Kahan p13 Example 1

Percentage Accurate: 99.9% → 99.9%

Time: 6.1s

Alternatives: 8

Speedup: 1.0×

Specification

Math

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

FPCore

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

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

FPCore

(FPCore (t)
 :precision binary64
 (let* ((t_1 (/ (* 2.0 t) (+ 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 * t) / (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 * t) / (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 * t) / (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 * t) / (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(Float64(2.0 * t) / 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 * t) / (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[(N[(2.0 * t), $MachinePrecision] / N[(1.0 + t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * t$95$1), $MachinePrecision]}, N[(N[(1.0 + t$95$2), $MachinePrecision] / N[(2.0 + t$95$2), $MachinePrecision]), $MachinePrecision]]]

Alternative 1: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 100.0%

   \[\frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{2 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}} \]
2. Add Preprocessing
3. Add Preprocessing

Alternative 2: 100.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.5 \cdot 10^{+19}:\\ \;\;\;\;0.8333333333333334\\ \mathbf{elif}\;t \leq 2 \cdot 10^{+14}:\\ \;\;\;\;0.5 + \frac{\frac{\frac{t \cdot t}{1 + t}}{1 + t}}{1 + \frac{\frac{2 \cdot \left(t \cdot t\right)}{1 + t}}{1 + t}}\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (<= t -1.5e+19)
   0.8333333333333334
   (if (<= t 2e+14)
     (+
      0.5
      (/
       (/ (/ (* t t) (+ 1.0 t)) (+ 1.0 t))
       (+ 1.0 (/ (/ (* 2.0 (* t t)) (+ 1.0 t)) (+ 1.0 t)))))
     0.8333333333333334)))

C

double code(double t) {
	double tmp;
	if (t <= -1.5e+19) {
		tmp = 0.8333333333333334;
	} else if (t <= 2e+14) {
		tmp = 0.5 + ((((t * t) / (1.0 + t)) / (1.0 + t)) / (1.0 + (((2.0 * (t * t)) / (1.0 + t)) / (1.0 + t))));
	} else {
		tmp = 0.8333333333333334;
	}
	return tmp;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-1.5d+19)) then
        tmp = 0.8333333333333334d0
    else if (t <= 2d+14) then
        tmp = 0.5d0 + ((((t * t) / (1.0d0 + t)) / (1.0d0 + t)) / (1.0d0 + (((2.0d0 * (t * t)) / (1.0d0 + t)) / (1.0d0 + t))))
    else
        tmp = 0.8333333333333334d0
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double tmp;
	if (t <= -1.5e+19) {
		tmp = 0.8333333333333334;
	} else if (t <= 2e+14) {
		tmp = 0.5 + ((((t * t) / (1.0 + t)) / (1.0 + t)) / (1.0 + (((2.0 * (t * t)) / (1.0 + t)) / (1.0 + t))));
	} else {
		tmp = 0.8333333333333334;
	}
	return tmp;
}

Python

def code(t):
	tmp = 0
	if t <= -1.5e+19:
		tmp = 0.8333333333333334
	elif t <= 2e+14:
		tmp = 0.5 + ((((t * t) / (1.0 + t)) / (1.0 + t)) / (1.0 + (((2.0 * (t * t)) / (1.0 + t)) / (1.0 + t))))
	else:
		tmp = 0.8333333333333334
	return tmp

Julia

function code(t)
	tmp = 0.0
	if (t <= -1.5e+19)
		tmp = 0.8333333333333334;
	elseif (t <= 2e+14)
		tmp = Float64(0.5 + Float64(Float64(Float64(Float64(t * t) / Float64(1.0 + t)) / Float64(1.0 + t)) / Float64(1.0 + Float64(Float64(Float64(2.0 * Float64(t * t)) / Float64(1.0 + t)) / Float64(1.0 + t)))));
	else
		tmp = 0.8333333333333334;
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	tmp = 0.0;
	if (t <= -1.5e+19)
		tmp = 0.8333333333333334;
	elseif (t <= 2e+14)
		tmp = 0.5 + ((((t * t) / (1.0 + t)) / (1.0 + t)) / (1.0 + (((2.0 * (t * t)) / (1.0 + t)) / (1.0 + t))));
	else
		tmp = 0.8333333333333334;
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := If[LessEqual[t, -1.5e+19], 0.8333333333333334, If[LessEqual[t, 2e+14], N[(0.5 + N[(N[(N[(N[(t * t), $MachinePrecision] / N[(1.0 + t), $MachinePrecision]), $MachinePrecision] / N[(1.0 + t), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(N[(N[(2.0 * N[(t * t), $MachinePrecision]), $MachinePrecision] / N[(1.0 + t), $MachinePrecision]), $MachinePrecision] / N[(1.0 + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.8333333333333334]]

