Hyperbolic tangent

Percentage Accurate: 8.9% → 100.0%

Time: 12.1s

Alternatives: 5

Speedup: 70.3×

• could not determine a ground truth

  1. x = -1.6701616035289183e+91

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: 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 10.9%

   \[\frac{e^{x} - e^{-x}}{e^{x} + e^{-x}} \]
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 rewrites 100.0%

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

Alternative 2: 97.4% accurate, 11.7× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\frac{x \cdot x}{-3 - \left(x \cdot x\right) \cdot 1.2}, x, x\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (fma (/ (* x x) (- -3.0 (* (* x x) 1.2))) x x))

C

double code(double x) {
	return fma(((x * x) / (-3.0 - ((x * x) * 1.2))), x, x);
}

Julia

function code(x)
	return fma(Float64(Float64(x * x) / Float64(-3.0 - Float64(Float64(x * x) * 1.2))), x, x)
end

Wolfram

code[x_] := N[(N[(N[(x * x), $MachinePrecision] / N[(-3.0 - N[(N[(x * x), $MachinePrecision] * 1.2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x + x), $MachinePrecision]

Derivation

1. Initial program 10.9%

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

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left(\frac{2}{15} \cdot {x}^{2} - \frac{1}{3}\right)\right)} \]
4. Step-by-step derivation

   1. +-commutative N/A

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

   2. distribute-rgt-in N/A

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

   3. *-lft-identity N/A

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

   4. *-commutative N/A

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

   5. associate-*r* N/A

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

   6. *-commutative N/A

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

   7. lower-fma.f64 N/A

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

   8. sub-neg N/A

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

   9. unpow2 N/A

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

   10. associate-*r* N/A

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

   11. *-commutative N/A

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

   12. metadata-eval N/A

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

   13. lower-fma.f64 N/A

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

   14. *-commutative N/A

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

   15. lower-*.f64 N/A

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

   16. lower-*.f64 N/A

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

   17. unpow2 N/A

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

   18. lower-*.f64 97.7

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

5. Applied rewrites 97.7%

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

   1. lift-*.f64 N/A

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

   2. +-commutative N/A

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

   3. flip-+ N/A

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

   4. lower-/.f64 N/A

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

   5. lower--.f64 N/A

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

   6. metadata-eval N/A

      \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{\frac{1}{9}} - \left(x \cdot \left(x \cdot \frac{2}{15}\right)\right) \cdot \left(x \cdot \left(x \cdot \frac{2}{15}\right)\right)}{\frac{-1}{3} - x \cdot \left(x \cdot \frac{2}{15}\right)}, x \cdot \left(x \cdot x\right), x\right) \]

   7. swap-sqr N/A

      \[\leadsto \mathsf{fma}\left(\frac{\frac{1}{9} - \color{blue}{\left(x \cdot x\right) \cdot \left(\left(x \cdot \frac{2}{15}\right) \cdot \left(x \cdot \frac{2}{15}\right)\right)}}{\frac{-1}{3} - x \cdot \left(x \cdot \frac{2}{15}\right)}, x \cdot \left(x \cdot x\right), x\right) \]

   8. lift-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(\frac{\frac{1}{9} - \color{blue}{\left(x \cdot x\right)} \cdot \left(\left(x \cdot \frac{2}{15}\right) \cdot \left(x \cdot \frac{2}{15}\right)\right)}{\frac{-1}{3} - x \cdot \left(x \cdot \frac{2}{15}\right)}, x \cdot \left(x \cdot x\right), x\right) \]

   9. lower-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(\frac{\frac{1}{9} - \color{blue}{\left(x \cdot x\right) \cdot \left(\left(x \cdot \frac{2}{15}\right) \cdot \left(x \cdot \frac{2}{15}\right)\right)}}{\frac{-1}{3} - x \cdot \left(x \cdot \frac{2}{15}\right)}, x \cdot \left(x \cdot x\right), x\right) \]

   10. lift-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(\frac{\frac{1}{9} - \left(x \cdot x\right) \cdot \left(\color{blue}{\left(x \cdot \frac{2}{15}\right)} \cdot \left(x \cdot \frac{2}{15}\right)\right)}{\frac{-1}{3} - x \cdot \left(x \cdot \frac{2}{15}\right)}, x \cdot \left(x \cdot x\right), x\right) \]

