Kahan p13 Example 1

Percentage Accurate: 100.0% → 100.0%

Time: 7.7s

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: 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 2: 99.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{2 \cdot t}{1 + t} \leq 10^{-5}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{t \cdot t}{1 + t}, \frac{4}{1 + t}, 1\right)}{2 + \frac{t \cdot t}{\mathsf{fma}\left(\mathsf{fma}\left(0.25, t, 0.5\right), t, 0.25\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{0.04938271604938271}{t} + 0.037037037037037035}{t} - 0.2222222222222222}{t} - -0.8333333333333334\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (<= (/ (* 2.0 t) (+ 1.0 t)) 1e-5)
   (/
    (fma (/ (* t t) (+ 1.0 t)) (/ 4.0 (+ 1.0 t)) 1.0)
    (+ 2.0 (/ (* t t) (fma (fma 0.25 t 0.5) t 0.25))))
   (-
    (/
     (-
      (/ (+ (/ 0.04938271604938271 t) 0.037037037037037035) t)
      0.2222222222222222)
     t)
    -0.8333333333333334)))

C

double code(double t) {
	double tmp;
	if (((2.0 * t) / (1.0 + t)) <= 1e-5) {
		tmp = fma(((t * t) / (1.0 + t)), (4.0 / (1.0 + t)), 1.0) / (2.0 + ((t * t) / fma(fma(0.25, t, 0.5), t, 0.25)));
	} else {
		tmp = (((((0.04938271604938271 / t) + 0.037037037037037035) / t) - 0.2222222222222222) / t) - -0.8333333333333334;
	}
	return tmp;
}

Julia

function code(t)
	tmp = 0.0
	if (Float64(Float64(2.0 * t) / Float64(1.0 + t)) <= 1e-5)
		tmp = Float64(fma(Float64(Float64(t * t) / Float64(1.0 + t)), Float64(4.0 / Float64(1.0 + t)), 1.0) / Float64(2.0 + Float64(Float64(t * t) / fma(fma(0.25, t, 0.5), t, 0.25))));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(0.04938271604938271 / t) + 0.037037037037037035) / t) - 0.2222222222222222) / t) - -0.8333333333333334);
	end
	return tmp
end

Wolfram

code[t_] := If[LessEqual[N[(N[(2.0 * t), $MachinePrecision] / N[(1.0 + t), $MachinePrecision]), $MachinePrecision], 1e-5], N[(N[(N[(N[(t * t), $MachinePrecision] / N[(1.0 + t), $MachinePrecision]), $MachinePrecision] * N[(4.0 / N[(1.0 + t), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / N[(2.0 + N[(N[(t * t), $MachinePrecision] / N[(N[(0.25 * t + 0.5), $MachinePrecision] * t + 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(N[(0.04938271604938271 / t), $MachinePrecision] + 0.037037037037037035), $MachinePrecision] / t), $MachinePrecision] - 0.2222222222222222), $MachinePrecision] / t), $MachinePrecision] - -0.8333333333333334), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) t)) < 1.00000000000000008e-5

   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. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. frac-times N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. swap-sqr N/A

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

      12. times-frac N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. metadata-eval 100.0

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

   4. Applied rewrites 100.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. frac-times N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. swap-sqr N/A

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

      8. metadata-eval N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. frac-times N/A

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

      12. clear-num N/A

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

      13. frac-times N/A

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

      14. lift-*.f64 N/A

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

      15. pow2 N/A

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

      16. metadata-eval N/A

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

      17. unpow-prod-down N/A

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

      18. *-commutative N/A

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

      19. *-lft-identity N/A

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

      20. pow2 N/A

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

      21. lift-*.f64 N/A

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

      22. lower-/.f64 N/A

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

   6. Applied rewrites 100.0%

      \[\leadsto \frac{\mathsf{fma}\left(\frac{t \cdot t}{1 + t}, \frac{4}{1 + t}, 1\right)}{2 + \color{blue}{\frac{t \cdot t}{\left(1 + t\right) \cdot \left(\left(1 + t\right) \cdot 0.25\right)}}} \]

