Kahan p13 Example 2

Percentage Accurate: 99.9% → 100.0%

Time: 13.6s

Alternatives: 8

Speedup: 1.9×

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: 100.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{2}{1 + t}\\ t_2 := t_1 \cdot \left(t_1 - 4\right)\\ \frac{5 + t_2}{t_2 + 6} \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (let* ((t_1 (/ 2.0 (+ 1.0 t))) (t_2 (* t_1 (- t_1 4.0))))
   (/ (+ 5.0 t_2) (+ t_2 6.0))))

C

double code(double t) {
	double t_1 = 2.0 / (1.0 + t);
	double t_2 = t_1 * (t_1 - 4.0);
	return (5.0 + t_2) / (t_2 + 6.0);
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    t_1 = 2.0d0 / (1.0d0 + t)
    t_2 = t_1 * (t_1 - 4.0d0)
    code = (5.0d0 + t_2) / (t_2 + 6.0d0)
end function

Java

public static double code(double t) {
	double t_1 = 2.0 / (1.0 + t);
	double t_2 = t_1 * (t_1 - 4.0);
	return (5.0 + t_2) / (t_2 + 6.0);
}

Python

def code(t):
	t_1 = 2.0 / (1.0 + t)
	t_2 = t_1 * (t_1 - 4.0)
	return (5.0 + t_2) / (t_2 + 6.0)

Julia

function code(t)
	t_1 = Float64(2.0 / Float64(1.0 + t))
	t_2 = Float64(t_1 * Float64(t_1 - 4.0))
	return Float64(Float64(5.0 + t_2) / Float64(t_2 + 6.0))
end

MATLAB

function tmp = code(t)
	t_1 = 2.0 / (1.0 + t);
	t_2 = t_1 * (t_1 - 4.0);
	tmp = (5.0 + t_2) / (t_2 + 6.0);
end

Wolfram

code[t_] := Block[{t$95$1 = N[(2.0 / N[(1.0 + t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * N[(t$95$1 - 4.0), $MachinePrecision]), $MachinePrecision]}, N[(N[(5.0 + t$95$2), $MachinePrecision] / N[(t$95$2 + 6.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 100.0%

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

3. Simplified 100.0%

   \[\leadsto \color{blue}{\frac{5 + \frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} - 4\right)}{6 + \frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} - 4\right)}} \]

4. Final simplification 100.0%

   \[\leadsto \frac{5 + \frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} - 4\right)}{\frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} - 4\right) + 6} \]

Alternative 2: 100.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{-8 + \frac{4}{1 + t}}{1 + t}\\ \frac{5 + t_1}{6 + t_1} \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (let* ((t_1 (/ (+ -8.0 (/ 4.0 (+ 1.0 t))) (+ 1.0 t))))
   (/ (+ 5.0 t_1) (+ 6.0 t_1))))

C

double code(double t) {
	double t_1 = (-8.0 + (4.0 / (1.0 + t))) / (1.0 + t);
	return (5.0 + t_1) / (6.0 + t_1);
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: t_1
    t_1 = ((-8.0d0) + (4.0d0 / (1.0d0 + t))) / (1.0d0 + t)
    code = (5.0d0 + t_1) / (6.0d0 + t_1)
end function

Java

public static double code(double t) {
	double t_1 = (-8.0 + (4.0 / (1.0 + t))) / (1.0 + t);
	return (5.0 + t_1) / (6.0 + t_1);
}

Python

def code(t):
	t_1 = (-8.0 + (4.0 / (1.0 + t))) / (1.0 + t)
	return (5.0 + t_1) / (6.0 + t_1)

Julia

function code(t)
	t_1 = Float64(Float64(-8.0 + Float64(4.0 / Float64(1.0 + t))) / Float64(1.0 + t))
	return Float64(Float64(5.0 + t_1) / Float64(6.0 + t_1))
end

MATLAB

function tmp = code(t)
	t_1 = (-8.0 + (4.0 / (1.0 + t))) / (1.0 + t);
	tmp = (5.0 + t_1) / (6.0 + t_1);
end

Wolfram

code[t_] := Block[{t$95$1 = N[(N[(-8.0 + N[(4.0 / N[(1.0 + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + t), $MachinePrecision]), $MachinePrecision]}, N[(N[(5.0 + t$95$1), $MachinePrecision] / N[(6.0 + t$95$1), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 100.0%

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

3. Simplified 100.0%

   \[\leadsto \color{blue}{\frac{5 + \frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} - 4\right)}{6 + \frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} - 4\right)}} \]
4. Step-by-step derivation

