Hyperbolic sine

Percentage Accurate: 54.8% → 100.0%

Time: 9.3s

Alternatives: 11

Speedup: 12.8×

• 49.8% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \frac{e^{x} - e^{-x}}{2} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ (- (exp x) (exp (- x))) 2.0))

C

double code(double x) {
	return (exp(x) - exp(-x)) / 2.0;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (exp(x) - exp(-x)) / 2.0d0
end function

Java

public static double code(double x) {
	return (Math.exp(x) - Math.exp(-x)) / 2.0;
}

Python

def code(x):
	return (math.exp(x) - math.exp(-x)) / 2.0

Julia

function code(x)
	return Float64(Float64(exp(x) - exp(Float64(-x))) / 2.0)
end

MATLAB

function tmp = code(x)
	tmp = (exp(x) - exp(-x)) / 2.0;
end

Wolfram

code[x_] := N[(N[(N[Exp[x], $MachinePrecision] - N[Exp[(-x)], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]

Initial Program: 54.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{e^{x} - e^{-x}}{2} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ (- (exp x) (exp (- x))) 2.0))

C

double code(double x) {
	return (exp(x) - exp(-x)) / 2.0;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (exp(x) - exp(-x)) / 2.0d0
end function

Java

public static double code(double x) {
	return (Math.exp(x) - Math.exp(-x)) / 2.0;
}

Python

def code(x):
	return (math.exp(x) - math.exp(-x)) / 2.0

Julia

function code(x)
	return Float64(Float64(exp(x) - exp(Float64(-x))) / 2.0)
end

MATLAB

function tmp = code(x)
	tmp = (exp(x) - exp(-x)) / 2.0;
end

Wolfram

code[x_] := N[(N[(N[Exp[x], $MachinePrecision] - N[Exp[(-x)], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]

Alternative 1: 100.0% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \sinh x \end{array} \]

FPCore

(FPCore (x) :precision binary64 (sinh x))

C

double code(double x) {
	return sinh(x);
}

Fortran

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

Java

public static double code(double x) {
	return Math.sinh(x);
}

Python

def code(x):
	return math.sinh(x)

Julia

function code(x)
	return sinh(x)
end

MATLAB

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

Wolfram

code[x_] := N[Sinh[x], $MachinePrecision]

Derivation

1. Initial program 54.7%

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

   1. lift-/.f64 N/A

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

   2. lift--.f64 N/A

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

   3. lift-exp.f64 N/A

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

   4. lift-exp.f64 N/A

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

   5. lift-neg.f64 N/A

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

   6. sinh-def N/A

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

   7. lower-sinh.f64 100.0

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

4. Applied rewrites 100.0%

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

Alternative 2: 88.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{x} - e^{-x} \leq 0.0002:\\ \;\;\;\;\mathsf{fma}\left(0.16666666666666666 \cdot x, x, 1\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot 0.0001984126984126984\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (- (exp x) (exp (- x))) 0.0002)
   (* (fma (* 0.16666666666666666 x) x 1.0) x)
   (* (* (* (* (* (* x x) x) x) (* x x)) 0.0001984126984126984) x)))

C

double code(double x) {
	double tmp;
	if ((exp(x) - exp(-x)) <= 0.0002) {
		tmp = fma((0.16666666666666666 * x), x, 1.0) * x;
	} else {
		tmp = (((((x * x) * x) * x) * (x * x)) * 0.0001984126984126984) * x;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(exp(x) - exp(Float64(-x))) <= 0.0002)
		tmp = Float64(fma(Float64(0.16666666666666666 * x), x, 1.0) * x);
	else
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(x * x) * x) * x) * Float64(x * x)) * 0.0001984126984126984) * x);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[(N[Exp[x], $MachinePrecision] - N[Exp[(-x)], $MachinePrecision]), $MachinePrecision], 0.0002], N[(N[(N[(0.16666666666666666 * x), $MachinePrecision] * x + 1.0), $MachinePrecision] * x), $MachinePrecision], N[(N[(N[(N[(N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] * 0.0001984126984126984), $MachinePrecision] * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (exp.f64 x) (exp.f64 (neg.f64 x))) < 2.0000000000000001e-4

   1. Initial program 40.0%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 88.8

         \[\leadsto \mathsf{fma}\left(\color{blue}{x \cdot x}, 0.16666666666666666, 1\right) \cdot x \]

   5. Applied rewrites 88.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot x, 0.16666666666666666, 1\right) \cdot x} \]
   6. Step-by-step derivation

   7. Applied rewrites 88.8%

      \[\leadsto \mathsf{fma}\left(0.16666666666666666 \cdot x, x, 1\right) \cdot x \]

   if 2.0000000000000001e-4 < (-.f64 (exp.f64 x) (exp.f64 (neg.f64 x)))

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right)\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right)\right)\right) \cdot x} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right)\right)\right) \cdot x} \]

      3. +-commutative N/A

         \[\leadsto \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right)\right) + 1\right)} \cdot x \]

      4. *-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right)\right) \cdot {x}^{2}} + 1\right) \cdot x \]

      5. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right), {x}^{2}, 1\right)} \cdot x \]

      6. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{{x}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right) + \frac{1}{6}}, {x}^{2}, 1\right) \cdot x \]

      7. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right) \cdot {x}^{2}} + \frac{1}{6}, {x}^{2}, 1\right) \cdot x \]

      8. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}, {x}^{2}, \frac{1}{6}\right)}, {x}^{2}, 1\right) \cdot x \]