   7. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. metadata-eval N/A

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

      4. lft-mult-inverse N/A

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

      5. associate-*l* N/A

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

      6. distribute-rgt-in N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. *-commutative N/A

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

      11. distribute-rgt-in N/A

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

      12. associate-*l* N/A

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

      13. lft-mult-inverse N/A

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

      14. metadata-eval N/A

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

      15. lower-fma.f64 100.0

          \[\leadsto \frac{\mathsf{fma}\left(\frac{t \cdot t}{1 + t}, \frac{4}{1 + t}, 1\right)}{2 + \frac{t \cdot t}{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(0.25, t, 0.5\right)}, t, 0.25\right)}} \]

   9. Applied rewrites 100.0%

      \[\leadsto \frac{\mathsf{fma}\left(\frac{t \cdot t}{1 + t}, \frac{4}{1 + t}, 1\right)}{2 + \frac{t \cdot t}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.25, t, 0.5\right), t, 0.25\right)}}} \]

   if 1.00000000000000008e-5 < (/.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. Applied rewrites 99.9%

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

Alternative 3: 99.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{2 \cdot t}{1 + t} \leq 10^{-5}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(t - 1, t, 1\right) \cdot t\right) \cdot t, \frac{4}{1 + t}, 1\right)}{2 + \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-16, t, 12\right), t, -8\right), t, 4\right) \cdot t\right) \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{0.04938271604938271}{t} + 0.037037037037037035}{t} - 0.2222222222222222}{t} - -0.8333333333333334\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (<= (/ (* 2.0 t) (+ 1.0 t)) 1e-5)
   (/
    (fma (* (* (fma (- t 1.0) t 1.0) t) t) (/ 4.0 (+ 1.0 t)) 1.0)
    (+ 2.0 (* (* (fma (fma (fma -16.0 t 12.0) t -8.0) t 4.0) t) t)))
   (-
    (/
     (-
      (/ (+ (/ 0.04938271604938271 t) 0.037037037037037035) t)
      0.2222222222222222)
     t)
    -0.8333333333333334)))

C

double code(double t) {
	double tmp;
	if (((2.0 * t) / (1.0 + t)) <= 1e-5) {
		tmp = fma(((fma((t - 1.0), t, 1.0) * t) * t), (4.0 / (1.0 + t)), 1.0) / (2.0 + ((fma(fma(fma(-16.0, t, 12.0), t, -8.0), t, 4.0) * t) * t));
	} else {
		tmp = (((((0.04938271604938271 / t) + 0.037037037037037035) / t) - 0.2222222222222222) / t) - -0.8333333333333334;
	}
	return tmp;
}

Julia

function code(t)
	tmp = 0.0
	if (Float64(Float64(2.0 * t) / Float64(1.0 + t)) <= 1e-5)
		tmp = Float64(fma(Float64(Float64(fma(Float64(t - 1.0), t, 1.0) * t) * t), Float64(4.0 / Float64(1.0 + t)), 1.0) / Float64(2.0 + Float64(Float64(fma(fma(fma(-16.0, t, 12.0), t, -8.0), t, 4.0) * t) * t)));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(0.04938271604938271 / t) + 0.037037037037037035) / t) - 0.2222222222222222) / t) - -0.8333333333333334);
	end
	return tmp
end

Wolfram

code[t_] := If[LessEqual[N[(N[(2.0 * t), $MachinePrecision] / N[(1.0 + t), $MachinePrecision]), $MachinePrecision], 1e-5], N[(N[(N[(N[(N[(N[(t - 1.0), $MachinePrecision] * t + 1.0), $MachinePrecision] * t), $MachinePrecision] * t), $MachinePrecision] * N[(4.0 / N[(1.0 + t), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / N[(2.0 + N[(N[(N[(N[(N[(-16.0 * t + 12.0), $MachinePrecision] * t + -8.0), $MachinePrecision] * t + 4.0), $MachinePrecision] * t), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(N[(0.04938271604938271 / t), $MachinePrecision] + 0.037037037037037035), $MachinePrecision] / t), $MachinePrecision] - 0.2222222222222222), $MachinePrecision] / t), $MachinePrecision] - -0.8333333333333334), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) t)) < 1.00000000000000008e-5

