Kahan p13 Example 2

Percentage Accurate: 100.0% → 100.0%

Time: 10.3s

Alternatives: 6

Speedup: 1.9×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 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, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{-4}{t + 1}\\ t_2 := 8 - t_1\\ \frac{\frac{8 + t_1}{t + 1} + -5}{\frac{\frac{64}{t_2} - \frac{\frac{t_1}{\left(t + 1\right) \cdot -0.25}}{t_2}}{t + 1} + -6} \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (let* ((t_1 (/ -4.0 (+ t 1.0))) (t_2 (- 8.0 t_1)))
   (/
    (+ (/ (+ 8.0 t_1) (+ t 1.0)) -5.0)
    (+
     (/ (- (/ 64.0 t_2) (/ (/ t_1 (* (+ t 1.0) -0.25)) t_2)) (+ t 1.0))
     -6.0))))

C

double code(double t) {
	double t_1 = -4.0 / (t + 1.0);
	double t_2 = 8.0 - t_1;
	return (((8.0 + t_1) / (t + 1.0)) + -5.0) / ((((64.0 / t_2) - ((t_1 / ((t + 1.0) * -0.25)) / t_2)) / (t + 1.0)) + -6.0);
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    t_1 = (-4.0d0) / (t + 1.0d0)
    t_2 = 8.0d0 - t_1
    code = (((8.0d0 + t_1) / (t + 1.0d0)) + (-5.0d0)) / ((((64.0d0 / t_2) - ((t_1 / ((t + 1.0d0) * (-0.25d0))) / t_2)) / (t + 1.0d0)) + (-6.0d0))
end function

Java

public static double code(double t) {
	double t_1 = -4.0 / (t + 1.0);
	double t_2 = 8.0 - t_1;
	return (((8.0 + t_1) / (t + 1.0)) + -5.0) / ((((64.0 / t_2) - ((t_1 / ((t + 1.0) * -0.25)) / t_2)) / (t + 1.0)) + -6.0);
}

Python

def code(t):
	t_1 = -4.0 / (t + 1.0)
	t_2 = 8.0 - t_1
	return (((8.0 + t_1) / (t + 1.0)) + -5.0) / ((((64.0 / t_2) - ((t_1 / ((t + 1.0) * -0.25)) / t_2)) / (t + 1.0)) + -6.0)

Julia

function code(t)
	t_1 = Float64(-4.0 / Float64(t + 1.0))
	t_2 = Float64(8.0 - t_1)
	return Float64(Float64(Float64(Float64(8.0 + t_1) / Float64(t + 1.0)) + -5.0) / Float64(Float64(Float64(Float64(64.0 / t_2) - Float64(Float64(t_1 / Float64(Float64(t + 1.0) * -0.25)) / t_2)) / Float64(t + 1.0)) + -6.0))
end

MATLAB

function tmp = code(t)
	t_1 = -4.0 / (t + 1.0);
	t_2 = 8.0 - t_1;
	tmp = (((8.0 + t_1) / (t + 1.0)) + -5.0) / ((((64.0 / t_2) - ((t_1 / ((t + 1.0) * -0.25)) / t_2)) / (t + 1.0)) + -6.0);
end

Wolfram

code[t_] := Block[{t$95$1 = N[(-4.0 / N[(t + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(8.0 - t$95$1), $MachinePrecision]}, N[(N[(N[(N[(8.0 + t$95$1), $MachinePrecision] / N[(t + 1.0), $MachinePrecision]), $MachinePrecision] + -5.0), $MachinePrecision] / N[(N[(N[(N[(64.0 / t$95$2), $MachinePrecision] - N[(N[(t$95$1 / N[(N[(t + 1.0), $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision] / N[(t + 1.0), $MachinePrecision]), $MachinePrecision] + -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 99.1%