   1. expm1-log1p-u 99.3%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{5 + \frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} - 4\right)}{6 + \frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} - 4\right)}\right)\right)} \]

   2. expm1-udef 99.3%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{5 + \frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} - 4\right)}{6 + \frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} - 4\right)}\right)} - 1} \]

5. Applied egg-rr 99.3%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\mathsf{fma}\left(\frac{2}{t + 1}, \frac{2}{t + 1} + -4, 5\right)}{\mathsf{fma}\left(\frac{2}{t + 1}, \frac{2}{t + 1} + -4, 6\right)}\right)} - 1} \]
6. Step-by-step derivation

   1. expm1-def 99.3%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\mathsf{fma}\left(\frac{2}{t + 1}, \frac{2}{t + 1} + -4, 5\right)}{\mathsf{fma}\left(\frac{2}{t + 1}, \frac{2}{t + 1} + -4, 6\right)}\right)\right)} \]

   2. expm1-log1p 100.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{2}{t + 1}, \frac{2}{t + 1} + -4, 5\right)}{\mathsf{fma}\left(\frac{2}{t + 1}, \frac{2}{t + 1} + -4, 6\right)}} \]

   3. fma-udef 100.0%

      \[\leadsto \frac{\color{blue}{\frac{2}{t + 1} \cdot \left(\frac{2}{t + 1} + -4\right) + 5}}{\mathsf{fma}\left(\frac{2}{t + 1}, \frac{2}{t + 1} + -4, 6\right)} \]

   4. associate-*l/ 100.0%

      \[\leadsto \frac{\color{blue}{\frac{2 \cdot \left(\frac{2}{t + 1} + -4\right)}{t + 1}} + 5}{\mathsf{fma}\left(\frac{2}{t + 1}, \frac{2}{t + 1} + -4, 6\right)} \]

   5. associate-*r/ 100.0%

      \[\leadsto \frac{\color{blue}{2 \cdot \frac{\frac{2}{t + 1} + -4}{t + 1}} + 5}{\mathsf{fma}\left(\frac{2}{t + 1}, \frac{2}{t + 1} + -4, 6\right)} \]

   6. +-commutative 100.0%

      \[\leadsto \frac{\color{blue}{5 + 2 \cdot \frac{\frac{2}{t + 1} + -4}{t + 1}}}{\mathsf{fma}\left(\frac{2}{t + 1}, \frac{2}{t + 1} + -4, 6\right)} \]

   7. associate-*r/ 100.0%

      \[\leadsto \frac{5 + \color{blue}{\frac{2 \cdot \left(\frac{2}{t + 1} + -4\right)}{t + 1}}}{\mathsf{fma}\left(\frac{2}{t + 1}, \frac{2}{t + 1} + -4, 6\right)} \]

   8. +-commutative 100.0%

      \[\leadsto \frac{5 + \frac{2 \cdot \color{blue}{\left(-4 + \frac{2}{t + 1}\right)}}{t + 1}}{\mathsf{fma}\left(\frac{2}{t + 1}, \frac{2}{t + 1} + -4, 6\right)} \]

   9. distribute-lft-in 100.0%

      \[\leadsto \frac{5 + \frac{\color{blue}{2 \cdot -4 + 2 \cdot \frac{2}{t + 1}}}{t + 1}}{\mathsf{fma}\left(\frac{2}{t + 1}, \frac{2}{t + 1} + -4, 6\right)} \]

   10. metadata-eval 100.0%

       \[\leadsto \frac{5 + \frac{\color{blue}{-8} + 2 \cdot \frac{2}{t + 1}}{t + 1}}{\mathsf{fma}\left(\frac{2}{t + 1}, \frac{2}{t + 1} + -4, 6\right)} \]

   11. associate-*r/ 100.0%

       \[\leadsto \frac{5 + \frac{-8 + \color{blue}{\frac{2 \cdot 2}{t + 1}}}{t + 1}}{\mathsf{fma}\left(\frac{2}{t + 1}, \frac{2}{t + 1} + -4, 6\right)} \]

   12. metadata-eval 100.0%

       \[\leadsto \frac{5 + \frac{-8 + \frac{\color{blue}{4}}{t + 1}}{t + 1}}{\mathsf{fma}\left(\frac{2}{t + 1}, \frac{2}{t + 1} + -4, 6\right)} \]

   13. fma-udef 100.0%

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

7. Simplified 100.0%

   \[\leadsto \color{blue}{\frac{5 + \frac{-8 + \frac{4}{t + 1}}{t + 1}}{6 + \frac{-8 + \frac{4}{t + 1}}{t + 1}}} \]