   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. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. frac-times N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. swap-sqr N/A

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

      12. times-frac N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. metadata-eval 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. sub-neg N/A

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

      10. *-commutative N/A

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

      11. metadata-eval N/A

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

      12. lower-fma.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-fma.f64 99.4

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

   7. Applied rewrites 99.4%

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

   8. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower--.f64 99.5

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

   10. Applied rewrites 99.5%

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

   if 1.00000000000000008e-5 < (/.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. Applied rewrites 99.9%

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

Alternative 4: 99.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{2 \cdot t}{1 + t} \leq 10^{-5}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(t - 1, t, 1\right) \cdot t\right) \cdot t, \frac{4}{1 + t}, 1\right)}{2 + \left(\mathsf{fma}\left(\mathsf{fma}\left(12, t, -8\right), t, 4\right) \cdot t\right) \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{0.04938271604938271}{t} + 0.037037037037037035}{t} - 0.2222222222222222}{t} - -0.8333333333333334\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (<= (/ (* 2.0 t) (+ 1.0 t)) 1e-5)
   (/
    (fma (* (* (fma (- t 1.0) t 1.0) t) t) (/ 4.0 (+ 1.0 t)) 1.0)
    (+ 2.0 (* (* (fma (fma 12.0 t -8.0) t 4.0) t) t)))
   (-
    (/
     (-
      (/ (+ (/ 0.04938271604938271 t) 0.037037037037037035) t)
      0.2222222222222222)
     t)
    -0.8333333333333334)))

C

double code(double t) {
	double tmp;
	if (((2.0 * t) / (1.0 + t)) <= 1e-5) {
		tmp = fma(((fma((t - 1.0), t, 1.0) * t) * t), (4.0 / (1.0 + t)), 1.0) / (2.0 + ((fma(fma(12.0, t, -8.0), t, 4.0) * t) * t));
	} else {
		tmp = (((((0.04938271604938271 / t) + 0.037037037037037035) / t) - 0.2222222222222222) / t) - -0.8333333333333334;
	}
	return tmp;
}

Julia

function code(t)
	tmp = 0.0
	if (Float64(Float64(2.0 * t) / Float64(1.0 + t)) <= 1e-5)
		tmp = Float64(fma(Float64(Float64(fma(Float64(t - 1.0), t, 1.0) * t) * t), Float64(4.0 / Float64(1.0 + t)), 1.0) / Float64(2.0 + Float64(Float64(fma(fma(12.0, t, -8.0), t, 4.0) * t) * t)));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(0.04938271604938271 / t) + 0.037037037037037035) / t) - 0.2222222222222222) / t) - -0.8333333333333334);
	end
	return tmp
end

Wolfram

code[t_] := If[LessEqual[N[(N[(2.0 * t), $MachinePrecision] / N[(1.0 + t), $MachinePrecision]), $MachinePrecision], 1e-5], N[(N[(N[(N[(N[(N[(t - 1.0), $MachinePrecision] * t + 1.0), $MachinePrecision] * t), $MachinePrecision] * t), $MachinePrecision] * N[(4.0 / N[(1.0 + t), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / N[(2.0 + N[(N[(N[(N[(12.0 * t + -8.0), $MachinePrecision] * t + 4.0), $MachinePrecision] * t), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(N[(0.04938271604938271 / t), $MachinePrecision] + 0.037037037037037035), $MachinePrecision] / t), $MachinePrecision] - 0.2222222222222222), $MachinePrecision] / t), $MachinePrecision] - -0.8333333333333334), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) t)) < 1.00000000000000008e-5

   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. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. frac-times N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. swap-sqr N/A

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

      12. times-frac N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. metadata-eval 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. sub-neg N/A