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

6. Applied egg-rr 99.1%

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

   1. expm1-def 99.2%

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

   2. expm1-log1p 100.0%

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

   3. *-lft-identity 100.0%

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

   4. metadata-eval 100.0%

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

   5. times-frac 100.0%

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

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. Step-by-step derivation

   1. flip-+ 100.0%

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

   2. div-sub 100.0%

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

   3. metadata-eval 100.0%

      \[\leadsto \frac{\frac{8 + \frac{-4}{t + 1}}{t + 1} + -5}{\frac{\frac{\color{blue}{64}}{8 - \frac{-4}{t + 1}} - \frac{\frac{-4}{t + 1} \cdot \frac{-4}{t + 1}}{8 - \frac{-4}{t + 1}}}{t + 1} + -6} \]

   4. +-commutative 100.0%

      \[\leadsto \frac{\frac{8 + \frac{-4}{t + 1}}{t + 1} + -5}{\frac{\frac{64}{8 - \frac{-4}{\color{blue}{1 + t}}} - \frac{\frac{-4}{t + 1} \cdot \frac{-4}{t + 1}}{8 - \frac{-4}{t + 1}}}{t + 1} + -6} \]

   5. pow2 100.0%

      \[\leadsto \frac{\frac{8 + \frac{-4}{t + 1}}{t + 1} + -5}{\frac{\frac{64}{8 - \frac{-4}{1 + t}} - \frac{\color{blue}{{\left(\frac{-4}{t + 1}\right)}^{2}}}{8 - \frac{-4}{t + 1}}}{t + 1} + -6} \]

   6. +-commutative 100.0%

      \[\leadsto \frac{\frac{8 + \frac{-4}{t + 1}}{t + 1} + -5}{\frac{\frac{64}{8 - \frac{-4}{1 + t}} - \frac{{\left(\frac{-4}{\color{blue}{1 + t}}\right)}^{2}}{8 - \frac{-4}{t + 1}}}{t + 1} + -6} \]

   7. +-commutative 100.0%

      \[\leadsto \frac{\frac{8 + \frac{-4}{t + 1}}{t + 1} + -5}{\frac{\frac{64}{8 - \frac{-4}{1 + t}} - \frac{{\left(\frac{-4}{1 + t}\right)}^{2}}{8 - \frac{-4}{\color{blue}{1 + t}}}}{t + 1} + -6} \]

10. Applied egg-rr 100.0%

    \[\leadsto \frac{\frac{8 + \frac{-4}{t + 1}}{t + 1} + -5}{\frac{\color{blue}{\frac{64}{8 - \frac{-4}{1 + t}} - \frac{{\left(\frac{-4}{1 + t}\right)}^{2}}{8 - \frac{-4}{1 + t}}}}{t + 1} + -6} \]
11. Step-by-step derivation

    1. unpow2 100.0%

       \[\leadsto \frac{\frac{8 + \frac{-4}{t + 1}}{t + 1} + -5}{\frac{\frac{64}{8 - \frac{-4}{1 + t}} - \frac{\color{blue}{\frac{-4}{1 + t} \cdot \frac{-4}{1 + t}}}{8 - \frac{-4}{1 + t}}}{t + 1} + -6} \]

    2. clear-num 100.0%

       \[\leadsto \frac{\frac{8 + \frac{-4}{t + 1}}{t + 1} + -5}{\frac{\frac{64}{8 - \frac{-4}{1 + t}} - \frac{\frac{-4}{1 + t} \cdot \color{blue}{\frac{1}{\frac{1 + t}{-4}}}}{8 - \frac{-4}{1 + t}}}{t + 1} + -6} \]

    3. un-div-inv 100.0%

       \[\leadsto \frac{\frac{8 + \frac{-4}{t + 1}}{t + 1} + -5}{\frac{\frac{64}{8 - \frac{-4}{1 + t}} - \frac{\color{blue}{\frac{\frac{-4}{1 + t}}{\frac{1 + t}{-4}}}}{8 - \frac{-4}{1 + t}}}{t + 1} + -6} \]

    4. div-inv 100.0%

       \[\leadsto \frac{\frac{8 + \frac{-4}{t + 1}}{t + 1} + -5}{\frac{\frac{64}{8 - \frac{-4}{1 + t}} - \frac{\frac{\frac{-4}{1 + t}}{\color{blue}{\left(1 + t\right) \cdot \frac{1}{-4}}}}{8 - \frac{-4}{1 + t}}}{t + 1} + -6} \]

