Kahan p13 Example 2

Percentage Accurate: 99.9% → 100.0%

Time: 7.5s

Alternatives: 8

Speedup: 1.6×

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. Add Preprocessing

5. Final simplification 100.0%

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

Alternative 2: 99.2% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{2}{1 + t}\\ \mathbf{if}\;t \leq -0.55:\\ \;\;\;\;0.8333333333333334 + \frac{-0.2222222222222222}{t}\\ \mathbf{elif}\;t \leq 1.85:\\ \;\;\;\;0.5 + \left(t \cdot t\right) \cdot \left(1 + t \cdot \left(t + -2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{5 + \frac{-8}{t}}{t\_1 \cdot \left(t\_1 + -4\right) + 6}\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (let* ((t_1 (/ 2.0 (+ 1.0 t))))
   (if (<= t -0.55)
     (+ 0.8333333333333334 (/ -0.2222222222222222 t))
     (if (<= t 1.85)
       (+ 0.5 (* (* t t) (+ 1.0 (* t (+ t -2.0)))))
       (/ (+ 5.0 (/ -8.0 t)) (+ (* t_1 (+ t_1 -4.0)) 6.0))))))

C

double code(double t) {
	double t_1 = 2.0 / (1.0 + t);
	double tmp;
	if (t <= -0.55) {
		tmp = 0.8333333333333334 + (-0.2222222222222222 / t);
	} else if (t <= 1.85) {
		tmp = 0.5 + ((t * t) * (1.0 + (t * (t + -2.0))));
	} else {
		tmp = (5.0 + (-8.0 / t)) / ((t_1 * (t_1 + -4.0)) + 6.0);
	}
	return tmp;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = 2.0d0 / (1.0d0 + t)
    if (t <= (-0.55d0)) then
        tmp = 0.8333333333333334d0 + ((-0.2222222222222222d0) / t)
    else if (t <= 1.85d0) then
        tmp = 0.5d0 + ((t * t) * (1.0d0 + (t * (t + (-2.0d0)))))
    else
        tmp = (5.0d0 + ((-8.0d0) / t)) / ((t_1 * (t_1 + (-4.0d0))) + 6.0d0)
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double t_1 = 2.0 / (1.0 + t);
	double tmp;
	if (t <= -0.55) {
		tmp = 0.8333333333333334 + (-0.2222222222222222 / t);
	} else if (t <= 1.85) {
		tmp = 0.5 + ((t * t) * (1.0 + (t * (t + -2.0))));
	} else {
		tmp = (5.0 + (-8.0 / t)) / ((t_1 * (t_1 + -4.0)) + 6.0);
	}
	return tmp;
}

Python

def code(t):
	t_1 = 2.0 / (1.0 + t)
	tmp = 0
	if t <= -0.55:
		tmp = 0.8333333333333334 + (-0.2222222222222222 / t)
	elif t <= 1.85:
		tmp = 0.5 + ((t * t) * (1.0 + (t * (t + -2.0))))
	else:
		tmp = (5.0 + (-8.0 / t)) / ((t_1 * (t_1 + -4.0)) + 6.0)
	return tmp

Julia

function code(t)
	t_1 = Float64(2.0 / Float64(1.0 + t))
	tmp = 0.0
	if (t <= -0.55)
		tmp = Float64(0.8333333333333334 + Float64(-0.2222222222222222 / t));
	elseif (t <= 1.85)
		tmp = Float64(0.5 + Float64(Float64(t * t) * Float64(1.0 + Float64(t * Float64(t + -2.0)))));
	else
		tmp = Float64(Float64(5.0 + Float64(-8.0 / t)) / Float64(Float64(t_1 * Float64(t_1 + -4.0)) + 6.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	t_1 = 2.0 / (1.0 + t);
	tmp = 0.0;
	if (t <= -0.55)
		tmp = 0.8333333333333334 + (-0.2222222222222222 / t);
	elseif (t <= 1.85)
		tmp = 0.5 + ((t * t) * (1.0 + (t * (t + -2.0))));
	else
		tmp = (5.0 + (-8.0 / t)) / ((t_1 * (t_1 + -4.0)) + 6.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := Block[{t$95$1 = N[(2.0 / N[(1.0 + t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -0.55], N[(0.8333333333333334 + N[(-0.2222222222222222 / t), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.85], N[(0.5 + N[(N[(t * t), $MachinePrecision] * N[(1.0 + N[(t * N[(t + -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(5.0 + N[(-8.0 / t), $MachinePrecision]), $MachinePrecision] / N[(N[(t$95$1 * N[(t$95$1 + -4.0), $MachinePrecision]), $MachinePrecision] + 6.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if 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. Add Preprocessing