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

      10. *-commutative N/A

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

      11. metadata-eval N/A

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

      12. lower-fma.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-fma.f64 99.4

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

   7. Applied rewrites 99.4%

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

   8. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower--.f64 99.5

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

   10. Applied rewrites 99.5%

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

   11. Taylor expanded in t around 0

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

       1. unpow2 N/A

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

       2. associate-*l* N/A

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

       3. *-commutative N/A

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

       4. lower-*.f64 N/A

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

       5. *-commutative N/A

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

       6. lower-*.f64 N/A

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

       7. +-commutative N/A

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

       8. *-commutative N/A

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

       9. lower-fma.f64 N/A

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

       10. sub-neg N/A

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

       11. metadata-eval N/A

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

       12. lower-fma.f64 99.4

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

   13. Applied rewrites 99.4%

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

   if 1.00000000000000008e-5 < (/.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. Applied rewrites 99.9%

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

Alternative 5: 99.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{2 \cdot t}{1 + t} \leq 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-2, t, 1\right), t \cdot t, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{0.04938271604938271}{t} + 0.037037037037037035}{t} - 0.2222222222222222}{t} - -0.8333333333333334\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (<= (/ (* 2.0 t) (+ 1.0 t)) 1e-5)
   (fma (fma -2.0 t 1.0) (* t t) 0.5)
   (-
    (/
     (-
      (/ (+ (/ 0.04938271604938271 t) 0.037037037037037035) t)
      0.2222222222222222)
     t)
    -0.8333333333333334)))

C

double code(double t) {
	double tmp;
	if (((2.0 * t) / (1.0 + t)) <= 1e-5) {
		tmp = fma(fma(-2.0, t, 1.0), (t * t), 0.5);
	} else {
		tmp = (((((0.04938271604938271 / t) + 0.037037037037037035) / t) - 0.2222222222222222) / t) - -0.8333333333333334;
	}
	return tmp;
}

Julia

function code(t)
	tmp = 0.0
	if (Float64(Float64(2.0 * t) / Float64(1.0 + t)) <= 1e-5)
		tmp = fma(fma(-2.0, t, 1.0), Float64(t * t), 0.5);
	else
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(0.04938271604938271 / t) + 0.037037037037037035) / t) - 0.2222222222222222) / t) - -0.8333333333333334);
	end
	return tmp
end

Wolfram

code[t_] := If[LessEqual[N[(N[(2.0 * t), $MachinePrecision] / N[(1.0 + t), $MachinePrecision]), $MachinePrecision], 1e-5], N[(N[(-2.0 * t + 1.0), $MachinePrecision] * N[(t * t), $MachinePrecision] + 0.5), $MachinePrecision], N[(N[(N[(N[(N[(N[(0.04938271604938271 / t), $MachinePrecision] + 0.037037037037037035), $MachinePrecision] / t), $MachinePrecision] - 0.2222222222222222), $MachinePrecision] / t), $MachinePrecision] - -0.8333333333333334), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) t)) < 1.00000000000000008e-5

   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. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 99.4

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

   5. Applied rewrites 99.4%

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

   if 1.00000000000000008e-5 < (/.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. Applied rewrites 99.9%

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

Alternative 6: 99.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{2 \cdot t}{1 + t} \leq 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-2, t, 1\right), t \cdot t, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222 - \frac{0.037037037037037035}{t}}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (<= (/ (* 2.0 t) (+ 1.0 t)) 1e-5)
   (fma (fma -2.0 t 1.0) (* t t) 0.5)
   (-
    0.8333333333333334
    (/ (- 0.2222222222222222 (/ 0.037037037037037035 t)) t))))

C

double code(double t) {
	double tmp;
	if (((2.0 * t) / (1.0 + t)) <= 1e-5) {
		tmp = fma(fma(-2.0, t, 1.0), (t * t), 0.5);
	} else {
		tmp = 0.8333333333333334 - ((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)) <= 1e-5)
		tmp = fma(fma(-2.0, t, 1.0), Float64(t * t), 0.5);
	else
		tmp = Float64(0.8333333333333334 - Float64(Float64(0.2222222222222222 - Float64(0.037037037037037035 / t)) / t));
	end
	return tmp
end

