Hyperbolic tangent

Percentage Accurate: 9.1% → 100.0%

Time: 9.9s

Alternatives: 7

Speedup: 70.3×

• could not determine a ground truth

  1. x = -1.622602649519976e+238

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 10.0%

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

   1. lift-/.f64 N/A

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

   2. lift--.f64 N/A

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

   3. lift-exp.f64 N/A

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

   4. lift-exp.f64 N/A

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

   5. lift-neg.f64 N/A

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

   6. lift-+.f64 N/A

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

   7. lift-exp.f64 N/A

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

   8. lift-exp.f64 N/A

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

   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: 97.1% accurate, 7.5× speedup

Math

\[\begin{array}{l} \\ \frac{0.5}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0010582010582010583, x \cdot x, -0.011111111111111112\right), x \cdot x, 0.16666666666666666\right), x \cdot x, 0.5\right)}{x}} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/
  0.5
  (/
   (fma
    (fma
     (fma 0.0010582010582010583 (* x x) -0.011111111111111112)
     (* x x)
     0.16666666666666666)
    (* x x)
    0.5)
   x)))

C

double code(double x) {
	return 0.5 / (fma(fma(fma(0.0010582010582010583, (x * x), -0.011111111111111112), (x * x), 0.16666666666666666), (x * x), 0.5) / x);
}

Julia

function code(x)
	return Float64(0.5 / Float64(fma(fma(fma(0.0010582010582010583, Float64(x * x), -0.011111111111111112), Float64(x * x), 0.16666666666666666), Float64(x * x), 0.5) / x))
end

Wolfram

code[x_] := N[(0.5 / N[(N[(N[(N[(0.0010582010582010583 * N[(x * x), $MachinePrecision] + -0.011111111111111112), $MachinePrecision] * N[(x * x), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * N[(x * x), $MachinePrecision] + 0.5), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 10.0%

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

   1. lift-/.f64 N/A

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

   2. div-inv N/A

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

   3. lift--.f64 N/A

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

   4. flip3-- N/A

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

   5. clear-num N/A

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

   6. lift-+.f64 N/A

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

   7. lift-exp.f64 N/A

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

   8. lift-exp.f64 N/A

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

   9. lift-neg.f64 N/A

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

   10. cosh-undef N/A

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

   11. associate-/r* N/A

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

   12. frac-times N/A

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

4. Applied rewrites 97.3%

   \[\leadsto \color{blue}{\frac{0.5}{\frac{0.5}{\sinh x} \cdot \cosh x}} \]

5. Taylor expanded in x around 0

   \[\leadsto \frac{\frac{1}{2}}{\color{blue}{\frac{\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{945} \cdot {x}^{2} - \frac{1}{90}\right)\right)}{x}}} \]
6. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{\frac{1}{2}}{\color{blue}{\frac{\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{945} \cdot {x}^{2} - \frac{1}{90}\right)\right)}{x}}} \]

   2. +-commutative N/A

      \[\leadsto \frac{\frac{1}{2}}{\frac{\color{blue}{{x}^{2} \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{945} \cdot {x}^{2} - \frac{1}{90}\right)\right) + \frac{1}{2}}}{x}} \]

   3. *-commutative N/A

      \[\leadsto \frac{\frac{1}{2}}{\frac{\color{blue}{\left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{945} \cdot {x}^{2} - \frac{1}{90}\right)\right) \cdot {x}^{2}} + \frac{1}{2}}{x}} \]

   4. lower-fma.f64 N/A

      \[\leadsto \frac{\frac{1}{2}}{\frac{\color{blue}{\mathsf{fma}\left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{945} \cdot {x}^{2} - \frac{1}{90}\right), {x}^{2}, \frac{1}{2}\right)}}{x}} \]

