Kahan p13 Example 1

Percentage Accurate: 99.9% → 99.9%

Time: 9.0s

Alternatives: 11

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 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: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[t_] := Block[{t$95$1 = N[(N[(N[(t * 2.0), $MachinePrecision] / N[(-1.0 - t), $MachinePrecision]), $MachinePrecision] * N[(N[(t * 2.0), $MachinePrecision] / N[(t - -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(N[(1.0 - t$95$1), $MachinePrecision] / N[(2.0 - t$95$1), $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. Final simplification 100.0%

   \[\leadsto \frac{1 - \frac{t \cdot 2}{-1 - t} \cdot \frac{t \cdot 2}{t - -1}}{2 - \frac{t \cdot 2}{-1 - t} \cdot \frac{t \cdot 2}{t - -1}} \]
4. Add Preprocessing

Alternative 2: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{t \cdot 2}{t - -1} \leq 1.9999999999997489:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{t \cdot t}{\left(-1 - t\right) \cdot \left(t - -1\right)}, -4, 1\right)}{\mathsf{fma}\left(\frac{t \cdot t}{\mathsf{fma}\left(-2 - t, t, -1\right)}, -4, 2\right)}\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222 - \frac{0.037037037037037035 + \frac{0.04938271604938271}{t}}{t}}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (<= (/ (* t 2.0) (- t -1.0)) 1.9999999999997489)
   (/
    (fma (/ (* t t) (* (- -1.0 t) (- t -1.0))) -4.0 1.0)
    (fma (/ (* t t) (fma (- -2.0 t) t -1.0)) -4.0 2.0))
   (-
    0.8333333333333334
    (/
     (-
      0.2222222222222222
      (/ (+ 0.037037037037037035 (/ 0.04938271604938271 t)) t))
     t))))

C

double code(double t) {
	double tmp;
	if (((t * 2.0) / (t - -1.0)) <= 1.9999999999997489) {
		tmp = fma(((t * t) / ((-1.0 - t) * (t - -1.0))), -4.0, 1.0) / fma(((t * t) / fma((-2.0 - t), t, -1.0)), -4.0, 2.0);
	} else {
		tmp = 0.8333333333333334 - ((0.2222222222222222 - ((0.037037037037037035 + (0.04938271604938271 / t)) / t)) / t);
	}
	return tmp;
}

Julia

function code(t)
	tmp = 0.0
	if (Float64(Float64(t * 2.0) / Float64(t - -1.0)) <= 1.9999999999997489)
		tmp = Float64(fma(Float64(Float64(t * t) / Float64(Float64(-1.0 - t) * Float64(t - -1.0))), -4.0, 1.0) / fma(Float64(Float64(t * t) / fma(Float64(-2.0 - t), t, -1.0)), -4.0, 2.0));
	else
		tmp = Float64(0.8333333333333334 - Float64(Float64(0.2222222222222222 - Float64(Float64(0.037037037037037035 + Float64(0.04938271604938271 / t)) / t)) / t));
	end
	return tmp
end

Wolfram

code[t_] := If[LessEqual[N[(N[(t * 2.0), $MachinePrecision] / N[(t - -1.0), $MachinePrecision]), $MachinePrecision], 1.9999999999997489], N[(N[(N[(N[(t * t), $MachinePrecision] / N[(N[(-1.0 - t), $MachinePrecision] * N[(t - -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -4.0 + 1.0), $MachinePrecision] / N[(N[(N[(t * t), $MachinePrecision] / N[(N[(-2.0 - t), $MachinePrecision] * t + -1.0), $MachinePrecision]), $MachinePrecision] * -4.0 + 2.0), $MachinePrecision]), $MachinePrecision], N[(0.8333333333333334 - N[(N[(0.2222222222222222 - N[(N[(0.037037037037037035 + N[(0.04938271604938271 / t), $MachinePrecision]), $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)) < 1.99999999999974887

   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

   4. Applied rewrites 100.0%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. metadata-eval N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

