Kahan p13 Example 3

Percentage Accurate: 100.0% → 100.0%

Time: 11.3s

Alternatives: 14

Speedup: 1.8×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

   \[1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lower-fma.f64 100.0

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

4. Applied rewrites 100.0%

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

6. Applied rewrites 100.0%

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

7. Final simplification 100.0%

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

Alternative 2: 99.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\frac{2}{t}}{1 - \frac{-1}{t}} \leq 2 \cdot 10^{-6}:\\ \;\;\;\;1 - \left(\frac{\frac{-0.037037037037037035 - \frac{0.04938271604938271}{t}}{t} - -0.2222222222222222}{t} + 0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{1}{\left(t \cdot t\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-16, t, 12\right), t, -8\right), t, 4\right) + 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (<= (/ (/ 2.0 t) (- 1.0 (/ -1.0 t))) 2e-6)
   (-
    1.0
    (+
     (/
      (-
       (/ (- -0.037037037037037035 (/ 0.04938271604938271 t)) t)
       -0.2222222222222222)
      t)
     0.16666666666666666))
   (-
    1.0
    (/ 1.0 (+ (* (* t t) (fma (fma (fma -16.0 t 12.0) t -8.0) t 4.0)) 2.0)))))

C

double code(double t) {
	double tmp;
	if (((2.0 / t) / (1.0 - (-1.0 / t))) <= 2e-6) {
		tmp = 1.0 - (((((-0.037037037037037035 - (0.04938271604938271 / t)) / t) - -0.2222222222222222) / t) + 0.16666666666666666);
	} else {
		tmp = 1.0 - (1.0 / (((t * t) * fma(fma(fma(-16.0, t, 12.0), t, -8.0), t, 4.0)) + 2.0));
	}
	return tmp;
}

Julia

function code(t)
	tmp = 0.0
	if (Float64(Float64(2.0 / t) / Float64(1.0 - Float64(-1.0 / t))) <= 2e-6)
		tmp = Float64(1.0 - Float64(Float64(Float64(Float64(Float64(-0.037037037037037035 - Float64(0.04938271604938271 / t)) / t) - -0.2222222222222222) / t) + 0.16666666666666666));
	else
		tmp = Float64(1.0 - Float64(1.0 / Float64(Float64(Float64(t * t) * fma(fma(fma(-16.0, t, 12.0), t, -8.0), t, 4.0)) + 2.0)));
	end
	return tmp
end

Wolfram

code[t_] := If[LessEqual[N[(N[(2.0 / t), $MachinePrecision] / N[(1.0 - N[(-1.0 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2e-6], N[(1.0 - N[(N[(N[(N[(N[(-0.037037037037037035 - N[(0.04938271604938271 / t), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision] - -0.2222222222222222), $MachinePrecision] / t), $MachinePrecision] + 0.16666666666666666), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(1.0 / N[(N[(N[(t * t), $MachinePrecision] * N[(N[(N[(-16.0 * t + 12.0), $MachinePrecision] * t + -8.0), $MachinePrecision] * t + 4.0), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (/.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) t))) < 1.99999999999999991e-6

   1. Initial program 100.0%

      \[1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto 1 - \color{blue}{\left(\frac{1}{6} + \left(-1 \cdot \frac{\frac{1}{27} + \frac{4}{81} \cdot \frac{1}{t}}{{t}^{2}} + \frac{2}{9} \cdot \frac{1}{t}\right)\right)} \]

   5. Applied rewrites 99.8%

      \[\leadsto 1 - \color{blue}{\left(0.16666666666666666 - \frac{-0.2222222222222222 - \frac{-0.037037037037037035 - \frac{0.04938271604938271}{t}}{t}}{t}\right)} \]

   if 1.99999999999999991e-6 < (/.f64 (/.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) t)))

   1. Initial program 100.0%

      \[1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. sub-neg N/A

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

      7. metadata-eval N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 98.6

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

   5. Applied rewrites 98.6%

      \[\leadsto 1 - \frac{1}{2 + \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-16, t, 12\right), t, -8\right), t, 4\right) \cdot \left(t \cdot t\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{2}{t}}{1 - \frac{-1}{t}} \leq 2 \cdot 10^{-6}:\\ \;\;\;\;1 - \left(\frac{\frac{-0.037037037037037035 - \frac{0.04938271604938271}{t}}{t} - -0.2222222222222222}{t} + 0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{1}{\left(t \cdot t\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-16, t, 12\right), t, -8\right), t, 4\right) + 2}\\ \end{array} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024230 
(FPCore (t)
  :name "Kahan p13 Example 3"
  :precision binary64
  (- 1.0 (/ 1.0 (+ 2.0 (* (- 2.0 (/ (/ 2.0 t) (+ 1.0 (/ 1.0 t)))) (- 2.0 (/ (/ 2.0 t) (+ 1.0 (/ 1.0 t)))))))))