Hyperbolic tangent

Percentage Accurate: 9.2% → 97.5%

Time: 6.3s

Alternatives: 7

Speedup: 409.0×

• Unsound rule application detected in e-graph. Results may not be sound.

Specification

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-x}\\ \frac{e^{x} - t_0}{e^{x} + t_0} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (exp (- x)))) (/ (- (exp x) t_0) (+ (exp x) t_0))))

C

double code(double x) {
	double t_0 = exp(-x);
	return (exp(x) - t_0) / (exp(x) + t_0);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    t_0 = exp(-x)
    code = (exp(x) - t_0) / (exp(x) + t_0)
end function

Java

public static double code(double x) {
	double t_0 = Math.exp(-x);
	return (Math.exp(x) - t_0) / (Math.exp(x) + t_0);
}

Python

def code(x):
	t_0 = math.exp(-x)
	return (math.exp(x) - t_0) / (math.exp(x) + t_0)

Julia

function code(x)
	t_0 = exp(Float64(-x))
	return Float64(Float64(exp(x) - t_0) / Float64(exp(x) + t_0))
end

MATLAB

function tmp = code(x)
	t_0 = exp(-x);
	tmp = (exp(x) - t_0) / (exp(x) + t_0);
end

Wolfram

code[x_] := Block[{t$95$0 = N[Exp[(-x)], $MachinePrecision]}, N[(N[(N[Exp[x], $MachinePrecision] - t$95$0), $MachinePrecision] / N[(N[Exp[x], $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]]

Initial Program: 9.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-x}\\ \frac{e^{x} - t_0}{e^{x} + t_0} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (exp (- x)))) (/ (- (exp x) t_0) (+ (exp x) t_0))))

C

double code(double x) {
	double t_0 = exp(-x);
	return (exp(x) - t_0) / (exp(x) + t_0);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    t_0 = exp(-x)
    code = (exp(x) - t_0) / (exp(x) + t_0)
end function

Java

public static double code(double x) {
	double t_0 = Math.exp(-x);
	return (Math.exp(x) - t_0) / (Math.exp(x) + t_0);
}

Python

def code(x):
	t_0 = math.exp(-x)
	return (math.exp(x) - t_0) / (math.exp(x) + t_0)

Julia

function code(x)
	t_0 = exp(Float64(-x))
	return Float64(Float64(exp(x) - t_0) / Float64(exp(x) + t_0))
end

MATLAB

function tmp = code(x)
	t_0 = exp(-x);
	tmp = (exp(x) - t_0) / (exp(x) + t_0);
end

Wolfram

code[x_] := Block[{t$95$0 = N[Exp[(-x)], $MachinePrecision]}, N[(N[(N[Exp[x], $MachinePrecision] - t$95$0), $MachinePrecision] / N[(N[Exp[x], $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]]

Alternative 1: 97.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \frac{2 \cdot x + \left(0.3333333333333333 \cdot {x}^{3} + 0.016666666666666666 \cdot {x}^{5}\right)}{2 + \left(x \cdot x + 0.08333333333333333 \cdot {x}^{4}\right)} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/
  (+
   (* 2.0 x)
   (+ (* 0.3333333333333333 (pow x 3.0)) (* 0.016666666666666666 (pow x 5.0))))
  (+ 2.0 (+ (* x x) (* 0.08333333333333333 (pow x 4.0))))))

C

double code(double x) {
	return ((2.0 * x) + ((0.3333333333333333 * pow(x, 3.0)) + (0.016666666666666666 * pow(x, 5.0)))) / (2.0 + ((x * x) + (0.08333333333333333 * pow(x, 4.0))));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = ((2.0d0 * x) + ((0.3333333333333333d0 * (x ** 3.0d0)) + (0.016666666666666666d0 * (x ** 5.0d0)))) / (2.0d0 + ((x * x) + (0.08333333333333333d0 * (x ** 4.0d0))))
end function

Java

public static double code(double x) {
	return ((2.0 * x) + ((0.3333333333333333 * Math.pow(x, 3.0)) + (0.016666666666666666 * Math.pow(x, 5.0)))) / (2.0 + ((x * x) + (0.08333333333333333 * Math.pow(x, 4.0))));
}

Python

def code(x):
	return ((2.0 * x) + ((0.3333333333333333 * math.pow(x, 3.0)) + (0.016666666666666666 * math.pow(x, 5.0)))) / (2.0 + ((x * x) + (0.08333333333333333 * math.pow(x, 4.0))))