   11. lift-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(\frac{\frac{1}{9} - \left(x \cdot x\right) \cdot \left(\left(x \cdot \frac{2}{15}\right) \cdot \color{blue}{\left(x \cdot \frac{2}{15}\right)}\right)}{\frac{-1}{3} - x \cdot \left(x \cdot \frac{2}{15}\right)}, x \cdot \left(x \cdot x\right), x\right) \]

   12. swap-sqr N/A

       \[\leadsto \mathsf{fma}\left(\frac{\frac{1}{9} - \left(x \cdot x\right) \cdot \color{blue}{\left(\left(x \cdot x\right) \cdot \left(\frac{2}{15} \cdot \frac{2}{15}\right)\right)}}{\frac{-1}{3} - x \cdot \left(x \cdot \frac{2}{15}\right)}, x \cdot \left(x \cdot x\right), x\right) \]

   13. lift-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(\frac{\frac{1}{9} - \left(x \cdot x\right) \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot \left(\frac{2}{15} \cdot \frac{2}{15}\right)\right)}{\frac{-1}{3} - x \cdot \left(x \cdot \frac{2}{15}\right)}, x \cdot \left(x \cdot x\right), x\right) \]

   14. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(\frac{\frac{1}{9} - \left(x \cdot x\right) \cdot \color{blue}{\left(\left(x \cdot x\right) \cdot \left(\frac{2}{15} \cdot \frac{2}{15}\right)\right)}}{\frac{-1}{3} - x \cdot \left(x \cdot \frac{2}{15}\right)}, x \cdot \left(x \cdot x\right), x\right) \]

   15. metadata-eval N/A

       \[\leadsto \mathsf{fma}\left(\frac{\frac{1}{9} - \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \color{blue}{\frac{4}{225}}\right)}{\frac{-1}{3} - x \cdot \left(x \cdot \frac{2}{15}\right)}, x \cdot \left(x \cdot x\right), x\right) \]

   16. lower--.f64 N/A

       \[\leadsto \mathsf{fma}\left(\frac{\frac{1}{9} - \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \frac{4}{225}\right)}{\color{blue}{\frac{-1}{3} - x \cdot \left(x \cdot \frac{2}{15}\right)}}, x \cdot \left(x \cdot x\right), x\right) \]

   17. lower-*.f64 97.7

       \[\leadsto \mathsf{fma}\left(\frac{0.1111111111111111 - \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot 0.017777777777777778\right)}{-0.3333333333333333 - \color{blue}{x \cdot \left(x \cdot 0.13333333333333333\right)}}, x \cdot \left(x \cdot x\right), x\right) \]

7. Applied rewrites 97.7%

   \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{0.1111111111111111 - \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot 0.017777777777777778\right)}{-0.3333333333333333 - x \cdot \left(x \cdot 0.13333333333333333\right)}}, x \cdot \left(x \cdot x\right), x\right) \]

8. Taylor expanded in x around 0

   \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{\frac{1}{9}}}{\frac{-1}{3} - x \cdot \left(x \cdot \frac{2}{15}\right)}, x \cdot \left(x \cdot x\right), x\right) \]
9. Step-by-step derivation

10. Applied rewrites 98.0%

    \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{0.1111111111111111}}{-0.3333333333333333 - x \cdot \left(x \cdot 0.13333333333333333\right)}, x \cdot \left(x \cdot x\right), x\right) \]
11. Step-by-step derivation

    1. lift-*.f64 N/A

       \[\leadsto \frac{\frac{1}{9}}{\frac{-1}{3} - x \cdot \color{blue}{\left(x \cdot \frac{2}{15}\right)}} \cdot \left(x \cdot \left(x \cdot x\right)\right) + x \]

    2. lift-*.f64 N/A

       \[\leadsto \frac{\frac{1}{9}}{\frac{-1}{3} - \color{blue}{x \cdot \left(x \cdot \frac{2}{15}\right)}} \cdot \left(x \cdot \left(x \cdot x\right)\right) + x \]

    3. lift--.f64 N/A

       \[\leadsto \frac{\frac{1}{9}}{\color{blue}{\frac{-1}{3} - x \cdot \left(x \cdot \frac{2}{15}\right)}} \cdot \left(x \cdot \left(x \cdot x\right)\right) + x \]

