Kahan p13 Example 2

Percentage Accurate: 100.0% → 100.0%

Time: 19.1s

Alternatives: 9

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: 100.0% 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, 0.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Simplified 100.0%

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

5. Final simplification 100.0%

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

Alternative 2: 99.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 8 - \frac{4}{t}\\ \mathbf{if}\;t \leq -0.58:\\ \;\;\;\;\frac{1}{1.2 + \frac{0.32}{t}}\\ \mathbf{elif}\;t \leq 0.78:\\ \;\;\;\;\frac{-5 + \frac{8 + \left(t \cdot 4 - 4\right)}{1 + t}}{-2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-5 + t\_1 \cdot \frac{1}{1 + t}}{-6 + \frac{t\_1}{1 + t}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (let* ((t_1 (- 8.0 (/ 4.0 t))))
   (if (<= t -0.58)
     (/ 1.0 (+ 1.2 (/ 0.32 t)))
     (if (<= t 0.78)
       (/ (+ -5.0 (/ (+ 8.0 (- (* t 4.0) 4.0)) (+ 1.0 t))) -2.0)
       (/ (+ -5.0 (* t_1 (/ 1.0 (+ 1.0 t)))) (+ -6.0 (/ t_1 (+ 1.0 t))))))))

C

double code(double t) {
	double t_1 = 8.0 - (4.0 / t);
	double tmp;
	if (t <= -0.58) {
		tmp = 1.0 / (1.2 + (0.32 / t));
	} else if (t <= 0.78) {
		tmp = (-5.0 + ((8.0 + ((t * 4.0) - 4.0)) / (1.0 + t))) / -2.0;
	} else {
		tmp = (-5.0 + (t_1 * (1.0 / (1.0 + t)))) / (-6.0 + (t_1 / (1.0 + 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)
    if (t <= (-0.58d0)) then
        tmp = 1.0d0 / (1.2d0 + (0.32d0 / t))
    else if (t <= 0.78d0) then
        tmp = ((-5.0d0) + ((8.0d0 + ((t * 4.0d0) - 4.0d0)) / (1.0d0 + t))) / (-2.0d0)
    else
        tmp = ((-5.0d0) + (t_1 * (1.0d0 / (1.0d0 + t)))) / ((-6.0d0) + (t_1 / (1.0d0 + t)))
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double t_1 = 8.0 - (4.0 / t);
	double tmp;
	if (t <= -0.58) {
		tmp = 1.0 / (1.2 + (0.32 / t));
	} else if (t <= 0.78) {
		tmp = (-5.0 + ((8.0 + ((t * 4.0) - 4.0)) / (1.0 + t))) / -2.0;
	} else {
		tmp = (-5.0 + (t_1 * (1.0 / (1.0 + t)))) / (-6.0 + (t_1 / (1.0 + t)));
	}
	return tmp;
}

Python

def code(t):
	t_1 = 8.0 - (4.0 / t)
	tmp = 0
	if t <= -0.58:
		tmp = 1.0 / (1.2 + (0.32 / t))
	elif t <= 0.78:
		tmp = (-5.0 + ((8.0 + ((t * 4.0) - 4.0)) / (1.0 + t))) / -2.0
	else:
		tmp = (-5.0 + (t_1 * (1.0 / (1.0 + t)))) / (-6.0 + (t_1 / (1.0 + t)))
	return tmp

Julia

function code(t)
	t_1 = Float64(8.0 - Float64(4.0 / t))
	tmp = 0.0
	if (t <= -0.58)
		tmp = Float64(1.0 / Float64(1.2 + Float64(0.32 / t)));
	elseif (t <= 0.78)
		tmp = Float64(Float64(-5.0 + Float64(Float64(8.0 + Float64(Float64(t * 4.0) - 4.0)) / Float64(1.0 + t))) / -2.0);
	else
		tmp = Float64(Float64(-5.0 + Float64(t_1 * Float64(1.0 / Float64(1.0 + t)))) / Float64(-6.0 + Float64(t_1 / Float64(1.0 + t))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	t_1 = 8.0 - (4.0 / t);
	tmp = 0.0;
	if (t <= -0.58)
		tmp = 1.0 / (1.2 + (0.32 / t));
	elseif (t <= 0.78)
		tmp = (-5.0 + ((8.0 + ((t * 4.0) - 4.0)) / (1.0 + t))) / -2.0;
	else
		tmp = (-5.0 + (t_1 * (1.0 / (1.0 + t)))) / (-6.0 + (t_1 / (1.0 + t)));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := Block[{t$95$1 = N[(8.0 - N[(4.0 / t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -0.58], N[(1.0 / N[(1.2 + N[(0.32 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 0.78], N[(N[(-5.0 + N[(N[(8.0 + N[(N[(t * 4.0), $MachinePrecision] - 4.0), $MachinePrecision]), $MachinePrecision] / N[(1.0 + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / -2.0), $MachinePrecision], N[(N[(-5.0 + N[(t$95$1 * N[(1.0 / N[(1.0 + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(-6.0 + N[(t$95$1 / N[(1.0 + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if t < -0.57999999999999996

