Kahan p13 Example 1

Percentage Accurate: 100.0% → 100.0%

Time: 7.6s

Alternatives: 12

Speedup: 1.2×

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, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   6. associate-/l* N/A

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

   7. *-commutative N/A

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

   8. associate-*l* N/A

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

   9. lower-fma.f64 N/A

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

   10. lower-/.f64 N/A

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

   11. lift-/.f64 N/A

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

   12. lift-*.f64 N/A

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

   13. associate-/l* N/A

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

   14. *-commutative N/A

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

   15. associate-*r* N/A

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

   16. *-commutative N/A

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

   17. associate-*l* N/A

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

   18. lower-*.f64 N/A

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

   19. lower-/.f64 N/A

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

   20. metadata-eval 100.0

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

4. Applied rewrites 100.0%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. pow2 N/A

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

   5. lift-/.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. associate-/l* N/A

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

   8. lift-/.f64 N/A

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

   9. *-commutative N/A

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

   10. unpow-prod-down N/A

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

   11. pow2 N/A

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

   12. metadata-eval N/A

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

   13. associate-*r* N/A

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

   14. lift-*.f64 N/A

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

   15. lower-fma.f64 100.0

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

   16. lift-*.f64 N/A

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

   17. *-commutative N/A

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

   18. lower-*.f64 100.0

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

6. Applied rewrites 100.0%

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

Alternative 2: 99.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{2 \cdot t}{1 + t} \leq 0.04:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(t - 2, t, 1\right), t \cdot t, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334 - {\left(t \cdot 4.5\right)}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (<= (/ (* 2.0 t) (+ 1.0 t)) 0.04)
   (fma (fma (- t 2.0) t 1.0) (* t t) 0.5)
   (- 0.8333333333333334 (pow (* t 4.5) -1.0))))

C

double code(double t) {
	double tmp;
	if (((2.0 * t) / (1.0 + t)) <= 0.04) {
		tmp = fma(fma((t - 2.0), t, 1.0), (t * t), 0.5);
	} else {
		tmp = 0.8333333333333334 - pow((t * 4.5), -1.0);
	}
	return tmp;
}

Julia

function code(t)
	tmp = 0.0
	if (Float64(Float64(2.0 * t) / Float64(1.0 + t)) <= 0.04)
		tmp = fma(fma(Float64(t - 2.0), t, 1.0), Float64(t * t), 0.5);
	else
		tmp = Float64(0.8333333333333334 - (Float64(t * 4.5) ^ -1.0));
	end
	return tmp
end

Wolfram

code[t_] := If[LessEqual[N[(N[(2.0 * t), $MachinePrecision] / N[(1.0 + t), $MachinePrecision]), $MachinePrecision], 0.04], N[(N[(N[(t - 2.0), $MachinePrecision] * t + 1.0), $MachinePrecision] * N[(t * t), $MachinePrecision] + 0.5), $MachinePrecision], N[(0.8333333333333334 - N[Power[N[(t * 4.5), $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) t)) < 0.0400000000000000008

   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. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower--.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 99.7

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

   5. Applied rewrites 99.7%

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

   if 0.0400000000000000008 < (/.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. 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

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-/l* N/A

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

      14. *-commutative N/A

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

      15. associate-*r* N/A

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

      16. *-commutative N/A

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

      17. associate-*l* N/A

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

      18. lower-*.f64 N/A

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

      19. lower-/.f64 N/A

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

      20. metadata-eval 100.0

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

   4. Applied rewrites 100.0%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. pow2 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/l* N/A

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

      8. lift-/.f64 N/A

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

      9. *-commutative N/A

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

      10. unpow-prod-down N/A

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

      11. pow2 N/A

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

      12. metadata-eval N/A

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

      13. associate-*r* N/A

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

      14. lift-*.f64 N/A

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

      15. lower-fma.f64 100.0

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 100.0

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

   6. Applied rewrites 100.0%

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

   7. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\frac{5}{6} - \frac{2}{9} \cdot \frac{1}{t}} \]
   8. 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 99.4

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

   9. Applied rewrites 99.4%

      \[\leadsto \color{blue}{0.8333333333333334 - \frac{0.2222222222222222}{t}} \]
   10. Step-by-step derivation

   11. Applied rewrites 99.4%

       \[\leadsto 0.8333333333333334 - \frac{1}{\color{blue}{t \cdot 4.5}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2 \cdot t}{1 + t} \leq 0.04:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(t - 2, t, 1\right), t \cdot t, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334 - {\left(t \cdot 4.5\right)}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{2 \cdot t}{1 + t} \leq 0.04:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-2, t, 1\right), t \cdot t, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334 - {\left(t \cdot 4.5\right)}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (<= (/ (* 2.0 t) (+ 1.0 t)) 0.04)
   (fma (fma -2.0 t 1.0) (* t t) 0.5)
   (- 0.8333333333333334 (pow (* t 4.5) -1.0))))