   5. Taylor expanded in t around inf

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

      1. sub-neg N/A

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

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{5}{6}, \color{blue}{\left(\mathsf{neg}\left(\frac{2}{9} \cdot \frac{1}{t}\right)\right)}\right) \]

      3. associate-*r/ N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{5}{6}, \left(\mathsf{neg}\left(\frac{\frac{2}{9} \cdot 1}{t}\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{5}{6}, \left(\mathsf{neg}\left(\frac{\frac{2}{9}}{t}\right)\right)\right) \]

      5. distribute-neg-frac N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{5}{6}, \left(\frac{\mathsf{neg}\left(\frac{2}{9}\right)}{\color{blue}{t}}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{5}{6}, \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\frac{2}{9}\right)\right), \color{blue}{t}\right)\right) \]

      7. metadata-eval 98.6%

         \[\leadsto \mathsf{+.f64}\left(\frac{5}{6}, \mathsf{/.f64}\left(\frac{-2}{9}, t\right)\right) \]

   7. Simplified 98.6%

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

   if -0.55000000000000004 < t < 1.8500000000000001

   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. Taylor expanded in t around 0

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

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left({t}^{2} \cdot \left(1 + t \cdot \left(t - 2\right)\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\left({t}^{2}\right), \color{blue}{\left(1 + t \cdot \left(t - 2\right)\right)}\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\left(t \cdot t\right), \left(\color{blue}{1} + t \cdot \left(t - 2\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \left(\color{blue}{1} + t \cdot \left(t - 2\right)\right)\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{+.f64}\left(1, \color{blue}{\left(t \cdot \left(t - 2\right)\right)}\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(t, \color{blue}{\left(t - 2\right)}\right)\right)\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(t, \left(t + \color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)\right)\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(t, \left(t + -2\right)\right)\right)\right)\right) \]

      9. +-lowering-+.f64 99.5%

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(t, \mathsf{+.f64}\left(t, \color{blue}{-2}\right)\right)\right)\right)\right) \]

   7. Simplified 99.5%

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

   if 1.8500000000000001 < 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. Taylor expanded in t around inf

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(5 - 8 \cdot \frac{1}{t}\right)}, \mathsf{+.f64}\left(6, \mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(1, t\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(1, t\right)\right), -4\right)\right)\right)\right) \]
   6. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(5 + \left(\mathsf{neg}\left(8 \cdot \frac{1}{t}\right)\right)\right), \mathsf{+.f64}\left(\color{blue}{6}, \mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(1, t\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(1, t\right)\right), -4\right)\right)\right)\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(5, \left(\mathsf{neg}\left(8 \cdot \frac{1}{t}\right)\right)\right), \mathsf{+.f64}\left(\color{blue}{6}, \mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(1, t\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(1, t\right)\right), -4\right)\right)\right)\right) \]

      3. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(5, \left(\mathsf{neg}\left(\frac{8 \cdot 1}{t}\right)\right)\right), \mathsf{+.f64}\left(6, \mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(1, t\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(1, t\right)\right), -4\right)\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(5, \left(\mathsf{neg}\left(\frac{8}{t}\right)\right)\right), \mathsf{+.f64}\left(6, \mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(1, t\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(1, t\right)\right), -4\right)\right)\right)\right) \]

      5. distribute-neg-frac N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(5, \left(\frac{\mathsf{neg}\left(8\right)}{t}\right)\right), \mathsf{+.f64}\left(6, \mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(1, t\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(1, t\right)\right), -4\right)\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(5, \left(\frac{-8}{t}\right)\right), \mathsf{+.f64}\left(6, \mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(1, t\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(1, t\right)\right), -4\right)\right)\right)\right) \]

      7. /-lowering-/.f64 100.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(5, \mathsf{/.f64}\left(-8, t\right)\right), \mathsf{+.f64}\left(6, \mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(1, t\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(1, t\right)\right), -4\right)\right)\right)\right) \]