   5. +-commutative N/A

      \[\leadsto \frac{\frac{1}{2}}{\frac{\mathsf{fma}\left(\color{blue}{{x}^{2} \cdot \left(\frac{1}{945} \cdot {x}^{2} - \frac{1}{90}\right) + \frac{1}{6}}, {x}^{2}, \frac{1}{2}\right)}{x}} \]

   6. *-commutative N/A

      \[\leadsto \frac{\frac{1}{2}}{\frac{\mathsf{fma}\left(\color{blue}{\left(\frac{1}{945} \cdot {x}^{2} - \frac{1}{90}\right) \cdot {x}^{2}} + \frac{1}{6}, {x}^{2}, \frac{1}{2}\right)}{x}} \]

   7. lower-fma.f64 N/A

      \[\leadsto \frac{\frac{1}{2}}{\frac{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{945} \cdot {x}^{2} - \frac{1}{90}, {x}^{2}, \frac{1}{6}\right)}, {x}^{2}, \frac{1}{2}\right)}{x}} \]

   8. sub-neg N/A

      \[\leadsto \frac{\frac{1}{2}}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{1}{945} \cdot {x}^{2} + \left(\mathsf{neg}\left(\frac{1}{90}\right)\right)}, {x}^{2}, \frac{1}{6}\right), {x}^{2}, \frac{1}{2}\right)}{x}} \]

   9. metadata-eval N/A

      \[\leadsto \frac{\frac{1}{2}}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{945} \cdot {x}^{2} + \color{blue}{\frac{-1}{90}}, {x}^{2}, \frac{1}{6}\right), {x}^{2}, \frac{1}{2}\right)}{x}} \]

   10. lower-fma.f64 N/A

       \[\leadsto \frac{\frac{1}{2}}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{945}, {x}^{2}, \frac{-1}{90}\right)}, {x}^{2}, \frac{1}{6}\right), {x}^{2}, \frac{1}{2}\right)}{x}} \]

   11. unpow2 N/A

       \[\leadsto \frac{\frac{1}{2}}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{945}, \color{blue}{x \cdot x}, \frac{-1}{90}\right), {x}^{2}, \frac{1}{6}\right), {x}^{2}, \frac{1}{2}\right)}{x}} \]

   12. lower-*.f64 N/A

       \[\leadsto \frac{\frac{1}{2}}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{945}, \color{blue}{x \cdot x}, \frac{-1}{90}\right), {x}^{2}, \frac{1}{6}\right), {x}^{2}, \frac{1}{2}\right)}{x}} \]

   13. unpow2 N/A

       \[\leadsto \frac{\frac{1}{2}}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{945}, x \cdot x, \frac{-1}{90}\right), \color{blue}{x \cdot x}, \frac{1}{6}\right), {x}^{2}, \frac{1}{2}\right)}{x}} \]

   14. lower-*.f64 N/A

       \[\leadsto \frac{\frac{1}{2}}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{945}, x \cdot x, \frac{-1}{90}\right), \color{blue}{x \cdot x}, \frac{1}{6}\right), {x}^{2}, \frac{1}{2}\right)}{x}} \]

   15. unpow2 N/A

       \[\leadsto \frac{\frac{1}{2}}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{945}, x \cdot x, \frac{-1}{90}\right), x \cdot x, \frac{1}{6}\right), \color{blue}{x \cdot x}, \frac{1}{2}\right)}{x}} \]

   16. lower-*.f64 97.3

       \[\leadsto \frac{0.5}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0010582010582010583, x \cdot x, -0.011111111111111112\right), x \cdot x, 0.16666666666666666\right), \color{blue}{x \cdot x}, 0.5\right)}{x}} \]

7. Applied rewrites 97.3%

   \[\leadsto \frac{0.5}{\color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0010582010582010583, x \cdot x, -0.011111111111111112\right), x \cdot x, 0.16666666666666666\right), x \cdot x, 0.5\right)}{x}}} \]
8. Add Preprocessing

