Kahan p13 Example 1

Percentage Accurate: 99.9% → 99.9%

Time: 9.2s

Alternatives: 9

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

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

Alternative 2: 99.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t \cdot 4}{\mathsf{fma}\left(t, 2 + t, 1\right)}\\ \mathbf{if}\;\frac{2 \cdot t}{1 + t} \leq 1.9999998:\\ \;\;\;\;\frac{\mathsf{fma}\left(t, t\_1, 1\right)}{\mathsf{fma}\left(t, t\_1, 2\right)}\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334 + \frac{\frac{0.037037037037037035 + \frac{0.04938271604938271}{t}}{t} + -0.2222222222222222}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (let* ((t_1 (/ (* t 4.0) (fma t (+ 2.0 t) 1.0))))
   (if (<= (/ (* 2.0 t) (+ 1.0 t)) 1.9999998)
     (/ (fma t t_1 1.0) (fma t t_1 2.0))
     (+
      0.8333333333333334
      (/
       (+
        (/ (+ 0.037037037037037035 (/ 0.04938271604938271 t)) t)
        -0.2222222222222222)
       t)))))

C

double code(double t) {
	double t_1 = (t * 4.0) / fma(t, (2.0 + t), 1.0);
	double tmp;
	if (((2.0 * t) / (1.0 + t)) <= 1.9999998) {
		tmp = fma(t, t_1, 1.0) / fma(t, t_1, 2.0);
	} else {
		tmp = 0.8333333333333334 + ((((0.037037037037037035 + (0.04938271604938271 / t)) / t) + -0.2222222222222222) / t);
	}
	return tmp;
}

Julia

function code(t)
	t_1 = Float64(Float64(t * 4.0) / fma(t, Float64(2.0 + t), 1.0))
	tmp = 0.0
	if (Float64(Float64(2.0 * t) / Float64(1.0 + t)) <= 1.9999998)
		tmp = Float64(fma(t, t_1, 1.0) / fma(t, t_1, 2.0));
	else
		tmp = Float64(0.8333333333333334 + Float64(Float64(Float64(Float64(0.037037037037037035 + Float64(0.04938271604938271 / t)) / t) + -0.2222222222222222) / t));
	end
	return tmp
end

Wolfram

code[t_] := Block[{t$95$1 = N[(N[(t * 4.0), $MachinePrecision] / N[(t * N[(2.0 + t), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(2.0 * t), $MachinePrecision] / N[(1.0 + t), $MachinePrecision]), $MachinePrecision], 1.9999998], N[(N[(t * t$95$1 + 1.0), $MachinePrecision] / N[(t * t$95$1 + 2.0), $MachinePrecision]), $MachinePrecision], N[(0.8333333333333334 + N[(N[(N[(N[(0.037037037037037035 + N[(0.04938271604938271 / t), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision] + -0.2222222222222222), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) t)) < 1.99999980000000011

   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

   4. Applied rewrites 100.0%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 100.0

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-/.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-/l* N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-/.f64 100.0

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

   6. Applied rewrites 100.0%

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

   7. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

      3. rgt-mult-inverse N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. *-rgt-identity N/A