    4. frac-2neg N/A

       \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(\frac{1}{9}\right)}{\mathsf{neg}\left(\left(\frac{-1}{3} - x \cdot \left(x \cdot \frac{2}{15}\right)\right)\right)}} \cdot \left(x \cdot \left(x \cdot x\right)\right) + x \]

    5. lift-*.f64 N/A

       \[\leadsto \frac{\mathsf{neg}\left(\frac{1}{9}\right)}{\mathsf{neg}\left(\left(\frac{-1}{3} - x \cdot \left(x \cdot \frac{2}{15}\right)\right)\right)} \cdot \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right) + x \]

    6. lift-*.f64 N/A

       \[\leadsto \frac{\mathsf{neg}\left(\frac{1}{9}\right)}{\mathsf{neg}\left(\left(\frac{-1}{3} - x \cdot \left(x \cdot \frac{2}{15}\right)\right)\right)} \cdot \color{blue}{\left(x \cdot \left(x \cdot x\right)\right)} + x \]

    7. frac-2neg N/A

       \[\leadsto \color{blue}{\frac{\frac{1}{9}}{\frac{-1}{3} - x \cdot \left(x \cdot \frac{2}{15}\right)}} \cdot \left(x \cdot \left(x \cdot x\right)\right) + x \]

    8. lift-/.f64 N/A

       \[\leadsto \color{blue}{\frac{\frac{1}{9}}{\frac{-1}{3} - x \cdot \left(x \cdot \frac{2}{15}\right)}} \cdot \left(x \cdot \left(x \cdot x\right)\right) + x \]

    9. lift-*.f64 N/A

       \[\leadsto \frac{\frac{1}{9}}{\frac{-1}{3} - x \cdot \left(x \cdot \frac{2}{15}\right)} \cdot \color{blue}{\left(x \cdot \left(x \cdot x\right)\right)} + x \]

    10. associate-*r* N/A

        \[\leadsto \color{blue}{\left(\frac{\frac{1}{9}}{\frac{-1}{3} - x \cdot \left(x \cdot \frac{2}{15}\right)} \cdot x\right) \cdot \left(x \cdot x\right)} + x \]

    11. lift-*.f64 N/A

        \[\leadsto \left(\frac{\frac{1}{9}}{\frac{-1}{3} - x \cdot \left(x \cdot \frac{2}{15}\right)} \cdot x\right) \cdot \color{blue}{\left(x \cdot x\right)} + x \]

    12. associate-*r* N/A

        \[\leadsto \color{blue}{\left(\left(\frac{\frac{1}{9}}{\frac{-1}{3} - x \cdot \left(x \cdot \frac{2}{15}\right)} \cdot x\right) \cdot x\right) \cdot x} + x \]

    13. lower-fma.f64 N/A

        \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\frac{\frac{1}{9}}{\frac{-1}{3} - x \cdot \left(x \cdot \frac{2}{15}\right)} \cdot x\right) \cdot x, x, x\right)} \]

12. Applied rewrites 98.0%

    \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x \cdot x}{-3 - \left(x \cdot x\right) \cdot 1.2}, x, x\right)} \]
13. Add Preprocessing

Alternative 3: 97.2% accurate, 15.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 10.9%

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

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left(\frac{2}{15} \cdot {x}^{2} - \frac{1}{3}\right)\right)} \]
4. Step-by-step derivation

   1. +-commutative N/A

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

   2. distribute-rgt-in N/A

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

   3. *-lft-identity N/A

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

   4. *-commutative N/A

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

   5. associate-*r* N/A

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

   6. *-commutative N/A

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

   7. lower-fma.f64 N/A

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

   8. sub-neg N/A

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

   9. unpow2 N/A

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

   10. associate-*r* N/A

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

   11. *-commutative N/A

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

   12. metadata-eval N/A

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

   13. lower-fma.f64 N/A

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

   14. *-commutative N/A

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

   15. lower-*.f64 N/A

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

   16. lower-*.f64 N/A

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

   17. unpow2 N/A

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

   18. lower-*.f64 97.7

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

5. Applied rewrites 97.7%

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

   1. lift-*.f64 N/A

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

   2. lift-fma.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. associate-*r* N/A