   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. Add Preprocessing
   5. Step-by-step derivation

      1. clear-num 100.0%

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

      2. inv-pow 100.0%

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

   6. Applied egg-rr 100.0%

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

      1. unpow-1 100.0%

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

      2. fma-udef 100.0%

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

      3. associate-*l/ 100.0%

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

      4. +-commutative 100.0%

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

      5. distribute-lft-in 100.0%

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

      6. metadata-eval 100.0%

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

      7. associate-*r/ 100.0%

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

      8. metadata-eval 100.0%

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

      9. fma-udef 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in t around inf 100.0%

      \[\leadsto \frac{1}{\color{blue}{1.2 + 0.32 \cdot \frac{1}{t}}} \]
   10. Step-by-step derivation

       1. associate-*r/ 100.0%

          \[\leadsto \frac{1}{1.2 + \color{blue}{\frac{0.32 \cdot 1}{t}}} \]

       2. metadata-eval 100.0%

          \[\leadsto \frac{1}{1.2 + \frac{\color{blue}{0.32}}{t}} \]

   11. Simplified 100.0%

       \[\leadsto \frac{1}{\color{blue}{1.2 + \frac{0.32}{t}}} \]

   if -0.57999999999999996 < t < 0.78000000000000003

   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. Add Preprocessing
   5. Step-by-step derivation

      1. expm1-log1p-u 100.0%

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

      2. expm1-udef 100.0%

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

   6. Applied egg-rr 100.0%

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

      1. expm1-def 100.0%

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

      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. *-lft-identity 100.0%

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

      4. metadata-eval 100.0%

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

      5. times-frac 100.0%

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

      6. neg-mul-1 100.0%

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

      7. neg-mul-1 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in t around 0 99.3%

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

   10. Taylor expanded in t around 0 99.3%

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

   if 0.78000000000000003 < 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. Add Preprocessing
   5. 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%

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

   6. Applied egg-rr 98.5%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\mathsf{fma}\left(\frac{2}{t + 1}, \frac{2}{t + 1} + -4, 5\right)}{\mathsf{fma}\left(\frac{2}{t + 1}, \frac{2}{t + 1} + -4, 6\right)}\right)} - 1} \]
   7. 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. *-lft-identity 100.0%

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

      4. metadata-eval 100.0%

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

      5. times-frac 100.0%

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

      6. neg-mul-1 100.0%

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

      7. neg-mul-1 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in t around inf 97.9%

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

       1. associate-*r/ 97.9%

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

       2. metadata-eval 97.9%

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

   11. Simplified 97.9%

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

   12. Taylor expanded in t around inf 98.0%

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

       1. associate-*r/ 97.9%

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

       2. metadata-eval 97.9%

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

   14. Simplified 98.0%

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

       1. div-inv 98.0%

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

   16. Applied egg-rr 98.0%

       \[\leadsto \frac{\color{blue}{\left(8 - \frac{4}{t}\right) \cdot \frac{1}{t + 1}} + -5}{\frac{8 - \frac{4}{t}}{t + 1} + -6} \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.58:\\ \;\;\;\;\frac{1}{1.2 + \frac{0.32}{t}}\\ \mathbf{elif}\;t \leq 0.78:\\ \;\;\;\;\frac{-5 + \frac{8 + \left(t \cdot 4 - 4\right)}{1 + t}}{-2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-5 + \left(8 - \frac{4}{t}\right) \cdot \frac{1}{1 + t}}{-6 + \frac{8 - \frac{4}{t}}{1 + t}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{8 - \frac{4}{t}}{1 + t}\\ \mathbf{if}\;t \leq -0.58:\\ \;\;\;\;\frac{1}{1.2 + \frac{0.32}{t}}\\ \mathbf{elif}\;t \leq 0.78:\\ \;\;\;\;\frac{-5 + \frac{8 + \left(t \cdot 4 - 4\right)}{1 + t}}{-2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-5 + t\_1}{-6 + t\_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (let* ((t_1 (/ (- 8.0 (/ 4.0 t)) (+ 1.0 t))))
   (if (<= t -0.58)
     (/ 1.0 (+ 1.2 (/ 0.32 t)))
     (if (<= t 0.78)
       (/ (+ -5.0 (/ (+ 8.0 (- (* t 4.0) 4.0)) (+ 1.0 t))) -2.0)
       (/ (+ -5.0 t_1) (+ -6.0 t_1))))))

