Kahan p13 Example 1

Percentage Accurate: 99.9% → 100.0%

Time: 7.9s

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: 99.9% 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, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   2. clear-num N/A

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

   3. lower-/.f64 N/A

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

   4. frac-2neg N/A

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

   5. lower-/.f64 N/A

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

4. Applied rewrites 100.0%

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

5. Final simplification 100.0%

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

Alternative 2: 98.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{2 \cdot t}{1 + t} \leq 2 \cdot 10^{-8}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-2, t, 1\right), t \cdot t, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(1.2 - \frac{-0.32}{t}\right) + \frac{\frac{0.032 - \frac{0.0768}{t}}{t}}{t}\right)}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (<= (/ (* 2.0 t) (+ 1.0 t)) 2e-8)
   (fma (fma -2.0 t 1.0) (* t t) 0.5)
   (pow (+ (- 1.2 (/ -0.32 t)) (/ (/ (- 0.032 (/ 0.0768 t)) t) t)) -1.0)))

C

double code(double t) {
	double tmp;
	if (((2.0 * t) / (1.0 + t)) <= 2e-8) {
		tmp = fma(fma(-2.0, t, 1.0), (t * t), 0.5);
	} else {
		tmp = pow(((1.2 - (-0.32 / t)) + (((0.032 - (0.0768 / t)) / t) / t)), -1.0);
	}
	return tmp;
}

Julia

function code(t)
	tmp = 0.0
	if (Float64(Float64(2.0 * t) / Float64(1.0 + t)) <= 2e-8)
		tmp = fma(fma(-2.0, t, 1.0), Float64(t * t), 0.5);
	else
		tmp = Float64(Float64(1.2 - Float64(-0.32 / t)) + Float64(Float64(Float64(0.032 - Float64(0.0768 / t)) / t) / t)) ^ -1.0;
	end
	return tmp
end

Wolfram

code[t_] := If[LessEqual[N[(N[(2.0 * t), $MachinePrecision] / N[(1.0 + t), $MachinePrecision]), $MachinePrecision], 2e-8], N[(N[(-2.0 * t + 1.0), $MachinePrecision] * N[(t * t), $MachinePrecision] + 0.5), $MachinePrecision], N[Power[N[(N[(1.2 - N[(-0.32 / t), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(0.032 - N[(0.0768 / t), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) t)) < 2e-8

   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 100.0

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

   5. Applied rewrites 100.0%

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

   if 2e-8 < (/.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

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. frac-2neg N/A

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

      5. lower-/.f64 N/A

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in t around inf

      \[\leadsto \frac{1}{\color{blue}{\left(\frac{6}{5} + \left(\frac{\frac{4}{125}}{{t}^{2}} + \frac{8}{25} \cdot \frac{1}{t}\right)\right) - \frac{48}{625} \cdot \frac{1}{{t}^{3}}}} \]
   6. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{6}{5} + \left(\left(\frac{\frac{4}{125}}{{t}^{2}} + \frac{8}{25} \cdot \frac{1}{t}\right) - \frac{48}{625} \cdot \frac{1}{{t}^{3}}\right)}} \]

      2. +-commutative N/A

         \[\leadsto \frac{1}{\frac{6}{5} + \left(\color{blue}{\left(\frac{8}{25} \cdot \frac{1}{t} + \frac{\frac{4}{125}}{{t}^{2}}\right)} - \frac{48}{625} \cdot \frac{1}{{t}^{3}}\right)} \]

      3. associate--l+ N/A

         \[\leadsto \frac{1}{\frac{6}{5} + \color{blue}{\left(\frac{8}{25} \cdot \frac{1}{t} + \left(\frac{\frac{4}{125}}{{t}^{2}} - \frac{48}{625} \cdot \frac{1}{{t}^{3}}\right)\right)}} \]

      4. cube-mult N/A

         \[\leadsto \frac{1}{\frac{6}{5} + \left(\frac{8}{25} \cdot \frac{1}{t} + \left(\frac{\frac{4}{125}}{{t}^{2}} - \frac{48}{625} \cdot \frac{1}{\color{blue}{t \cdot \left(t \cdot t\right)}}\right)\right)} \]

      5. unpow2 N/A

         \[\leadsto \frac{1}{\frac{6}{5} + \left(\frac{8}{25} \cdot \frac{1}{t} + \left(\frac{\frac{4}{125}}{{t}^{2}} - \frac{48}{625} \cdot \frac{1}{t \cdot \color{blue}{{t}^{2}}}\right)\right)} \]