   7. Simplified 100.0%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.55:\\ \;\;\;\;0.8333333333333334 + \frac{-0.2222222222222222}{t}\\ \mathbf{elif}\;t \leq 1.85:\\ \;\;\;\;0.5 + \left(t \cdot t\right) \cdot \left(1 + t \cdot \left(t + -2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{5 + \frac{-8}{t}}{\frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} + -4\right) + 6}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.3% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 0.8333333333333334 + \frac{-0.2222222222222222}{t}\\ \mathbf{if}\;t \leq -0.55:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 0.75:\\ \;\;\;\;0.5 + \left(t \cdot t\right) \cdot \left(1 + t \cdot \left(t + -2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (let* ((t_1 (+ 0.8333333333333334 (/ -0.2222222222222222 t))))
   (if (<= t -0.55)
     t_1
     (if (<= t 0.75) (+ 0.5 (* (* t t) (+ 1.0 (* t (+ t -2.0))))) t_1))))

C

double code(double t) {
	double t_1 = 0.8333333333333334 + (-0.2222222222222222 / t);
	double tmp;
	if (t <= -0.55) {
		tmp = t_1;
	} else if (t <= 0.75) {
		tmp = 0.5 + ((t * t) * (1.0 + (t * (t + -2.0))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = 0.8333333333333334d0 + ((-0.2222222222222222d0) / t)
    if (t <= (-0.55d0)) then
        tmp = t_1
    else if (t <= 0.75d0) then
        tmp = 0.5d0 + ((t * t) * (1.0d0 + (t * (t + (-2.0d0)))))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double t_1 = 0.8333333333333334 + (-0.2222222222222222 / t);
	double tmp;
	if (t <= -0.55) {
		tmp = t_1;
	} else if (t <= 0.75) {
		tmp = 0.5 + ((t * t) * (1.0 + (t * (t + -2.0))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(t):
	t_1 = 0.8333333333333334 + (-0.2222222222222222 / t)
	tmp = 0
	if t <= -0.55:
		tmp = t_1
	elif t <= 0.75:
		tmp = 0.5 + ((t * t) * (1.0 + (t * (t + -2.0))))
	else:
		tmp = t_1
	return tmp

Julia

function code(t)
	t_1 = Float64(0.8333333333333334 + Float64(-0.2222222222222222 / t))
	tmp = 0.0
	if (t <= -0.55)
		tmp = t_1;
	elseif (t <= 0.75)
		tmp = Float64(0.5 + Float64(Float64(t * t) * Float64(1.0 + Float64(t * Float64(t + -2.0)))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	t_1 = 0.8333333333333334 + (-0.2222222222222222 / t);
	tmp = 0.0;
	if (t <= -0.55)
		tmp = t_1;
	elseif (t <= 0.75)
		tmp = 0.5 + ((t * t) * (1.0 + (t * (t + -2.0))));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := Block[{t$95$1 = N[(0.8333333333333334 + N[(-0.2222222222222222 / t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -0.55], t$95$1, If[LessEqual[t, 0.75], N[(0.5 + N[(N[(t * t), $MachinePrecision] * N[(1.0 + N[(t * N[(t + -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -0.55000000000000004 or 0.75 < 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. Taylor expanded in t around inf

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

      1. sub-neg N/A

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

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{5}{6}, \color{blue}{\left(\mathsf{neg}\left(\frac{2}{9} \cdot \frac{1}{t}\right)\right)}\right) \]

      3. associate-*r/ N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{5}{6}, \left(\mathsf{neg}\left(\frac{\frac{2}{9} \cdot 1}{t}\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{5}{6}, \left(\mathsf{neg}\left(\frac{\frac{2}{9}}{t}\right)\right)\right) \]

      5. distribute-neg-frac N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{5}{6}, \left(\frac{\mathsf{neg}\left(\frac{2}{9}\right)}{\color{blue}{t}}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{5}{6}, \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\frac{2}{9}\right)\right), \color{blue}{t}\right)\right) \]

      7. metadata-eval 99.4%

         \[\leadsto \mathsf{+.f64}\left(\frac{5}{6}, \mathsf{/.f64}\left(\frac{-2}{9}, t\right)\right) \]

   7. Simplified 99.4%

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

   if -0.55000000000000004 < t < 0.75

   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. Taylor expanded in t around 0