C

double code(double t) {
	double t_1 = (8.0 - (4.0 / t)) / (1.0 + t);
	double tmp;
	if (t <= -0.58) {
		tmp = 1.0 / (1.2 + (0.32 / t));
	} else if (t <= 0.78) {
		tmp = (-5.0 + ((8.0 + ((t * 4.0) - 4.0)) / (1.0 + t))) / -2.0;
	} else {
		tmp = (-5.0 + t_1) / (-6.0 + t_1);
	}
	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 <= (-0.58d0)) then
        tmp = 1.0d0 / (1.2d0 + (0.32d0 / t))
    else if (t <= 0.78d0) then
        tmp = ((-5.0d0) + ((8.0d0 + ((t * 4.0d0) - 4.0d0)) / (1.0d0 + t))) / (-2.0d0)
    else
        tmp = ((-5.0d0) + t_1) / ((-6.0d0) + t_1)
    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 <= -0.58) {
		tmp = 1.0 / (1.2 + (0.32 / t));
	} else if (t <= 0.78) {
		tmp = (-5.0 + ((8.0 + ((t * 4.0) - 4.0)) / (1.0 + t))) / -2.0;
	} else {
		tmp = (-5.0 + t_1) / (-6.0 + t_1);
	}
	return tmp;
}

Python

def code(t):
	t_1 = (8.0 - (4.0 / t)) / (1.0 + t)
	tmp = 0
	if t <= -0.58:
		tmp = 1.0 / (1.2 + (0.32 / t))
	elif t <= 0.78:
		tmp = (-5.0 + ((8.0 + ((t * 4.0) - 4.0)) / (1.0 + t))) / -2.0
	else:
		tmp = (-5.0 + t_1) / (-6.0 + t_1)
	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 <= -0.58)
		tmp = Float64(1.0 / Float64(1.2 + Float64(0.32 / t)));
	elseif (t <= 0.78)
		tmp = Float64(Float64(-5.0 + Float64(Float64(8.0 + Float64(Float64(t * 4.0) - 4.0)) / Float64(1.0 + t))) / -2.0);
	else
		tmp = Float64(Float64(-5.0 + t_1) / Float64(-6.0 + t_1));
	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 <= -0.58)
		tmp = 1.0 / (1.2 + (0.32 / t));
	elseif (t <= 0.78)
		tmp = (-5.0 + ((8.0 + ((t * 4.0) - 4.0)) / (1.0 + t))) / -2.0;
	else
		tmp = (-5.0 + t_1) / (-6.0 + t_1);
	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, -0.58], N[(1.0 / N[(1.2 + N[(0.32 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 0.78], N[(N[(-5.0 + N[(N[(8.0 + N[(N[(t * 4.0), $MachinePrecision] - 4.0), $MachinePrecision]), $MachinePrecision] / N[(1.0 + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / -2.0), $MachinePrecision], N[(N[(-5.0 + t$95$1), $MachinePrecision] / N[(-6.0 + t$95$1), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if t < -0.57999999999999996

   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. Add Preprocessing
   5. Step-by-step derivation

      1. clear-num 100.0%

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

      2. inv-pow 100.0%

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

   6. Applied egg-rr 100.0%

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

      1. unpow-1 100.0%

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

      2. fma-udef 100.0%

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

      3. associate-*l/ 100.0%

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

      4. +-commutative 100.0%

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

      5. distribute-lft-in 100.0%

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

      6. metadata-eval 100.0%

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

      7. associate-*r/ 100.0%

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

      8. metadata-eval 100.0%

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

      9. fma-udef 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in t around inf 100.0%

      \[\leadsto \frac{1}{\color{blue}{1.2 + 0.32 \cdot \frac{1}{t}}} \]
   10. Step-by-step derivation