      6. associate-/r* N/A

         \[\leadsto \frac{1}{\frac{6}{5} + \left(\frac{8}{25} \cdot \frac{1}{t} + \left(\frac{\frac{4}{125}}{{t}^{2}} - \frac{48}{625} \cdot \color{blue}{\frac{\frac{1}{t}}{{t}^{2}}}\right)\right)} \]

      7. associate-*r/ N/A

         \[\leadsto \frac{1}{\frac{6}{5} + \left(\frac{8}{25} \cdot \frac{1}{t} + \left(\frac{\frac{4}{125}}{{t}^{2}} - \color{blue}{\frac{\frac{48}{625} \cdot \frac{1}{t}}{{t}^{2}}}\right)\right)} \]

      8. div-sub N/A

         \[\leadsto \frac{1}{\frac{6}{5} + \left(\frac{8}{25} \cdot \frac{1}{t} + \color{blue}{\frac{\frac{4}{125} - \frac{48}{625} \cdot \frac{1}{t}}{{t}^{2}}}\right)} \]

      9. unpow2 N/A

         \[\leadsto \frac{1}{\frac{6}{5} + \left(\frac{8}{25} \cdot \frac{1}{t} + \frac{\frac{4}{125} - \frac{48}{625} \cdot \frac{1}{t}}{\color{blue}{t \cdot t}}\right)} \]

      10. associate-/l/ N/A

          \[\leadsto \frac{1}{\frac{6}{5} + \left(\frac{8}{25} \cdot \frac{1}{t} + \color{blue}{\frac{\frac{\frac{4}{125} - \frac{48}{625} \cdot \frac{1}{t}}{t}}{t}}\right)} \]

      11. +-commutative N/A

          \[\leadsto \frac{1}{\frac{6}{5} + \color{blue}{\left(\frac{\frac{\frac{4}{125} - \frac{48}{625} \cdot \frac{1}{t}}{t}}{t} + \frac{8}{25} \cdot \frac{1}{t}\right)}} \]

   7. Applied rewrites 99.5%

      \[\leadsto \frac{1}{\color{blue}{1.2 - \frac{-0.32 - \frac{0.032 - \frac{0.0768}{t}}{t}}{t}}} \]
   8. Step-by-step derivation

   9. Applied rewrites 99.5%

      \[\leadsto \frac{1}{\left(1.2 - \frac{-0.32}{t}\right) + \color{blue}{\frac{\frac{0.032 - \frac{0.0768}{t}}{t}}{t}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2 \cdot t}{1 + t} \leq 2 \cdot 10^{-8}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-2, t, 1\right), t \cdot t, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(1.2 - \frac{-0.32}{t}\right) + \frac{\frac{0.032 - \frac{0.0768}{t}}{t}}{t}\right)}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{2 \cdot t}{1 + t} \leq 2 \cdot 10^{-8}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-2, t, 1\right), t \cdot t, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(1.2 - \frac{-0.32 - \frac{0.032 - \frac{0.0768}{t}}{t}}{t}\right)}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (<= (/ (* 2.0 t) (+ 1.0 t)) 2e-8)
   (fma (fma -2.0 t 1.0) (* t t) 0.5)
   (pow (- 1.2 (/ (- -0.32 (/ (- 0.032 (/ 0.0768 t)) t)) t)) -1.0)))

C

double code(double t) {
	double tmp;
	if (((2.0 * t) / (1.0 + t)) <= 2e-8) {
		tmp = fma(fma(-2.0, t, 1.0), (t * t), 0.5);
	} else {
		tmp = pow((1.2 - ((-0.32 - ((0.032 - (0.0768 / t)) / t)) / t)), -1.0);
	}
	return tmp;
}

Julia

function code(t)
	tmp = 0.0
	if (Float64(Float64(2.0 * t) / Float64(1.0 + t)) <= 2e-8)
		tmp = fma(fma(-2.0, t, 1.0), Float64(t * t), 0.5);
	else
		tmp = Float64(1.2 - Float64(Float64(-0.32 - Float64(Float64(0.032 - Float64(0.0768 / t)) / t)) / t)) ^ -1.0;
	end
	return tmp
end