Alternative 3: 96.8% accurate, 9.4× speedup

Math

\[\begin{array}{l} \\ \frac{0.5}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.011111111111111112, x \cdot x, 0.16666666666666666\right), x \cdot x, 0.5\right)}{x}} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/
  0.5
  (/
   (fma (fma -0.011111111111111112 (* x x) 0.16666666666666666) (* x x) 0.5)
   x)))

C

double code(double x) {
	return 0.5 / (fma(fma(-0.011111111111111112, (x * x), 0.16666666666666666), (x * x), 0.5) / x);
}

Julia

function code(x)
	return Float64(0.5 / Float64(fma(fma(-0.011111111111111112, Float64(x * x), 0.16666666666666666), Float64(x * x), 0.5) / x))
end

Wolfram

code[x_] := N[(0.5 / N[(N[(N[(-0.011111111111111112 * N[(x * x), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * N[(x * x), $MachinePrecision] + 0.5), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 10.0%

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

   1. lift-/.f64 N/A

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

   2. div-inv N/A

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

   3. lift--.f64 N/A

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

   4. flip3-- N/A

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

   5. clear-num N/A

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

   6. lift-+.f64 N/A

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

   7. lift-exp.f64 N/A

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

   8. lift-exp.f64 N/A

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

   9. lift-neg.f64 N/A

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

   10. cosh-undef N/A

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

   11. associate-/r* N/A

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

   12. frac-times N/A

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

4. Applied rewrites 97.3%

   \[\leadsto \color{blue}{\frac{0.5}{\frac{0.5}{\sinh x} \cdot \cosh x}} \]

5. Taylor expanded in x around 0

   \[\leadsto \frac{\frac{1}{2}}{\color{blue}{\frac{\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{6} + \frac{-1}{90} \cdot {x}^{2}\right)}{x}}} \]
6. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{\frac{1}{2}}{\color{blue}{\frac{\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{6} + \frac{-1}{90} \cdot {x}^{2}\right)}{x}}} \]

   2. +-commutative N/A

      \[\leadsto \frac{\frac{1}{2}}{\frac{\color{blue}{{x}^{2} \cdot \left(\frac{1}{6} + \frac{-1}{90} \cdot {x}^{2}\right) + \frac{1}{2}}}{x}} \]

   3. *-commutative N/A

      \[\leadsto \frac{\frac{1}{2}}{\frac{\color{blue}{\left(\frac{1}{6} + \frac{-1}{90} \cdot {x}^{2}\right) \cdot {x}^{2}} + \frac{1}{2}}{x}} \]

   4. lower-fma.f64 N/A

      \[\leadsto \frac{\frac{1}{2}}{\frac{\color{blue}{\mathsf{fma}\left(\frac{1}{6} + \frac{-1}{90} \cdot {x}^{2}, {x}^{2}, \frac{1}{2}\right)}}{x}} \]

   5. +-commutative N/A

      \[\leadsto \frac{\frac{1}{2}}{\frac{\mathsf{fma}\left(\color{blue}{\frac{-1}{90} \cdot {x}^{2} + \frac{1}{6}}, {x}^{2}, \frac{1}{2}\right)}{x}} \]

   6. lower-fma.f64 N/A

      \[\leadsto \frac{\frac{1}{2}}{\frac{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{-1}{90}, {x}^{2}, \frac{1}{6}\right)}, {x}^{2}, \frac{1}{2}\right)}{x}} \]

   7. unpow2 N/A

      \[\leadsto \frac{\frac{1}{2}}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{90}, \color{blue}{x \cdot x}, \frac{1}{6}\right), {x}^{2}, \frac{1}{2}\right)}{x}} \]

   8. lower-*.f64 N/A

      \[\leadsto \frac{\frac{1}{2}}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{90}, \color{blue}{x \cdot x}, \frac{1}{6}\right), {x}^{2}, \frac{1}{2}\right)}{x}} \]

   9. unpow2 N/A

      \[\leadsto \frac{\frac{1}{2}}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{90}, x \cdot x, \frac{1}{6}\right), \color{blue}{x \cdot x}, \frac{1}{2}\right)}{x}} \]

