17. +-lowering-+.f64 100.0

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

   9. Simplified 100.0%

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

   10. Taylor expanded in t around 0

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

       1. +-commutative N/A

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

       2. +-commutative N/A

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

       3. distribute-lft-in N/A

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

       4. metadata-eval N/A

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

       5. lft-mult-inverse N/A

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

       6. associate-*l* N/A

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

       7. distribute-lft-in N/A

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

       8. *-lft-identity N/A

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

       9. distribute-rgt-in N/A

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

       10. *-commutative N/A

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

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

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

       12. *-commutative N/A

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

       13. +-commutative N/A

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

       14. distribute-rgt-in N/A

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

       15. associate-*l* N/A

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

       16. lft-mult-inverse N/A

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

       17. metadata-eval N/A

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

       18. *-lft-identity N/A

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

       19. +-lowering-+.f64 100.0

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

   12. Simplified 100.0%

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

   if 1.98999999999999999 < (/.f64 (*.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) t))

   1. Initial program 100.0%

      \[\frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{2 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

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

   5. Simplified 100.0%

      \[\leadsto \color{blue}{0.8333333333333334 + \frac{\frac{0.037037037037037035 + \frac{0.04938271604938271}{t}}{t} + -0.2222222222222222}{t}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

   \[\frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{2 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}} \]
2. Add Preprocessing
3. Add Preprocessing

Alternative 3: 99.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

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

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

   4. Applied egg-rr 100.0%

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

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

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

      10. +-commutative N/A

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

      11. associate-/l* N/A

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

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

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

      3. rgt-mult-inverse N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. *-rgt-identity N/A

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

      7. distribute-lft-in N/A

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

      8. +-commutative N/A

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

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

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

      10. distribute-lft-in N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. rgt-mult-inverse N/A

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

      14. metadata-eval N/A

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

      15. *-rgt-identity N/A

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

      16. +-commutative N/A

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

      17. +-lowering-+.f64 100.0

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

   9. Simplified 100.0%

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

   10. Taylor expanded in t around 0

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

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

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

       2. +-commutative N/A

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

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

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

       4. sub-neg N/A

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

       5. metadata-eval N/A

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

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

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

       7. +-commutative N/A

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

       8. *-commutative N/A

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

       9. accelerator-lowering-fma.f64 98.7

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

   12. Simplified 98.7%

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

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

   1. Initial program 100.0%

      \[\frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{2 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

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

   5. Simplified 99.5%

      \[\leadsto \color{blue}{0.8333333333333334 + \frac{\frac{0.037037037037037035 + \frac{0.04938271604938271}{t}}{t} + -0.2222222222222222}{t}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 99.5% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

      \[\frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{2 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

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

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

      6. *-commutative N/A

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

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

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

      8. +-commutative N/A

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

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

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

      10. sub-neg N/A

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

      11. metadata-eval N/A

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

      12. +-lowering-+.f64 98.6

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

   5. Simplified 98.6%

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

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

   1. Initial program 100.0%

      \[\frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{2 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

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

   5. Simplified 99.5%

      \[\leadsto \color{blue}{0.8333333333333334 + \frac{\frac{0.037037037037037035 + \frac{0.04938271604938271}{t}}{t} + -0.2222222222222222}{t}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 99.4% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

      \[\frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{2 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

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

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

      6. *-commutative N/A

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

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

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

      8. +-commutative N/A

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

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

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

      10. sub-neg N/A

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

      11. metadata-eval N/A

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

      12. +-lowering-+.f64 98.6

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

   5. Simplified 98.6%

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

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

   1. Initial program 100.0%

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

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

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

   4. Applied egg-rr 51.9%

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

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

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

      10. +-commutative N/A

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

      11. associate-/l* N/A

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

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

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

   6. Applied egg-rr 61.3%

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

   7. Taylor expanded in t around -inf

      \[\leadsto \color{blue}{\frac{5}{6} + -1 \cdot \frac{\frac{2}{9} + -1 \cdot \frac{\frac{1}{27} + \frac{4}{81} \cdot \frac{1}{t}}{t}}{t}} \]
   8. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

         \[\leadsto \color{blue}{\frac{5}{6} - \frac{\frac{2}{9} + -1 \cdot \frac{\frac{1}{27} + \frac{4}{81} \cdot \frac{1}{t}}{t}}{t}} \]

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

         \[\leadsto \color{blue}{\frac{5}{6} - \frac{\frac{2}{9} + -1 \cdot \frac{\frac{1}{27} + \frac{4}{81} \cdot \frac{1}{t}}{t}}{t}} \]