C

double code(double t) {
	double tmp;
	if (((2.0 * t) / (1.0 + t)) <= 0.04) {
		tmp = fma(fma(-2.0, t, 1.0), (t * t), 0.5);
	} else {
		tmp = 0.8333333333333334 - pow((t * 4.5), -1.0);
	}
	return tmp;
}

Julia

function code(t)
	tmp = 0.0
	if (Float64(Float64(2.0 * t) / Float64(1.0 + t)) <= 0.04)
		tmp = fma(fma(-2.0, t, 1.0), Float64(t * t), 0.5);
	else
		tmp = Float64(0.8333333333333334 - (Float64(t * 4.5) ^ -1.0));
	end
	return tmp
end

Wolfram

code[t_] := If[LessEqual[N[(N[(2.0 * t), $MachinePrecision] / N[(1.0 + t), $MachinePrecision]), $MachinePrecision], 0.04], N[(N[(-2.0 * t + 1.0), $MachinePrecision] * N[(t * t), $MachinePrecision] + 0.5), $MachinePrecision], N[(0.8333333333333334 - N[Power[N[(t * 4.5), $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) t)) < 0.0400000000000000008

   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 99.7

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

   5. Applied rewrites 99.7%

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

   if 0.0400000000000000008 < (/.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. 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

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-/l* N/A

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

      14. *-commutative N/A

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

      15. associate-*r* N/A

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

      16. *-commutative N/A

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

      17. associate-*l* N/A

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

      18. lower-*.f64 N/A

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

      19. lower-/.f64 N/A

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

      20. metadata-eval 100.0

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

   4. Applied rewrites 100.0%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. pow2 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/l* N/A

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

      8. lift-/.f64 N/A

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

      9. *-commutative N/A

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

      10. unpow-prod-down N/A

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

      11. pow2 N/A

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

      12. metadata-eval N/A

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

      13. associate-*r* N/A

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

      14. lift-*.f64 N/A

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

      15. lower-fma.f64 100.0

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 100.0

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

   6. Applied rewrites 100.0%

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

   7. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\frac{5}{6} - \frac{2}{9} \cdot \frac{1}{t}} \]
   8. 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 99.4

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

   9. Applied rewrites 99.4%

      \[\leadsto \color{blue}{0.8333333333333334 - \frac{0.2222222222222222}{t}} \]
   10. Step-by-step derivation

   11. Applied rewrites 99.4%

       \[\leadsto 0.8333333333333334 - \frac{1}{\color{blue}{t \cdot 4.5}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2 \cdot t}{1 + t} \leq 0.04:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-2, t, 1\right), t \cdot t, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334 - {\left(t \cdot 4.5\right)}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{2 \cdot t}{1 + t} \leq 0.04:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{t}{1 + t}, \left(\left(\mathsf{fma}\left(t, t, 1\right) \cdot \left(1 - t\right)\right) \cdot t\right) \cdot 4, 1\right)}{2 + \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-16, t, 12\right), t, -8\right), t, 4\right) \cdot t\right) \cdot t}\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222 - \frac{\frac{0.04938271604938271}{t} + 0.037037037037037035}{t}}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (<= (/ (* 2.0 t) (+ 1.0 t)) 0.04)
   (/
    (fma (/ t (+ 1.0 t)) (* (* (* (fma t t 1.0) (- 1.0 t)) t) 4.0) 1.0)
    (+ 2.0 (* (* (fma (fma (fma -16.0 t 12.0) t -8.0) t 4.0) t) t)))
   (-
    0.8333333333333334
    (/
     (-
      0.2222222222222222
      (/ (+ (/ 0.04938271604938271 t) 0.037037037037037035) t))
     t))))

C

double code(double t) {
	double tmp;
	if (((2.0 * t) / (1.0 + t)) <= 0.04) {
		tmp = fma((t / (1.0 + t)), (((fma(t, t, 1.0) * (1.0 - t)) * t) * 4.0), 1.0) / (2.0 + ((fma(fma(fma(-16.0, t, 12.0), t, -8.0), t, 4.0) * t) * t));
	} else {
		tmp = 0.8333333333333334 - ((0.2222222222222222 - (((0.04938271604938271 / t) + 0.037037037037037035) / t)) / t);
	}
	return tmp;
}