Wolfram

code[t_] := If[LessEqual[N[(N[(2.0 * t), $MachinePrecision] / N[(1.0 + t), $MachinePrecision]), $MachinePrecision], 1e-5], N[(N[(-2.0 * t + 1.0), $MachinePrecision] * N[(t * t), $MachinePrecision] + 0.5), $MachinePrecision], N[(0.8333333333333334 - N[(N[(0.2222222222222222 - N[(0.037037037037037035 / t), $MachinePrecision]), $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)) < 1.00000000000000008e-5

   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. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 99.4

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

   5. Applied rewrites 99.4%

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

   if 1.00000000000000008e-5 < (/.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

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. +-commutative N/A

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

      4. associate--r- N/A

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. unpow2 N/A

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

      8. associate-/r* N/A

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

      9. metadata-eval N/A

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

      10. associate-*r/ N/A

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

      11. div-sub N/A

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

      12. lower--.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower--.f64 N/A

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

      15. associate-*r/ N/A

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

      16. metadata-eval N/A

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

      17. lower-/.f64 99.7

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

   5. Applied rewrites 99.7%

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

Alternative 7: 98.8% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{2 \cdot t}{1 + t} \leq 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-2, t, 1\right), t \cdot t, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (<= (/ (* 2.0 t) (+ 1.0 t)) 1e-5)
   (fma (fma -2.0 t 1.0) (* t t) 0.5)
   (- 0.8333333333333334 (/ 0.2222222222222222 t))))

C

double code(double t) {
	double tmp;
	if (((2.0 * t) / (1.0 + t)) <= 1e-5) {
		tmp = fma(fma(-2.0, t, 1.0), (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)) <= 1e-5)
		tmp = fma(fma(-2.0, t, 1.0), Float64(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], 1e-5], N[(N[(-2.0 * t + 1.0), $MachinePrecision] * N[(t * t), $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)) < 1.00000000000000008e-5

   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. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 99.4

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

   5. Applied rewrites 99.4%

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

   if 1.00000000000000008e-5 < (/.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. lower--.f64 N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. lower-/.f64 99.5

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

   5. Applied rewrites 99.5%

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

Alternative 8: 98.8% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{2 \cdot t}{1 + t} \leq 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(t, t, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (<= (/ (* 2.0 t) (+ 1.0 t)) 1e-5)
   (fma t t 0.5)
   (- 0.8333333333333334 (/ 0.2222222222222222 t))))

C

double code(double t) {
	double tmp;
	if (((2.0 * t) / (1.0 + t)) <= 1e-5) {
		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)) <= 1e-5)
		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], 1e-5], 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)) < 1.00000000000000008e-5

   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. lower-fma.f64 99.3

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

   5. Applied rewrites 99.3%

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

   if 1.00000000000000008e-5 < (/.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. lower--.f64 N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. lower-/.f64 99.5

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

   5. Applied rewrites 99.5%

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

Alternative 9: 98.3% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{2 \cdot t}{1 + t} \leq 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(t, t, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (<= (/ (* 2.0 t) (+ 1.0 t)) 1e-5) (fma t t 0.5) 0.8333333333333334))

C

double code(double t) {
	double tmp;
	if (((2.0 * t) / (1.0 + t)) <= 1e-5) {
		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)) <= 1e-5)
		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], 1e-5], 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)) < 1.00000000000000008e-5

   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. lower-fma.f64 99.3

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

   5. Applied rewrites 99.3%

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

   if 1.00000000000000008e-5 < (/.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. Applied rewrites 98.8%

      \[\leadsto \color{blue}{0.8333333333333334} \]
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. Applied rewrites 98.9%

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

   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. Applied rewrites 98.8%

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

Alternative 11: 59.4% 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. Applied rewrites 58.0%

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

Reproduce

herbie shell --seed 2024325 
(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))))))