Julia

function code(x)
	return Float64(Float64(Float64(2.0 * x) + Float64(Float64(0.3333333333333333 * (x ^ 3.0)) + Float64(0.016666666666666666 * (x ^ 5.0)))) / Float64(2.0 + Float64(Float64(x * x) + Float64(0.08333333333333333 * (x ^ 4.0)))))
end

MATLAB

function tmp = code(x)
	tmp = ((2.0 * x) + ((0.3333333333333333 * (x ^ 3.0)) + (0.016666666666666666 * (x ^ 5.0)))) / (2.0 + ((x * x) + (0.08333333333333333 * (x ^ 4.0))));
end

Wolfram

code[x_] := N[(N[(N[(2.0 * x), $MachinePrecision] + N[(N[(0.3333333333333333 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision] + N[(0.016666666666666666 * N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 + N[(N[(x * x), $MachinePrecision] + N[(0.08333333333333333 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 10.5%

   \[\frac{e^{x} - e^{-x}}{e^{x} + e^{-x}} \]

2. Taylor expanded in x around 0 9.2%

   \[\leadsto \frac{e^{x} - e^{-x}}{\color{blue}{2 + \left({x}^{2} + 0.08333333333333333 \cdot {x}^{4}\right)}} \]
3. Step-by-step derivation

   1. unpow2 9.2%

      \[\leadsto \frac{e^{x} - e^{-x}}{2 + \left(\color{blue}{x \cdot x} + 0.08333333333333333 \cdot {x}^{4}\right)} \]

4. Simplified 9.2%

   \[\leadsto \frac{e^{x} - e^{-x}}{\color{blue}{2 + \left(x \cdot x + 0.08333333333333333 \cdot {x}^{4}\right)}} \]

5. Taylor expanded in x around 0 96.3%

   \[\leadsto \frac{\color{blue}{2 \cdot x + \left(0.3333333333333333 \cdot {x}^{3} + 0.016666666666666666 \cdot {x}^{5}\right)}}{2 + \left(x \cdot x + 0.08333333333333333 \cdot {x}^{4}\right)} \]

6. Final simplification 96.3%

   \[\leadsto \frac{2 \cdot x + \left(0.3333333333333333 \cdot {x}^{3} + 0.016666666666666666 \cdot {x}^{5}\right)}{2 + \left(x \cdot x + 0.08333333333333333 \cdot {x}^{4}\right)} \]

Alternative 2: 97.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \frac{2 \cdot x + 0.3333333333333333 \cdot {x}^{3}}{2 + \left(x \cdot x + 0.08333333333333333 \cdot {x}^{4}\right)} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/
  (+ (* 2.0 x) (* 0.3333333333333333 (pow x 3.0)))
  (+ 2.0 (+ (* x x) (* 0.08333333333333333 (pow x 4.0))))))

C

double code(double x) {
	return ((2.0 * x) + (0.3333333333333333 * pow(x, 3.0))) / (2.0 + ((x * x) + (0.08333333333333333 * pow(x, 4.0))));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = ((2.0d0 * x) + (0.3333333333333333d0 * (x ** 3.0d0))) / (2.0d0 + ((x * x) + (0.08333333333333333d0 * (x ** 4.0d0))))
end function

Java

public static double code(double x) {
	return ((2.0 * x) + (0.3333333333333333 * Math.pow(x, 3.0))) / (2.0 + ((x * x) + (0.08333333333333333 * Math.pow(x, 4.0))));
}

Python

def code(x):
	return ((2.0 * x) + (0.3333333333333333 * math.pow(x, 3.0))) / (2.0 + ((x * x) + (0.08333333333333333 * math.pow(x, 4.0))))

Julia

function code(x)
	return Float64(Float64(Float64(2.0 * x) + Float64(0.3333333333333333 * (x ^ 3.0))) / Float64(2.0 + Float64(Float64(x * x) + Float64(0.08333333333333333 * (x ^ 4.0)))))
end

MATLAB

function tmp = code(x)
	tmp = ((2.0 * x) + (0.3333333333333333 * (x ^ 3.0))) / (2.0 + ((x * x) + (0.08333333333333333 * (x ^ 4.0))));
end