8. Final simplification 100.0%

   \[\leadsto \frac{5 + \frac{-8 + \frac{4}{1 + t}}{1 + t}}{6 + \frac{-8 + \frac{4}{1 + t}}{1 + t}} \]

Alternative 3: 99.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{-8 + \frac{4}{t}}{1 + t}\\ \mathbf{if}\;t \leq -0.52:\\ \;\;\;\;\frac{5 + t_1}{6 + t_1}\\ \mathbf{elif}\;t \leq 0.55:\\ \;\;\;\;\frac{5 + \left(2 + t \cdot -2\right) \cdot \left(t \cdot -2 - 2\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{5 + \frac{-8}{t}}{6 + \frac{-8}{t}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (let* ((t_1 (/ (+ -8.0 (/ 4.0 t)) (+ 1.0 t))))
   (if (<= t -0.52)
     (/ (+ 5.0 t_1) (+ 6.0 t_1))
     (if (<= t 0.55)
       (/ (+ 5.0 (* (+ 2.0 (* t -2.0)) (- (* t -2.0) 2.0))) 2.0)
       (/ (+ 5.0 (/ -8.0 t)) (+ 6.0 (/ -8.0 t)))))))

C

double code(double t) {
	double t_1 = (-8.0 + (4.0 / t)) / (1.0 + t);
	double tmp;
	if (t <= -0.52) {
		tmp = (5.0 + t_1) / (6.0 + t_1);
	} else if (t <= 0.55) {
		tmp = (5.0 + ((2.0 + (t * -2.0)) * ((t * -2.0) - 2.0))) / 2.0;
	} else {
		tmp = (5.0 + (-8.0 / t)) / (6.0 + (-8.0 / t));
	}
	return tmp;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = ((-8.0d0) + (4.0d0 / t)) / (1.0d0 + t)
    if (t <= (-0.52d0)) then
        tmp = (5.0d0 + t_1) / (6.0d0 + t_1)
    else if (t <= 0.55d0) then
        tmp = (5.0d0 + ((2.0d0 + (t * (-2.0d0))) * ((t * (-2.0d0)) - 2.0d0))) / 2.0d0
    else
        tmp = (5.0d0 + ((-8.0d0) / t)) / (6.0d0 + ((-8.0d0) / t))
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double t_1 = (-8.0 + (4.0 / t)) / (1.0 + t);
	double tmp;
	if (t <= -0.52) {
		tmp = (5.0 + t_1) / (6.0 + t_1);
	} else if (t <= 0.55) {
		tmp = (5.0 + ((2.0 + (t * -2.0)) * ((t * -2.0) - 2.0))) / 2.0;
	} else {
		tmp = (5.0 + (-8.0 / t)) / (6.0 + (-8.0 / t));
	}
	return tmp;
}

Python

def code(t):
	t_1 = (-8.0 + (4.0 / t)) / (1.0 + t)
	tmp = 0
	if t <= -0.52:
		tmp = (5.0 + t_1) / (6.0 + t_1)
	elif t <= 0.55:
		tmp = (5.0 + ((2.0 + (t * -2.0)) * ((t * -2.0) - 2.0))) / 2.0
	else:
		tmp = (5.0 + (-8.0 / t)) / (6.0 + (-8.0 / t))
	return tmp

Julia

function code(t)
	t_1 = Float64(Float64(-8.0 + Float64(4.0 / t)) / Float64(1.0 + t))
	tmp = 0.0
	if (t <= -0.52)
		tmp = Float64(Float64(5.0 + t_1) / Float64(6.0 + t_1));
	elseif (t <= 0.55)
		tmp = Float64(Float64(5.0 + Float64(Float64(2.0 + Float64(t * -2.0)) * Float64(Float64(t * -2.0) - 2.0))) / 2.0);
	else
		tmp = Float64(Float64(5.0 + Float64(-8.0 / t)) / Float64(6.0 + Float64(-8.0 / t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	t_1 = (-8.0 + (4.0 / t)) / (1.0 + t);
	tmp = 0.0;
	if (t <= -0.52)
		tmp = (5.0 + t_1) / (6.0 + t_1);
	elseif (t <= 0.55)
		tmp = (5.0 + ((2.0 + (t * -2.0)) * ((t * -2.0) - 2.0))) / 2.0;
	else
		tmp = (5.0 + (-8.0 / t)) / (6.0 + (-8.0 / t));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := Block[{t$95$1 = N[(N[(-8.0 + N[(4.0 / t), $MachinePrecision]), $MachinePrecision] / N[(1.0 + t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -0.52], N[(N[(5.0 + t$95$1), $MachinePrecision] / N[(6.0 + t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 0.55], N[(N[(5.0 + N[(N[(2.0 + N[(t * -2.0), $MachinePrecision]), $MachinePrecision] * N[(N[(t * -2.0), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(5.0 + N[(-8.0 / t), $MachinePrecision]), $MachinePrecision] / N[(6.0 + N[(-8.0 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if t < -0.52000000000000002

