Kahan p13 Example 2

Percentage Accurate: 99.9% → 99.9%

Time: 1.2min

Alternatives: 7

Speedup: N/A×

Specification

Math

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

FPCore

(FPCore (t)
 :precision binary64
 (let* ((t_1 (- 2.0 (/ (/ 2.0 t) (+ 1.0 (/ 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 - ((2.0 / t) / (1.0 + (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 - ((2.0d0 / t) / (1.0d0 + (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 - ((2.0 / t) / (1.0 + (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 - ((2.0 / t) / (1.0 + (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(2.0 - Float64(Float64(2.0 / t) / Float64(1.0 + 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 - ((2.0 / t) / (1.0 + (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[(2.0 - N[(N[(2.0 / t), $MachinePrecision] / N[(1.0 + N[(1.0 / t), $MachinePrecision]), $MachinePrecision]), $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

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

FPCore

(FPCore (t)
 :precision binary64
 (let* ((t_1 (- 2.0 (/ (/ 2.0 t) (+ 1.0 (/ 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 - ((2.0 / t) / (1.0 + (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 - ((2.0d0 / t) / (1.0d0 + (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 - ((2.0 / t) / (1.0 + (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 - ((2.0 / t) / (1.0 + (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(2.0 - Float64(Float64(2.0 / t) / Float64(1.0 + 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 - ((2.0 / t) / (1.0 + (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[(2.0 - N[(N[(2.0 / t), $MachinePrecision] / N[(1.0 + N[(1.0 / t), $MachinePrecision]), $MachinePrecision]), $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, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 1 + \frac{1}{t}\\ t_2 := 2 - \frac{\frac{2}{t}}{t\_1}\\ \frac{1 + \mathsf{fma}\left(\frac{-2}{t}, \frac{1}{t\_1}, 2\right) \cdot t\_2}{2 + t\_2 \cdot \left(2 - \frac{2}{1 + t}\right)} \end{array} \end{array} \]

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. sub-neg 100.0%

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

   2. +-commutative 100.0%

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

   3. div-inv 100.0%

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

   4. distribute-lft-neg-in 100.0%

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

   5. fma-define 100.0%

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

   6. distribute-neg-frac 100.0%

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

   7. metadata-eval 100.0%

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

4. Applied egg-rr 100.0%

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

   1. add-log-exp 100.0%

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

   2. *-un-lft-identity 100.0%

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

   3. log-prod 100.0%

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

   4. metadata-eval 100.0%

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

   5. add-log-exp 100.0%

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

   6. associate-/l/ 100.0%

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

   7. *-commutative 100.0%

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

6. Applied egg-rr 100.0%

   \[\leadsto \frac{1 + \mathsf{fma}\left(\frac{-2}{t}, \frac{1}{1 + \frac{1}{t}}, 2\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \color{blue}{\left(0 + \frac{2}{t \cdot \left(1 + \frac{1}{t}\right)}\right)}\right)} \]
7. Step-by-step derivation

   1. +-lft-identity 100.0%

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

   2. distribute-lft-in 100.0%

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

   3. *-rgt-identity 100.0%

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

   4. rgt-mult-inverse 100.0%

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

8. Simplified 100.0%

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

9. Final simplification 100.0%

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

Alternative 2: 99.9% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

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 + (t_1 * t_1)) / (2.0d0 + (t_1 * (2.0d0 - (2.0d0 / (1.0d0 + t)))))
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 + (t_1 * t_1)) / (2.0 + (t_1 * (2.0 - (2.0 / (1.0 + t)))));
}

Python

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

Julia

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

MATLAB

function tmp = code(t)
	t_1 = 2.0 - ((2.0 / t) / (1.0 + (1.0 / t)));
	tmp = (1.0 + (t_1 * t_1)) / (2.0 + (t_1 * (2.0 - (2.0 / (1.0 + t)))));
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[(N[(1.0 + N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision] / N[(2.0 + N[(t$95$1 * N[(2.0 - N[(2.0 / N[(1.0 + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 100.0%

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

   1. add-log-exp 100.0%

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

   2. *-un-lft-identity 100.0%

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

   3. log-prod 100.0%

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

   4. metadata-eval 100.0%

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

   5. add-log-exp 100.0%

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

   6. associate-/l/ 100.0%

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

   7. *-commutative 100.0%

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

4. Applied egg-rr 100.0%

   \[\leadsto \frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \color{blue}{\left(0 + \frac{2}{t \cdot \left(1 + \frac{1}{t}\right)}\right)}\right)} \]
5. Step-by-step derivation

