Hyperbolic tangent

Percentage Accurate: 8.9% → 97.5%

Time: 8.0s

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: 8.9% 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, 1.9× speedup

Math

\[\begin{array}{l} \\ x + \left(-0.3333333333333333 \cdot {x}^{3} + 0.13333333333333333 \cdot {x}^{5}\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (+
  x
  (+ (* -0.3333333333333333 (pow x 3.0)) (* 0.13333333333333333 (pow x 5.0)))))

C

double code(double x) {
	return x + ((-0.3333333333333333 * pow(x, 3.0)) + (0.13333333333333333 * pow(x, 5.0)));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = x + (((-0.3333333333333333d0) * (x ** 3.0d0)) + (0.13333333333333333d0 * (x ** 5.0d0)))
end function

Java

public static double code(double x) {
	return x + ((-0.3333333333333333 * Math.pow(x, 3.0)) + (0.13333333333333333 * Math.pow(x, 5.0)));
}

Python

def code(x):
	return x + ((-0.3333333333333333 * math.pow(x, 3.0)) + (0.13333333333333333 * math.pow(x, 5.0)))

Julia

function code(x)
	return Float64(x + Float64(Float64(-0.3333333333333333 * (x ^ 3.0)) + Float64(0.13333333333333333 * (x ^ 5.0))))
end

MATLAB

function tmp = code(x)
	tmp = x + ((-0.3333333333333333 * (x ^ 3.0)) + (0.13333333333333333 * (x ^ 5.0)));
end

Wolfram

code[x_] := N[(x + N[(N[(-0.3333333333333333 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision] + N[(0.13333333333333333 * N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 7.6%

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

2. Taylor expanded in x around 0 99.1%

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

3. Final simplification 99.1%

   \[\leadsto x + \left(-0.3333333333333333 \cdot {x}^{3} + 0.13333333333333333 \cdot {x}^{5}\right) \]

Alternative 2: 97.5% accurate, 24.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 7.6%

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

2. Taylor expanded in x around 0 6.9%

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

   1. unpow2 6.9%

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

4. Simplified 6.9%

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

5. Taylor expanded in x around 0 99.0%

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

   1. unpow3 99.0%

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

   2. unpow2 99.0%

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

   3. associate-*r* 99.0%

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

   4. distribute-rgt-out 99.0%

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

   5. *-commutative 99.0%

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

   6. unpow2 99.0%

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

   7. associate-*l* 99.0%

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

   8. fma-def 99.0%

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

7. Simplified 99.0%

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

   1. fma-udef 99.0%

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

   2. distribute-rgt-in 99.0%

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

9. Applied egg-rr 99.0%

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

10. Final simplification 99.0%

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

Alternative 3: 97.5% accurate, 27.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 7.6%

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

2. Taylor expanded in x around 0 6.9%

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

   1. unpow2 6.9%

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

4. Simplified 6.9%

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

5. Taylor expanded in x around 0 99.0%

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

   1. unpow3 99.0%

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

   2. unpow2 99.0%

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

   3. associate-*r* 99.0%

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

   4. distribute-rgt-out 99.0%

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

   5. *-commutative 99.0%

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

   6. unpow2 99.0%

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

   7. associate-*l* 99.0%

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

   8. fma-def 99.0%

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

7. Simplified 99.0%

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

   1. fma-udef 99.0%

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

9. Applied egg-rr 99.0%

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

10. Final simplification 99.0%

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

Alternative 4: 97.0% accurate, 45.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 7.6%

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

2. Taylor expanded in x around 0 6.9%

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

   1. unpow2 6.9%

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

4. Simplified 6.9%

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

5. Taylor expanded in x around 0 98.8%

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

6. Final simplification 98.8%

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

Alternative 5: 3.9% accurate, 409.0× speedup

Math

\[\begin{array}{l} \\ 0.012345679012345678 \end{array} \]

FPCore

(FPCore (x) :precision binary64 0.012345679012345678)

C

double code(double x) {
	return 0.012345679012345678;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 0.012345679012345678d0
end function

Java

public static double code(double x) {
	return 0.012345679012345678;
}

Python

def code(x):
	return 0.012345679012345678

Julia

function code(x)
	return 0.012345679012345678
end

MATLAB

function tmp = code(x)
	tmp = 0.012345679012345678;
end

Wolfram

code[x_] := 0.012345679012345678

Derivation

1. Initial program 7.6%

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

2. Taylor expanded in x around 0 98.9%

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

   1. *-commutative 98.9%

      \[\leadsto x + \color{blue}{{x}^{3} \cdot -0.3333333333333333} \]

4. Simplified 98.9%

   \[\leadsto \color{blue}{x + {x}^{3} \cdot -0.3333333333333333} \]

6. Applied egg-rr 4.0%

   \[\leadsto x + \color{blue}{0.012345679012345678} \]

7. Taylor expanded in x around 0 3.9%

   \[\leadsto \color{blue}{0.012345679012345678} \]

8. Final simplification 3.9%

   \[\leadsto 0.012345679012345678 \]

Alternative 6: 96.9% 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 7.6%

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

2. Taylor expanded in x around 0 98.8%

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

3. Final simplification 98.8%

   \[\leadsto x \]

Reproduce

herbie shell --seed 2023278 
(FPCore (x)
  :name "Hyperbolic tangent"
  :precision binary64
  (/ (- (exp x) (exp (- x))) (+ (exp x) (exp (- x)))))