Hyperbolic tangent

Percentage Accurate: 9.4% → 97.0%

Time: 8.2s

Alternatives: 3

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: 9.4% 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: 97.0% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

function code(x)
	return Float64(Float64(Float64(x * 2.0) + Float64(0.3333333333333333 * (x ^ 3.0))) / fma(x, x, 2.0))
end

Wolfram

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

Derivation

1. Initial program 8.9%

   \[\frac{e^{x} - e^{-x}}{e^{x} + e^{-x}} \]
2. Add Preprocessing

3. Taylor expanded in x around 0 7.4%

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

   1. +-commutative 7.4%

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

   2. unpow2 7.4%

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

   3. fma-define 7.4%

      \[\leadsto \frac{e^{x} - e^{-x}}{\color{blue}{\mathsf{fma}\left(x, x, 2\right)}} \]

5. Simplified 7.4%

   \[\leadsto \frac{e^{x} - e^{-x}}{\color{blue}{\mathsf{fma}\left(x, x, 2\right)}} \]

6. Taylor expanded in x around 0 96.4%

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

   1. distribute-rgt-in 96.4%

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

   2. *-commutative 96.4%

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

   3. associate-*l* 96.4%

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

   4. unpow2 96.4%

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

   5. unpow3 96.4%

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

8. Applied egg-rr 96.4%

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

9. Final simplification 96.4%

   \[\leadsto \frac{x \cdot 2 + 0.3333333333333333 \cdot {x}^{3}}{\mathsf{fma}\left(x, x, 2\right)} \]
10. Add Preprocessing

Alternative 2: 97.0% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 8.9%

   \[\frac{e^{x} - e^{-x}}{e^{x} + e^{-x}} \]
2. Add Preprocessing

3. Taylor expanded in x around 0 7.4%

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

   1. +-commutative 7.4%

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

   2. unpow2 7.4%

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

   3. fma-define 7.4%

      \[\leadsto \frac{e^{x} - e^{-x}}{\color{blue}{\mathsf{fma}\left(x, x, 2\right)}} \]

5. Simplified 7.4%

   \[\leadsto \frac{e^{x} - e^{-x}}{\color{blue}{\mathsf{fma}\left(x, x, 2\right)}} \]

6. Taylor expanded in x around 0 96.4%

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

7. Final simplification 96.4%

   \[\leadsto \frac{x \cdot \left(2 + 0.3333333333333333 \cdot {x}^{2}\right)}{\mathsf{fma}\left(x, x, 2\right)} \]
8. Add Preprocessing

Alternative 3: 96.4% 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 8.9%

   \[\frac{e^{x} - e^{-x}}{e^{x} + e^{-x}} \]
2. Add Preprocessing

3. Taylor expanded in x around 0 96.2%

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

4. Final simplification 96.2%

   \[\leadsto x \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024076 
(FPCore (x)
  :name "Hyperbolic tangent"
  :precision binary64
  (/ (- (exp x) (exp (- x))) (+ (exp x) (exp (- x)))))