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

      7. distribute-lft-in N/A

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

      8. +-commutative N/A

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

      9. associate-*l* N/A

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

      10. unpow2 N/A

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

      11. unpow2 N/A

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

      12. associate-*l* N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

   9. Applied rewrites 100.0%

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

   10. Taylor expanded in t around 0

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

       1. +-commutative N/A

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

       2. metadata-eval N/A

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

       3. lft-mult-inverse N/A

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

       4. associate-*l* N/A

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

       5. distribute-lft1-in N/A

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

       6. +-commutative N/A

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

       7. lower-fma.f64 N/A

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

       8. +-commutative N/A

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

       9. distribute-rgt1-in N/A

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

       10. associate-*l* N/A

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

       11. lft-mult-inverse N/A

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

       12. metadata-eval N/A

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

       13. lower-+.f64 100.0

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

   12. Applied rewrites 100.0%

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

   if 1.99999980000000011 < (/.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 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2 \cdot t}{1 + t} \leq 1.9999998:\\ \;\;\;\;\frac{\mathsf{fma}\left(t, \frac{t \cdot 4}{\mathsf{fma}\left(t, 2 + t, 1\right)}, 1\right)}{\mathsf{fma}\left(t, \frac{t \cdot 4}{\mathsf{fma}\left(t, 2 + t, 1\right)}, 2\right)}\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334 + \frac{\frac{0.037037037037037035 + \frac{0.04938271604938271}{t}}{t} + -0.2222222222222222}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.1% accurate, 1.5× speedup

Math

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

FPCore

(FPCore (t)
 :precision binary64
 (if (<= (/ (* 2.0 t) (+ 1.0 t)) 1e-8)
   (fma (* t t) (fma t -2.0 1.0) 0.5)
   (+
    0.8333333333333334
    (/
     (+
      (/ (+ 0.037037037037037035 (/ 0.04938271604938271 t)) t)
      -0.2222222222222222)
     t))))

C

double code(double t) {
	double tmp;
	if (((2.0 * t) / (1.0 + t)) <= 1e-8) {
		tmp = fma((t * t), fma(t, -2.0, 1.0), 0.5);
	} else {
		tmp = 0.8333333333333334 + ((((0.037037037037037035 + (0.04938271604938271 / t)) / t) + -0.2222222222222222) / t);
	}
	return tmp;
}

Julia

function code(t)
	tmp = 0.0
	if (Float64(Float64(2.0 * t) / Float64(1.0 + t)) <= 1e-8)
		tmp = fma(Float64(t * t), fma(t, -2.0, 1.0), 0.5);
	else
		tmp = Float64(0.8333333333333334 + Float64(Float64(Float64(Float64(0.037037037037037035 + Float64(0.04938271604938271 / t)) / t) + -0.2222222222222222) / t));
	end
	return tmp
end

Wolfram

code[t_] := If[LessEqual[N[(N[(2.0 * t), $MachinePrecision] / N[(1.0 + t), $MachinePrecision]), $MachinePrecision], 1e-8], N[(N[(t * t), $MachinePrecision] * N[(t * -2.0 + 1.0), $MachinePrecision] + 0.5), $MachinePrecision], N[(0.8333333333333334 + N[(N[(N[(N[(0.037037037037037035 + N[(0.04938271604938271 / t), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision] + -0.2222222222222222), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) t)) < 1e-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. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 100.0

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

   5. Applied rewrites 100.0%

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

   if 1e-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 99.3%

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

Alternative 4: 99.1% accurate, 2.0× speedup

Math

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

FPCore

(FPCore (t)
 :precision binary64
 (if (<= (/ (* 2.0 t) (+ 1.0 t)) 1e-8)
   (fma (* t t) (fma t -2.0 1.0) 0.5)
   (+
    0.8333333333333334
    (/ (fma t -0.2222222222222222 0.037037037037037035) (* t t)))))

C

double code(double t) {
	double tmp;
	if (((2.0 * t) / (1.0 + t)) <= 1e-8) {
		tmp = fma((t * t), fma(t, -2.0, 1.0), 0.5);
	} else {
		tmp = 0.8333333333333334 + (fma(t, -0.2222222222222222, 0.037037037037037035) / (t * t));
	}
	return tmp;
}

Julia

function code(t)
	tmp = 0.0
	if (Float64(Float64(2.0 * t) / Float64(1.0 + t)) <= 1e-8)
		tmp = fma(Float64(t * t), fma(t, -2.0, 1.0), 0.5);
	else
		tmp = Float64(0.8333333333333334 + Float64(fma(t, -0.2222222222222222, 0.037037037037037035) / Float64(t * t)));
	end
	return tmp
end