   10. lower-*.f64 96.9

       \[\leadsto \frac{0.5}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.011111111111111112, x \cdot x, 0.16666666666666666\right), \color{blue}{x \cdot x}, 0.5\right)}{x}} \]

7. Applied rewrites 96.9%

   \[\leadsto \frac{0.5}{\color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.011111111111111112, x \cdot x, 0.16666666666666666\right), x \cdot x, 0.5\right)}{x}}} \]
8. Add Preprocessing

Alternative 4: 96.9% accurate, 14.6× speedup

Math

\[\begin{array}{l} \\ \frac{0.5}{\mathsf{fma}\left(0.16666666666666666, x, \frac{0.5}{x}\right)} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ 0.5 (fma 0.16666666666666666 x (/ 0.5 x))))

C

double code(double x) {
	return 0.5 / fma(0.16666666666666666, x, (0.5 / x));
}

Julia

function code(x)
	return Float64(0.5 / fma(0.16666666666666666, x, Float64(0.5 / x)))
end

Wolfram

code[x_] := N[(0.5 / N[(0.16666666666666666 * x + N[(0.5 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 10.0%

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

   1. lift-/.f64 N/A

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

   2. div-inv N/A

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

   3. lift--.f64 N/A

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

   4. flip3-- N/A

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

   5. clear-num N/A

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

   6. lift-+.f64 N/A

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

   7. lift-exp.f64 N/A

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

   8. lift-exp.f64 N/A

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

   9. lift-neg.f64 N/A

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

   10. cosh-undef N/A

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

   11. associate-/r* N/A

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

   12. frac-times N/A

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

4. Applied rewrites 97.3%

   \[\leadsto \color{blue}{\frac{0.5}{\frac{0.5}{\sinh x} \cdot \cosh x}} \]

5. Taylor expanded in x around 0

   \[\leadsto \frac{\frac{1}{2}}{\color{blue}{\frac{\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{945} \cdot {x}^{2} - \frac{1}{90}\right)\right)}{x}}} \]
6. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{\frac{1}{2}}{\color{blue}{\frac{\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{945} \cdot {x}^{2} - \frac{1}{90}\right)\right)}{x}}} \]

   2. +-commutative N/A

      \[\leadsto \frac{\frac{1}{2}}{\frac{\color{blue}{{x}^{2} \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{945} \cdot {x}^{2} - \frac{1}{90}\right)\right) + \frac{1}{2}}}{x}} \]

   3. *-commutative N/A

      \[\leadsto \frac{\frac{1}{2}}{\frac{\color{blue}{\left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{945} \cdot {x}^{2} - \frac{1}{90}\right)\right) \cdot {x}^{2}} + \frac{1}{2}}{x}} \]

   4. lower-fma.f64 N/A

      \[\leadsto \frac{\frac{1}{2}}{\frac{\color{blue}{\mathsf{fma}\left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{945} \cdot {x}^{2} - \frac{1}{90}\right), {x}^{2}, \frac{1}{2}\right)}}{x}} \]

   5. +-commutative N/A

      \[\leadsto \frac{\frac{1}{2}}{\frac{\mathsf{fma}\left(\color{blue}{{x}^{2} \cdot \left(\frac{1}{945} \cdot {x}^{2} - \frac{1}{90}\right) + \frac{1}{6}}, {x}^{2}, \frac{1}{2}\right)}{x}} \]

   6. *-commutative N/A

      \[\leadsto \frac{\frac{1}{2}}{\frac{\mathsf{fma}\left(\color{blue}{\left(\frac{1}{945} \cdot {x}^{2} - \frac{1}{90}\right) \cdot {x}^{2}} + \frac{1}{6}, {x}^{2}, \frac{1}{2}\right)}{x}} \]