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

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left({t}^{2} \cdot \left(1 + t \cdot \left(t - 2\right)\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\left({t}^{2}\right), \color{blue}{\left(1 + t \cdot \left(t - 2\right)\right)}\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\left(t \cdot t\right), \left(\color{blue}{1} + t \cdot \left(t - 2\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \left(\color{blue}{1} + t \cdot \left(t - 2\right)\right)\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{+.f64}\left(1, \color{blue}{\left(t \cdot \left(t - 2\right)\right)}\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(t, \color{blue}{\left(t - 2\right)}\right)\right)\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(t, \left(t + \color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)\right)\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(t, \left(t + -2\right)\right)\right)\right)\right) \]

      9. +-lowering-+.f64 99.5%

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(t, \mathsf{+.f64}\left(t, \color{blue}{-2}\right)\right)\right)\right)\right) \]

   7. Simplified 99.5%

      \[\leadsto \color{blue}{0.5 + \left(t \cdot t\right) \cdot \left(1 + t \cdot \left(t + -2\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 99.2% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 0.8333333333333334 + \frac{-0.2222222222222222}{t}\\ \mathbf{if}\;t \leq -0.58:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 0.56:\\ \;\;\;\;0.5 + t \cdot \left(t \cdot \left(1 + t \cdot -2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (let* ((t_1 (+ 0.8333333333333334 (/ -0.2222222222222222 t))))
   (if (<= t -0.58)
     t_1
     (if (<= t 0.56) (+ 0.5 (* t (* t (+ 1.0 (* t -2.0))))) t_1))))

C

double code(double t) {
	double t_1 = 0.8333333333333334 + (-0.2222222222222222 / t);
	double tmp;
	if (t <= -0.58) {
		tmp = t_1;
	} else if (t <= 0.56) {
		tmp = 0.5 + (t * (t * (1.0 + (t * -2.0))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = 0.8333333333333334d0 + ((-0.2222222222222222d0) / t)
    if (t <= (-0.58d0)) then
        tmp = t_1
    else if (t <= 0.56d0) then
        tmp = 0.5d0 + (t * (t * (1.0d0 + (t * (-2.0d0)))))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double t_1 = 0.8333333333333334 + (-0.2222222222222222 / t);
	double tmp;
	if (t <= -0.58) {
		tmp = t_1;
	} else if (t <= 0.56) {
		tmp = 0.5 + (t * (t * (1.0 + (t * -2.0))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(t):
	t_1 = 0.8333333333333334 + (-0.2222222222222222 / t)
	tmp = 0
	if t <= -0.58:
		tmp = t_1
	elif t <= 0.56:
		tmp = 0.5 + (t * (t * (1.0 + (t * -2.0))))
	else:
		tmp = t_1
	return tmp

Julia

function code(t)
	t_1 = Float64(0.8333333333333334 + Float64(-0.2222222222222222 / t))
	tmp = 0.0
	if (t <= -0.58)
		tmp = t_1;
	elseif (t <= 0.56)
		tmp = Float64(0.5 + Float64(t * Float64(t * Float64(1.0 + Float64(t * -2.0)))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	t_1 = 0.8333333333333334 + (-0.2222222222222222 / t);
	tmp = 0.0;
	if (t <= -0.58)
		tmp = t_1;
	elseif (t <= 0.56)
		tmp = 0.5 + (t * (t * (1.0 + (t * -2.0))));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := Block[{t$95$1 = N[(0.8333333333333334 + N[(-0.2222222222222222 / t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -0.58], t$95$1, If[LessEqual[t, 0.56], N[(0.5 + N[(t * N[(t * N[(1.0 + N[(t * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -0.57999999999999996 or 0.56000000000000005 < 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. Taylor expanded in t around inf

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

      1. sub-neg N/A

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

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{5}{6}, \color{blue}{\left(\mathsf{neg}\left(\frac{2}{9} \cdot \frac{1}{t}\right)\right)}\right) \]

      3. associate-*r/ N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{5}{6}, \left(\mathsf{neg}\left(\frac{\frac{2}{9} \cdot 1}{t}\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{5}{6}, \left(\mathsf{neg}\left(\frac{\frac{2}{9}}{t}\right)\right)\right) \]