Derivation

1. Split input into 2 regimes

2. if t < -1.5e19 or 2e14 < t

   1. Initial program 100.0%

      \[\frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{2 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}} \]

   3. Simplified 49.7%

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

   5. Taylor expanded in t around inf

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

   7. Simplified 100.0%

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

   if -1.5e19 < t < 2e14

   1. Initial program 100.0%

      \[\frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{2 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}} \]

   3. Simplified 100.0%

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

Alternative 3: 99.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -0.8:\\ \;\;\;\;0.8333333333333334 + \frac{-0.2222222222222222}{t}\\ \mathbf{elif}\;t \leq 0.58:\\ \;\;\;\;0.5 + t \cdot t\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222 + \frac{-0.037037037037037035 + \frac{-0.04938271604938271}{t}}{t}}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (<= t -0.8)
   (+ 0.8333333333333334 (/ -0.2222222222222222 t))
   (if (<= t 0.58)
     (+ 0.5 (* t t))
     (-
      0.8333333333333334
      (/
       (+
        0.2222222222222222
        (/ (+ -0.037037037037037035 (/ -0.04938271604938271 t)) t))
       t)))))

C

double code(double t) {
	double tmp;
	if (t <= -0.8) {
		tmp = 0.8333333333333334 + (-0.2222222222222222 / t);
	} else if (t <= 0.58) {
		tmp = 0.5 + (t * t);
	} else {
		tmp = 0.8333333333333334 - ((0.2222222222222222 + ((-0.037037037037037035 + (-0.04938271604938271 / t)) / t)) / t);
	}
	return tmp;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-0.8d0)) then
        tmp = 0.8333333333333334d0 + ((-0.2222222222222222d0) / t)
    else if (t <= 0.58d0) then
        tmp = 0.5d0 + (t * t)
    else
        tmp = 0.8333333333333334d0 - ((0.2222222222222222d0 + (((-0.037037037037037035d0) + ((-0.04938271604938271d0) / t)) / t)) / t)
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double tmp;
	if (t <= -0.8) {
		tmp = 0.8333333333333334 + (-0.2222222222222222 / t);
	} else if (t <= 0.58) {
		tmp = 0.5 + (t * t);
	} else {
		tmp = 0.8333333333333334 - ((0.2222222222222222 + ((-0.037037037037037035 + (-0.04938271604938271 / t)) / t)) / t);
	}
	return tmp;
}

Python

def code(t):
	tmp = 0
	if t <= -0.8:
		tmp = 0.8333333333333334 + (-0.2222222222222222 / t)
	elif t <= 0.58:
		tmp = 0.5 + (t * t)
	else:
		tmp = 0.8333333333333334 - ((0.2222222222222222 + ((-0.037037037037037035 + (-0.04938271604938271 / t)) / t)) / t)
	return tmp

Julia

function code(t)
	tmp = 0.0
	if (t <= -0.8)
		tmp = Float64(0.8333333333333334 + Float64(-0.2222222222222222 / t));
	elseif (t <= 0.58)
		tmp = Float64(0.5 + Float64(t * t));
	else
		tmp = Float64(0.8333333333333334 - Float64(Float64(0.2222222222222222 + Float64(Float64(-0.037037037037037035 + Float64(-0.04938271604938271 / t)) / t)) / t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	tmp = 0.0;
	if (t <= -0.8)
		tmp = 0.8333333333333334 + (-0.2222222222222222 / t);
	elseif (t <= 0.58)
		tmp = 0.5 + (t * t);
	else
		tmp = 0.8333333333333334 - ((0.2222222222222222 + ((-0.037037037037037035 + (-0.04938271604938271 / t)) / t)) / t);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := If[LessEqual[t, -0.8], N[(0.8333333333333334 + N[(-0.2222222222222222 / t), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 0.58], N[(0.5 + N[(t * t), $MachinePrecision]), $MachinePrecision], N[(0.8333333333333334 - N[(N[(0.2222222222222222 + N[(N[(-0.037037037037037035 + N[(-0.04938271604938271 / t), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -0.80000000000000004