Wolfram

code[x_] := N[(N[(N[(2.0 * x), $MachinePrecision] + N[(0.3333333333333333 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 + N[(N[(x * x), $MachinePrecision] + N[(0.08333333333333333 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 10.5%

   \[\frac{e^{x} - e^{-x}}{e^{x} + e^{-x}} \]

2. Taylor expanded in x around 0 9.2%

   \[\leadsto \frac{e^{x} - e^{-x}}{\color{blue}{2 + \left({x}^{2} + 0.08333333333333333 \cdot {x}^{4}\right)}} \]
3. Step-by-step derivation

   1. unpow2 9.2%

      \[\leadsto \frac{e^{x} - e^{-x}}{2 + \left(\color{blue}{x \cdot x} + 0.08333333333333333 \cdot {x}^{4}\right)} \]

4. Simplified 9.2%

   \[\leadsto \frac{e^{x} - e^{-x}}{\color{blue}{2 + \left(x \cdot x + 0.08333333333333333 \cdot {x}^{4}\right)}} \]

5. Taylor expanded in x around 0 96.2%

   \[\leadsto \frac{\color{blue}{2 \cdot x + 0.3333333333333333 \cdot {x}^{3}}}{2 + \left(x \cdot x + 0.08333333333333333 \cdot {x}^{4}\right)} \]

6. Final simplification 96.2%

   \[\leadsto \frac{2 \cdot x + 0.3333333333333333 \cdot {x}^{3}}{2 + \left(x \cdot x + 0.08333333333333333 \cdot {x}^{4}\right)} \]

Alternative 3: 97.3% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \frac{2}{2 + x \cdot x} \cdot \left(x + {x}^{3} \cdot 0.16666666666666666\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (* (/ 2.0 (+ 2.0 (* x x))) (+ x (* (pow x 3.0) 0.16666666666666666))))

C

double code(double x) {
	return (2.0 / (2.0 + (x * x))) * (x + (pow(x, 3.0) * 0.16666666666666666));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (2.0d0 / (2.0d0 + (x * x))) * (x + ((x ** 3.0d0) * 0.16666666666666666d0))
end function

Java

public static double code(double x) {
	return (2.0 / (2.0 + (x * x))) * (x + (Math.pow(x, 3.0) * 0.16666666666666666));
}

Python

def code(x):
	return (2.0 / (2.0 + (x * x))) * (x + (math.pow(x, 3.0) * 0.16666666666666666))

Julia

function code(x)
	return Float64(Float64(2.0 / Float64(2.0 + Float64(x * x))) * Float64(x + Float64((x ^ 3.0) * 0.16666666666666666)))
end

MATLAB

function tmp = code(x)
	tmp = (2.0 / (2.0 + (x * x))) * (x + ((x ^ 3.0) * 0.16666666666666666));
end

Wolfram

code[x_] := N[(N[(2.0 / N[(2.0 + N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(x + N[(N[Power[x, 3.0], $MachinePrecision] * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 10.5%

   \[\frac{e^{x} - e^{-x}}{e^{x} + e^{-x}} \]

2. Taylor expanded in x around 0 9.2%

   \[\leadsto \frac{e^{x} - e^{-x}}{\color{blue}{2 + \left({x}^{2} + 0.08333333333333333 \cdot {x}^{4}\right)}} \]
3. Step-by-step derivation

   1. unpow2 9.2%

      \[\leadsto \frac{e^{x} - e^{-x}}{2 + \left(\color{blue}{x \cdot x} + 0.08333333333333333 \cdot {x}^{4}\right)} \]

4. Simplified 9.2%

   \[\leadsto \frac{e^{x} - e^{-x}}{\color{blue}{2 + \left(x \cdot x + 0.08333333333333333 \cdot {x}^{4}\right)}} \]
5. Step-by-step derivation

   1. div-inv 9.2%

      \[\leadsto \color{blue}{\left(e^{x} - e^{-x}\right) \cdot \frac{1}{2 + \left(x \cdot x + 0.08333333333333333 \cdot {x}^{4}\right)}} \]

   2. sinh-undef 95.9%

      \[\leadsto \color{blue}{\left(2 \cdot \sinh x\right)} \cdot \frac{1}{2 + \left(x \cdot x + 0.08333333333333333 \cdot {x}^{4}\right)} \]

   3. fma-def 95.9%

      \[\leadsto \left(2 \cdot \sinh x\right) \cdot \frac{1}{2 + \color{blue}{\mathsf{fma}\left(x, x, 0.08333333333333333 \cdot {x}^{4}\right)}} \]