Wolfram

code[t_] := If[LessEqual[N[(N[(2.0 * t), $MachinePrecision] / N[(1.0 + t), $MachinePrecision]), $MachinePrecision], 2e-8], N[(N[(-2.0 * t + 1.0), $MachinePrecision] * N[(t * t), $MachinePrecision] + 0.5), $MachinePrecision], N[Power[N[(1.2 - N[(N[(-0.32 - N[(N[(0.032 - N[(0.0768 / t), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) t)) < 2e-8

   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 100.0

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

   5. Applied rewrites 100.0%

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

   if 2e-8 < (/.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

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. frac-2neg N/A

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

      5. lower-/.f64 N/A

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in t around inf

      \[\leadsto \frac{1}{\color{blue}{\left(\frac{6}{5} + \left(\frac{\frac{4}{125}}{{t}^{2}} + \frac{8}{25} \cdot \frac{1}{t}\right)\right) - \frac{48}{625} \cdot \frac{1}{{t}^{3}}}} \]
   6. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{6}{5} + \left(\left(\frac{\frac{4}{125}}{{t}^{2}} + \frac{8}{25} \cdot \frac{1}{t}\right) - \frac{48}{625} \cdot \frac{1}{{t}^{3}}\right)}} \]

      2. +-commutative N/A

         \[\leadsto \frac{1}{\frac{6}{5} + \left(\color{blue}{\left(\frac{8}{25} \cdot \frac{1}{t} + \frac{\frac{4}{125}}{{t}^{2}}\right)} - \frac{48}{625} \cdot \frac{1}{{t}^{3}}\right)} \]

      3. associate--l+ N/A

         \[\leadsto \frac{1}{\frac{6}{5} + \color{blue}{\left(\frac{8}{25} \cdot \frac{1}{t} + \left(\frac{\frac{4}{125}}{{t}^{2}} - \frac{48}{625} \cdot \frac{1}{{t}^{3}}\right)\right)}} \]

      4. cube-mult N/A

         \[\leadsto \frac{1}{\frac{6}{5} + \left(\frac{8}{25} \cdot \frac{1}{t} + \left(\frac{\frac{4}{125}}{{t}^{2}} - \frac{48}{625} \cdot \frac{1}{\color{blue}{t \cdot \left(t \cdot t\right)}}\right)\right)} \]

      5. unpow2 N/A

         \[\leadsto \frac{1}{\frac{6}{5} + \left(\frac{8}{25} \cdot \frac{1}{t} + \left(\frac{\frac{4}{125}}{{t}^{2}} - \frac{48}{625} \cdot \frac{1}{t \cdot \color{blue}{{t}^{2}}}\right)\right)} \]

      6. associate-/r* N/A

         \[\leadsto \frac{1}{\frac{6}{5} + \left(\frac{8}{25} \cdot \frac{1}{t} + \left(\frac{\frac{4}{125}}{{t}^{2}} - \frac{48}{625} \cdot \color{blue}{\frac{\frac{1}{t}}{{t}^{2}}}\right)\right)} \]

      7. associate-*r/ N/A

         \[\leadsto \frac{1}{\frac{6}{5} + \left(\frac{8}{25} \cdot \frac{1}{t} + \left(\frac{\frac{4}{125}}{{t}^{2}} - \color{blue}{\frac{\frac{48}{625} \cdot \frac{1}{t}}{{t}^{2}}}\right)\right)} \]

      8. div-sub N/A

         \[\leadsto \frac{1}{\frac{6}{5} + \left(\frac{8}{25} \cdot \frac{1}{t} + \color{blue}{\frac{\frac{4}{125} - \frac{48}{625} \cdot \frac{1}{t}}{{t}^{2}}}\right)} \]

      9. unpow2 N/A

         \[\leadsto \frac{1}{\frac{6}{5} + \left(\frac{8}{25} \cdot \frac{1}{t} + \frac{\frac{4}{125} - \frac{48}{625} \cdot \frac{1}{t}}{\color{blue}{t \cdot t}}\right)} \]

      10. associate-/l/ N/A

          \[\leadsto \frac{1}{\frac{6}{5} + \left(\frac{8}{25} \cdot \frac{1}{t} + \color{blue}{\frac{\frac{\frac{4}{125} - \frac{48}{625} \cdot \frac{1}{t}}{t}}{t}}\right)} \]

      11. +-commutative N/A

          \[\leadsto \frac{1}{\frac{6}{5} + \color{blue}{\left(\frac{\frac{\frac{4}{125} - \frac{48}{625} \cdot \frac{1}{t}}{t}}{t} + \frac{8}{25} \cdot \frac{1}{t}\right)}} \]