   1. Initial program 100.0%

      \[\frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{2 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}} \]

   3. Simplified 51.6%

      \[\leadsto \color{blue}{0.5 + \frac{\frac{\frac{t \cdot t}{1 + t}}{1 + t}}{1 + \frac{\frac{2 \cdot \left(t \cdot t\right)}{1 + t}}{1 + t}}} \]
   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 -0.80000000000000004 < t < 0.57999999999999996

   1. Initial program 100.0%

      \[\frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{2 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{0.5 + \frac{\frac{\frac{t \cdot t}{1 + t}}{1 + t}}{1 + \frac{\frac{2 \cdot \left(t \cdot t\right)}{1 + t}}{1 + t}}} \]
   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 100.0%

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

   7. Simplified 100.0%

      \[\leadsto \color{blue}{0.5 + t \cdot t} \]

   if 0.57999999999999996 < t

   1. Initial program 100.0%

      \[\frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{2 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}} \]

   3. Simplified 53.0%

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

   5. Taylor expanded in t around -inf

      \[\leadsto \color{blue}{\frac{5}{6} + -1 \cdot \frac{\frac{2}{9} + -1 \cdot \frac{\frac{1}{27} + \frac{4}{81} \cdot \frac{1}{t}}{t}}{t}} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \frac{5}{6} + \left(\mathsf{neg}\left(\frac{\frac{2}{9} + -1 \cdot \frac{\frac{1}{27} + \frac{4}{81} \cdot \frac{1}{t}}{t}}{t}\right)\right) \]

      2. unsub-neg N/A

         \[\leadsto \frac{5}{6} - \color{blue}{\frac{\frac{2}{9} + -1 \cdot \frac{\frac{1}{27} + \frac{4}{81} \cdot \frac{1}{t}}{t}}{t}} \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\frac{5}{6}, \color{blue}{\left(\frac{\frac{2}{9} + -1 \cdot \frac{\frac{1}{27} + \frac{4}{81} \cdot \frac{1}{t}}{t}}{t}\right)}\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\frac{5}{6}, \mathsf{/.f64}\left(\left(\frac{2}{9} + -1 \cdot \frac{\frac{1}{27} + \frac{4}{81} \cdot \frac{1}{t}}{t}\right), \color{blue}{t}\right)\right) \]

   7. Simplified 100.0%

      \[\leadsto \color{blue}{0.8333333333333334 - \frac{0.2222222222222222 + \frac{-0.037037037037037035 + \frac{-0.04938271604938271}{t}}{t}}{t}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 4: 99.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -0.8:\\ \;\;\;\;0.8333333333333334 + \frac{-0.2222222222222222}{t}\\ \mathbf{elif}\;t \leq 0.235:\\ \;\;\;\;0.5 + t \cdot t\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222 + \frac{-0.037037037037037035}{t}}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (<= t -0.8)
   (+ 0.8333333333333334 (/ -0.2222222222222222 t))
   (if (<= t 0.235)
     (+ 0.5 (* t t))
     (-
      0.8333333333333334
      (/ (+ 0.2222222222222222 (/ -0.037037037037037035 t)) t)))))

C

double code(double t) {
	double tmp;
	if (t <= -0.8) {
		tmp = 0.8333333333333334 + (-0.2222222222222222 / t);
	} else if (t <= 0.235) {
		tmp = 0.5 + (t * t);
	} else {
		tmp = 0.8333333333333334 - ((0.2222222222222222 + (-0.037037037037037035 / t)) / t);
	}
	return tmp;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-0.8d0)) then
        tmp = 0.8333333333333334d0 + ((-0.2222222222222222d0) / t)
    else if (t <= 0.235d0) then
        tmp = 0.5d0 + (t * t)
    else
        tmp = 0.8333333333333334d0 - ((0.2222222222222222d0 + ((-0.037037037037037035d0) / t)) / t)
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double tmp;
	if (t <= -0.8) {
		tmp = 0.8333333333333334 + (-0.2222222222222222 / t);
	} else if (t <= 0.235) {
		tmp = 0.5 + (t * t);
	} else {
		tmp = 0.8333333333333334 - ((0.2222222222222222 + (-0.037037037037037035 / t)) / t);
	}
	return tmp;
}