   1. +-lft-identity 100.0%

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

   2. distribute-lft-in 100.0%

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

   3. *-rgt-identity 100.0%

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

   4. rgt-mult-inverse 100.0%

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

6. Simplified 100.0%

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

7. Final simplification 100.0%

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

Alternative 3: 98.0% accurate, 1.6× speedup

Math

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

FPCore

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

C

double code(double t) {
	double t_1 = 2.0 * (2.0 - ((2.0 / t) / (1.0 + (1.0 / t))));
	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 = 2.0d0 * (2.0d0 - ((2.0d0 / t) / (1.0d0 + (1.0d0 / t))))
    code = (1.0d0 + t_1) / (2.0d0 + t_1)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

3. Taylor expanded in t around inf 98.3%

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

4. Taylor expanded in t around inf 98.3%

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

5. Final simplification 98.3%

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

Alternative 4: 97.3% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \left(\left(-4 + \frac{4}{1 + t}\right) + -1\right) \cdot \frac{-0.5}{3 + \frac{2}{-1 - t}} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (* (+ (+ -4.0 (/ 4.0 (+ 1.0 t))) -1.0) (/ -0.5 (+ 3.0 (/ 2.0 (- -1.0 t))))))

C

double code(double t) {
	return ((-4.0 + (4.0 / (1.0 + t))) + -1.0) * (-0.5 / (3.0 + (2.0 / (-1.0 - t))));
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    code = (((-4.0d0) + (4.0d0 / (1.0d0 + t))) + (-1.0d0)) * ((-0.5d0) / (3.0d0 + (2.0d0 / ((-1.0d0) - t))))
end function

Java

public static double code(double t) {
	return ((-4.0 + (4.0 / (1.0 + t))) + -1.0) * (-0.5 / (3.0 + (2.0 / (-1.0 - t))));
}

Python

def code(t):
	return ((-4.0 + (4.0 / (1.0 + t))) + -1.0) * (-0.5 / (3.0 + (2.0 / (-1.0 - t))))

Julia

function code(t)
	return Float64(Float64(Float64(-4.0 + Float64(4.0 / Float64(1.0 + t))) + -1.0) * Float64(-0.5 / Float64(3.0 + Float64(2.0 / Float64(-1.0 - t)))))
end

MATLAB

function tmp = code(t)
	tmp = ((-4.0 + (4.0 / (1.0 + t))) + -1.0) * (-0.5 / (3.0 + (2.0 / (-1.0 - t))));
end

Wolfram

code[t_] := N[(N[(N[(-4.0 + N[(4.0 / N[(1.0 + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision] * N[(-0.5 / N[(3.0 + N[(2.0 / N[(-1.0 - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in t around inf 98.3%

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

4. Taylor expanded in t around inf 98.3%

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

   1. *-un-lft-identity 98.3%

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

   2. +-commutative 98.3%

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

   3. *-commutative 98.3%

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

   4. fma-define 98.3%

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

   5. sub-neg 98.3%

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

   6. distribute-neg-frac 98.3%

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

   7. distribute-neg-frac 98.3%

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

   8. metadata-eval 98.3%

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

   9. +-commutative 98.3%

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

   10. *-commutative 98.3%

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

   11. fma-define 98.3%

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

6. Applied egg-rr 98.3%

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

8. Simplified 97.6%

   \[\leadsto \color{blue}{\left(\left(-4 + \frac{4}{t + 1}\right) + -1\right) \cdot \frac{-0.5}{3 - \frac{2}{t + 1}}} \]

9. Final simplification 97.6%

   \[\leadsto \left(\left(-4 + \frac{4}{1 + t}\right) + -1\right) \cdot \frac{-0.5}{3 + \frac{2}{-1 - t}} \]
10. Add Preprocessing

Alternative 5: 98.6% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -0.44 \lor \neg \left(t \leq 0.33\right):\\ \;\;\;\;0.8333333333333334 - \frac{0.1111111111111111}{t}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (or (<= t -0.44) (not (<= t 0.33)))
   (- 0.8333333333333334 (/ 0.1111111111111111 t))
   0.5))

C

double code(double t) {
	double tmp;
	if ((t <= -0.44) || !(t <= 0.33)) {
		tmp = 0.8333333333333334 - (0.1111111111111111 / t);
	} else {
		tmp = 0.5;
	}
	return tmp;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((t <= (-0.44d0)) .or. (.not. (t <= 0.33d0))) then
        tmp = 0.8333333333333334d0 - (0.1111111111111111d0 / t)
    else
        tmp = 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double tmp;
	if ((t <= -0.44) || !(t <= 0.33)) {
		tmp = 0.8333333333333334 - (0.1111111111111111 / t);
	} else {
		tmp = 0.5;
	}
	return tmp;
}

Python

def code(t):
	tmp = 0
	if (t <= -0.44) or not (t <= 0.33):
		tmp = 0.8333333333333334 - (0.1111111111111111 / t)
	else:
		tmp = 0.5
	return tmp