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

   7. lift-*.f64 N/A

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

   8. associate-*r* N/A

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

   9. lower-fma.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 97.7

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

   13. lift-fma.f64 N/A

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

   14. lift-*.f64 N/A

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

   15. associate-*r* N/A

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

   16. lift-*.f64 N/A

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

   17. lower-fma.f64 97.7

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

7. Applied rewrites 97.7%

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

8. Final simplification 97.7%

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

Alternative 4: 96.8% 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 10.9%

   \[\frac{e^{x} - e^{-x}}{e^{x} + e^{-x}} \]
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 96.9

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

5. Applied rewrites 96.9%

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

6. Final simplification 96.9%

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

Alternative 5: 16.5% accurate, 70.3× speedup

Math

\[\begin{array}{l} \\ x \cdot 0.16666666666666666 \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* x 0.16666666666666666))

C

double code(double x) {
	return x * 0.16666666666666666;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = x * 0.16666666666666666d0
end function

Java

public static double code(double x) {
	return x * 0.16666666666666666;
}

Python

def code(x):
	return x * 0.16666666666666666

Julia

function code(x)
	return Float64(x * 0.16666666666666666)
end

MATLAB

function tmp = code(x)
	tmp = x * 0.16666666666666666;
end

Wolfram

code[x_] := N[(x * 0.16666666666666666), $MachinePrecision]

Derivation

1. Initial program 10.9%

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

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left(\frac{2}{15} \cdot {x}^{2} - \frac{1}{3}\right)\right)} \]
4. Step-by-step derivation

   1. +-commutative N/A

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

   2. distribute-rgt-in N/A

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

   3. *-lft-identity N/A

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

   4. *-commutative N/A

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

   5. associate-*r* N/A

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

   6. *-commutative N/A

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

   7. lower-fma.f64 N/A

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

   8. sub-neg N/A

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

   9. unpow2 N/A

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

   10. associate-*r* N/A

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

   11. *-commutative N/A

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

   12. metadata-eval N/A

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

   13. lower-fma.f64 N/A

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

   14. *-commutative N/A

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

   15. lower-*.f64 N/A

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

   16. lower-*.f64 N/A

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

   17. unpow2 N/A

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

   18. lower-*.f64 97.7

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

5. Applied rewrites 97.7%

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

   1. lift-*.f64 N/A

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

   2. +-commutative N/A

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

   3. flip-+ N/A

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

   4. lower-/.f64 N/A

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

   5. lower--.f64 N/A

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

   6. metadata-eval N/A

      \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{\frac{1}{9}} - \left(x \cdot \left(x \cdot \frac{2}{15}\right)\right) \cdot \left(x \cdot \left(x \cdot \frac{2}{15}\right)\right)}{\frac{-1}{3} - x \cdot \left(x \cdot \frac{2}{15}\right)}, x \cdot \left(x \cdot x\right), x\right) \]

   7. swap-sqr N/A

      \[\leadsto \mathsf{fma}\left(\frac{\frac{1}{9} - \color{blue}{\left(x \cdot x\right) \cdot \left(\left(x \cdot \frac{2}{15}\right) \cdot \left(x \cdot \frac{2}{15}\right)\right)}}{\frac{-1}{3} - x \cdot \left(x \cdot \frac{2}{15}\right)}, x \cdot \left(x \cdot x\right), x\right) \]

   8. lift-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(\frac{\frac{1}{9} - \color{blue}{\left(x \cdot x\right)} \cdot \left(\left(x \cdot \frac{2}{15}\right) \cdot \left(x \cdot \frac{2}{15}\right)\right)}{\frac{-1}{3} - x \cdot \left(x \cdot \frac{2}{15}\right)}, x \cdot \left(x \cdot x\right), x\right) \]

   9. lower-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(\frac{\frac{1}{9} - \color{blue}{\left(x \cdot x\right) \cdot \left(\left(x \cdot \frac{2}{15}\right) \cdot \left(x \cdot \frac{2}{15}\right)\right)}}{\frac{-1}{3} - x \cdot \left(x \cdot \frac{2}{15}\right)}, x \cdot \left(x \cdot x\right), x\right) \]