Python

def code(t):
	tmp = 0
	if t <= -0.8:
		tmp = 0.8333333333333334 + (-0.2222222222222222 / t)
	elif t <= 0.235:
		tmp = 0.5 + (t * t)
	else:
		tmp = 0.8333333333333334 - ((0.2222222222222222 + (-0.037037037037037035 / t)) / t)
	return tmp

Julia

function code(t)
	tmp = 0.0
	if (t <= -0.8)
		tmp = Float64(0.8333333333333334 + Float64(-0.2222222222222222 / t));
	elseif (t <= 0.235)
		tmp = Float64(0.5 + Float64(t * t));
	else
		tmp = Float64(0.8333333333333334 - Float64(Float64(0.2222222222222222 + Float64(-0.037037037037037035 / t)) / t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	tmp = 0.0;
	if (t <= -0.8)
		tmp = 0.8333333333333334 + (-0.2222222222222222 / t);
	elseif (t <= 0.235)
		tmp = 0.5 + (t * t);
	else
		tmp = 0.8333333333333334 - ((0.2222222222222222 + (-0.037037037037037035 / t)) / t);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := If[LessEqual[t, -0.8], N[(0.8333333333333334 + N[(-0.2222222222222222 / t), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 0.235], N[(0.5 + N[(t * t), $MachinePrecision]), $MachinePrecision], N[(0.8333333333333334 - N[(N[(0.2222222222222222 + N[(-0.037037037037037035 / t), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -0.80000000000000004

   1. Initial program 100.0%

      \[\frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{2 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}} \]

   3. Simplified 51.6%

      \[\leadsto \color{blue}{0.5 + \frac{\frac{\frac{t \cdot t}{1 + t}}{1 + t}}{1 + \frac{\frac{2 \cdot \left(t \cdot t\right)}{1 + t}}{1 + t}}} \]
   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 -0.80000000000000004 < t < 0.23499999999999999

   1. Initial program 100.0%

      \[\frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{2 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{0.5 + \frac{\frac{\frac{t \cdot t}{1 + t}}{1 + t}}{1 + \frac{\frac{2 \cdot \left(t \cdot t\right)}{1 + t}}{1 + t}}} \]
   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 100.0%

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

   7. Simplified 100.0%

      \[\leadsto \color{blue}{0.5 + t \cdot t} \]

   if 0.23499999999999999 < t

   1. Initial program 100.0%

      \[\frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{2 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}} \]

   3. Simplified 53.0%

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

   5. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\left(\frac{5}{6} + \frac{\frac{1}{27}}{{t}^{2}}\right) - \frac{2}{9} \cdot \frac{1}{t}} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\frac{\frac{1}{27}}{{t}^{2}} + \frac{5}{6}\right) - \color{blue}{\frac{2}{9}} \cdot \frac{1}{t} \]

      2. associate--l+ N/A

         \[\leadsto \frac{\frac{1}{27}}{{t}^{2}} + \color{blue}{\left(\frac{5}{6} - \frac{2}{9} \cdot \frac{1}{t}\right)} \]

      3. +-commutative N/A

         \[\leadsto \left(\frac{5}{6} - \frac{2}{9} \cdot \frac{1}{t}\right) + \color{blue}{\frac{\frac{1}{27}}{{t}^{2}}} \]

      4. associate--r- N/A

         \[\leadsto \frac{5}{6} - \color{blue}{\left(\frac{2}{9} \cdot \frac{1}{t} - \frac{\frac{1}{27}}{{t}^{2}}\right)} \]

      5. associate-*r/ N/A

         \[\leadsto \frac{5}{6} - \left(\frac{\frac{2}{9} \cdot 1}{t} - \frac{\color{blue}{\frac{1}{27}}}{{t}^{2}}\right) \]

      6. metadata-eval N/A

         \[\leadsto \frac{5}{6} - \left(\frac{\frac{2}{9}}{t} - \frac{\frac{1}{27}}{{t}^{2}}\right) \]