Julia

function code(t)
	tmp = 0.0
	if ((t <= -0.44) || !(t <= 0.33))
		tmp = Float64(0.8333333333333334 - Float64(0.1111111111111111 / t));
	else
		tmp = 0.5;
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	tmp = 0.0;
	if ((t <= -0.44) || ~((t <= 0.33)))
		tmp = 0.8333333333333334 - (0.1111111111111111 / t);
	else
		tmp = 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := If[Or[LessEqual[t, -0.44], N[Not[LessEqual[t, 0.33]], $MachinePrecision]], N[(0.8333333333333334 - N[(0.1111111111111111 / t), $MachinePrecision]), $MachinePrecision], 0.5]

Derivation

1. Split input into 2 regimes

2. if t < -0.440000000000000002 or 0.330000000000000016 < t

   1. Initial program 100.0%

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

   3. Taylor expanded in t around inf 97.4%

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

   4. Taylor expanded in t around inf 97.5%

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

   5. Taylor expanded in t around inf 97.5%

      \[\leadsto \color{blue}{0.8333333333333334 - 0.1111111111111111 \cdot \frac{1}{t}} \]
   6. Step-by-step derivation

      1. associate-*r/ 97.5%

         \[\leadsto 0.8333333333333334 - \color{blue}{\frac{0.1111111111111111 \cdot 1}{t}} \]

      2. metadata-eval 97.5%

         \[\leadsto 0.8333333333333334 - \frac{\color{blue}{0.1111111111111111}}{t} \]

   7. Simplified 97.5%

      \[\leadsto \color{blue}{0.8333333333333334 - \frac{0.1111111111111111}{t}} \]

   if -0.440000000000000002 < t < 0.330000000000000016

   1. Initial program 100.0%

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

   3. Taylor expanded in t around inf 99.0%

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

   4. Taylor expanded in t around 0 99.6%

      \[\leadsto \color{blue}{0.5} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.44 \lor \neg \left(t \leq 0.33\right):\\ \;\;\;\;0.8333333333333334 - \frac{0.1111111111111111}{t}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 98.6% accurate, 4.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -0.34:\\ \;\;\;\;0.8333333333333334\\ \mathbf{elif}\;t \leq 1:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (<= t -0.34) 0.8333333333333334 (if (<= t 1.0) 0.5 0.8333333333333334)))

C

double code(double t) {
	double tmp;
	if (t <= -0.34) {
		tmp = 0.8333333333333334;
	} else if (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 (t <= (-0.34d0)) then
        tmp = 0.8333333333333334d0
    else if (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 (t <= -0.34) {
		tmp = 0.8333333333333334;
	} else if (t <= 1.0) {
		tmp = 0.5;
	} else {
		tmp = 0.8333333333333334;
	}
	return tmp;
}

Python

def code(t):
	tmp = 0
	if t <= -0.34:
		tmp = 0.8333333333333334
	elif t <= 1.0:
		tmp = 0.5
	else:
		tmp = 0.8333333333333334
	return tmp

Julia

function code(t)
	tmp = 0.0
	if (t <= -0.34)
		tmp = 0.8333333333333334;
	elseif (t <= 1.0)
		tmp = 0.5;
	else
		tmp = 0.8333333333333334;
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	tmp = 0.0;
	if (t <= -0.34)
		tmp = 0.8333333333333334;
	elseif (t <= 1.0)
		tmp = 0.5;
	else
		tmp = 0.8333333333333334;
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := If[LessEqual[t, -0.34], 0.8333333333333334, If[LessEqual[t, 1.0], 0.5, 0.8333333333333334]]

Derivation

1. Split input into 2 regimes

2. if t < -0.340000000000000024 or 1 < t

   1. Initial program 100.0%

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

   3. Taylor expanded in t around inf 97.4%

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

   4. Taylor expanded in t around inf 97.5%

      \[\leadsto \color{blue}{0.8333333333333334} \]

   if -0.340000000000000024 < t < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in t around inf 99.0%

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

   4. Taylor expanded in t around 0 99.6%

      \[\leadsto \color{blue}{0.5} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.34:\\ \;\;\;\;0.8333333333333334\\ \mathbf{elif}\;t \leq 1:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 59.6% accurate, 51.0× 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%

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

3. Taylor expanded in t around inf 98.3%

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

4. Taylor expanded in t around 0 62.5%

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

5. Final simplification 62.5%

   \[\leadsto 0.5 \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024053 
(FPCore (t)
  :name "Kahan p13 Example 2"
  :precision binary64
  (/ (+ 1.0 (* (- 2.0 (/ (/ 2.0 t) (+ 1.0 (/ 1.0 t)))) (- 2.0 (/ (/ 2.0 t) (+ 1.0 (/ 1.0 t)))))) (+ 2.0 (* (- 2.0 (/ (/ 2.0 t) (+ 1.0 (/ 1.0 t)))) (- 2.0 (/ (/ 2.0 t) (+ 1.0 (/ 1.0 t))))))))