      5. distribute-neg-frac N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{5}{6}, \left(\frac{\mathsf{neg}\left(\frac{2}{9}\right)}{\color{blue}{t}}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{5}{6}, \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\frac{2}{9}\right)\right), \color{blue}{t}\right)\right) \]

      7. metadata-eval 99.4%

         \[\leadsto \mathsf{+.f64}\left(\frac{5}{6}, \mathsf{/.f64}\left(\frac{-2}{9}, t\right)\right) \]

   7. Simplified 99.4%

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

   if -0.57999999999999996 < t < 0.56000000000000005

   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. Taylor expanded in t around 0

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

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left({t}^{2} \cdot \left(1 + -2 \cdot t\right)\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(\left(t \cdot t\right) \cdot \left(\color{blue}{1} + -2 \cdot t\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(t \cdot \color{blue}{\left(t \cdot \left(1 + -2 \cdot t\right)\right)}\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(t \cdot \left(\left(1 + -2 \cdot t\right) \cdot \color{blue}{t}\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(t, \color{blue}{\left(\left(1 + -2 \cdot t\right) \cdot t\right)}\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(t, \left(t \cdot \color{blue}{\left(1 + -2 \cdot t\right)}\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \color{blue}{\left(1 + -2 \cdot t\right)}\right)\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \mathsf{+.f64}\left(1, \color{blue}{\left(-2 \cdot t\right)}\right)\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \mathsf{+.f64}\left(1, \left(t \cdot \color{blue}{-2}\right)\right)\right)\right)\right) \]

      10. *-lowering-*.f64 99.5%

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(t, \color{blue}{-2}\right)\right)\right)\right)\right) \]

   7. Simplified 99.5%

      \[\leadsto \color{blue}{0.5 + t \cdot \left(t \cdot \left(1 + t \cdot -2\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 99.1% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 0.8333333333333334 + \frac{-0.2222222222222222}{t}\\ \mathbf{if}\;t \leq -0.79:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 0.56:\\ \;\;\;\;0.5 + t \cdot t\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (let* ((t_1 (+ 0.8333333333333334 (/ -0.2222222222222222 t))))
   (if (<= t -0.79) t_1 (if (<= t 0.56) (+ 0.5 (* t t)) t_1))))

C

double code(double t) {
	double t_1 = 0.8333333333333334 + (-0.2222222222222222 / t);
	double tmp;
	if (t <= -0.79) {
		tmp = t_1;
	} else if (t <= 0.56) {
		tmp = 0.5 + (t * t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = 0.8333333333333334d0 + ((-0.2222222222222222d0) / t)
    if (t <= (-0.79d0)) then
        tmp = t_1
    else if (t <= 0.56d0) then
        tmp = 0.5d0 + (t * t)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double t_1 = 0.8333333333333334 + (-0.2222222222222222 / t);
	double tmp;
	if (t <= -0.79) {
		tmp = t_1;
	} else if (t <= 0.56) {
		tmp = 0.5 + (t * t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(t):
	t_1 = 0.8333333333333334 + (-0.2222222222222222 / t)
	tmp = 0
	if t <= -0.79:
		tmp = t_1
	elif t <= 0.56:
		tmp = 0.5 + (t * t)
	else:
		tmp = t_1
	return tmp

Julia

function code(t)
	t_1 = Float64(0.8333333333333334 + Float64(-0.2222222222222222 / t))
	tmp = 0.0
	if (t <= -0.79)
		tmp = t_1;
	elseif (t <= 0.56)
		tmp = Float64(0.5 + Float64(t * t));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	t_1 = 0.8333333333333334 + (-0.2222222222222222 / t);
	tmp = 0.0;
	if (t <= -0.79)
		tmp = t_1;
	elseif (t <= 0.56)
		tmp = 0.5 + (t * t);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := Block[{t$95$1 = N[(0.8333333333333334 + N[(-0.2222222222222222 / t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -0.79], t$95$1, If[LessEqual[t, 0.56], N[(0.5 + N[(t * t), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -0.79000000000000004 or 0.56000000000000005 < 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. Taylor expanded in t around inf

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

      1. sub-neg N/A

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

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{5}{6}, \color{blue}{\left(\mathsf{neg}\left(\frac{2}{9} \cdot \frac{1}{t}\right)\right)}\right) \]

