Hyperbolic tangent

Percentage Accurate: 9.2% → 100.0%

Time: 10.3s

Alternatives: 4

Speedup: 422.0×

• could not determine a ground truth

  1. x = -1.3159480069190486e+95

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: 100.0% accurate, 4.2× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 (tanh x))

C

double code(double x) {
	return tanh(x);
}

Fortran

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

Java

public static double code(double x) {
	return Math.tanh(x);
}

Python

def code(x):
	return math.tanh(x)

Julia

function code(x)
	return tanh(x)
end

MATLAB

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

Wolfram

code[x_] := N[Tanh[x], $MachinePrecision]

Derivation

1. Initial program 8.0%

   \[\frac{e^{x} - e^{-x}}{e^{x} + e^{-x}} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. tanh-undef N/A

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

   2. lower-tanh.f64 100.0

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

4. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{\tanh x} \]
5. Add Preprocessing

Alternative 2: 96.9% accurate, 15.1× speedup

Math

\[\begin{array}{l} \\ x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.13333333333333333, -0.3333333333333333\right), 1\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (* x (fma (* x x) (fma x (* x 0.13333333333333333) -0.3333333333333333) 1.0)))

C

double code(double x) {
	return x * fma((x * x), fma(x, (x * 0.13333333333333333), -0.3333333333333333), 1.0);
}

Julia

function code(x)
	return Float64(x * fma(Float64(x * x), fma(x, Float64(x * 0.13333333333333333), -0.3333333333333333), 1.0))
end

Wolfram

code[x_] := N[(x * N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * 0.13333333333333333), $MachinePrecision] + -0.3333333333333333), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 8.0%

   \[\frac{e^{x} - e^{-x}}{e^{x} + e^{-x}} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. tanh-undef N/A

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

   2. lower-tanh.f64 100.0

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

4. Applied egg-rr 100.0%

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

5. Taylor expanded in x around 0

   \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left(\frac{2}{15} \cdot {x}^{2} - \frac{1}{3}\right)\right)} \]

7. Simplified 97.3%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x, x \cdot \left(x \cdot \mathsf{fma}\left(x, x \cdot 0.13333333333333333, -0.3333333333333333\right)\right), x\right)} \]
8. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \color{blue}{\left(x \cdot \frac{2}{15}\right)} + \frac{-1}{3}\right)\right)\right) + x \]

   2. lift-fma.f64 N/A

      \[\leadsto x \cdot \left(x \cdot \left(x \cdot \color{blue}{\mathsf{fma}\left(x, x \cdot \frac{2}{15}, \frac{-1}{3}\right)}\right)\right) + x \]

   3. lift-*.f64 N/A

      \[\leadsto x \cdot \left(x \cdot \color{blue}{\left(x \cdot \mathsf{fma}\left(x, x \cdot \frac{2}{15}, \frac{-1}{3}\right)\right)}\right) + x \]

   4. lift-*.f64 N/A

      \[\leadsto x \cdot \color{blue}{\left(x \cdot \left(x \cdot \mathsf{fma}\left(x, x \cdot \frac{2}{15}, \frac{-1}{3}\right)\right)\right)} + x \]

9. Applied egg-rr 97.3%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.13333333333333333, -0.3333333333333333\right), 1\right) \cdot x} \]

10. Final simplification 97.3%

    \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.13333333333333333, -0.3333333333333333\right), 1\right) \]
11. Add Preprocessing

Alternative 3: 96.5% accurate, 24.8× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(x, \left(x \cdot x\right) \cdot -0.3333333333333333, x\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (fma x (* (* x x) -0.3333333333333333) x))

C

double code(double x) {
	return fma(x, ((x * x) * -0.3333333333333333), x);
}

Julia

function code(x)
	return fma(x, Float64(Float64(x * x) * -0.3333333333333333), x)
end

Wolfram

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

Derivation

1. Initial program 8.0%

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

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{x \cdot \left(1 + \frac{-1}{3} \cdot {x}^{2}\right)} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto x \cdot \color{blue}{\left(\frac{-1}{3} \cdot {x}^{2} + 1\right)} \]

   2. distribute-lft-in N/A

      \[\leadsto \color{blue}{x \cdot \left(\frac{-1}{3} \cdot {x}^{2}\right) + x \cdot 1} \]

   3. *-rgt-identity N/A

      \[\leadsto x \cdot \left(\frac{-1}{3} \cdot {x}^{2}\right) + \color{blue}{x} \]

   4. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, \frac{-1}{3} \cdot {x}^{2}, x\right)} \]

   5. lower-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(x, \color{blue}{\frac{-1}{3} \cdot {x}^{2}}, x\right) \]

   6. unpow2 N/A

      \[\leadsto \mathsf{fma}\left(x, \frac{-1}{3} \cdot \color{blue}{\left(x \cdot x\right)}, x\right) \]

   7. lower-*.f64 96.8

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

5. Simplified 96.8%

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

6. Final simplification 96.8%

   \[\leadsto \mathsf{fma}\left(x, \left(x \cdot x\right) \cdot -0.3333333333333333, x\right) \]
7. Add Preprocessing

Alternative 4: 96.4% accurate, 422.0× 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 8.0%

   \[\frac{e^{x} - e^{-x}}{e^{x} + e^{-x}} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. tanh-undef N/A

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

   2. lower-tanh.f64 100.0

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

4. Applied egg-rr 100.0%

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

5. Taylor expanded in x around 0

   \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left(\frac{2}{15} \cdot {x}^{2} - \frac{1}{3}\right)\right)} \]

7. Simplified 97.3%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x, x \cdot \left(x \cdot \mathsf{fma}\left(x, x \cdot 0.13333333333333333, -0.3333333333333333\right)\right), x\right)} \]
8. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \color{blue}{\left(x \cdot \frac{2}{15}\right)} + \frac{-1}{3}\right)\right)\right) + x \]

   2. lift-fma.f64 N/A

      \[\leadsto x \cdot \left(x \cdot \left(x \cdot \color{blue}{\mathsf{fma}\left(x, x \cdot \frac{2}{15}, \frac{-1}{3}\right)}\right)\right) + x \]

   3. lift-*.f64 N/A

      \[\leadsto x \cdot \left(x \cdot \color{blue}{\left(x \cdot \mathsf{fma}\left(x, x \cdot \frac{2}{15}, \frac{-1}{3}\right)\right)}\right) + x \]

   4. lift-*.f64 N/A

      \[\leadsto x \cdot \color{blue}{\left(x \cdot \left(x \cdot \mathsf{fma}\left(x, x \cdot \frac{2}{15}, \frac{-1}{3}\right)\right)\right)} + x \]

9. Applied egg-rr 97.3%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.13333333333333333, -0.3333333333333333\right), 1\right) \cdot x} \]

10. Taylor expanded in x around 0

    \[\leadsto \color{blue}{1} \cdot x \]
11. Step-by-step derivation

12. Simplified 96.6%

    \[\leadsto \color{blue}{1} \cdot x \]

13. Final simplification 96.6%

    \[\leadsto x \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2024207 
(FPCore (x)
  :name "Hyperbolic tangent"
  :precision binary64
  (/ (- (exp x) (exp (- x))) (+ (exp x) (exp (- x)))))