6. Applied egg-rr 95.9%

   \[\leadsto \color{blue}{\left(2 \cdot \sinh x\right) \cdot \frac{1}{2 + \mathsf{fma}\left(x, x, 0.08333333333333333 \cdot {x}^{4}\right)}} \]
7. Step-by-step derivation

   1. associate-*r/ 95.9%

      \[\leadsto \color{blue}{\frac{\left(2 \cdot \sinh x\right) \cdot 1}{2 + \mathsf{fma}\left(x, x, 0.08333333333333333 \cdot {x}^{4}\right)}} \]

   2. *-rgt-identity 95.9%

      \[\leadsto \frac{\color{blue}{2 \cdot \sinh x}}{2 + \mathsf{fma}\left(x, x, 0.08333333333333333 \cdot {x}^{4}\right)} \]

   3. associate-/l* 95.4%

      \[\leadsto \color{blue}{\frac{2}{\frac{2 + \mathsf{fma}\left(x, x, 0.08333333333333333 \cdot {x}^{4}\right)}{\sinh x}}} \]

   4. associate-/r/ 95.9%

      \[\leadsto \color{blue}{\frac{2}{2 + \mathsf{fma}\left(x, x, 0.08333333333333333 \cdot {x}^{4}\right)} \cdot \sinh x} \]

   5. fma-udef 95.9%

      \[\leadsto \frac{2}{2 + \color{blue}{\left(x \cdot x + 0.08333333333333333 \cdot {x}^{4}\right)}} \cdot \sinh x \]

   6. +-commutative 95.9%

      \[\leadsto \frac{2}{2 + \color{blue}{\left(0.08333333333333333 \cdot {x}^{4} + x \cdot x\right)}} \cdot \sinh x \]

   7. *-commutative 95.9%

      \[\leadsto \frac{2}{2 + \left(\color{blue}{{x}^{4} \cdot 0.08333333333333333} + x \cdot x\right)} \cdot \sinh x \]

   8. fma-def 95.9%

      \[\leadsto \frac{2}{2 + \color{blue}{\mathsf{fma}\left({x}^{4}, 0.08333333333333333, x \cdot x\right)}} \cdot \sinh x \]

8. Simplified 95.9%

   \[\leadsto \color{blue}{\frac{2}{2 + \mathsf{fma}\left({x}^{4}, 0.08333333333333333, x \cdot x\right)} \cdot \sinh x} \]

9. Taylor expanded in x around 0 95.8%

   \[\leadsto \frac{2}{2 + \color{blue}{{x}^{2}}} \cdot \sinh x \]
10. Step-by-step derivation

    1. unpow2 95.8%

       \[\leadsto \frac{2}{2 + \color{blue}{x \cdot x}} \cdot \sinh x \]

11. Simplified 95.8%

    \[\leadsto \frac{2}{2 + \color{blue}{x \cdot x}} \cdot \sinh x \]

12. Taylor expanded in x around 0 96.2%

    \[\leadsto \frac{2}{2 + x \cdot x} \cdot \color{blue}{\left(0.16666666666666666 \cdot {x}^{3} + x\right)} \]

13. Final simplification 96.2%

    \[\leadsto \frac{2}{2 + x \cdot x} \cdot \left(x + {x}^{3} \cdot 0.16666666666666666\right) \]

Alternative 4: 97.1% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \frac{2}{2 + x \cdot x} \cdot \sinh x \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* (/ 2.0 (+ 2.0 (* x x))) (sinh x)))

C

double code(double x) {
	return (2.0 / (2.0 + (x * x))) * sinh(x);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (2.0d0 / (2.0d0 + (x * x))) * sinh(x)
end function

Java

public static double code(double x) {
	return (2.0 / (2.0 + (x * x))) * Math.sinh(x);
}

Python

def code(x):
	return (2.0 / (2.0 + (x * x))) * math.sinh(x)

Julia

function code(x)
	return Float64(Float64(2.0 / Float64(2.0 + Float64(x * x))) * sinh(x))
end

MATLAB

function tmp = code(x)
	tmp = (2.0 / (2.0 + (x * x))) * sinh(x);
end

Wolfram

code[x_] := N[(N[(2.0 / N[(2.0 + N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sinh[x], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 10.5%

   \[\frac{e^{x} - e^{-x}}{e^{x} + e^{-x}} \]