   1. Initial program 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{5 + \frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} - 4\right)}{6 + \frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} - 4\right)}} \]
   4. Step-by-step derivation

      1. expm1-log1p-u 98.5%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{5 + \frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} - 4\right)}{6 + \frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} - 4\right)}\right)\right)} \]

      2. expm1-udef 98.5%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{5 + \frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} - 4\right)}{6 + \frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} - 4\right)}\right)} - 1} \]

   5. Applied egg-rr 98.5%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\mathsf{fma}\left(\frac{2}{t + 1}, \frac{2}{t + 1} + -4, 5\right)}{\mathsf{fma}\left(\frac{2}{t + 1}, \frac{2}{t + 1} + -4, 6\right)}\right)} - 1} \]
   6. Step-by-step derivation

      1. expm1-def 98.5%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\mathsf{fma}\left(\frac{2}{t + 1}, \frac{2}{t + 1} + -4, 5\right)}{\mathsf{fma}\left(\frac{2}{t + 1}, \frac{2}{t + 1} + -4, 6\right)}\right)\right)} \]

      2. expm1-log1p 100.0%

         \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{2}{t + 1}, \frac{2}{t + 1} + -4, 5\right)}{\mathsf{fma}\left(\frac{2}{t + 1}, \frac{2}{t + 1} + -4, 6\right)}} \]

      3. fma-udef 100.0%

         \[\leadsto \frac{\color{blue}{\frac{2}{t + 1} \cdot \left(\frac{2}{t + 1} + -4\right) + 5}}{\mathsf{fma}\left(\frac{2}{t + 1}, \frac{2}{t + 1} + -4, 6\right)} \]

      4. associate-*l/ 100.0%

         \[\leadsto \frac{\color{blue}{\frac{2 \cdot \left(\frac{2}{t + 1} + -4\right)}{t + 1}} + 5}{\mathsf{fma}\left(\frac{2}{t + 1}, \frac{2}{t + 1} + -4, 6\right)} \]

      5. associate-*r/ 100.0%

         \[\leadsto \frac{\color{blue}{2 \cdot \frac{\frac{2}{t + 1} + -4}{t + 1}} + 5}{\mathsf{fma}\left(\frac{2}{t + 1}, \frac{2}{t + 1} + -4, 6\right)} \]

      6. +-commutative 100.0%

         \[\leadsto \frac{\color{blue}{5 + 2 \cdot \frac{\frac{2}{t + 1} + -4}{t + 1}}}{\mathsf{fma}\left(\frac{2}{t + 1}, \frac{2}{t + 1} + -4, 6\right)} \]

      7. associate-*r/ 100.0%

         \[\leadsto \frac{5 + \color{blue}{\frac{2 \cdot \left(\frac{2}{t + 1} + -4\right)}{t + 1}}}{\mathsf{fma}\left(\frac{2}{t + 1}, \frac{2}{t + 1} + -4, 6\right)} \]

      8. +-commutative 100.0%

         \[\leadsto \frac{5 + \frac{2 \cdot \color{blue}{\left(-4 + \frac{2}{t + 1}\right)}}{t + 1}}{\mathsf{fma}\left(\frac{2}{t + 1}, \frac{2}{t + 1} + -4, 6\right)} \]

      9. distribute-lft-in 100.0%

         \[\leadsto \frac{5 + \frac{\color{blue}{2 \cdot -4 + 2 \cdot \frac{2}{t + 1}}}{t + 1}}{\mathsf{fma}\left(\frac{2}{t + 1}, \frac{2}{t + 1} + -4, 6\right)} \]

      10. metadata-eval 100.0%

          \[\leadsto \frac{5 + \frac{\color{blue}{-8} + 2 \cdot \frac{2}{t + 1}}{t + 1}}{\mathsf{fma}\left(\frac{2}{t + 1}, \frac{2}{t + 1} + -4, 6\right)} \]

      11. associate-*r/ 100.0%

          \[\leadsto \frac{5 + \frac{-8 + \color{blue}{\frac{2 \cdot 2}{t + 1}}}{t + 1}}{\mathsf{fma}\left(\frac{2}{t + 1}, \frac{2}{t + 1} + -4, 6\right)} \]