       1. associate-*r/ 100.0%

          \[\leadsto \frac{1}{1.2 + \color{blue}{\frac{0.32 \cdot 1}{t}}} \]

       2. metadata-eval 100.0%

          \[\leadsto \frac{1}{1.2 + \frac{\color{blue}{0.32}}{t}} \]

   11. Simplified 100.0%

       \[\leadsto \frac{1}{\color{blue}{1.2 + \frac{0.32}{t}}} \]

   if -0.57999999999999996 < t < 0.78000000000000003

   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. Add Preprocessing
   5. Step-by-step derivation

      1. expm1-log1p-u 100.0%

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

      2. expm1-udef 100.0%

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

   6. Applied egg-rr 100.0%

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

      1. expm1-def 100.0%

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

      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. *-lft-identity 100.0%

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

      4. metadata-eval 100.0%

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

      5. times-frac 100.0%

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

      6. neg-mul-1 100.0%

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

      7. neg-mul-1 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in t around 0 99.3%

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

   10. Taylor expanded in t around 0 99.3%

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

   if 0.78000000000000003 < 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. Add Preprocessing
   5. 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%

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

   6. Applied egg-rr 98.5%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\mathsf{fma}\left(\frac{2}{t + 1}, \frac{2}{t + 1} + -4, 5\right)}{\mathsf{fma}\left(\frac{2}{t + 1}, \frac{2}{t + 1} + -4, 6\right)}\right)} - 1} \]
   7. 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. *-lft-identity 100.0%

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

      4. metadata-eval 100.0%

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

      5. times-frac 100.0%

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

      6. neg-mul-1 100.0%

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

      7. neg-mul-1 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in t around inf 97.9%

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

       1. associate-*r/ 97.9%

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

       2. metadata-eval 97.9%

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

   11. Simplified 97.9%

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

   12. Taylor expanded in t around inf 98.0%

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

       1. associate-*r/ 97.9%

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

       2. metadata-eval 97.9%

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

   14. Simplified 98.0%

       \[\leadsto \frac{\frac{\color{blue}{8 - \frac{4}{t}}}{t + 1} + -5}{\frac{8 - \frac{4}{t}}{t + 1} + -6} \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.58:\\ \;\;\;\;\frac{1}{1.2 + \frac{0.32}{t}}\\ \mathbf{elif}\;t \leq 0.78:\\ \;\;\;\;\frac{-5 + \frac{8 + \left(t \cdot 4 - 4\right)}{1 + t}}{-2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-5 + \frac{8 - \frac{4}{t}}{1 + t}}{-6 + \frac{8 - \frac{4}{t}}{1 + t}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 100.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{8 + \frac{-4}{1 + t}}{1 + t}\\ \frac{t\_1 + -5}{t\_1 + -6} \end{array} \end{array} \]

FPCore

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

C

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

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 = (t_1 + (-5.0d0)) / (t_1 + (-6.0d0))
end function

Java

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

Python

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

Julia

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

MATLAB

function tmp = code(t)
	t_1 = (8.0 + (-4.0 / (1.0 + t))) / (1.0 + t);
	tmp = (t_1 + -5.0) / (t_1 + -6.0);
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[(t$95$1 + -5.0), $MachinePrecision] / N[(t$95$1 + -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. Add Preprocessing
5. Step-by-step derivation

   1. expm1-log1p-u 99.2%

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

   2. expm1-udef 99.2%

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

6. Applied egg-rr 99.2%

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

   1. expm1-def 99.2%

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

   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. *-lft-identity 100.0%

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

   4. metadata-eval 100.0%

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

   5. times-frac 100.0%

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

   6. neg-mul-1 100.0%

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

   7. neg-mul-1 100.0%

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

8. Simplified 100.0%

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

9. Final simplification 100.0%

   \[\leadsto \frac{\frac{8 + \frac{-4}{1 + t}}{1 + t} + -5}{\frac{8 + \frac{-4}{1 + t}}{1 + t} + -6} \]
10. Add Preprocessing

Alternative 5: 99.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -0.58 \lor \neg \left(t \leq 0.4\right):\\ \;\;\;\;\frac{1}{1.2 + \frac{0.32}{t}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-5 + \frac{8 + \left(t \cdot 4 - 4\right)}{1 + t}}{-2}\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (or (<= t -0.58) (not (<= t 0.4)))
   (/ 1.0 (+ 1.2 (/ 0.32 t)))
   (/ (+ -5.0 (/ (+ 8.0 (- (* t 4.0) 4.0)) (+ 1.0 t))) -2.0)))