      3. associate-*r/ N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{5}{6}, \left(\mathsf{neg}\left(\frac{\frac{2}{9} \cdot 1}{t}\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{5}{6}, \left(\mathsf{neg}\left(\frac{\frac{2}{9}}{t}\right)\right)\right) \]

      5. distribute-neg-frac N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{5}{6}, \left(\frac{\mathsf{neg}\left(\frac{2}{9}\right)}{\color{blue}{t}}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{5}{6}, \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\frac{2}{9}\right)\right), \color{blue}{t}\right)\right) \]

      7. metadata-eval 100.0%

         \[\leadsto \mathsf{+.f64}\left(\frac{5}{6}, \mathsf{/.f64}\left(\frac{-2}{9}, t\right)\right) \]

   7. Simplified 100.0%

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

   if -0.79000000000000004 < t < 0.56000000000000005

   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. Taylor expanded in t around 0

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

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left({t}^{2}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(t \cdot \color{blue}{t}\right)\right) \]

      3. *-lowering-*.f64 98.9%

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(t, \color{blue}{t}\right)\right) \]

   7. Simplified 98.9%

      \[\leadsto \color{blue}{0.5 + t \cdot t} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 98.5% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -0.9:\\ \;\;\;\;0.8333333333333334\\ \mathbf{elif}\;t \leq 0.56:\\ \;\;\;\;0.5 + t \cdot t\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (<= t -0.9)
   0.8333333333333334
   (if (<= t 0.56) (+ 0.5 (* t t)) 0.8333333333333334)))

C

double code(double t) {
	double tmp;
	if (t <= -0.9) {
		tmp = 0.8333333333333334;
	} else if (t <= 0.56) {
		tmp = 0.5 + (t * t);
	} else {
		tmp = 0.8333333333333334;
	}
	return tmp;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-0.9d0)) then
        tmp = 0.8333333333333334d0
    else if (t <= 0.56d0) then
        tmp = 0.5d0 + (t * t)
    else
        tmp = 0.8333333333333334d0
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double tmp;
	if (t <= -0.9) {
		tmp = 0.8333333333333334;
	} else if (t <= 0.56) {
		tmp = 0.5 + (t * t);
	} else {
		tmp = 0.8333333333333334;
	}
	return tmp;
}

Python

def code(t):
	tmp = 0
	if t <= -0.9:
		tmp = 0.8333333333333334
	elif t <= 0.56:
		tmp = 0.5 + (t * t)
	else:
		tmp = 0.8333333333333334
	return tmp

Julia

function code(t)
	tmp = 0.0
	if (t <= -0.9)
		tmp = 0.8333333333333334;
	elseif (t <= 0.56)
		tmp = Float64(0.5 + Float64(t * t));
	else
		tmp = 0.8333333333333334;
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	tmp = 0.0;
	if (t <= -0.9)
		tmp = 0.8333333333333334;
	elseif (t <= 0.56)
		tmp = 0.5 + (t * t);
	else
		tmp = 0.8333333333333334;
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := If[LessEqual[t, -0.9], 0.8333333333333334, If[LessEqual[t, 0.56], N[(0.5 + N[(t * t), $MachinePrecision]), $MachinePrecision], 0.8333333333333334]]

Derivation

1. Split input into 2 regimes

2. if t < -0.900000000000000022 or 0.56000000000000005 < 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. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\frac{5}{6}} \]
   6. Step-by-step derivation

   7. Simplified 99.4%

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

   if -0.900000000000000022 < t < 0.56000000000000005

   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. Taylor expanded in t around 0

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

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left({t}^{2}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(t \cdot \color{blue}{t}\right)\right) \]

      3. *-lowering-*.f64 98.9%

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(t, \color{blue}{t}\right)\right) \]

   7. Simplified 98.9%

      \[\leadsto \color{blue}{0.5 + t \cdot t} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 98.3% 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%

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

   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. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\frac{5}{6}} \]
   6. Step-by-step derivation

   7. Simplified 98.8%

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

   if -0.330000000000000016 < t < 1

   1. Initial program 100.0%

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

   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. Taylor expanded in t around 0

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

   7. Simplified 99.4%

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

Alternative 8: 59.3% 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%

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

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. Taylor expanded in t around 0

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

7. Simplified 59.5%

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

Reproduce

herbie shell --seed 2024138 
(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))))))))