      12. metadata-eval 100.0%

          \[\leadsto \frac{5 + \frac{-8 + \frac{\color{blue}{4}}{t + 1}}{t + 1}}{\mathsf{fma}\left(\frac{2}{t + 1}, \frac{2}{t + 1} + -4, 6\right)} \]

      13. fma-udef 100.0%

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

   7. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{5 + \frac{-8 + \frac{4}{t + 1}}{t + 1}}{6 + \frac{-8 + \frac{4}{t + 1}}{t + 1}}} \]

   8. Taylor expanded in t around inf 100.0%

      \[\leadsto \frac{5 + \frac{-8 + \frac{4}{t + 1}}{t + 1}}{6 + \frac{\color{blue}{4 \cdot \frac{1}{t} - 8}}{t + 1}} \]
   9. Step-by-step derivation

      1. sub-neg 100.0%

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

      2. associate-*r/ 100.0%

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

      3. metadata-eval 100.0%

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

      4. metadata-eval 100.0%

         \[\leadsto \frac{5 + \frac{-8 + \frac{4}{t + 1}}{t + 1}}{6 + \frac{\frac{4}{t} + \color{blue}{-8}}{t + 1}} \]

   10. Simplified 100.0%

       \[\leadsto \frac{5 + \frac{-8 + \frac{4}{t + 1}}{t + 1}}{6 + \frac{\color{blue}{\frac{4}{t} + -8}}{t + 1}} \]

   11. Taylor expanded in t around inf 100.0%

       \[\leadsto \frac{5 + \frac{\color{blue}{4 \cdot \frac{1}{t} - 8}}{t + 1}}{6 + \frac{\frac{4}{t} + -8}{t + 1}} \]
   12. Step-by-step derivation

       1. sub-neg 100.0%

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

       2. associate-*r/ 100.0%

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

       3. metadata-eval 100.0%

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

       4. metadata-eval 100.0%

          \[\leadsto \frac{5 + \frac{-8 + \frac{4}{t + 1}}{t + 1}}{6 + \frac{\frac{4}{t} + \color{blue}{-8}}{t + 1}} \]

   13. Simplified 100.0%

       \[\leadsto \frac{5 + \frac{\color{blue}{\frac{4}{t} + -8}}{t + 1}}{6 + \frac{\frac{4}{t} + -8}{t + 1}} \]

   if -0.52000000000000002 < t < 0.55000000000000004

   1. Initial program 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{5 + \frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} - 4\right)}{6 + \frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} - 4\right)}} \]

   4. Taylor expanded in t around 0 98.2%

      \[\leadsto \frac{5 + \frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} - 4\right)}{6 + \color{blue}{-4}} \]

   5. Taylor expanded in t around 0 98.3%

      \[\leadsto \frac{5 + \frac{2}{1 + t} \cdot \color{blue}{\left(-2 \cdot t - 2\right)}}{6 + -4} \]

   6. Taylor expanded in t around 0 98.3%

      \[\leadsto \frac{5 + \color{blue}{\left(2 + -2 \cdot t\right)} \cdot \left(-2 \cdot t - 2\right)}{6 + -4} \]
   7. Step-by-step derivation

      1. *-commutative 98.3%

         \[\leadsto \frac{5 + \left(2 + \color{blue}{t \cdot -2}\right) \cdot \left(-2 \cdot t - 2\right)}{6 + -4} \]

   8. Simplified 98.3%

      \[\leadsto \frac{5 + \color{blue}{\left(2 + t \cdot -2\right)} \cdot \left(-2 \cdot t - 2\right)}{6 + -4} \]

   if 0.55000000000000004 < t

   1. Initial program 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{5 + \frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} - 4\right)}{6 + \frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} - 4\right)}} \]

   4. Taylor expanded in t around inf 100.0%

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

   5. Taylor expanded in t around inf 100.0%

      \[\leadsto \frac{5 + \frac{-8}{t}}{6 + \color{blue}{\frac{-8}{t}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.52:\\ \;\;\;\;\frac{5 + \frac{-8 + \frac{4}{t}}{1 + t}}{6 + \frac{-8 + \frac{4}{t}}{1 + t}}\\ \mathbf{elif}\;t \leq 0.55:\\ \;\;\;\;\frac{5 + \left(2 + t \cdot -2\right) \cdot \left(t \cdot -2 - 2\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{5 + \frac{-8}{t}}{6 + \frac{-8}{t}}\\ \end{array} \]