      7. unpow2 N/A

         \[\leadsto \frac{5}{6} - \left(\frac{\frac{2}{9}}{t} - \frac{\frac{1}{27}}{t \cdot \color{blue}{t}}\right) \]

      8. associate-/r* N/A

         \[\leadsto \frac{5}{6} - \left(\frac{\frac{2}{9}}{t} - \frac{\frac{\frac{1}{27}}{t}}{\color{blue}{t}}\right) \]

      9. metadata-eval N/A

         \[\leadsto \frac{5}{6} - \left(\frac{\frac{2}{9}}{t} - \frac{\frac{\frac{1}{27} \cdot 1}{t}}{t}\right) \]

      10. associate-*r/ N/A

          \[\leadsto \frac{5}{6} - \left(\frac{\frac{2}{9}}{t} - \frac{\frac{1}{27} \cdot \frac{1}{t}}{t}\right) \]

      11. div-sub N/A

          \[\leadsto \frac{5}{6} - \frac{\frac{2}{9} - \frac{1}{27} \cdot \frac{1}{t}}{\color{blue}{t}} \]

      12. --lowering--.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\frac{5}{6}, \color{blue}{\left(\frac{\frac{2}{9} - \frac{1}{27} \cdot \frac{1}{t}}{t}\right)}\right) \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\frac{5}{6}, \mathsf{/.f64}\left(\left(\frac{2}{9} - \frac{1}{27} \cdot \frac{1}{t}\right), \color{blue}{t}\right)\right) \]

      14. sub-neg N/A

          \[\leadsto \mathsf{\_.f64}\left(\frac{5}{6}, \mathsf{/.f64}\left(\left(\frac{2}{9} + \left(\mathsf{neg}\left(\frac{1}{27} \cdot \frac{1}{t}\right)\right)\right), t\right)\right) \]

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

          \[\leadsto \mathsf{\_.f64}\left(\frac{5}{6}, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{2}{9}, \left(\mathsf{neg}\left(\frac{1}{27} \cdot \frac{1}{t}\right)\right)\right), t\right)\right) \]

      16. associate-*r/ N/A

          \[\leadsto \mathsf{\_.f64}\left(\frac{5}{6}, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{2}{9}, \left(\mathsf{neg}\left(\frac{\frac{1}{27} \cdot 1}{t}\right)\right)\right), t\right)\right) \]

      17. metadata-eval N/A

          \[\leadsto \mathsf{\_.f64}\left(\frac{5}{6}, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{2}{9}, \left(\mathsf{neg}\left(\frac{\frac{1}{27}}{t}\right)\right)\right), t\right)\right) \]

      18. distribute-neg-frac N/A

          \[\leadsto \mathsf{\_.f64}\left(\frac{5}{6}, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{2}{9}, \left(\frac{\mathsf{neg}\left(\frac{1}{27}\right)}{t}\right)\right), t\right)\right) \]

      19. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\frac{5}{6}, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{2}{9}, \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\frac{1}{27}\right)\right), t\right)\right), t\right)\right) \]

      20. metadata-eval 99.9%

          \[\leadsto \mathsf{\_.f64}\left(\frac{5}{6}, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{2}{9}, \mathsf{/.f64}\left(\frac{-1}{27}, t\right)\right), t\right)\right) \]

   7. Simplified 99.9%

      \[\leadsto \color{blue}{0.8333333333333334 - \frac{0.2222222222222222 + \frac{-0.037037037037037035}{t}}{t}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 5: 99.3% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 0.8333333333333334 + \frac{-0.2222222222222222}{t}\\ \mathbf{if}\;t \leq -0.8:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 0.58:\\ \;\;\;\;0.5 + t \cdot t\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (let* ((t_1 (+ 0.8333333333333334 (/ -0.2222222222222222 t))))
   (if (<= t -0.8) t_1 (if (<= t 0.58) (+ 0.5 (* t t)) t_1))))