   6. Applied rewrites 100.0%

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in t around 0

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

       1. sub-neg N/A

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

       2. *-commutative N/A

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

       3. mul-1-neg N/A

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

       4. neg-sub0 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{t \cdot t}{\left(-1 - t\right) \cdot \left(t + 1\right)}, -4, 1\right)}{\mathsf{fma}\left(\frac{t \cdot t}{\left(\color{blue}{\left(0 - t\right)} - 2\right) \cdot t + \left(\mathsf{neg}\left(1\right)\right)}, -4, 2\right)} \]

       5. associate--l- N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{t \cdot t}{\left(-1 - t\right) \cdot \left(t + 1\right)}, -4, 1\right)}{\mathsf{fma}\left(\frac{t \cdot t}{\color{blue}{\left(0 - \left(t + 2\right)\right)} \cdot t + \left(\mathsf{neg}\left(1\right)\right)}, -4, 2\right)} \]

       6. *-lft-identity N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{t \cdot t}{\left(-1 - t\right) \cdot \left(t + 1\right)}, -4, 1\right)}{\mathsf{fma}\left(\frac{t \cdot t}{\left(0 - \left(\color{blue}{1 \cdot t} + 2\right)\right) \cdot t + \left(\mathsf{neg}\left(1\right)\right)}, -4, 2\right)} \]

       7. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{t \cdot t}{\left(-1 - t\right) \cdot \left(t + 1\right)}, -4, 1\right)}{\mathsf{fma}\left(\frac{t \cdot t}{\left(0 - \left(1 \cdot t + \color{blue}{2 \cdot 1}\right)\right) \cdot t + \left(\mathsf{neg}\left(1\right)\right)}, -4, 2\right)} \]

       8. lft-mult-inverse N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{t \cdot t}{\left(-1 - t\right) \cdot \left(t + 1\right)}, -4, 1\right)}{\mathsf{fma}\left(\frac{t \cdot t}{\left(0 - \left(1 \cdot t + 2 \cdot \color{blue}{\left(\frac{1}{t} \cdot t\right)}\right)\right) \cdot t + \left(\mathsf{neg}\left(1\right)\right)}, -4, 2\right)} \]

       9. associate-*l* N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{t \cdot t}{\left(-1 - t\right) \cdot \left(t + 1\right)}, -4, 1\right)}{\mathsf{fma}\left(\frac{t \cdot t}{\left(0 - \left(1 \cdot t + \color{blue}{\left(2 \cdot \frac{1}{t}\right) \cdot t}\right)\right) \cdot t + \left(\mathsf{neg}\left(1\right)\right)}, -4, 2\right)} \]

       10. distribute-rgt-in N/A

           \[\leadsto \frac{\mathsf{fma}\left(\frac{t \cdot t}{\left(-1 - t\right) \cdot \left(t + 1\right)}, -4, 1\right)}{\mathsf{fma}\left(\frac{t \cdot t}{\left(0 - \color{blue}{t \cdot \left(1 + 2 \cdot \frac{1}{t}\right)}\right) \cdot t + \left(\mathsf{neg}\left(1\right)\right)}, -4, 2\right)} \]

       11. neg-sub0 N/A

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

       12. metadata-eval N/A

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

       13. lower-fma.f64 N/A

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

   11. Applied rewrites 100.0%

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

   if 1.99999999999974887 < (/.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 99.6%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t \cdot 2}{t - -1} \leq 1.9999999999997489:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{t \cdot t}{\left(-1 - t\right) \cdot \left(t - -1\right)}, -4, 1\right)}{\mathsf{fma}\left(\frac{t \cdot t}{\mathsf{fma}\left(-2 - t, t, -1\right)}, -4, 2\right)}\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222 - \frac{0.037037037037037035 + \frac{0.04938271604938271}{t}}{t}}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{t \cdot 2}{t - -1} \leq 5 \cdot 10^{-6}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{t \cdot t}{\mathsf{fma}\left(t - -2, t, 1\right)}, -4, -1\right)}{-2 - \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-16, t, 12\right), t, -8\right), t, 4\right) \cdot t\right) \cdot t}\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222 - \frac{0.037037037037037035 + \frac{0.04938271604938271}{t}}{t}}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (<= (/ (* t 2.0) (- t -1.0)) 5e-6)
   (/
    (fma (/ (* t t) (fma (- t -2.0) t 1.0)) -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
      (/ (+ 0.037037037037037035 (/ 0.04938271604938271 t)) t))
     t))))