Alternative 4: 99.0% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -0.6:\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222}{t}\\ \mathbf{elif}\;t \leq 0.55:\\ \;\;\;\;\frac{5 + \left(2 + t \cdot -2\right) \cdot \left(t \cdot -2 - 2\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{5 + \frac{-8}{t}}{6 + \frac{-8}{t}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (<= t -0.6)
   (- 0.8333333333333334 (/ 0.2222222222222222 t))
   (if (<= t 0.55)
     (/ (+ 5.0 (* (+ 2.0 (* t -2.0)) (- (* t -2.0) 2.0))) 2.0)
     (/ (+ 5.0 (/ -8.0 t)) (+ 6.0 (/ -8.0 t))))))

C

double code(double t) {
	double tmp;
	if (t <= -0.6) {
		tmp = 0.8333333333333334 - (0.2222222222222222 / t);
	} else if (t <= 0.55) {
		tmp = (5.0 + ((2.0 + (t * -2.0)) * ((t * -2.0) - 2.0))) / 2.0;
	} else {
		tmp = (5.0 + (-8.0 / t)) / (6.0 + (-8.0 / t));
	}
	return tmp;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-0.6d0)) then
        tmp = 0.8333333333333334d0 - (0.2222222222222222d0 / t)
    else if (t <= 0.55d0) then
        tmp = (5.0d0 + ((2.0d0 + (t * (-2.0d0))) * ((t * (-2.0d0)) - 2.0d0))) / 2.0d0
    else
        tmp = (5.0d0 + ((-8.0d0) / t)) / (6.0d0 + ((-8.0d0) / t))
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double tmp;
	if (t <= -0.6) {
		tmp = 0.8333333333333334 - (0.2222222222222222 / t);
	} else if (t <= 0.55) {
		tmp = (5.0 + ((2.0 + (t * -2.0)) * ((t * -2.0) - 2.0))) / 2.0;
	} else {
		tmp = (5.0 + (-8.0 / t)) / (6.0 + (-8.0 / t));
	}
	return tmp;
}

Python

def code(t):
	tmp = 0
	if t <= -0.6:
		tmp = 0.8333333333333334 - (0.2222222222222222 / t)
	elif t <= 0.55:
		tmp = (5.0 + ((2.0 + (t * -2.0)) * ((t * -2.0) - 2.0))) / 2.0
	else:
		tmp = (5.0 + (-8.0 / t)) / (6.0 + (-8.0 / t))
	return tmp

Julia

function code(t)
	tmp = 0.0
	if (t <= -0.6)
		tmp = Float64(0.8333333333333334 - Float64(0.2222222222222222 / t));
	elseif (t <= 0.55)
		tmp = Float64(Float64(5.0 + Float64(Float64(2.0 + Float64(t * -2.0)) * Float64(Float64(t * -2.0) - 2.0))) / 2.0);
	else
		tmp = Float64(Float64(5.0 + Float64(-8.0 / t)) / Float64(6.0 + Float64(-8.0 / t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	tmp = 0.0;
	if (t <= -0.6)
		tmp = 0.8333333333333334 - (0.2222222222222222 / t);
	elseif (t <= 0.55)
		tmp = (5.0 + ((2.0 + (t * -2.0)) * ((t * -2.0) - 2.0))) / 2.0;
	else
		tmp = (5.0 + (-8.0 / t)) / (6.0 + (-8.0 / t));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := If[LessEqual[t, -0.6], N[(0.8333333333333334 - N[(0.2222222222222222 / t), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 0.55], N[(N[(5.0 + N[(N[(2.0 + N[(t * -2.0), $MachinePrecision]), $MachinePrecision] * N[(N[(t * -2.0), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(5.0 + N[(-8.0 / t), $MachinePrecision]), $MachinePrecision] / N[(6.0 + N[(-8.0 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -0.599999999999999978

   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. Taylor expanded in t around inf 99.6%

      \[\leadsto \color{blue}{0.8333333333333334 - 0.2222222222222222 \cdot \frac{1}{t}} \]
   3. Step-by-step derivation

      1. associate-*r/ 99.6%

         \[\leadsto 0.8333333333333334 - \color{blue}{\frac{0.2222222222222222 \cdot 1}{t}} \]

      2. metadata-eval 99.6%

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

   4. Simplified 99.6%

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

   if -0.599999999999999978 < t < 0.55000000000000004

   1. Initial program 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{5 + \frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} - 4\right)}{6 + \frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} - 4\right)}} \]

   4. Taylor expanded in t around 0 98.2%

      \[\leadsto \frac{5 + \frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} - 4\right)}{6 + \color{blue}{-4}} \]

   5. Taylor expanded in t around 0 98.3%

      \[\leadsto \frac{5 + \frac{2}{1 + t} \cdot \color{blue}{\left(-2 \cdot t - 2\right)}}{6 + -4} \]