C

double code(double t) {
	double tmp;
	if ((t <= -0.58) || !(t <= 0.4)) {
		tmp = 1.0 / (1.2 + (0.32 / t));
	} else {
		tmp = (-5.0 + ((8.0 + ((t * 4.0) - 4.0)) / (1.0 + t))) / -2.0;
	}
	return tmp;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((t <= (-0.58d0)) .or. (.not. (t <= 0.4d0))) then
        tmp = 1.0d0 / (1.2d0 + (0.32d0 / t))
    else
        tmp = ((-5.0d0) + ((8.0d0 + ((t * 4.0d0) - 4.0d0)) / (1.0d0 + t))) / (-2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double tmp;
	if ((t <= -0.58) || !(t <= 0.4)) {
		tmp = 1.0 / (1.2 + (0.32 / t));
	} else {
		tmp = (-5.0 + ((8.0 + ((t * 4.0) - 4.0)) / (1.0 + t))) / -2.0;
	}
	return tmp;
}

Python

def code(t):
	tmp = 0
	if (t <= -0.58) or not (t <= 0.4):
		tmp = 1.0 / (1.2 + (0.32 / t))
	else:
		tmp = (-5.0 + ((8.0 + ((t * 4.0) - 4.0)) / (1.0 + t))) / -2.0
	return tmp

Julia

function code(t)
	tmp = 0.0
	if ((t <= -0.58) || !(t <= 0.4))
		tmp = Float64(1.0 / Float64(1.2 + Float64(0.32 / t)));
	else
		tmp = Float64(Float64(-5.0 + Float64(Float64(8.0 + Float64(Float64(t * 4.0) - 4.0)) / Float64(1.0 + t))) / -2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	tmp = 0.0;
	if ((t <= -0.58) || ~((t <= 0.4)))
		tmp = 1.0 / (1.2 + (0.32 / t));
	else
		tmp = (-5.0 + ((8.0 + ((t * 4.0) - 4.0)) / (1.0 + t))) / -2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := If[Or[LessEqual[t, -0.58], N[Not[LessEqual[t, 0.4]], $MachinePrecision]], N[(1.0 / N[(1.2 + N[(0.32 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-5.0 + N[(N[(8.0 + N[(N[(t * 4.0), $MachinePrecision] - 4.0), $MachinePrecision]), $MachinePrecision] / N[(1.0 + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / -2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -0.57999999999999996 or 0.40000000000000002 < 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. Add Preprocessing
   5. Step-by-step derivation

      1. clear-num 100.0%

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

      2. inv-pow 100.0%

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

   6. Applied egg-rr 100.0%

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

      1. unpow-1 100.0%

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

      2. fma-udef 100.0%

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

      3. associate-*l/ 100.0%

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

      4. +-commutative 100.0%

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

      5. distribute-lft-in 100.0%

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

      6. metadata-eval 100.0%

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

      7. associate-*r/ 100.0%

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

      8. metadata-eval 100.0%

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

      9. fma-udef 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in t around inf 99.1%

      \[\leadsto \frac{1}{\color{blue}{1.2 + 0.32 \cdot \frac{1}{t}}} \]
   10. Step-by-step derivation

       1. associate-*r/ 99.1%

          \[\leadsto \frac{1}{1.2 + \color{blue}{\frac{0.32 \cdot 1}{t}}} \]

       2. metadata-eval 99.1%

          \[\leadsto \frac{1}{1.2 + \frac{\color{blue}{0.32}}{t}} \]

   11. Simplified 99.1%

       \[\leadsto \frac{1}{\color{blue}{1.2 + \frac{0.32}{t}}} \]

   if -0.57999999999999996 < t < 0.40000000000000002

   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. Add Preprocessing
   5. Step-by-step derivation

      1. expm1-log1p-u 100.0%

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

      2. expm1-udef 100.0%

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

   6. Applied egg-rr 100.0%

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

      1. expm1-def 100.0%

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

      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. *-lft-identity 100.0%

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

      4. metadata-eval 100.0%

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

      5. times-frac 100.0%

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

      6. neg-mul-1 100.0%

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

      7. neg-mul-1 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in t around 0 99.3%