C

double code(double t) {
	double tmp;
	if (((t * 2.0) / (t - -1.0)) <= 5e-6) {
		tmp = fma(((t * t) / fma((t - -2.0), t, 1.0)), -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 - ((0.037037037037037035 + (0.04938271604938271 / t)) / t)) / t);
	}
	return tmp;
}

Julia

function code(t)
	tmp = 0.0
	if (Float64(Float64(t * 2.0) / Float64(t - -1.0)) <= 5e-6)
		tmp = Float64(fma(Float64(Float64(t * t) / fma(Float64(t - -2.0), t, 1.0)), -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(0.037037037037037035 + Float64(0.04938271604938271 / t)) / t)) / t));
	end
	return tmp
end

Wolfram

code[t_] := If[LessEqual[N[(N[(t * 2.0), $MachinePrecision] / N[(t - -1.0), $MachinePrecision]), $MachinePrecision], 5e-6], N[(N[(N[(N[(t * t), $MachinePrecision] / N[(N[(t - -2.0), $MachinePrecision] * t + 1.0), $MachinePrecision]), $MachinePrecision] * -4.0 + -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[(0.037037037037037035 + N[(0.04938271604938271 / t), $MachinePrecision]), $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)) < 5.00000000000000041e-6

   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

   4. Applied rewrites 100.0%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. metadata-eval N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

   6. Applied rewrites 100.0%

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

   7. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. sub-neg N/A

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

      10. *-commutative N/A

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

      11. metadata-eval N/A

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

      12. lower-fma.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-fma.f64 99.5

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

   9. Applied rewrites 99.5%

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

   10. Taylor expanded in t around 0

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

       1. +-commutative N/A

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

       2. +-commutative N/A

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

       3. distribute-lft-in N/A

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

       4. metadata-eval N/A

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

       5. lft-mult-inverse N/A

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

       6. associate-*l* N/A

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

       7. *-commutative N/A

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

       8. distribute-rgt-out N/A

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

       9. *-lft-identity N/A

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

       10. distribute-rgt-in N/A

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

       11. *-commutative N/A

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

       12. lower-fma.f64 N/A

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

       13. distribute-rgt-in N/A

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

       14. *-lft-identity N/A

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

       15. associate-*l* N/A

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

       16. lft-mult-inverse N/A

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

       17. metadata-eval N/A

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

       18. metadata-eval N/A

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

       19. sub-neg N/A

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

       20. lower--.f64 99.5

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

   12. Applied rewrites 99.5%

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

   if 5.00000000000000041e-6 < (/.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. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t \cdot 2}{t - -1} \leq 5 \cdot 10^{-6}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{t \cdot t}{\mathsf{fma}\left(t - -2, t, 1\right)}, -4, -1\right)}{-2 - \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-16, t, 12\right), t, -8\right), t, 4\right) \cdot t\right) \cdot t}\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222 - \frac{0.037037037037037035 + \frac{0.04938271604938271}{t}}{t}}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{t \cdot 2}{t - -1} \leq 5 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(t - 2, t, 1\right), t \cdot t, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222 - \frac{0.037037037037037035 + \frac{0.04938271604938271}{t}}{t}}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (<= (/ (* t 2.0) (- t -1.0)) 5e-6)
   (fma (fma (- t 2.0) t 1.0) (* t t) 0.5)
   (-
    0.8333333333333334
    (/
     (-
      0.2222222222222222
      (/ (+ 0.037037037037037035 (/ 0.04938271604938271 t)) t))
     t))))

C

double code(double t) {
	double tmp;
	if (((t * 2.0) / (t - -1.0)) <= 5e-6) {
		tmp = fma(fma((t - 2.0), t, 1.0), (t * t), 0.5);
	} else {
		tmp = 0.8333333333333334 - ((0.2222222222222222 - ((0.037037037037037035 + (0.04938271604938271 / t)) / t)) / t);
	}
	return tmp;
}