    5. metadata-eval 100.0%

       \[\leadsto \frac{\frac{8 + \frac{-4}{t + 1}}{t + 1} + -5}{\frac{\frac{64}{8 - \frac{-4}{1 + t}} - \frac{\frac{\frac{-4}{1 + t}}{\left(1 + t\right) \cdot \color{blue}{-0.25}}}{8 - \frac{-4}{1 + t}}}{t + 1} + -6} \]

12. Applied egg-rr 100.0%

    \[\leadsto \frac{\frac{8 + \frac{-4}{t + 1}}{t + 1} + -5}{\frac{\frac{64}{8 - \frac{-4}{1 + t}} - \frac{\color{blue}{\frac{\frac{-4}{1 + t}}{\left(1 + t\right) \cdot -0.25}}}{8 - \frac{-4}{1 + t}}}{t + 1} + -6} \]

13. Final simplification 100.0%

    \[\leadsto \frac{\frac{8 + \frac{-4}{t + 1}}{t + 1} + -5}{\frac{\frac{64}{8 - \frac{-4}{t + 1}} - \frac{\frac{\frac{-4}{t + 1}}{\left(t + 1\right) \cdot -0.25}}{8 - \frac{-4}{t + 1}}}{t + 1} + -6} \]
14. Add Preprocessing

Alternative 2: 100.0% accurate, 1.6× speedup

Math

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

FPCore

(FPCore (t)
 :precision binary64
 (let* ((t_1 (/ -2.0 (+ t 1.0))) (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 / (t + 1.0);
	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) / (t + 1.0d0)
    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 / (t + 1.0);
	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 / (t + 1.0)
	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(t + 1.0))
	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 / (t + 1.0);
	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[(t + 1.0), $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%

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

Alternative 3: 100.0% accurate, 1.9× speedup

Math

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

FPCore

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

C

double code(double t) {
	double t_1 = (8.0 + (-4.0 / (t + 1.0))) / (t + 1.0);
	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) / (t + 1.0d0))) / (t + 1.0d0)
    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 / (t + 1.0))) / (t + 1.0);
	return (t_1 + -5.0) / (t_1 + -6.0);
}

Python

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

Julia

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

MATLAB

function tmp = code(t)
	t_1 = (8.0 + (-4.0 / (t + 1.0))) / (t + 1.0);
	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[(t + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t + 1.0), $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 99.1%

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

6. Applied egg-rr 99.1%

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

   1. expm1-def 99.2%

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

   2. expm1-log1p 100.0%

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

   3. *-lft-identity 100.0%

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

   4. metadata-eval 100.0%

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

   5. times-frac 100.0%

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

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%

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

Alternative 4: 99.0% accurate, 5.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -0.49 \lor \neg \left(t \leq 0.65\right):\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222}{t}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (or (<= t -0.49) (not (<= t 0.65)))
   (- 0.8333333333333334 (/ 0.2222222222222222 t))
   0.5))

C

double code(double t) {
	double tmp;
	if ((t <= -0.49) || !(t <= 0.65)) {
		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 <= 0.65d0))) 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 <= 0.65)) {
		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 <= 0.65):
		tmp = 0.8333333333333334 - (0.2222222222222222 / t)
	else:
		tmp = 0.5
	return tmp

Julia

function code(t)
	tmp = 0.0
	if ((t <= -0.49) || !(t <= 0.65))
		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 <= 0.65)))
		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, 0.65]], $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 0.650000000000000022 < 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 98.9%

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

      1. associate-*r/ 98.9%

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

      2. metadata-eval 98.9%

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

   7. Simplified 98.9%

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

   if -0.48999999999999999 < t < 0.650000000000000022

   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 99.3%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.49 \lor \neg \left(t \leq 0.65\right):\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222}{t}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.5% accurate, 10.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if t < -0.340000000000000024 or 1 < t

   1. Initial program 100.0%

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

   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 98.1%

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

   if -0.340000000000000024 < t < 1

   1. Initial program 100.0%

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

   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 99.3%

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

4. Final simplification 98.6%

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

Alternative 6: 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 55.7%

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

6. Final simplification 55.7%

   \[\leadsto 0.5 \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024010 
(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))))))))