2. Taylor expanded in x around 0 9.2%

   \[\leadsto \frac{e^{x} - e^{-x}}{\color{blue}{2 + \left({x}^{2} + 0.08333333333333333 \cdot {x}^{4}\right)}} \]
3. Step-by-step derivation

   1. unpow2 9.2%

      \[\leadsto \frac{e^{x} - e^{-x}}{2 + \left(\color{blue}{x \cdot x} + 0.08333333333333333 \cdot {x}^{4}\right)} \]

4. Simplified 9.2%

   \[\leadsto \frac{e^{x} - e^{-x}}{\color{blue}{2 + \left(x \cdot x + 0.08333333333333333 \cdot {x}^{4}\right)}} \]
5. Step-by-step derivation

   1. div-inv 9.2%

      \[\leadsto \color{blue}{\left(e^{x} - e^{-x}\right) \cdot \frac{1}{2 + \left(x \cdot x + 0.08333333333333333 \cdot {x}^{4}\right)}} \]

   2. sinh-undef 95.9%

      \[\leadsto \color{blue}{\left(2 \cdot \sinh x\right)} \cdot \frac{1}{2 + \left(x \cdot x + 0.08333333333333333 \cdot {x}^{4}\right)} \]

   3. fma-def 95.9%

      \[\leadsto \left(2 \cdot \sinh x\right) \cdot \frac{1}{2 + \color{blue}{\mathsf{fma}\left(x, x, 0.08333333333333333 \cdot {x}^{4}\right)}} \]

6. Applied egg-rr 95.9%

   \[\leadsto \color{blue}{\left(2 \cdot \sinh x\right) \cdot \frac{1}{2 + \mathsf{fma}\left(x, x, 0.08333333333333333 \cdot {x}^{4}\right)}} \]
7. Step-by-step derivation

   1. associate-*r/ 95.9%

      \[\leadsto \color{blue}{\frac{\left(2 \cdot \sinh x\right) \cdot 1}{2 + \mathsf{fma}\left(x, x, 0.08333333333333333 \cdot {x}^{4}\right)}} \]

   2. *-rgt-identity 95.9%

      \[\leadsto \frac{\color{blue}{2 \cdot \sinh x}}{2 + \mathsf{fma}\left(x, x, 0.08333333333333333 \cdot {x}^{4}\right)} \]

   3. associate-/l* 95.4%

      \[\leadsto \color{blue}{\frac{2}{\frac{2 + \mathsf{fma}\left(x, x, 0.08333333333333333 \cdot {x}^{4}\right)}{\sinh x}}} \]

   4. associate-/r/ 95.9%

      \[\leadsto \color{blue}{\frac{2}{2 + \mathsf{fma}\left(x, x, 0.08333333333333333 \cdot {x}^{4}\right)} \cdot \sinh x} \]

   5. fma-udef 95.9%

      \[\leadsto \frac{2}{2 + \color{blue}{\left(x \cdot x + 0.08333333333333333 \cdot {x}^{4}\right)}} \cdot \sinh x \]

   6. +-commutative 95.9%

      \[\leadsto \frac{2}{2 + \color{blue}{\left(0.08333333333333333 \cdot {x}^{4} + x \cdot x\right)}} \cdot \sinh x \]

   7. *-commutative 95.9%

      \[\leadsto \frac{2}{2 + \left(\color{blue}{{x}^{4} \cdot 0.08333333333333333} + x \cdot x\right)} \cdot \sinh x \]

   8. fma-def 95.9%

      \[\leadsto \frac{2}{2 + \color{blue}{\mathsf{fma}\left({x}^{4}, 0.08333333333333333, x \cdot x\right)}} \cdot \sinh x \]

8. Simplified 95.9%

   \[\leadsto \color{blue}{\frac{2}{2 + \mathsf{fma}\left({x}^{4}, 0.08333333333333333, x \cdot x\right)} \cdot \sinh x} \]

9. Taylor expanded in x around 0 95.8%

   \[\leadsto \frac{2}{2 + \color{blue}{{x}^{2}}} \cdot \sinh x \]
10. Step-by-step derivation

    1. unpow2 95.8%

       \[\leadsto \frac{2}{2 + \color{blue}{x \cdot x}} \cdot \sinh x \]

11. Simplified 95.8%

    \[\leadsto \frac{2}{2 + \color{blue}{x \cdot x}} \cdot \sinh x \]