   7. lower-fma.f64 N/A

      \[\leadsto \frac{\frac{1}{2}}{\frac{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{945} \cdot {x}^{2} - \frac{1}{90}, {x}^{2}, \frac{1}{6}\right)}, {x}^{2}, \frac{1}{2}\right)}{x}} \]

   8. sub-neg N/A

      \[\leadsto \frac{\frac{1}{2}}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{1}{945} \cdot {x}^{2} + \left(\mathsf{neg}\left(\frac{1}{90}\right)\right)}, {x}^{2}, \frac{1}{6}\right), {x}^{2}, \frac{1}{2}\right)}{x}} \]

   9. metadata-eval N/A

      \[\leadsto \frac{\frac{1}{2}}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{945} \cdot {x}^{2} + \color{blue}{\frac{-1}{90}}, {x}^{2}, \frac{1}{6}\right), {x}^{2}, \frac{1}{2}\right)}{x}} \]

   10. lower-fma.f64 N/A

       \[\leadsto \frac{\frac{1}{2}}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{945}, {x}^{2}, \frac{-1}{90}\right)}, {x}^{2}, \frac{1}{6}\right), {x}^{2}, \frac{1}{2}\right)}{x}} \]

   11. unpow2 N/A

       \[\leadsto \frac{\frac{1}{2}}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{945}, \color{blue}{x \cdot x}, \frac{-1}{90}\right), {x}^{2}, \frac{1}{6}\right), {x}^{2}, \frac{1}{2}\right)}{x}} \]

   12. lower-*.f64 N/A

       \[\leadsto \frac{\frac{1}{2}}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{945}, \color{blue}{x \cdot x}, \frac{-1}{90}\right), {x}^{2}, \frac{1}{6}\right), {x}^{2}, \frac{1}{2}\right)}{x}} \]

   13. unpow2 N/A

       \[\leadsto \frac{\frac{1}{2}}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{945}, x \cdot x, \frac{-1}{90}\right), \color{blue}{x \cdot x}, \frac{1}{6}\right), {x}^{2}, \frac{1}{2}\right)}{x}} \]

   14. lower-*.f64 N/A

       \[\leadsto \frac{\frac{1}{2}}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{945}, x \cdot x, \frac{-1}{90}\right), \color{blue}{x \cdot x}, \frac{1}{6}\right), {x}^{2}, \frac{1}{2}\right)}{x}} \]

   15. unpow2 N/A

       \[\leadsto \frac{\frac{1}{2}}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{945}, x \cdot x, \frac{-1}{90}\right), x \cdot x, \frac{1}{6}\right), \color{blue}{x \cdot x}, \frac{1}{2}\right)}{x}} \]

   16. lower-*.f64 97.3

       \[\leadsto \frac{0.5}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0010582010582010583, x \cdot x, -0.011111111111111112\right), x \cdot x, 0.16666666666666666\right), \color{blue}{x \cdot x}, 0.5\right)}{x}} \]

7. Applied rewrites 97.3%

   \[\leadsto \frac{0.5}{\color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0010582010582010583, x \cdot x, -0.011111111111111112\right), x \cdot x, 0.16666666666666666\right), x \cdot x, 0.5\right)}{x}}} \]

8. Taylor expanded in x around 0

   \[\leadsto \frac{\frac{1}{2}}{\color{blue}{\frac{\frac{1}{2} + \frac{1}{6} \cdot {x}^{2}}{x}}} \]
9. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{\frac{1}{2}}{\frac{\color{blue}{\frac{1}{6} \cdot {x}^{2} + \frac{1}{2}}}{x}} \]

   2. unpow2 N/A

      \[\leadsto \frac{\frac{1}{2}}{\frac{\frac{1}{6} \cdot \color{blue}{\left(x \cdot x\right)} + \frac{1}{2}}{x}} \]

   3. associate-*r* N/A

      \[\leadsto \frac{\frac{1}{2}}{\frac{\color{blue}{\left(\frac{1}{6} \cdot x\right) \cdot x} + \frac{1}{2}}{x}} \]

