Hyperbolic tangent

Percentage Accurate: 8.8% → 97.3%

Time: 6.4s

Alternatives: 6

Speedup: 409.0×

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: 8.8% 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: 97.3% accurate, 3.4× speedup

Math

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

FPCore

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

C

double code(double x) {
	return ((2.0 * x) + (0.3333333333333333 * (x * (x * x)))) / (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 * (x * x)))) / (2.0d0 + ((x * x) + (0.08333333333333333d0 * (x ** 4.0d0))))
end function

Java

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

Python

def code(x):
	return ((2.0 * x) + (0.3333333333333333 * (x * (x * x)))) / (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 * Float64(x * Float64(x * x)))) / 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 * (x * x)))) / (2.0 + ((x * x) + (0.08333333333333333 * (x ^ 4.0))));
end

Wolfram

code[x_] := N[(N[(N[(2.0 * x), $MachinePrecision] + N[(0.3333333333333333 * N[(x * N[(x * x), $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 9.9%

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

2. Taylor expanded in x around 0 96.6%

   \[\leadsto \frac{\color{blue}{2 \cdot x + 0.3333333333333333 \cdot {x}^{3}}}{e^{x} + e^{-x}} \]
3. Step-by-step derivation

   1. unpow3 96.6%

      \[\leadsto \frac{2 \cdot x + 0.3333333333333333 \cdot \color{blue}{\left(\left(x \cdot x\right) \cdot x\right)}}{e^{x} + e^{-x}} \]

4. Applied egg-rr 96.6%

   \[\leadsto \frac{2 \cdot x + 0.3333333333333333 \cdot \color{blue}{\left(\left(x \cdot x\right) \cdot x\right)}}{e^{x} + e^{-x}} \]

5. Taylor expanded in x around 0 96.8%

   \[\leadsto \frac{2 \cdot x + 0.3333333333333333 \cdot \left(\left(x \cdot x\right) \cdot x\right)}{\color{blue}{2 + \left({x}^{2} + 0.08333333333333333 \cdot {x}^{4}\right)}} \]
6. Step-by-step derivation

   1. unpow2 96.8%

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

7. Simplified 96.8%

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

8. Final simplification 96.8%

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

Alternative 2: 97.3% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \left(x \cdot \left(x \cdot x\right)\right) \cdot -0.3333333333333333 + \left(x + 0.13333333333333333 \cdot {x}^{5}\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (+
  (* (* x (* x x)) -0.3333333333333333)
  (+ x (* 0.13333333333333333 (pow x 5.0)))))

C

double code(double x) {
	return ((x * (x * x)) * -0.3333333333333333) + (x + (0.13333333333333333 * pow(x, 5.0)));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = ((x * (x * x)) * (-0.3333333333333333d0)) + (x + (0.13333333333333333d0 * (x ** 5.0d0)))
end function

Java

public static double code(double x) {
	return ((x * (x * x)) * -0.3333333333333333) + (x + (0.13333333333333333 * Math.pow(x, 5.0)));
}

Python

def code(x):
	return ((x * (x * x)) * -0.3333333333333333) + (x + (0.13333333333333333 * math.pow(x, 5.0)))

Julia

function code(x)
	return Float64(Float64(Float64(x * Float64(x * x)) * -0.3333333333333333) + Float64(x + Float64(0.13333333333333333 * (x ^ 5.0))))
end

MATLAB

function tmp = code(x)
	tmp = ((x * (x * x)) * -0.3333333333333333) + (x + (0.13333333333333333 * (x ^ 5.0)));
end

Wolfram

code[x_] := N[(N[(N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision] * -0.3333333333333333), $MachinePrecision] + N[(x + N[(0.13333333333333333 * N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 9.9%

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

2. Taylor expanded in x around 0 96.8%

   \[\leadsto \color{blue}{-0.3333333333333333 \cdot {x}^{3} + \left(0.13333333333333333 \cdot {x}^{5} + x\right)} \]
3. Step-by-step derivation

   1. unpow3 96.6%

      \[\leadsto \frac{2 \cdot x + 0.3333333333333333 \cdot \color{blue}{\left(\left(x \cdot x\right) \cdot x\right)}}{e^{x} + e^{-x}} \]

