Hyperbolic tangent

Percentage Accurate: 9.1% → 97.2%

Time: 19.8s

Alternatives: 7

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: 9.1% 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: 97.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ x + {x}^{3} \cdot \mathsf{fma}\left(0.13333059146197387, {x}^{2}, -0.3333333333333333\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (+
  x
  (* (pow x 3.0) (fma 0.13333059146197387 (pow x 2.0) -0.3333333333333333))))

C

double code(double x) {
	return x + (pow(x, 3.0) * fma(0.13333059146197387, pow(x, 2.0), -0.3333333333333333));
}

Julia

function code(x)
	return Float64(x + Float64((x ^ 3.0) * fma(0.13333059146197387, (x ^ 2.0), -0.3333333333333333)))
end

Wolfram

code[x_] := N[(x + N[(N[Power[x, 3.0], $MachinePrecision] * N[(0.13333059146197387 * N[Power[x, 2.0], $MachinePrecision] + -0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 9.2%

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

3. Taylor expanded in x around 0 98.4%

   \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(0.13333333333333333 + -0.05396825396825397 \cdot {x}^{2}\right) - 0.3333333333333333\right)\right)} \]

5. Applied egg-rr 98.4%

   \[\leadsto x \cdot \left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(0.13333333333333333 + -0.05396825396825397 \cdot \color{blue}{5.080526342529086 \cdot 10^{-5}}\right) - 0.3333333333333333\right)\right) \]

6. Taylor expanded in x around 0 98.4%

   \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left(0.13333059146197387 \cdot {x}^{2} - 0.3333333333333333\right)\right)} \]
7. Step-by-step derivation

   1. distribute-lft-in 98.4%

      \[\leadsto \color{blue}{x \cdot 1 + x \cdot \left({x}^{2} \cdot \left(0.13333059146197387 \cdot {x}^{2} - 0.3333333333333333\right)\right)} \]

   2. *-rgt-identity 98.4%

      \[\leadsto \color{blue}{x} + x \cdot \left({x}^{2} \cdot \left(0.13333059146197387 \cdot {x}^{2} - 0.3333333333333333\right)\right) \]

   3. *-commutative 98.4%

      \[\leadsto x + x \cdot \left({x}^{2} \cdot \left(\color{blue}{{x}^{2} \cdot 0.13333059146197387} - 0.3333333333333333\right)\right) \]

   4. sub-neg 98.4%

      \[\leadsto x + x \cdot \left({x}^{2} \cdot \color{blue}{\left({x}^{2} \cdot 0.13333059146197387 + \left(-0.3333333333333333\right)\right)}\right) \]

   5. metadata-eval 98.4%

      \[\leadsto x + x \cdot \left({x}^{2} \cdot \left({x}^{2} \cdot 0.13333059146197387 + \color{blue}{-0.3333333333333333}\right)\right) \]

   6. fma-define 98.4%

      \[\leadsto x + x \cdot \left({x}^{2} \cdot \color{blue}{\mathsf{fma}\left({x}^{2}, 0.13333059146197387, -0.3333333333333333\right)}\right) \]

   7. associate-*r* 98.4%

      \[\leadsto x + \color{blue}{\left(x \cdot {x}^{2}\right) \cdot \mathsf{fma}\left({x}^{2}, 0.13333059146197387, -0.3333333333333333\right)} \]

   8. unpow2 98.4%

      \[\leadsto x + \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right) \cdot \mathsf{fma}\left({x}^{2}, 0.13333059146197387, -0.3333333333333333\right) \]

   9. cube-mult 98.4%

      \[\leadsto x + \color{blue}{{x}^{3}} \cdot \mathsf{fma}\left({x}^{2}, 0.13333059146197387, -0.3333333333333333\right) \]

   10. fma-define 98.4%

       \[\leadsto x + {x}^{3} \cdot \color{blue}{\left({x}^{2} \cdot 0.13333059146197387 + -0.3333333333333333\right)} \]

   11. *-commutative 98.4%

       \[\leadsto x + {x}^{3} \cdot \left(\color{blue}{0.13333059146197387 \cdot {x}^{2}} + -0.3333333333333333\right) \]

   12. fma-undefine 98.4%

       \[\leadsto x + {x}^{3} \cdot \color{blue}{\mathsf{fma}\left(0.13333059146197387, {x}^{2}, -0.3333333333333333\right)} \]

8. Simplified 98.4%

   \[\leadsto \color{blue}{x + {x}^{3} \cdot \mathsf{fma}\left(0.13333059146197387, {x}^{2}, -0.3333333333333333\right)} \]
9. Add Preprocessing

Alternative 2: 97.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ x \cdot \left(1 + {x}^{2} \cdot \left({x}^{2} \cdot 0.13333333333333333 - 0.3333333333333333\right)\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  x
  (+
   1.0
   (*
    (pow x 2.0)
    (- (* (pow x 2.0) 0.13333333333333333) 0.3333333333333333)))))

C

double code(double x) {
	return x * (1.0 + (pow(x, 2.0) * ((pow(x, 2.0) * 0.13333333333333333) - 0.3333333333333333)));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = x * (1.0d0 + ((x ** 2.0d0) * (((x ** 2.0d0) * 0.13333333333333333d0) - 0.3333333333333333d0)))
end function

Java

public static double code(double x) {
	return x * (1.0 + (Math.pow(x, 2.0) * ((Math.pow(x, 2.0) * 0.13333333333333333) - 0.3333333333333333)));
}

Python

def code(x):
	return x * (1.0 + (math.pow(x, 2.0) * ((math.pow(x, 2.0) * 0.13333333333333333) - 0.3333333333333333)))