   10. lift-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(\frac{\frac{1}{9} - \left(x \cdot x\right) \cdot \left(\color{blue}{\left(x \cdot \frac{2}{15}\right)} \cdot \left(x \cdot \frac{2}{15}\right)\right)}{\frac{-1}{3} - x \cdot \left(x \cdot \frac{2}{15}\right)}, x \cdot \left(x \cdot x\right), x\right) \]

   11. lift-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(\frac{\frac{1}{9} - \left(x \cdot x\right) \cdot \left(\left(x \cdot \frac{2}{15}\right) \cdot \color{blue}{\left(x \cdot \frac{2}{15}\right)}\right)}{\frac{-1}{3} - x \cdot \left(x \cdot \frac{2}{15}\right)}, x \cdot \left(x \cdot x\right), x\right) \]

   12. swap-sqr N/A

       \[\leadsto \mathsf{fma}\left(\frac{\frac{1}{9} - \left(x \cdot x\right) \cdot \color{blue}{\left(\left(x \cdot x\right) \cdot \left(\frac{2}{15} \cdot \frac{2}{15}\right)\right)}}{\frac{-1}{3} - x \cdot \left(x \cdot \frac{2}{15}\right)}, x \cdot \left(x \cdot x\right), x\right) \]

   13. lift-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(\frac{\frac{1}{9} - \left(x \cdot x\right) \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot \left(\frac{2}{15} \cdot \frac{2}{15}\right)\right)}{\frac{-1}{3} - x \cdot \left(x \cdot \frac{2}{15}\right)}, x \cdot \left(x \cdot x\right), x\right) \]

   14. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(\frac{\frac{1}{9} - \left(x \cdot x\right) \cdot \color{blue}{\left(\left(x \cdot x\right) \cdot \left(\frac{2}{15} \cdot \frac{2}{15}\right)\right)}}{\frac{-1}{3} - x \cdot \left(x \cdot \frac{2}{15}\right)}, x \cdot \left(x \cdot x\right), x\right) \]

   15. metadata-eval N/A

       \[\leadsto \mathsf{fma}\left(\frac{\frac{1}{9} - \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \color{blue}{\frac{4}{225}}\right)}{\frac{-1}{3} - x \cdot \left(x \cdot \frac{2}{15}\right)}, x \cdot \left(x \cdot x\right), x\right) \]

   16. lower--.f64 N/A

       \[\leadsto \mathsf{fma}\left(\frac{\frac{1}{9} - \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \frac{4}{225}\right)}{\color{blue}{\frac{-1}{3} - x \cdot \left(x \cdot \frac{2}{15}\right)}}, x \cdot \left(x \cdot x\right), x\right) \]

   17. lower-*.f64 97.7

       \[\leadsto \mathsf{fma}\left(\frac{0.1111111111111111 - \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot 0.017777777777777778\right)}{-0.3333333333333333 - \color{blue}{x \cdot \left(x \cdot 0.13333333333333333\right)}}, x \cdot \left(x \cdot x\right), x\right) \]

7. Applied rewrites 97.7%

   \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{0.1111111111111111 - \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot 0.017777777777777778\right)}{-0.3333333333333333 - x \cdot \left(x \cdot 0.13333333333333333\right)}}, x \cdot \left(x \cdot x\right), x\right) \]

8. Taylor expanded in x around 0

   \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{\frac{1}{9}}}{\frac{-1}{3} - x \cdot \left(x \cdot \frac{2}{15}\right)}, x \cdot \left(x \cdot x\right), x\right) \]
9. Step-by-step derivation

10. Applied rewrites 98.0%

    \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{0.1111111111111111}}{-0.3333333333333333 - x \cdot \left(x \cdot 0.13333333333333333\right)}, x \cdot \left(x \cdot x\right), x\right) \]

11. Taylor expanded in x around inf

    \[\leadsto \color{blue}{\frac{1}{6} \cdot x} \]
12. Step-by-step derivation

    1. *-commutative N/A

       \[\leadsto \color{blue}{x \cdot \frac{1}{6}} \]

    2. lower-*.f64 16.6

       \[\leadsto \color{blue}{x \cdot 0.16666666666666666} \]

13. Applied rewrites 16.6%

    \[\leadsto \color{blue}{x \cdot 0.16666666666666666} \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2024221 
(FPCore (x)
  :name "Hyperbolic tangent"
  :precision binary64
  (/ (- (exp x) (exp (- x))) (+ (exp x) (exp (- x)))))