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

         \[\leadsto \frac{5}{6} - \color{blue}{\frac{\frac{2}{9} + -1 \cdot \frac{\frac{1}{27} + \frac{4}{81} \cdot \frac{1}{t}}{t}}{t}} \]

   9. Simplified 99.5%

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

   10. Taylor expanded in t around inf

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

       1. div-sub N/A

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

       2. metadata-eval N/A

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

       3. rgt-mult-inverse N/A

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

       4. associate-*r* N/A

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

       5. *-commutative N/A

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

       6. div-sub N/A

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

       7. associate-*r/ N/A

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

       8. metadata-eval N/A

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

       9. associate-*r/ N/A

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

       10. *-commutative N/A

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

       11. associate-*r/ N/A

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

       12. metadata-eval N/A

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

       13. div-sub N/A

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

       14. associate-/r* N/A

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

       15. unpow2 N/A

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

       16. /-lowering-/.f64 N/A

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

       17. sub-neg N/A

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

       18. *-commutative N/A

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

       19. metadata-eval N/A

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

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

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

       21. unpow2 N/A

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

       22. *-lowering-*.f64 99.3

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

   12. Simplified 99.3%

       \[\leadsto 0.8333333333333334 - \color{blue}{\frac{\mathsf{fma}\left(t, 0.2222222222222222, -0.037037037037037035\right)}{t \cdot t}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 99.3% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

      \[\frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{2 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

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

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

      6. *-commutative N/A

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

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

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

      8. +-commutative N/A

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

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

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

      10. sub-neg N/A

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

      11. metadata-eval N/A

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

      12. +-lowering-+.f64 98.6

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

   5. Simplified 98.6%

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

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

   1. Initial program 100.0%

      \[\frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{2 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}} \]
   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 98.8

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

   5. Simplified 98.8%

      \[\leadsto \color{blue}{0.8333333333333334 + \frac{-0.2222222222222222}{t}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 99.3% accurate, 2.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

      \[\frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{2 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}} \]
   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. accelerator-lowering-fma.f64 N/A

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

      3. unpow2 N/A

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

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

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. accelerator-lowering-fma.f64 98.4

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

   5. Simplified 98.4%

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

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

   1. Initial program 100.0%

      \[\frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{2 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}} \]
   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 98.8

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

   5. Simplified 98.8%

      \[\leadsto \color{blue}{0.8333333333333334 + \frac{-0.2222222222222222}{t}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 99.2% accurate, 2.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

      \[\frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{2 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}} \]
   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 97.9

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

   5. Simplified 97.9%

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

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

   1. Initial program 100.0%

      \[\frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{2 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}} \]
   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 98.8

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

   5. Simplified 98.8%

      \[\leadsto \color{blue}{0.8333333333333334 + \frac{-0.2222222222222222}{t}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 98.6% accurate, 3.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

      \[\frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{2 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}} \]
   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 97.9

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

   5. Simplified 97.9%

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

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

   1. Initial program 100.0%

      \[\frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{2 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

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

   5. Simplified 98.0%

      \[\leadsto \color{blue}{0.8333333333333334} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 98.4% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Python

def code(t):
	tmp = 0
	if ((2.0 * t) / (1.0 + t)) <= 1.0:
		tmp = 0.5
	else:
		tmp = 0.8333333333333334
	return tmp

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

      \[\frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{2 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

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

   5. Simplified 97.5%

      \[\leadsto \color{blue}{0.5} \]

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

   1. Initial program 100.0%

      \[\frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{2 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

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

   5. Simplified 98.0%

      \[\leadsto \color{blue}{0.8333333333333334} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 59.0% accurate, 104.0× speedup

Math

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

FPCore

(FPCore (t) :precision binary64 0.5)

C

double code(double t) {
	return 0.5;
}

Fortran

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

Java

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

Python

def code(t):
	return 0.5

Julia

function code(t)
	return 0.5
end

MATLAB

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

Wolfram

code[t_] := 0.5

Derivation

1. Initial program 100.0%

   \[\frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{2 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}} \]
2. Add Preprocessing

3. Taylor expanded in t around 0

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

5. Simplified 57.1%

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

Reproduce

herbie shell --seed 2024195 
(FPCore (t)
  :name "Kahan p13 Example 1"
  :precision binary64
  (/ (+ 1.0 (* (/ (* 2.0 t) (+ 1.0 t)) (/ (* 2.0 t) (+ 1.0 t)))) (+ 2.0 (* (/ (* 2.0 t) (+ 1.0 t)) (/ (* 2.0 t) (+ 1.0 t))))))