Julia

function code(x)
	return Float64(x * Float64(1.0 + Float64((x ^ 2.0) * Float64(Float64((x ^ 2.0) * 0.13333333333333333) - 0.3333333333333333))))
end

MATLAB

function tmp = code(x)
	tmp = x * (1.0 + ((x ^ 2.0) * (((x ^ 2.0) * 0.13333333333333333) - 0.3333333333333333)));
end

Wolfram

code[x_] := N[(x * N[(1.0 + N[(N[Power[x, 2.0], $MachinePrecision] * N[(N[(N[Power[x, 2.0], $MachinePrecision] * 0.13333333333333333), $MachinePrecision] - 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 9.2%

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

3. Taylor expanded in x around 0 98.4%

   \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left(0.13333333333333333 \cdot {x}^{2} - 0.3333333333333333\right)\right)} \]

4. Final simplification 98.4%

   \[\leadsto x \cdot \left(1 + {x}^{2} \cdot \left({x}^{2} \cdot 0.13333333333333333 - 0.3333333333333333\right)\right) \]
5. Add Preprocessing

Alternative 3: 97.2% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 9.2%

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

3. Taylor expanded in x around 0 98.2%

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

   1. *-commutative 98.2%

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

5. Simplified 98.2%

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

6. Taylor expanded in x around 0 98.2%

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

   1. +-commutative 98.2%

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

   2. unpow2 98.2%

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

   3. fma-define 98.2%

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

8. Simplified 98.2%

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

Alternative 4: 96.8% accurate, 3.9× speedup

Math

\[\begin{array}{l} \\ x + {x}^{3} \cdot -0.3333333333333333 \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 9.2%

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

3. Taylor expanded in x around 0 98.4%

   \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(0.13333333333333333 + -0.05396825396825397 \cdot {x}^{2}\right) - 0.3333333333333333\right)\right)} \]

4. Taylor expanded in x around 0 98.0%

   \[\leadsto \color{blue}{x \cdot \left(1 + -0.3333333333333333 \cdot {x}^{2}\right)} \]
5. Step-by-step derivation

   1. distribute-lft-in 98.0%

      \[\leadsto \color{blue}{x \cdot 1 + x \cdot \left(-0.3333333333333333 \cdot {x}^{2}\right)} \]

   2. *-rgt-identity 98.0%

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

   3. *-commutative 98.0%

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

   4. associate-*r* 98.0%

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

   5. unpow2 98.0%

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

   6. cube-mult 98.0%

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

6. Simplified 98.0%

   \[\leadsto \color{blue}{x + {x}^{3} \cdot -0.3333333333333333} \]
7. Add Preprocessing

Alternative 5: 96.7% accurate, 24.1× speedup

Math

\[\begin{array}{l} \\ \frac{x \cdot 2}{\left(1 + x \cdot \left(1 + x \cdot 0.5\right)\right) + \left(1 - x\right)} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/ (* x 2.0) (+ (+ 1.0 (* x (+ 1.0 (* x 0.5)))) (- 1.0 x))))

C

double code(double x) {
	return (x * 2.0) / ((1.0 + (x * (1.0 + (x * 0.5)))) + (1.0 - x));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (x * 2.0d0) / ((1.0d0 + (x * (1.0d0 + (x * 0.5d0)))) + (1.0d0 - x))
end function

Java

public static double code(double x) {
	return (x * 2.0) / ((1.0 + (x * (1.0 + (x * 0.5)))) + (1.0 - x));
}

Python

def code(x):
	return (x * 2.0) / ((1.0 + (x * (1.0 + (x * 0.5)))) + (1.0 - x))

Julia

function code(x)
	return Float64(Float64(x * 2.0) / Float64(Float64(1.0 + Float64(x * Float64(1.0 + Float64(x * 0.5)))) + Float64(1.0 - x)))
end

MATLAB

function tmp = code(x)
	tmp = (x * 2.0) / ((1.0 + (x * (1.0 + (x * 0.5)))) + (1.0 - x));
end

Wolfram

code[x_] := N[(N[(x * 2.0), $MachinePrecision] / N[(N[(1.0 + N[(x * N[(1.0 + N[(x * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 9.2%

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

3. Taylor expanded in x around 0 97.4%

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

   1. *-commutative 97.4%

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

5. Simplified 97.4%

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

6. Taylor expanded in x around 0 97.6%

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

   1. mul-1-neg 97.6%

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

   2. unsub-neg 97.6%

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

8. Simplified 97.6%

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

9. Taylor expanded in x around 0 97.6%

   \[\leadsto \frac{x \cdot 2}{\color{blue}{\left(1 + x \cdot \left(1 + 0.5 \cdot x\right)\right)} + \left(1 - x\right)} \]
10. Step-by-step derivation

    1. *-commutative 97.6%

       \[\leadsto \frac{x \cdot 2}{\left(1 + x \cdot \left(1 + \color{blue}{x \cdot 0.5}\right)\right) + \left(1 - x\right)} \]

11. Simplified 97.6%

    \[\leadsto \frac{x \cdot 2}{\color{blue}{\left(1 + x \cdot \left(1 + x \cdot 0.5\right)\right)} + \left(1 - x\right)} \]
12. Add Preprocessing

Alternative 6: 96.6% 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 9.2%

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

3. Taylor expanded in x around 0 97.5%

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

Alternative 7: 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 9.2%

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

4. Applied egg-rr 4.1%

   \[\leadsto \color{blue}{1.5} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024143 
(FPCore (x)
  :name "Hyperbolic tangent"
  :precision binary64
  (/ (- (exp x) (exp (- x))) (+ (exp x) (exp (- x)))))