   4. metadata-eval N/A

      \[\leadsto \frac{\frac{1}{2}}{\frac{\left(\frac{1}{6} \cdot x\right) \cdot x + \color{blue}{\frac{1}{2} \cdot 1}}{x}} \]

   5. lft-mult-inverse N/A

      \[\leadsto \frac{\frac{1}{2}}{\frac{\left(\frac{1}{6} \cdot x\right) \cdot x + \frac{1}{2} \cdot \color{blue}{\left(\frac{1}{{x}^{2}} \cdot {x}^{2}\right)}}{x}} \]

   6. associate-*l* N/A

      \[\leadsto \frac{\frac{1}{2}}{\frac{\left(\frac{1}{6} \cdot x\right) \cdot x + \color{blue}{\left(\frac{1}{2} \cdot \frac{1}{{x}^{2}}\right) \cdot {x}^{2}}}{x}} \]

   7. unpow2 N/A

      \[\leadsto \frac{\frac{1}{2}}{\frac{\left(\frac{1}{6} \cdot x\right) \cdot x + \left(\frac{1}{2} \cdot \frac{1}{{x}^{2}}\right) \cdot \color{blue}{\left(x \cdot x\right)}}{x}} \]

   8. associate-*r* N/A

      \[\leadsto \frac{\frac{1}{2}}{\frac{\left(\frac{1}{6} \cdot x\right) \cdot x + \color{blue}{\left(\left(\frac{1}{2} \cdot \frac{1}{{x}^{2}}\right) \cdot x\right) \cdot x}}{x}} \]

   9. distribute-rgt-in N/A

      \[\leadsto \frac{\frac{1}{2}}{\frac{\color{blue}{x \cdot \left(\frac{1}{6} \cdot x + \left(\frac{1}{2} \cdot \frac{1}{{x}^{2}}\right) \cdot x\right)}}{x}} \]

   10. distribute-rgt-in N/A

       \[\leadsto \frac{\frac{1}{2}}{\frac{x \cdot \color{blue}{\left(x \cdot \left(\frac{1}{6} + \frac{1}{2} \cdot \frac{1}{{x}^{2}}\right)\right)}}{x}} \]

   11. *-commutative N/A

       \[\leadsto \frac{\frac{1}{2}}{\frac{\color{blue}{\left(x \cdot \left(\frac{1}{6} + \frac{1}{2} \cdot \frac{1}{{x}^{2}}\right)\right) \cdot x}}{x}} \]

   12. associate-/l* N/A

       \[\leadsto \frac{\frac{1}{2}}{\color{blue}{\left(x \cdot \left(\frac{1}{6} + \frac{1}{2} \cdot \frac{1}{{x}^{2}}\right)\right) \cdot \frac{x}{x}}} \]

   13. *-inverses N/A

       \[\leadsto \frac{\frac{1}{2}}{\left(x \cdot \left(\frac{1}{6} + \frac{1}{2} \cdot \frac{1}{{x}^{2}}\right)\right) \cdot \color{blue}{1}} \]

   14. *-rgt-identity N/A

       \[\leadsto \frac{\frac{1}{2}}{\color{blue}{x \cdot \left(\frac{1}{6} + \frac{1}{2} \cdot \frac{1}{{x}^{2}}\right)}} \]

   15. distribute-rgt-in N/A

       \[\leadsto \frac{\frac{1}{2}}{\color{blue}{\frac{1}{6} \cdot x + \left(\frac{1}{2} \cdot \frac{1}{{x}^{2}}\right) \cdot x}} \]

   16. associate-*r/ N/A

       \[\leadsto \frac{\frac{1}{2}}{\frac{1}{6} \cdot x + \color{blue}{\frac{\frac{1}{2} \cdot 1}{{x}^{2}}} \cdot x} \]

   17. metadata-eval N/A

       \[\leadsto \frac{\frac{1}{2}}{\frac{1}{6} \cdot x + \frac{\color{blue}{\frac{1}{2}}}{{x}^{2}} \cdot x} \]