Julia

function code(t)
	tmp = 0.0
	if (Float64(Float64(t * 2.0) / Float64(t - -1.0)) <= 5e-6)
		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(0.037037037037037035 + Float64(0.04938271604938271 / t)) / t)) / t));
	end
	return tmp
end

Wolfram

code[t_] := If[LessEqual[N[(N[(t * 2.0), $MachinePrecision] / N[(t - -1.0), $MachinePrecision]), $MachinePrecision], 5e-6], 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[(0.037037037037037035 + N[(0.04938271604938271 / t), $MachinePrecision]), $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)) < 5.00000000000000041e-6

   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 99.5

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

   5. Applied rewrites 99.5%

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

   if 5.00000000000000041e-6 < (/.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. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t \cdot 2}{t - -1} \leq 5 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(t - 2, t, 1\right), t \cdot t, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222 - \frac{0.037037037037037035 + \frac{0.04938271604938271}{t}}{t}}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{t \cdot 2}{t - -1} \leq 5 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(t - 2, t, 1\right), t \cdot t, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334 - \frac{\mathsf{fma}\left(0.2222222222222222, t, -0.037037037037037035\right)}{t \cdot t}\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (<= (/ (* t 2.0) (- t -1.0)) 5e-6)
   (fma (fma (- t 2.0) t 1.0) (* t t) 0.5)
   (-
    0.8333333333333334
    (/ (fma 0.2222222222222222 t -0.037037037037037035) (* t t)))))

C

double code(double t) {
	double tmp;
	if (((t * 2.0) / (t - -1.0)) <= 5e-6) {
		tmp = fma(fma((t - 2.0), t, 1.0), (t * t), 0.5);
	} else {
		tmp = 0.8333333333333334 - (fma(0.2222222222222222, t, -0.037037037037037035) / (t * t));
	}
	return tmp;
}

Julia

function code(t)
	tmp = 0.0
	if (Float64(Float64(t * 2.0) / Float64(t - -1.0)) <= 5e-6)
		tmp = fma(fma(Float64(t - 2.0), t, 1.0), Float64(t * t), 0.5);
	else
		tmp = Float64(0.8333333333333334 - Float64(fma(0.2222222222222222, t, -0.037037037037037035) / Float64(t * t)));
	end
	return tmp
end

Wolfram

code[t_] := If[LessEqual[N[(N[(t * 2.0), $MachinePrecision] / N[(t - -1.0), $MachinePrecision]), $MachinePrecision], 5e-6], 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 * t + -0.037037037037037035), $MachinePrecision] / N[(t * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) t)) < 5.00000000000000041e-6

   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 99.5

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

   5. Applied rewrites 99.5%

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

   if 5.00000000000000041e-6 < (/.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 0

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

   5. Applied rewrites 19.7%

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

   6. Taylor expanded in t around inf

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

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

      14. *-inverses N/A

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

      15. associate-/l* N/A

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

      16. associate-*r/ N/A

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

      17. metadata-eval N/A

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

      18. div-sub N/A

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

      19. associate-/r* N/A

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

      20. unpow2 N/A

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

      21. lower-/.f64 N/A

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

   8. Applied rewrites 99.3%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t \cdot 2}{t - -1} \leq 5 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(t - 2, t, 1\right), t \cdot t, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334 - \frac{\mathsf{fma}\left(0.2222222222222222, t, -0.037037037037037035\right)}{t \cdot t}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.0% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{t \cdot 2}{t - -1} \leq 5 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(t - 2, t, 1\right), t \cdot t, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (<= (/ (* t 2.0) (- t -1.0)) 5e-6)
   (fma (fma (- t 2.0) t 1.0) (* t t) 0.5)
   (- 0.8333333333333334 (/ 0.2222222222222222 t))))