C

double code(double t) {
	double t_1 = 0.8333333333333334 + (-0.2222222222222222 / t);
	double tmp;
	if (t <= -0.8) {
		tmp = t_1;
	} else if (t <= 0.58) {
		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 <= (-0.8d0)) then
        tmp = t_1
    else if (t <= 0.58d0) 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 <= -0.8) {
		tmp = t_1;
	} else if (t <= 0.58) {
		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 <= -0.8:
		tmp = t_1
	elif t <= 0.58:
		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 <= -0.8)
		tmp = t_1;
	elseif (t <= 0.58)
		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 <= -0.8)
		tmp = t_1;
	elseif (t <= 0.58)
		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, -0.8], t$95$1, If[LessEqual[t, 0.58], N[(0.5 + N[(t * t), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -0.80000000000000004 or 0.57999999999999996 < t

   1. Initial program 100.0%

      \[\frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{2 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}} \]

   3. Simplified 52.2%

      \[\leadsto \color{blue}{0.5 + \frac{\frac{\frac{t \cdot t}{1 + t}}{1 + t}}{1 + \frac{\frac{2 \cdot \left(t \cdot t\right)}{1 + t}}{1 + t}}} \]
   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 99.8%

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

   7. Simplified 99.8%

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

   if -0.80000000000000004 < t < 0.57999999999999996

   1. Initial program 100.0%

      \[\frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{2 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{0.5 + \frac{\frac{\frac{t \cdot t}{1 + t}}{1 + t}}{1 + \frac{\frac{2 \cdot \left(t \cdot t\right)}{1 + t}}{1 + t}}} \]
   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 100.0%

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

   7. Simplified 100.0%

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

Alternative 6: 98.8% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -0.9:\\ \;\;\;\;0.8333333333333334\\ \mathbf{elif}\;t \leq 0.58:\\ \;\;\;\;0.5 + t \cdot t\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (<= t -0.9)
   0.8333333333333334
   (if (<= t 0.58) (+ 0.5 (* t t)) 0.8333333333333334)))

C

double code(double t) {
	double tmp;
	if (t <= -0.9) {
		tmp = 0.8333333333333334;
	} else if (t <= 0.58) {
		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 <= 0.58d0) 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 <= 0.58) {
		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 <= 0.58:
		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 <= 0.58)
		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 <= 0.58)
		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, 0.58], N[(0.5 + N[(t * t), $MachinePrecision]), $MachinePrecision], 0.8333333333333334]]

Derivation

1. Split input into 2 regimes

2. if t < -0.900000000000000022 or 0.57999999999999996 < t

   1. Initial program 100.0%

      \[\frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{2 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}} \]

   3. Simplified 52.2%

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

   5. Taylor expanded in t around inf

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

   7. Simplified 98.9%

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

   if -0.900000000000000022 < t < 0.57999999999999996

   1. Initial program 100.0%

      \[\frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{2 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{0.5 + \frac{\frac{\frac{t \cdot t}{1 + t}}{1 + t}}{1 + \frac{\frac{2 \cdot \left(t \cdot t\right)}{1 + t}}{1 + t}}} \]
   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 100.0%

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

   7. Simplified 100.0%

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

Alternative 7: 98.6% accurate, 3.2× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-0.34d0)) then
        tmp = 0.8333333333333334d0
    else if (t <= 1.0d0) then
        tmp = 0.5d0
    else
        tmp = 0.8333333333333334d0
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if t < -0.340000000000000024 or 1 < t

   1. Initial program 100.0%

      \[\frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{2 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}} \]

   3. Simplified 52.2%

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

   5. Taylor expanded in t around inf

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

   7. Simplified 98.9%

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

   if -0.340000000000000024 < t < 1

   1. Initial program 100.0%

      \[\frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{2 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}} \]

   3. Simplified 100.0%

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

   5. Taylor expanded in t around 0

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

   7. Simplified 99.8%

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

Alternative 8: 59.6% accurate, 35.0× 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%

   \[\frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{2 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}} \]

3. Simplified 77.8%

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

5. Taylor expanded in t around 0

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

7. Simplified 62.5%

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

Reproduce

herbie shell --seed 2024155 
(FPCore (t)
  :name "Kahan p13 Example 1"
  :precision binary64
  (/ (+ 1.0 (* (/ (* 2.0 t) (+ 1.0 t)) (/ (* 2.0 t) (+ 1.0 t)))) (+ 2.0 (* (/ (* 2.0 t) (+ 1.0 t)) (/ (* 2.0 t) (+ 1.0 t))))))