   18. associate-*l/ N/A

       \[\leadsto \frac{\frac{1}{2}}{\frac{1}{6} \cdot x + \color{blue}{\frac{\frac{1}{2} \cdot x}{{x}^{2}}}} \]

   19. unpow2 N/A

       \[\leadsto \frac{\frac{1}{2}}{\frac{1}{6} \cdot x + \frac{\frac{1}{2} \cdot x}{\color{blue}{x \cdot x}}} \]

   20. times-frac N/A

       \[\leadsto \frac{\frac{1}{2}}{\frac{1}{6} \cdot x + \color{blue}{\frac{\frac{1}{2}}{x} \cdot \frac{x}{x}}} \]

   21. *-inverses N/A

       \[\leadsto \frac{\frac{1}{2}}{\frac{1}{6} \cdot x + \frac{\frac{1}{2}}{x} \cdot \color{blue}{1}} \]

10. Applied rewrites 96.9%

    \[\leadsto \frac{0.5}{\color{blue}{\mathsf{fma}\left(0.16666666666666666, x, \frac{0.5}{x}\right)}} \]
11. Add Preprocessing

Alternative 5: 96.8% accurate, 24.8× speedup

Math

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

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

Wolfram

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

Derivation

1. Initial program 10.0%

   \[\frac{e^{x} - e^{-x}}{e^{x} + e^{-x}} \]
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. *-commutative N/A

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

   4. associate-*r* N/A

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

   5. *-rgt-identity N/A

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

   6. lower-fma.f64 N/A

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

   7. *-commutative N/A

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

   8. pow-plus N/A

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

   9. lower-pow.f64 N/A

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

   10. metadata-eval 96.8

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

5. Applied rewrites 96.8%

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

7. Applied rewrites 96.8%

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

Alternative 6: 96.8% accurate, 24.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 10.0%

   \[\frac{e^{x} - e^{-x}}{e^{x} + e^{-x}} \]
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. *-commutative N/A

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

   4. associate-*r* N/A

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

   5. *-rgt-identity N/A

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

   6. lower-fma.f64 N/A

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

   7. *-commutative N/A

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

   8. pow-plus N/A

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

   9. lower-pow.f64 N/A

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

   10. metadata-eval 96.8

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

5. Applied rewrites 96.8%

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

7. Applied rewrites 96.8%

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

Alternative 7: 96.6% accurate, 70.3× speedup

Math

\[\begin{array}{l} \\ 1 \cdot x \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* 1.0 x))

C

double code(double x) {
	return 1.0 * x;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 1.0d0 * x
end function

Java

public static double code(double x) {
	return 1.0 * x;
}

Python

def code(x):
	return 1.0 * x

Julia

function code(x)
	return Float64(1.0 * x)
end

MATLAB

function tmp = code(x)
	tmp = 1.0 * x;
end

Wolfram

code[x_] := N[(1.0 * x), $MachinePrecision]

Derivation

1. Initial program 10.0%

   \[\frac{e^{x} - e^{-x}}{e^{x} + e^{-x}} \]
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. *-commutative N/A

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

   4. associate-*r* N/A

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

   5. *-rgt-identity N/A

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

   6. lower-fma.f64 N/A

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

   7. *-commutative N/A

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

   8. pow-plus N/A

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

   9. lower-pow.f64 N/A

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

   10. metadata-eval 96.8

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

5. Applied rewrites 96.8%

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

7. Applied rewrites 96.8%

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

8. Taylor expanded in x around 0

   \[\leadsto 1 \cdot x \]
9. Step-by-step derivation

10. Applied rewrites 96.3%

    \[\leadsto 1 \cdot x \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024248 
(FPCore (x)
  :name "Hyperbolic tangent"
  :precision binary64
  (/ (- (exp x) (exp (- x))) (+ (exp x) (exp (- x)))))