Julia

function code(t)
	tmp = 0.0
	if (Float64(Float64(2.0 * t) / Float64(1.0 + t)) <= 0.04)
		tmp = Float64(fma(Float64(t / Float64(1.0 + t)), Float64(Float64(Float64(fma(t, t, 1.0) * Float64(1.0 - t)) * t) * 4.0), 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(0.8333333333333334 - Float64(Float64(0.2222222222222222 - Float64(Float64(Float64(0.04938271604938271 / t) + 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], 0.04], N[(N[(N[(t / N[(1.0 + t), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(t * t + 1.0), $MachinePrecision] * N[(1.0 - t), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision] * 4.0), $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[(0.8333333333333334 - N[(N[(0.2222222222222222 - N[(N[(N[(0.04938271604938271 / t), $MachinePrecision] + 0.037037037037037035), $MachinePrecision] / 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)) < 0.0400000000000000008

   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

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-/l* N/A

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

      14. *-commutative N/A

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

      15. associate-*r* N/A

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

      16. *-commutative N/A

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

      17. associate-*l* N/A

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

      18. lower-*.f64 N/A

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

      19. lower-/.f64 N/A

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

      20. metadata-eval 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in t around 0

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

      1. unpow2 N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. sub-neg N/A

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

      11. *-commutative N/A

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

      12. metadata-eval N/A

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

      13. lower-fma.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-fma.f64 99.8

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

   7. Applied rewrites 99.8%

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

   8. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. sub-neg N/A

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

      7. *-commutative N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 N/A

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

      10. mul-1-neg N/A

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

      11. unsub-neg N/A

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

      12. lower--.f64 99.8

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

   10. Applied rewrites 99.8%

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

   11. Taylor expanded in t around 0

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

   13. Applied rewrites 99.8%

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

   if 0.0400000000000000008 < (/.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 99.5%

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

Alternative 5: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t}{1 + t}\\ \mathbf{if}\;\frac{2 \cdot t}{1 + t} \leq 0.04:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_1, t\_1 \cdot 4, 1\right)}{2 + \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-16, t, 12\right), t, -8\right), t, 4\right) \cdot t\right) \cdot t}\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222 - \frac{\frac{0.04938271604938271}{t} + 0.037037037037037035}{t}}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (let* ((t_1 (/ t (+ 1.0 t))))
   (if (<= (/ (* 2.0 t) (+ 1.0 t)) 0.04)
     (/
      (fma t_1 (* t_1 4.0) 1.0)
      (+ 2.0 (* (* (fma (fma (fma -16.0 t 12.0) t -8.0) t 4.0) t) t)))
     (-
      0.8333333333333334
      (/
       (-
        0.2222222222222222
        (/ (+ (/ 0.04938271604938271 t) 0.037037037037037035) t))
       t)))))

C

double code(double t) {
	double t_1 = t / (1.0 + t);
	double tmp;
	if (((2.0 * t) / (1.0 + t)) <= 0.04) {
		tmp = fma(t_1, (t_1 * 4.0), 1.0) / (2.0 + ((fma(fma(fma(-16.0, t, 12.0), t, -8.0), t, 4.0) * t) * t));
	} else {
		tmp = 0.8333333333333334 - ((0.2222222222222222 - (((0.04938271604938271 / t) + 0.037037037037037035) / t)) / t);
	}
	return tmp;
}

Julia

function code(t)
	t_1 = Float64(t / Float64(1.0 + t))
	tmp = 0.0
	if (Float64(Float64(2.0 * t) / Float64(1.0 + t)) <= 0.04)
		tmp = Float64(fma(t_1, Float64(t_1 * 4.0), 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(0.8333333333333334 - Float64(Float64(0.2222222222222222 - Float64(Float64(Float64(0.04938271604938271 / t) + 0.037037037037037035) / t)) / t));
	end
	return tmp
end

Wolfram

code[t_] := Block[{t$95$1 = N[(t / N[(1.0 + t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(2.0 * t), $MachinePrecision] / N[(1.0 + t), $MachinePrecision]), $MachinePrecision], 0.04], N[(N[(t$95$1 * N[(t$95$1 * 4.0), $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[(0.8333333333333334 - N[(N[(0.2222222222222222 - N[(N[(N[(0.04938271604938271 / t), $MachinePrecision] + 0.037037037037037035), $MachinePrecision] / 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)) < 0.0400000000000000008

   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

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-/l* N/A

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

      14. *-commutative N/A

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

      15. associate-*r* N/A

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

      16. *-commutative N/A

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

      17. associate-*l* N/A

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

      18. lower-*.f64 N/A

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

      19. lower-/.f64 N/A

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

      20. metadata-eval 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in t around 0

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

      1. unpow2 N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. sub-neg N/A

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

      11. *-commutative N/A

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

      12. metadata-eval N/A

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

      13. lower-fma.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-fma.f64 99.8

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

   7. Applied rewrites 99.8%

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

   if 0.0400000000000000008 < (/.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 99.5%