12. Final simplification 95.8%

    \[\leadsto \frac{2}{2 + x \cdot x} \cdot \sinh x \]

Alternative 5: 96.8% accurate, 45.4× speedup

Math

\[\begin{array}{l} \\ \frac{x + x}{2 + x \cdot x} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ (+ x x) (+ 2.0 (* x x))))

C

double code(double x) {
	return (x + x) / (2.0 + (x * x));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (x + x) / (2.0d0 + (x * x))
end function

Java

public static double code(double x) {
	return (x + x) / (2.0 + (x * x));
}

Python

def code(x):
	return (x + x) / (2.0 + (x * x))

Julia

function code(x)
	return Float64(Float64(x + x) / Float64(2.0 + Float64(x * x)))
end

MATLAB

function tmp = code(x)
	tmp = (x + x) / (2.0 + (x * x));
end

Wolfram

code[x_] := N[(N[(x + x), $MachinePrecision] / N[(2.0 + N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 10.5%

   \[\frac{e^{x} - e^{-x}}{e^{x} + e^{-x}} \]

2. Taylor expanded in x around 0 9.1%

   \[\leadsto \frac{e^{x} - e^{-x}}{\color{blue}{2 + {x}^{2}}} \]
3. Step-by-step derivation

   1. unpow2 9.1%

      \[\leadsto \frac{e^{x} - e^{-x}}{2 + \color{blue}{x \cdot x}} \]

4. Simplified 9.1%

   \[\leadsto \frac{e^{x} - e^{-x}}{\color{blue}{2 + x \cdot x}} \]

5. Taylor expanded in x around 0 95.8%

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

   1. count-2 95.8%

      \[\leadsto \frac{\color{blue}{x + x}}{2 + x \cdot x} \]

7. Simplified 95.8%

   \[\leadsto \frac{\color{blue}{x + x}}{2 + x \cdot x} \]

8. Final simplification 95.8%

   \[\leadsto \frac{x + x}{2 + x \cdot x} \]

Alternative 6: 4.0% accurate, 409.0× speedup

Math

\[\begin{array}{l} \\ 1.5 \end{array} \]

FPCore

(FPCore (x) :precision binary64 1.5)

C

double code(double x) {
	return 1.5;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 1.5d0
end function

Java

public static double code(double x) {
	return 1.5;
}

Python

def code(x):
	return 1.5

Julia

function code(x)
	return 1.5
end

MATLAB

function tmp = code(x)
	tmp = 1.5;
end

Wolfram

code[x_] := 1.5

Derivation

1. Initial program 10.5%

   \[\frac{e^{x} - e^{-x}}{e^{x} + e^{-x}} \]

2. Taylor expanded in x around 0 9.1%

   \[\leadsto \frac{e^{x} - e^{-x}}{\color{blue}{2 + {x}^{2}}} \]
3. Step-by-step derivation

   1. unpow2 9.1%

      \[\leadsto \frac{e^{x} - e^{-x}}{2 + \color{blue}{x \cdot x}} \]

4. Simplified 9.1%

   \[\leadsto \frac{e^{x} - e^{-x}}{\color{blue}{2 + x \cdot x}} \]

6. Applied egg-rr 3.8%

   \[\leadsto \frac{\color{blue}{3}}{2 + x \cdot x} \]

7. Taylor expanded in x around 0 3.9%

   \[\leadsto \color{blue}{1.5} \]

8. Final simplification 3.9%

   \[\leadsto 1.5 \]

Alternative 7: 96.7% accurate, 409.0× speedup

Math

\[\begin{array}{l} \\ x \end{array} \]

FPCore

(FPCore (x) :precision binary64 x)

C

double code(double x) {
	return x;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = x
end function

Java

public static double code(double x) {
	return x;
}

Python

def code(x):
	return x

Julia

function code(x)
	return x
end

MATLAB

function tmp = code(x)
	tmp = x;
end

Wolfram

code[x_] := x

Derivation

1. Initial program 10.5%

   \[\frac{e^{x} - e^{-x}}{e^{x} + e^{-x}} \]

2. Taylor expanded in x around 0 95.7%

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

3. Final simplification 95.7%

   \[\leadsto x \]

Reproduce

herbie shell --seed 2023175 
(FPCore (x)
  :name "Hyperbolic tangent"
  :precision binary64
  (/ (- (exp x) (exp (- x))) (+ (exp x) (exp (- x)))))