   7. Applied rewrites 99.5%

      \[\leadsto \frac{1}{\color{blue}{1.2 - \frac{-0.32 - \frac{0.032 - \frac{0.0768}{t}}{t}}{t}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2 \cdot t}{1 + t} \leq 2 \cdot 10^{-8}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-2, t, 1\right), t \cdot t, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(1.2 - \frac{-0.32 - \frac{0.032 - \frac{0.0768}{t}}{t}}{t}\right)}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{2 \cdot t}{1 + t} \leq 2 \cdot 10^{-8}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-2, t, 1\right), t \cdot t, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{0.32}{t} + 1.2\right)}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (<= (/ (* 2.0 t) (+ 1.0 t)) 2e-8)
   (fma (fma -2.0 t 1.0) (* t t) 0.5)
   (pow (+ (/ 0.32 t) 1.2) -1.0)))

C

double code(double t) {
	double tmp;
	if (((2.0 * t) / (1.0 + t)) <= 2e-8) {
		tmp = fma(fma(-2.0, t, 1.0), (t * t), 0.5);
	} else {
		tmp = pow(((0.32 / t) + 1.2), -1.0);
	}
	return tmp;
}

Julia

function code(t)
	tmp = 0.0
	if (Float64(Float64(2.0 * t) / Float64(1.0 + t)) <= 2e-8)
		tmp = fma(fma(-2.0, t, 1.0), Float64(t * t), 0.5);
	else
		tmp = Float64(Float64(0.32 / t) + 1.2) ^ -1.0;
	end
	return tmp
end

Wolfram

code[t_] := If[LessEqual[N[(N[(2.0 * t), $MachinePrecision] / N[(1.0 + t), $MachinePrecision]), $MachinePrecision], 2e-8], N[(N[(-2.0 * t + 1.0), $MachinePrecision] * N[(t * t), $MachinePrecision] + 0.5), $MachinePrecision], N[Power[N[(N[(0.32 / t), $MachinePrecision] + 1.2), $MachinePrecision], -1.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) t)) < 2e-8

   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 100.0

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

   5. Applied rewrites 100.0%

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

   if 2e-8 < (/.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

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. frac-2neg N/A

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

      5. lower-/.f64 N/A

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in t around inf

      \[\leadsto \frac{1}{\color{blue}{\frac{6}{5} + \frac{8}{25} \cdot \frac{1}{t}}} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{8}{25} \cdot \frac{1}{t} + \frac{6}{5}}} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{8}{25} \cdot \frac{1}{t} + \frac{6}{5}}} \]

      3. associate-*r/ N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{\frac{8}{25} \cdot 1}{t}} + \frac{6}{5}} \]

      4. metadata-eval N/A

         \[\leadsto \frac{1}{\frac{\color{blue}{\frac{8}{25}}}{t} + \frac{6}{5}} \]

      5. lower-/.f64 98.9

         \[\leadsto \frac{1}{\color{blue}{\frac{0.32}{t}} + 1.2} \]

   7. Applied rewrites 98.9%

      \[\leadsto \frac{1}{\color{blue}{\frac{0.32}{t} + 1.2}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2 \cdot t}{1 + t} \leq 2 \cdot 10^{-8}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-2, t, 1\right), t \cdot t, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{0.32}{t} + 1.2\right)}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 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. unpow2 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 6: 99.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{2 \cdot t}{1 + t} \leq 2 \cdot 10^{-8}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-2, t, 1\right), t \cdot t, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222 - \frac{\frac{0.04938271604938271}{t} - -0.037037037037037035}{t}}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (<= (/ (* 2.0 t) (+ 1.0 t)) 2e-8)
   (fma (fma -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)) <= 2e-8) {
		tmp = fma(fma(-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)) <= 2e-8)
		tmp = fma(fma(-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], 2e-8], N[(N[(-2.0 * 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)) < 2e-8

   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 100.0

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

   5. Applied rewrites 100.0%

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

   if 2e-8 < (/.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 99.4%

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

Alternative 7: 98.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{2 \cdot t}{1 + t} \leq 2 \cdot 10^{-8}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-2, t, 1\right), t \cdot t, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222 - \frac{0.037037037037037035}{t}}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (<= (/ (* 2.0 t) (+ 1.0 t)) 2e-8)
   (fma (fma -2.0 t 1.0) (* t t) 0.5)
   (-
    0.8333333333333334
    (/ (- 0.2222222222222222 (/ 0.037037037037037035 t)) t))))