   6. Taylor expanded in t around 0 98.3%

      \[\leadsto \frac{5 + \color{blue}{\left(2 + -2 \cdot t\right)} \cdot \left(-2 \cdot t - 2\right)}{6 + -4} \]
   7. Step-by-step derivation

      1. *-commutative 98.3%

         \[\leadsto \frac{5 + \left(2 + \color{blue}{t \cdot -2}\right) \cdot \left(-2 \cdot t - 2\right)}{6 + -4} \]

   8. Simplified 98.3%

      \[\leadsto \frac{5 + \color{blue}{\left(2 + t \cdot -2\right)} \cdot \left(-2 \cdot t - 2\right)}{6 + -4} \]

   if 0.55000000000000004 < t

   1. Initial program 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{5 + \frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} - 4\right)}{6 + \frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} - 4\right)}} \]

   4. Taylor expanded in t around inf 100.0%

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

   5. Taylor expanded in t around inf 100.0%

      \[\leadsto \frac{5 + \frac{-8}{t}}{6 + \color{blue}{\frac{-8}{t}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.6:\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222}{t}\\ \mathbf{elif}\;t \leq 0.55:\\ \;\;\;\;\frac{5 + \left(2 + t \cdot -2\right) \cdot \left(t \cdot -2 - 2\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{5 + \frac{-8}{t}}{6 + \frac{-8}{t}}\\ \end{array} \]

Alternative 5: 99.1% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -0.49:\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222}{t}\\ \mathbf{elif}\;t \leq 2:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{5 + \frac{-8}{t}}{6 + \frac{-8}{t}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (<= t -0.49)
   (- 0.8333333333333334 (/ 0.2222222222222222 t))
   (if (<= t 2.0) 0.5 (/ (+ 5.0 (/ -8.0 t)) (+ 6.0 (/ -8.0 t))))))

C

double code(double t) {
	double tmp;
	if (t <= -0.49) {
		tmp = 0.8333333333333334 - (0.2222222222222222 / t);
	} else if (t <= 2.0) {
		tmp = 0.5;
	} else {
		tmp = (5.0 + (-8.0 / t)) / (6.0 + (-8.0 / t));
	}
	return tmp;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-0.49d0)) then
        tmp = 0.8333333333333334d0 - (0.2222222222222222d0 / t)
    else if (t <= 2.0d0) then
        tmp = 0.5d0
    else
        tmp = (5.0d0 + ((-8.0d0) / t)) / (6.0d0 + ((-8.0d0) / t))
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double tmp;
	if (t <= -0.49) {
		tmp = 0.8333333333333334 - (0.2222222222222222 / t);
	} else if (t <= 2.0) {
		tmp = 0.5;
	} else {
		tmp = (5.0 + (-8.0 / t)) / (6.0 + (-8.0 / t));
	}
	return tmp;
}

Python

def code(t):
	tmp = 0
	if t <= -0.49:
		tmp = 0.8333333333333334 - (0.2222222222222222 / t)
	elif t <= 2.0:
		tmp = 0.5
	else:
		tmp = (5.0 + (-8.0 / t)) / (6.0 + (-8.0 / t))
	return tmp

Julia

function code(t)
	tmp = 0.0
	if (t <= -0.49)
		tmp = Float64(0.8333333333333334 - Float64(0.2222222222222222 / t));
	elseif (t <= 2.0)
		tmp = 0.5;
	else
		tmp = Float64(Float64(5.0 + Float64(-8.0 / t)) / Float64(6.0 + Float64(-8.0 / t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	tmp = 0.0;
	if (t <= -0.49)
		tmp = 0.8333333333333334 - (0.2222222222222222 / t);
	elseif (t <= 2.0)
		tmp = 0.5;
	else
		tmp = (5.0 + (-8.0 / t)) / (6.0 + (-8.0 / t));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := If[LessEqual[t, -0.49], N[(0.8333333333333334 - N[(0.2222222222222222 / t), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 2.0], 0.5, N[(N[(5.0 + N[(-8.0 / t), $MachinePrecision]), $MachinePrecision] / N[(6.0 + N[(-8.0 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -0.48999999999999999

   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. Taylor expanded in t around inf 99.6%

      \[\leadsto \color{blue}{0.8333333333333334 - 0.2222222222222222 \cdot \frac{1}{t}} \]
   3. Step-by-step derivation

      1. associate-*r/ 99.6%

         \[\leadsto 0.8333333333333334 - \color{blue}{\frac{0.2222222222222222 \cdot 1}{t}} \]

