Hyperbolic tangent

Percentage Accurate: 9.1% → 100.0%

Time: 12.7s

Alternatives: 6

Speedup: 24.8×

• could not determine a ground truth

  1. x = 1.6966326689921299e+208

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: 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 7.0%

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

   1. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{e^{x} - e^{\mathsf{neg}\left(x\right)}}{e^{x} + e^{\mathsf{neg}\left(x\right)}}} \]

   2. lift--.f64 N/A

      \[\leadsto \frac{\color{blue}{e^{x} - e^{\mathsf{neg}\left(x\right)}}}{e^{x} + e^{\mathsf{neg}\left(x\right)}} \]

   3. lift-exp.f64 N/A

      \[\leadsto \frac{\color{blue}{e^{x}} - e^{\mathsf{neg}\left(x\right)}}{e^{x} + e^{\mathsf{neg}\left(x\right)}} \]

   4. lift-exp.f64 N/A

      \[\leadsto \frac{e^{x} - \color{blue}{e^{\mathsf{neg}\left(x\right)}}}{e^{x} + e^{\mathsf{neg}\left(x\right)}} \]

   5. lift-neg.f64 N/A

      \[\leadsto \frac{e^{x} - e^{\color{blue}{\mathsf{neg}\left(x\right)}}}{e^{x} + e^{\mathsf{neg}\left(x\right)}} \]

   6. lift-+.f64 N/A

      \[\leadsto \frac{e^{x} - e^{\mathsf{neg}\left(x\right)}}{\color{blue}{e^{x} + e^{\mathsf{neg}\left(x\right)}}} \]

   7. lift-exp.f64 N/A

      \[\leadsto \frac{e^{x} - e^{\mathsf{neg}\left(x\right)}}{\color{blue}{e^{x}} + e^{\mathsf{neg}\left(x\right)}} \]

   8. lift-exp.f64 N/A

      \[\leadsto \frac{e^{x} - e^{\mathsf{neg}\left(x\right)}}{e^{x} + \color{blue}{e^{\mathsf{neg}\left(x\right)}}} \]

   9. lift-neg.f64 N/A

      \[\leadsto \frac{e^{x} - e^{\mathsf{neg}\left(x\right)}}{e^{x} + e^{\color{blue}{\mathsf{neg}\left(x\right)}}} \]

   10. tanh-undef N/A

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

   11. 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: 97.3% accurate, 10.8× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\left(x \cdot x\right) \cdot \frac{0.1111111111111111}{\mathsf{fma}\left(x \cdot x, -0.13333333333333333, -0.3333333333333333\right)}, x, x\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (fma
  (*
   (* x x)
   (/
    0.1111111111111111
    (fma (* x x) -0.13333333333333333 -0.3333333333333333)))
  x
  x))

C

double code(double x) {
	return fma(((x * x) * (0.1111111111111111 / fma((x * x), -0.13333333333333333, -0.3333333333333333))), x, x);
}

Julia

function code(x)
	return fma(Float64(Float64(x * x) * Float64(0.1111111111111111 / fma(Float64(x * x), -0.13333333333333333, -0.3333333333333333))), x, x)
end

Wolfram

code[x_] := N[(N[(N[(x * x), $MachinePrecision] * N[(0.1111111111111111 / N[(N[(x * x), $MachinePrecision] * -0.13333333333333333 + -0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x + x), $MachinePrecision]

Derivation

1. Initial program 7.0%

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

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left(\frac{2}{15} \cdot {x}^{2} - \frac{1}{3}\right)\right)} \]
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 98.7

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

5. Applied rewrites 98.7%

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

7. Applied rewrites 98.7%

   \[\leadsto \mathsf{fma}\left(\frac{1}{\frac{-0.3333333333333333 - x \cdot \left(x \cdot 0.13333333333333333\right)}{0.1111111111111111 - 0.017777777777777778 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)}}, \color{blue}{x} \cdot \left(x \cdot x\right), x\right) \]

8. Taylor expanded in x around 0

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

10. Applied rewrites 98.8%

    \[\leadsto \mathsf{fma}\left(\frac{1}{\frac{-0.3333333333333333 - x \cdot \left(x \cdot 0.13333333333333333\right)}{0.1111111111111111}}, x \cdot \left(x \cdot x\right), x\right) \]
11. Step-by-step derivation

12. Applied rewrites 98.8%

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

13. Final simplification 98.8%

    \[\leadsto \mathsf{fma}\left(\left(x \cdot x\right) \cdot \frac{0.1111111111111111}{\mathsf{fma}\left(x \cdot x, -0.13333333333333333, -0.3333333333333333\right)}, x, x\right) \]
14. Add Preprocessing

Alternative 3: 97.1% accurate, 15.1× speedup

Math

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

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(Float64(x * x) * fma(x, Float64(x * 0.13333333333333333), -0.3333333333333333)), x, x)
end

Wolfram

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

Derivation

1. Initial program 7.0%

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

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left(\frac{2}{15} \cdot {x}^{2} - \frac{1}{3}\right)\right)} \]
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 98.7

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

5. Applied rewrites 98.7%

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

7. Applied rewrites 98.7%

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

8. Final simplification 98.7%

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

Alternative 4: 96.5% accurate, 15.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 7.0%

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

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left(\frac{2}{15} \cdot {x}^{2} - \frac{1}{3}\right)\right)} \]
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 98.7

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

5. Applied rewrites 98.7%

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

6. Taylor expanded in x around inf

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

8. Applied rewrites 98.6%

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

Alternative 5: 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 7.0%

   \[\frac{e^{x} - e^{-x}}{e^{x} + e^{-x}} \]
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 98.6

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

5. Applied rewrites 98.6%

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

6. Final simplification 98.6%

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

Alternative 6: 5.0% accurate, 26.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

public static double code(double x) {
	return -0.3333333333333333 * (x * (x * x));
}

Python

def code(x):
	return -0.3333333333333333 * (x * (x * x))

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 7.0%

   \[\frac{e^{x} - e^{-x}}{e^{x} + e^{-x}} \]
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 98.6

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

5. Applied rewrites 98.6%

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

6. Taylor expanded in x around inf

   \[\leadsto \frac{-1}{3} \cdot \color{blue}{{x}^{3}} \]
7. Step-by-step derivation

8. Applied rewrites 5.1%

   \[\leadsto -0.3333333333333333 \cdot \color{blue}{\left(x \cdot \left(x \cdot x\right)\right)} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024227 
(FPCore (x)
  :name "Hyperbolic tangent"
  :precision binary64
  (/ (- (exp x) (exp (- x))) (+ (exp x) (exp (- x)))))