C

double code(double t) {
	double tmp;
	if (((t * 2.0) / (t - -1.0)) <= 5e-6) {
		tmp = fma(fma((t - 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(t * 2.0) / Float64(t - -1.0)) <= 5e-6)
		tmp = fma(fma(Float64(t - 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[(t * 2.0), $MachinePrecision] / N[(t - -1.0), $MachinePrecision]), $MachinePrecision], 5e-6], N[(N[(N[(t - 2.0), $MachinePrecision] * 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)) < 5.00000000000000041e-6

   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 99.5

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

   5. Applied rewrites 99.5%

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

   if 5.00000000000000041e-6 < (/.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 99.1

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

   5. Applied rewrites 99.1%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t \cdot 2}{t - -1} \leq 5 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(t - 2, t, 1\right), t \cdot t, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 99.0% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{t \cdot 2}{t - -1} \leq 5 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-2, t, 1\right), t \cdot t, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (<= (/ (* t 2.0) (- t -1.0)) 5e-6)
   (fma (fma -2.0 t 1.0) (* t t) 0.5)
   (- 0.8333333333333334 (/ 0.2222222222222222 t))))

C

double code(double t) {
	double tmp;
	if (((t * 2.0) / (t - -1.0)) <= 5e-6) {
		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(t * 2.0) / Float64(t - -1.0)) <= 5e-6)
		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[(t * 2.0), $MachinePrecision] / N[(t - -1.0), $MachinePrecision]), $MachinePrecision], 5e-6], 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)) < 5.00000000000000041e-6

   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 99.5

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

   5. Applied rewrites 99.5%

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

   if 5.00000000000000041e-6 < (/.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 99.1

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

   5. Applied rewrites 99.1%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t \cdot 2}{t - -1} \leq 5 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-2, t, 1\right), t \cdot t, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 99.0% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{t \cdot 2}{t - -1} \leq 5 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(t, t, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (<= (/ (* t 2.0) (- t -1.0)) 5e-6)
   (fma t t 0.5)
   (- 0.8333333333333334 (/ 0.2222222222222222 t))))

C

double code(double t) {
	double tmp;
	if (((t * 2.0) / (t - -1.0)) <= 5e-6) {
		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(t * 2.0) / Float64(t - -1.0)) <= 5e-6)
		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[(t * 2.0), $MachinePrecision] / N[(t - -1.0), $MachinePrecision]), $MachinePrecision], 5e-6], 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)) < 5.00000000000000041e-6

   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 99.4

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

   5. Applied rewrites 99.4%

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

   if 5.00000000000000041e-6 < (/.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 99.1

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

   5. Applied rewrites 99.1%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t \cdot 2}{t - -1} \leq 5 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(t, t, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 98.5% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{t \cdot 2}{t - -1} \leq 5 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(t, t, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (<= (/ (* t 2.0) (- t -1.0)) 5e-6) (fma t t 0.5) 0.8333333333333334))

C

double code(double t) {
	double tmp;
	if (((t * 2.0) / (t - -1.0)) <= 5e-6) {
		tmp = fma(t, t, 0.5);
	} else {
		tmp = 0.8333333333333334;
	}
	return tmp;
}

Julia

function code(t)
	tmp = 0.0
	if (Float64(Float64(t * 2.0) / Float64(t - -1.0)) <= 5e-6)
		tmp = fma(t, t, 0.5);
	else
		tmp = 0.8333333333333334;
	end
	return tmp
end

Wolfram

code[t_] := If[LessEqual[N[(N[(t * 2.0), $MachinePrecision] / N[(t - -1.0), $MachinePrecision]), $MachinePrecision], 5e-6], 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)) < 5.00000000000000041e-6

   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 99.4

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

   5. Applied rewrites 99.4%

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

   if 5.00000000000000041e-6 < (/.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 97.9%

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

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t \cdot 2}{t - -1} \leq 5 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(t, t, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 98.5% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[t_] := If[LessEqual[N[(N[(t * 2.0), $MachinePrecision] / N[(t - -1.0), $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 99.2%

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

   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 97.9%

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

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t \cdot 2}{t - -1} \leq 1:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 59.1% 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 56.0%

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

Reproduce

herbie shell --seed 2024240 
(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))))))