Hyperbolic tangent

Percentage Accurate: 9.1% → 97.3%

Time: 9.2s

Alternatives: 6

Speedup: 409.0×

• could not determine a ground truth

  1. x = 1.4998777877931673e-7

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: 97.3% 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 9.0%

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

3. Taylor expanded in x around 0 7.7%

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

   1. +-commutative 7.7%

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

   2. unpow2 7.7%

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

   3. fma-define 7.7%

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

5. Simplified 7.7%

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

6. Taylor expanded in x around 0 96.6%

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

   1. distribute-rgt-in 96.6%

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

   2. fma-define 96.6%

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

   3. associate-*l* 96.6%

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

   4. pow-plus 96.6%

      \[\leadsto \frac{\mathsf{fma}\left(2, x, 0.3333333333333333 \cdot \color{blue}{{x}^{\left(2 + 1\right)}}\right)}{\mathsf{fma}\left(x, x, 2\right)} \]

   5. metadata-eval 96.6%

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

8. Simplified 96.6%

   \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(2, x, 0.3333333333333333 \cdot {x}^{3}\right)}}{\mathsf{fma}\left(x, x, 2\right)} \]
9. Step-by-step derivation

   1. fma-undefine 96.6%

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

   2. *-commutative 96.6%

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

10. Applied egg-rr 96.6%

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

11. Final simplification 96.6%

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

Alternative 2: 97.3% 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 9.0%

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

3. Taylor expanded in x around 0 7.7%

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

   1. +-commutative 7.7%

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

   2. unpow2 7.7%

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

   3. fma-define 7.7%

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

5. Simplified 7.7%

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

6. Taylor expanded in x around 0 96.6%

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

7. Final simplification 96.6%

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

Alternative 3: 97.3% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

function code(x)
	return fma(Float64(Float64(0.13333333333333333 * Float64(x * x)) - 0.3333333333333333), (x ^ 3.0), x)
end

Wolfram

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

Derivation

1. Initial program 9.0%

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

3. Taylor expanded in x around 0 96.5%

   \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left(0.13333333333333333 \cdot {x}^{2} - 0.3333333333333333\right)\right)} \]
4. Step-by-step derivation

   1. +-commutative 96.5%

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

   2. distribute-rgt-in 96.6%

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

   3. *-commutative 96.6%

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

   4. associate-*l* 96.6%

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

   5. *-lft-identity 96.6%

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

   6. fma-define 96.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.13333333333333333 \cdot {x}^{2} - 0.3333333333333333, {x}^{2} \cdot x, x\right)} \]

   7. *-commutative 96.6%

      \[\leadsto \mathsf{fma}\left(\color{blue}{{x}^{2} \cdot 0.13333333333333333} - 0.3333333333333333, {x}^{2} \cdot x, x\right) \]

   8. fma-neg 96.6%

      \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left({x}^{2}, 0.13333333333333333, -0.3333333333333333\right)}, {x}^{2} \cdot x, x\right) \]

   9. metadata-eval 96.6%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left({x}^{2}, 0.13333333333333333, \color{blue}{-0.3333333333333333}\right), {x}^{2} \cdot x, x\right) \]

   10. pow-plus 96.6%

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

   11. metadata-eval 96.6%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left({x}^{2}, 0.13333333333333333, -0.3333333333333333\right), {x}^{\color{blue}{3}}, x\right) \]

5. Simplified 96.6%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left({x}^{2}, 0.13333333333333333, -0.3333333333333333\right), {x}^{3}, x\right)} \]

6. Taylor expanded in x around 0 96.6%

   \[\leadsto \mathsf{fma}\left(\color{blue}{0.13333333333333333 \cdot {x}^{2} - 0.3333333333333333}, {x}^{3}, x\right) \]
7. Step-by-step derivation

   1. unpow2 96.6%

      \[\leadsto \mathsf{fma}\left(0.13333333333333333 \cdot \color{blue}{\left(x \cdot x\right)} - 0.3333333333333333, {x}^{3}, x\right) \]

8. Applied egg-rr 96.6%

   \[\leadsto \mathsf{fma}\left(0.13333333333333333 \cdot \color{blue}{\left(x \cdot x\right)} - 0.3333333333333333, {x}^{3}, x\right) \]

9. Final simplification 96.6%

   \[\leadsto \mathsf{fma}\left(0.13333333333333333 \cdot \left(x \cdot x\right) - 0.3333333333333333, {x}^{3}, x\right) \]
10. Add Preprocessing

Alternative 4: 96.8% accurate, 3.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 9.0%

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

3. Taylor expanded in x around 0 7.7%

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

   1. +-commutative 7.7%

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

   2. unpow2 7.7%

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

   3. fma-define 7.7%

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

5. Simplified 7.7%

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

6. Taylor expanded in x around 0 96.3%

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

7. Final simplification 96.3%

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

Alternative 5: 4.0% accurate, 409.0× speedup

Math

\[\begin{array}{l} \\ 0.75 \end{array} \]

FPCore

(FPCore (x) :precision binary64 0.75)

C

double code(double x) {
	return 0.75;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 0.75d0
end function

Java

public static double code(double x) {
	return 0.75;
}

Python

def code(x):
	return 0.75

Julia

function code(x)
	return 0.75
end

MATLAB

function tmp = code(x)
	tmp = 0.75;
end

Wolfram

code[x_] := 0.75

Derivation

1. Initial program 9.0%

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

3. Taylor expanded in x around 0 7.7%

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

   1. +-commutative 7.7%

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

   2. unpow2 7.7%

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

   3. fma-define 7.7%

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

5. Simplified 7.7%

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

7. Applied egg-rr 4.1%

   \[\leadsto \frac{\color{blue}{1.5}}{\mathsf{fma}\left(x, x, 2\right)} \]

8. Taylor expanded in x around 0 4.3%

   \[\leadsto \color{blue}{0.75} \]

9. Final simplification 4.3%

   \[\leadsto 0.75 \]
10. Add Preprocessing

Alternative 6: 96.7% 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 9.0%

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

3. Taylor expanded in x around 0 96.3%

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

4. Final simplification 96.3%

   \[\leadsto x \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024073 
(FPCore (x)
  :name "Hyperbolic tangent"
  :precision binary64
  (/ (- (exp x) (exp (- x))) (+ (exp x) (exp (- x)))))