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

   10. Taylor expanded in t around 0 99.3%

       \[\leadsto \frac{\frac{8 + \color{blue}{\left(4 \cdot t - 4\right)}}{t + 1} + -5}{4 + -6} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.58 \lor \neg \left(t \leq 0.4\right):\\ \;\;\;\;\frac{1}{1.2 + \frac{0.32}{t}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-5 + \frac{8 + \left(t \cdot 4 - 4\right)}{1 + t}}{-2}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.1% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -0.58 \lor \neg \left(t \leq 0.4\right):\\ \;\;\;\;\frac{1}{1.2 + \frac{0.32}{t}}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (or (<= t -0.58) (not (<= t 0.4))) (/ 1.0 (+ 1.2 (/ 0.32 t))) 0.5))

C

double code(double t) {
	double tmp;
	if ((t <= -0.58) || !(t <= 0.4)) {
		tmp = 1.0 / (1.2 + (0.32 / t));
	} else {
		tmp = 0.5;
	}
	return tmp;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((t <= (-0.58d0)) .or. (.not. (t <= 0.4d0))) then
        tmp = 1.0d0 / (1.2d0 + (0.32d0 / t))
    else
        tmp = 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double tmp;
	if ((t <= -0.58) || !(t <= 0.4)) {
		tmp = 1.0 / (1.2 + (0.32 / t));
	} else {
		tmp = 0.5;
	}
	return tmp;
}

Python

def code(t):
	tmp = 0
	if (t <= -0.58) or not (t <= 0.4):
		tmp = 1.0 / (1.2 + (0.32 / t))
	else:
		tmp = 0.5
	return tmp

Julia

function code(t)
	tmp = 0.0
	if ((t <= -0.58) || !(t <= 0.4))
		tmp = Float64(1.0 / Float64(1.2 + Float64(0.32 / t)));
	else
		tmp = 0.5;
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	tmp = 0.0;
	if ((t <= -0.58) || ~((t <= 0.4)))
		tmp = 1.0 / (1.2 + (0.32 / t));
	else
		tmp = 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := If[Or[LessEqual[t, -0.58], N[Not[LessEqual[t, 0.4]], $MachinePrecision]], N[(1.0 / N[(1.2 + N[(0.32 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5]

Derivation

1. Split input into 2 regimes

2. if t < -0.57999999999999996 or 0.40000000000000002 < 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. Add Preprocessing
   5. Step-by-step derivation

      1. clear-num 100.0%

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

      2. inv-pow 100.0%

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

   6. Applied egg-rr 100.0%

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

      1. unpow-1 100.0%

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

      2. fma-udef 100.0%

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

      3. associate-*l/ 100.0%

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

      4. +-commutative 100.0%

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

      5. distribute-lft-in 100.0%

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

      6. metadata-eval 100.0%

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

      7. associate-*r/ 100.0%

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

      8. metadata-eval 100.0%

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

      9. fma-udef 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in t around inf 99.1%

      \[\leadsto \frac{1}{\color{blue}{1.2 + 0.32 \cdot \frac{1}{t}}} \]
   10. Step-by-step derivation

       1. associate-*r/ 99.1%

          \[\leadsto \frac{1}{1.2 + \color{blue}{\frac{0.32 \cdot 1}{t}}} \]

       2. metadata-eval 99.1%

          \[\leadsto \frac{1}{1.2 + \frac{\color{blue}{0.32}}{t}} \]

   11. Simplified 99.1%

       \[\leadsto \frac{1}{\color{blue}{1.2 + \frac{0.32}{t}}} \]

   if -0.57999999999999996 < t < 0.40000000000000002

   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 0 99.3%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.58 \lor \neg \left(t \leq 0.4\right):\\ \;\;\;\;\frac{1}{1.2 + \frac{0.32}{t}}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 99.1% accurate, 3.4× 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. Add Preprocessing

   3. Taylor expanded in t around inf 99.0%

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

      1. associate-*r/ 99.0%

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

      2. metadata-eval 99.0%

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

   5. Simplified 99.0%

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

   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. Add Preprocessing

   3. Taylor expanded in t around 0 99.3%

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

4. Final simplification 99.2%

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

Alternative 8: 98.6% accurate, 4.6× 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. Add Preprocessing

   3. Taylor expanded in t around inf 98.7%

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

   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. Add Preprocessing

   3. Taylor expanded in t around 0 99.3%

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

4. Final simplification 99.0%

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

Alternative 9: 59.5% 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 0 59.5%

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

4. Final simplification 59.5%

   \[\leadsto 0.5 \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024036 
(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))))))))