4. Applied egg-rr 96.8%

   \[\leadsto -0.3333333333333333 \cdot \color{blue}{\left(\left(x \cdot x\right) \cdot x\right)} + \left(0.13333333333333333 \cdot {x}^{5} + x\right) \]

5. Final simplification 96.8%

   \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot -0.3333333333333333 + \left(x + 0.13333333333333333 \cdot {x}^{5}\right) \]

Alternative 3: 97.2% accurate, 24.1× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (/ (+ (* 2.0 x) (* 0.3333333333333333 (* x (* x x)))) (+ 2.0 (* x x))))

C

double code(double x) {
	return ((2.0 * x) + (0.3333333333333333 * (x * (x * x)))) / (2.0 + (x * x));
}

Fortran

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

Java

public static double code(double x) {
	return ((2.0 * x) + (0.3333333333333333 * (x * (x * x)))) / (2.0 + (x * x));
}

Python

def code(x):
	return ((2.0 * x) + (0.3333333333333333 * (x * (x * x)))) / (2.0 + (x * x))

Julia

function code(x)
	return Float64(Float64(Float64(2.0 * x) + Float64(0.3333333333333333 * Float64(x * Float64(x * x)))) / Float64(2.0 + Float64(x * x)))
end

MATLAB

function tmp = code(x)
	tmp = ((2.0 * x) + (0.3333333333333333 * (x * (x * x)))) / (2.0 + (x * x));
end

Wolfram

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

Derivation

1. Initial program 9.9%

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

2. Taylor expanded in x around 0 96.6%

   \[\leadsto \frac{\color{blue}{2 \cdot x + 0.3333333333333333 \cdot {x}^{3}}}{e^{x} + e^{-x}} \]
3. Step-by-step derivation

   1. unpow3 96.6%

      \[\leadsto \frac{2 \cdot x + 0.3333333333333333 \cdot \color{blue}{\left(\left(x \cdot x\right) \cdot x\right)}}{e^{x} + e^{-x}} \]

4. Applied egg-rr 96.6%

   \[\leadsto \frac{2 \cdot x + 0.3333333333333333 \cdot \color{blue}{\left(\left(x \cdot x\right) \cdot x\right)}}{e^{x} + e^{-x}} \]

5. Taylor expanded in x around 0 96.8%

   \[\leadsto \frac{2 \cdot x + 0.3333333333333333 \cdot \left(\left(x \cdot x\right) \cdot x\right)}{\color{blue}{2 + {x}^{2}}} \]
6. Step-by-step derivation

   1. unpow2 96.4%

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

7. Simplified 96.8%

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

8. Final simplification 96.8%

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

Alternative 4: 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 9.9%

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

2. Taylor expanded in x around 0 96.2%

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

   1. count-2 96.2%

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

4. Simplified 96.2%

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

5. Taylor expanded in x around 0 96.4%

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

   1. unpow2 96.4%

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

7. Simplified 96.4%

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

8. Final simplification 96.4%

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

Alternative 5: 3.9% accurate, 409.0× speedup

Math

\[\begin{array}{l} \\ 4.5 \end{array} \]

FPCore

(FPCore (x) :precision binary64 4.5)

C

double code(double x) {
	return 4.5;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 4.5d0
end function

Java

public static double code(double x) {
	return 4.5;
}

Python

def code(x):
	return 4.5

Julia

function code(x)
	return 4.5
end

MATLAB

function tmp = code(x)
	tmp = 4.5;
end

Wolfram

code[x_] := 4.5

Derivation

1. Initial program 9.9%

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

3. Applied egg-rr 3.8%

   \[\leadsto \frac{\color{blue}{9}}{e^{x} + e^{-x}} \]

4. Taylor expanded in x around 0 3.9%

   \[\leadsto \color{blue}{4.5} \]

5. Final simplification 3.9%

   \[\leadsto 4.5 \]

Alternative 6: 96.8% 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 9.9%

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

2. Taylor expanded in x around 0 96.4%

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

3. Final simplification 96.4%

   \[\leadsto x \]

Reproduce

herbie shell --seed 2023229 
(FPCore (x)
  :name "Hyperbolic tangent"
  :precision binary64
  (/ (- (exp x) (exp (- x))) (+ (exp x) (exp (- x)))))