      2. metadata-eval 99.6%

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

   4. Simplified 99.6%

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

   if -0.48999999999999999 < t < 2

   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. Taylor expanded in t around 0 98.3%

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

   if 2 < t

   1. Initial program 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{5 + \frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} - 4\right)}{6 + \frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} - 4\right)}} \]

   4. Taylor expanded in t around inf 100.0%

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

   5. Taylor expanded in t around inf 100.0%

      \[\leadsto \frac{5 + \frac{-8}{t}}{6 + \color{blue}{\frac{-8}{t}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.49:\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222}{t}\\ \mathbf{elif}\;t \leq 2:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{5 + \frac{-8}{t}}{6 + \frac{-8}{t}}\\ \end{array} \]

Alternative 6: 99.1% accurate, 5.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -0.49 \lor \neg \left(t \leq 0.66\right):\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222}{t}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (or (<= t -0.49) (not (<= t 0.66)))
   (- 0.8333333333333334 (/ 0.2222222222222222 t))
   0.5))

C

double code(double t) {
	double tmp;
	if ((t <= -0.49) || !(t <= 0.66)) {
		tmp = 0.8333333333333334 - (0.2222222222222222 / t);
	} else {
		tmp = 0.5;
	}
	return tmp;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((t <= (-0.49d0)) .or. (.not. (t <= 0.66d0))) then
        tmp = 0.8333333333333334d0 - (0.2222222222222222d0 / t)
    else
        tmp = 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double tmp;
	if ((t <= -0.49) || !(t <= 0.66)) {
		tmp = 0.8333333333333334 - (0.2222222222222222 / t);
	} else {
		tmp = 0.5;
	}
	return tmp;
}

Python

def code(t):
	tmp = 0
	if (t <= -0.49) or not (t <= 0.66):
		tmp = 0.8333333333333334 - (0.2222222222222222 / t)
	else:
		tmp = 0.5
	return tmp

Julia

function code(t)
	tmp = 0.0
	if ((t <= -0.49) || !(t <= 0.66))
		tmp = Float64(0.8333333333333334 - Float64(0.2222222222222222 / t));
	else
		tmp = 0.5;
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	tmp = 0.0;
	if ((t <= -0.49) || ~((t <= 0.66)))
		tmp = 0.8333333333333334 - (0.2222222222222222 / t);
	else
		tmp = 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := If[Or[LessEqual[t, -0.49], N[Not[LessEqual[t, 0.66]], $MachinePrecision]], N[(0.8333333333333334 - N[(0.2222222222222222 / t), $MachinePrecision]), $MachinePrecision], 0.5]

Derivation

1. Split input into 2 regimes

2. if t < -0.48999999999999999 or 0.660000000000000031 < 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. Taylor expanded in t around inf 99.8%

      \[\leadsto \color{blue}{0.8333333333333334 - 0.2222222222222222 \cdot \frac{1}{t}} \]
   3. Step-by-step derivation

      1. associate-*r/ 99.8%

         \[\leadsto 0.8333333333333334 - \color{blue}{\frac{0.2222222222222222 \cdot 1}{t}} \]

      2. metadata-eval 99.8%

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

   4. Simplified 99.8%

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

   if -0.48999999999999999 < t < 0.660000000000000031

   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. Taylor expanded in t around 0 98.3%

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

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.49 \lor \neg \left(t \leq 0.66\right):\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222}{t}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]

Alternative 7: 98.5% accurate, 10.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -0.33:\\ \;\;\;\;0.8333333333333334\\ \mathbf{elif}\;t \leq 1:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (<= t -0.33) 0.8333333333333334 (if (<= t 1.0) 0.5 0.8333333333333334)))

C

double code(double t) {
	double tmp;
	if (t <= -0.33) {
		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 <= (-0.33d0)) 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 <= -0.33) {
		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 <= -0.33:
		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 <= -0.33)
		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 <= -0.33)
		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, -0.33], 0.8333333333333334, If[LessEqual[t, 1.0], 0.5, 0.8333333333333334]]

Derivation

1. Split input into 2 regimes

2. if t < -0.330000000000000016 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. Taylor expanded in t around inf 98.2%

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

   if -0.330000000000000016 < 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. Taylor expanded in t around 0 98.8%

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

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.33:\\ \;\;\;\;0.8333333333333334\\ \mathbf{elif}\;t \leq 1:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334\\ \end{array} \]

Alternative 8: 59.1% 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. Taylor expanded in t around 0 62.9%

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

3. Final simplification 62.9%

   \[\leadsto 0.5 \]

Reproduce

herbie shell --seed 2023320 
(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))))))))