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

Alternative 6: 99.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{2 \cdot t}{1 + t} \leq 0.04:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(t - 2, t, 1\right), t \cdot t, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222 - \frac{\frac{0.04938271604938271}{t} + 0.037037037037037035}{t}}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (<= (/ (* 2.0 t) (+ 1.0 t)) 0.04)
   (fma (fma (- t 2.0) t 1.0) (* t t) 0.5)
   (-
    0.8333333333333334
    (/
     (-
      0.2222222222222222
      (/ (+ (/ 0.04938271604938271 t) 0.037037037037037035) t))
     t))))

C

double code(double t) {
	double tmp;
	if (((2.0 * t) / (1.0 + t)) <= 0.04) {
		tmp = fma(fma((t - 2.0), t, 1.0), (t * t), 0.5);
	} else {
		tmp = 0.8333333333333334 - ((0.2222222222222222 - (((0.04938271604938271 / t) + 0.037037037037037035) / t)) / t);
	}
	return tmp;
}

Julia

function code(t)
	tmp = 0.0
	if (Float64(Float64(2.0 * t) / Float64(1.0 + t)) <= 0.04)
		tmp = fma(fma(Float64(t - 2.0), t, 1.0), Float64(t * t), 0.5);
	else
		tmp = Float64(0.8333333333333334 - Float64(Float64(0.2222222222222222 - Float64(Float64(Float64(0.04938271604938271 / t) + 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], 0.04], N[(N[(N[(t - 2.0), $MachinePrecision] * t + 1.0), $MachinePrecision] * N[(t * t), $MachinePrecision] + 0.5), $MachinePrecision], N[(0.8333333333333334 - N[(N[(0.2222222222222222 - N[(N[(N[(0.04938271604938271 / t), $MachinePrecision] + 0.037037037037037035), $MachinePrecision] / 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)) < 0.0400000000000000008

   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. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower--.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 99.7

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

   5. Applied rewrites 99.7%

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

   if 0.0400000000000000008 < (/.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 99.5%

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

Alternative 7: 99.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{2 \cdot t}{1 + t} \leq 0.04:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(t - 2, t, 1\right), t \cdot t, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222 - \frac{0.037037037037037035}{t}}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (<= (/ (* 2.0 t) (+ 1.0 t)) 0.04)
   (fma (fma (- t 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)) <= 0.04) {
		tmp = fma(fma((t - 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)) <= 0.04)
		tmp = fma(fma(Float64(t - 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], 0.04], N[(N[(N[(t - 2.0), $MachinePrecision] * 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)) < 0.0400000000000000008

   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. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower--.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 99.7

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

   5. Applied rewrites 99.7%

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

   if 0.0400000000000000008 < (/.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 99.4

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

   5. Applied rewrites 99.4%

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

Alternative 8: 99.3% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{2 \cdot t}{1 + t} \leq 0.04:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-2, t, 1\right), t \cdot t, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (<= (/ (* 2.0 t) (+ 1.0 t)) 0.04)
   (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)) <= 0.04) {
		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)) <= 0.04)
		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], 0.04], 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)) < 0.0400000000000000008

   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 99.7

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

   5. Applied rewrites 99.7%

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

   if 0.0400000000000000008 < (/.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 99.4

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

   5. Applied rewrites 99.4%

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

Alternative 9: 99.3% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{2 \cdot t}{1 + t} \leq 0.04:\\ \;\;\;\;\mathsf{fma}\left(t, t, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (<= (/ (* 2.0 t) (+ 1.0 t)) 0.04)
   (fma t t 0.5)
   (- 0.8333333333333334 (/ 0.2222222222222222 t))))

C

double code(double t) {
	double tmp;
	if (((2.0 * t) / (1.0 + t)) <= 0.04) {
		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)) <= 0.04)
		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], 0.04], 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)) < 0.0400000000000000008

   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 99.5

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

   5. Applied rewrites 99.5%

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

   if 0.0400000000000000008 < (/.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 99.4

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

   5. Applied rewrites 99.4%

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

Alternative 10: 98.7% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{2 \cdot t}{1 + t} \leq 0.04:\\ \;\;\;\;\mathsf{fma}\left(t, t, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (<= (/ (* 2.0 t) (+ 1.0 t)) 0.04) (fma t t 0.5) 0.8333333333333334))

C

double code(double t) {
	double tmp;
	if (((2.0 * t) / (1.0 + t)) <= 0.04) {
		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)) <= 0.04)
		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], 0.04], 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)) < 0.0400000000000000008

   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 99.5

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

   5. Applied rewrites 99.5%

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

   if 0.0400000000000000008 < (/.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 98.3%

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

Alternative 11: 98.6% 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 99.3%

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

   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 98.3%

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

Alternative 12: 58.6% 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 63.2%

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

Reproduce

herbie shell --seed 2024315 
(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))))))