C

double code(double t) {
	double tmp;
	if (((2.0 * t) / (1.0 + t)) <= 2e-8) {
		tmp = fma(fma(-2.0, t, 1.0), (t * t), 0.5);
	} else {
		tmp = 0.8333333333333334 - ((0.2222222222222222 - (0.037037037037037035 / t)) / t);
	}
	return tmp;
}

Julia

function code(t)
	tmp = 0.0
	if (Float64(Float64(2.0 * t) / Float64(1.0 + t)) <= 2e-8)
		tmp = fma(fma(-2.0, t, 1.0), Float64(t * t), 0.5);
	else
		tmp = Float64(0.8333333333333334 - Float64(Float64(0.2222222222222222 - Float64(0.037037037037037035 / t)) / t));
	end
	return tmp
end

Wolfram

code[t_] := If[LessEqual[N[(N[(2.0 * t), $MachinePrecision] / N[(1.0 + t), $MachinePrecision]), $MachinePrecision], 2e-8], N[(N[(-2.0 * t + 1.0), $MachinePrecision] * N[(t * t), $MachinePrecision] + 0.5), $MachinePrecision], N[(0.8333333333333334 - N[(N[(0.2222222222222222 - N[(0.037037037037037035 / t), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) t)) < 2e-8

   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 100.0

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

   5. Applied rewrites 100.0%

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

   if 2e-8 < (/.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 99.1

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

   5. Applied rewrites 99.1%

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

Alternative 8: 98.7% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{2 \cdot t}{1 + t} \leq 2 \cdot 10^{-8}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-2, t, 1\right), t \cdot t, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (<= (/ (* 2.0 t) (+ 1.0 t)) 2e-8)
   (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)) <= 2e-8) {
		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)) <= 2e-8)
		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], 2e-8], 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)) < 2e-8

   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 100.0

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

   5. Applied rewrites 100.0%

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

   if 2e-8 < (/.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 98.8

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

   5. Applied rewrites 98.8%

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

Alternative 9: 98.7% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{2 \cdot t}{1 + t} \leq 2 \cdot 10^{-8}:\\ \;\;\;\;\mathsf{fma}\left(t, t, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (<= (/ (* 2.0 t) (+ 1.0 t)) 2e-8)
   (fma t t 0.5)
   (- 0.8333333333333334 (/ 0.2222222222222222 t))))

C

double code(double t) {
	double tmp;
	if (((2.0 * t) / (1.0 + t)) <= 2e-8) {
		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)) <= 2e-8)
		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], 2e-8], 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)) < 2e-8

   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 99.9

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

   5. Applied rewrites 99.9%

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

   if 2e-8 < (/.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 98.8

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

   5. Applied rewrites 98.8%

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

Alternative 10: 98.2% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{2 \cdot t}{1 + t} \leq 2 \cdot 10^{-8}:\\ \;\;\;\;\mathsf{fma}\left(t, t, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (<= (/ (* 2.0 t) (+ 1.0 t)) 2e-8) (fma t t 0.5) 0.8333333333333334))

C

double code(double t) {
	double tmp;
	if (((2.0 * t) / (1.0 + t)) <= 2e-8) {
		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)) <= 2e-8)
		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], 2e-8], 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)) < 2e-8

   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 99.9

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

   5. Applied rewrites 99.9%

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

   if 2e-8 < (/.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 97.8%

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

Alternative 11: 98.4% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: tmp
    if (((2.0d0 * t) / (1.0d0 + t)) <= 1.0d0) then
        tmp = 0.5d0
    else
        tmp = 0.8333333333333334d0
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in t around 0

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

   5. Applied rewrites 99.7%

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

   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 97.8%

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

Alternative 12: 59.3% accurate, 104.0× speedup

Math

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

FPCore

(FPCore (t) :precision binary64 0.5)

C

double code(double t) {
	return 0.5;
}

Fortran

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

Java

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

Python

def code(t):
	return 0.5

Julia

function code(t)
	return 0.5
end

MATLAB

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

Wolfram

code[t_] := 0.5

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in t around 0

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

5. Applied rewrites 58.7%

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

Reproduce

herbie shell --seed 2024312 
(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))))))