Wolfram

code[t_] := If[LessEqual[N[(N[(2.0 * t), $MachinePrecision] / N[(1.0 + t), $MachinePrecision]), $MachinePrecision], 1e-8], N[(N[(t * t), $MachinePrecision] * N[(t * -2.0 + 1.0), $MachinePrecision] + 0.5), $MachinePrecision], N[(0.8333333333333334 + N[(N[(t * -0.2222222222222222 + 0.037037037037037035), $MachinePrecision] / N[(t * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) t)) < 1e-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. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 100.0

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

   5. Applied rewrites 100.0%

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

   if 1e-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

   4. Applied rewrites 46.1%

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

   5. Taylor expanded in t around inf

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

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

      13. mul-1-neg N/A

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

      14. lower-+.f64 N/A

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

      15. associate-*r/ N/A

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

   7. Applied rewrites 99.0%

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

   8. Taylor expanded in t around 0

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

   10. Applied rewrites 99.0%

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

Alternative 5: 98.9% accurate, 2.4× speedup

Math

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

FPCore

(FPCore (t)
 :precision binary64
 (if (<= (/ (* 2.0 t) (+ 1.0 t)) 1e-8)
   (fma (* t t) (fma t -2.0 1.0) 0.5)
   (+ 0.8333333333333334 (/ -0.2222222222222222 t))))

C

double code(double t) {
	double tmp;
	if (((2.0 * t) / (1.0 + t)) <= 1e-8) {
		tmp = fma((t * t), fma(t, -2.0, 1.0), 0.5);
	} else {
		tmp = 0.8333333333333334 + (-0.2222222222222222 / t);
	}
	return tmp;
}

Julia

function code(t)
	tmp = 0.0
	if (Float64(Float64(2.0 * t) / Float64(1.0 + t)) <= 1e-8)
		tmp = fma(Float64(t * t), fma(t, -2.0, 1.0), 0.5);
	else
		tmp = Float64(0.8333333333333334 + Float64(-0.2222222222222222 / t));
	end
	return tmp
end

Wolfram

code[t_] := If[LessEqual[N[(N[(2.0 * t), $MachinePrecision] / N[(1.0 + t), $MachinePrecision]), $MachinePrecision], 1e-8], N[(N[(t * t), $MachinePrecision] * N[(t * -2.0 + 1.0), $MachinePrecision] + 0.5), $MachinePrecision], N[(0.8333333333333334 + N[(-0.2222222222222222 / t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) t)) < 1e-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. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 100.0

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

   5. Applied rewrites 100.0%

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

   if 1e-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. sub-neg N/A

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

      2. lower-+.f64 N/A

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

      5. distribute-neg-frac N/A

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

      6. lower-/.f64 N/A

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

      7. metadata-eval 98.5

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

   5. Applied rewrites 98.5%

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

Alternative 6: 98.8% accurate, 2.6× speedup

Math

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

FPCore

(FPCore (t)
 :precision binary64
 (if (<= (/ (* 2.0 t) (+ 1.0 t)) 1e-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)) <= 1e-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)) <= 1e-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], 1e-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)) < 1e-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 1e-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. sub-neg N/A

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

      2. lower-+.f64 N/A

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

      5. distribute-neg-frac N/A

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

      6. lower-/.f64 N/A

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

      7. metadata-eval 98.5

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

   5. Applied rewrites 98.5%

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

Alternative 7: 98.5% accurate, 3.2× speedup

Math

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

FPCore

(FPCore (t)
 :precision binary64
 (if (<= (/ (* 2.0 t) (+ 1.0 t)) 1e-8) (fma t t 0.5) 0.8333333333333334))

C

double code(double t) {
	double tmp;
	if (((2.0 * t) / (1.0 + t)) <= 1e-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)) <= 1e-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], 1e-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)) < 1e-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 1e-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 96.6%

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

Alternative 8: 98.6% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in t around 0

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

   5. Applied rewrites 99.6%

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

   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 96.6%

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

Alternative 9: 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 63.7%

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

Reproduce

herbie shell --seed 2024233 
(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))))))