      9. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{1}{5040} \cdot {x}^{2} + \frac{1}{120}}, {x}^{2}, \frac{1}{6}\right), {x}^{2}, 1\right) \cdot x \]

      10. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{5040}, {x}^{2}, \frac{1}{120}\right)}, {x}^{2}, \frac{1}{6}\right), {x}^{2}, 1\right) \cdot x \]

      11. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{5040}, \color{blue}{x \cdot x}, \frac{1}{120}\right), {x}^{2}, \frac{1}{6}\right), {x}^{2}, 1\right) \cdot x \]

      12. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{5040}, \color{blue}{x \cdot x}, \frac{1}{120}\right), {x}^{2}, \frac{1}{6}\right), {x}^{2}, 1\right) \cdot x \]

      13. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{5040}, x \cdot x, \frac{1}{120}\right), \color{blue}{x \cdot x}, \frac{1}{6}\right), {x}^{2}, 1\right) \cdot x \]

      14. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{5040}, x \cdot x, \frac{1}{120}\right), \color{blue}{x \cdot x}, \frac{1}{6}\right), {x}^{2}, 1\right) \cdot x \]

      15. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{5040}, x \cdot x, \frac{1}{120}\right), x \cdot x, \frac{1}{6}\right), \color{blue}{x \cdot x}, 1\right) \cdot x \]

      16. lower-*.f64 86.4

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0001984126984126984, x \cdot x, 0.008333333333333333\right), x \cdot x, 0.16666666666666666\right), \color{blue}{x \cdot x}, 1\right) \cdot x \]

   5. Applied rewrites 86.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0001984126984126984, x \cdot x, 0.008333333333333333\right), x \cdot x, 0.16666666666666666\right), x \cdot x, 1\right) \cdot x} \]

   6. Taylor expanded in x around inf

      \[\leadsto \left(\frac{1}{5040} \cdot {x}^{6}\right) \cdot x \]
   7. Step-by-step derivation

   8. Applied rewrites 86.4%

      \[\leadsto \left(\left(\left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot 0.0001984126984126984\right) \cdot x\right) \cdot x\right) \cdot x \]
   9. Step-by-step derivation

   10. Applied rewrites 86.4%

       \[\leadsto \left(0.0001984126984126984 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) \cdot x \]
3. Recombined 2 regimes into one program.

4. Final simplification 88.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{x} - e^{-x} \leq 0.0002:\\ \;\;\;\;\mathsf{fma}\left(0.16666666666666666 \cdot x, x, 1\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot 0.0001984126984126984\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 86.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{x} - e^{-x} \leq 0.0002:\\ \;\;\;\;\mathsf{fma}\left(0.16666666666666666 \cdot x, x, 1\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(0.008333333333333333 \cdot x\right) \cdot \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (- (exp x) (exp (- x))) 0.0002)
   (* (fma (* 0.16666666666666666 x) x 1.0) x)
   (* (* 0.008333333333333333 x) (* (* (* x x) x) x))))

C

double code(double x) {
	double tmp;
	if ((exp(x) - exp(-x)) <= 0.0002) {
		tmp = fma((0.16666666666666666 * x), x, 1.0) * x;
	} else {
		tmp = (0.008333333333333333 * x) * (((x * x) * x) * x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(exp(x) - exp(Float64(-x))) <= 0.0002)
		tmp = Float64(fma(Float64(0.16666666666666666 * x), x, 1.0) * x);
	else
		tmp = Float64(Float64(0.008333333333333333 * x) * Float64(Float64(Float64(x * x) * x) * x));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[(N[Exp[x], $MachinePrecision] - N[Exp[(-x)], $MachinePrecision]), $MachinePrecision], 0.0002], N[(N[(N[(0.16666666666666666 * x), $MachinePrecision] * x + 1.0), $MachinePrecision] * x), $MachinePrecision], N[(N[(0.008333333333333333 * x), $MachinePrecision] * N[(N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (exp.f64 x) (exp.f64 (neg.f64 x))) < 2.0000000000000001e-4

   1. Initial program 40.0%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 88.8

         \[\leadsto \mathsf{fma}\left(\color{blue}{x \cdot x}, 0.16666666666666666, 1\right) \cdot x \]

   5. Applied rewrites 88.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot x, 0.16666666666666666, 1\right) \cdot x} \]
   6. Step-by-step derivation

   7. Applied rewrites 88.8%

      \[\leadsto \mathsf{fma}\left(0.16666666666666666 \cdot x, x, 1\right) \cdot x \]

   if 2.0000000000000001e-4 < (-.f64 (exp.f64 x) (exp.f64 (neg.f64 x)))

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right)\right) \cdot x} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right)\right) \cdot x} \]

      3. +-commutative N/A

         \[\leadsto \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right) + 1\right)} \cdot x \]

      4. *-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right) \cdot {x}^{2}} + 1\right) \cdot x \]

      5. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}, {x}^{2}, 1\right)} \cdot x \]

      6. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{120} \cdot {x}^{2} + \frac{1}{6}}, {x}^{2}, 1\right) \cdot x \]

      7. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{120}, {x}^{2}, \frac{1}{6}\right)}, {x}^{2}, 1\right) \cdot x \]

      8. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{120}, \color{blue}{x \cdot x}, \frac{1}{6}\right), {x}^{2}, 1\right) \cdot x \]

      9. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{120}, \color{blue}{x \cdot x}, \frac{1}{6}\right), {x}^{2}, 1\right) \cdot x \]

      10. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{120}, x \cdot x, \frac{1}{6}\right), \color{blue}{x \cdot x}, 1\right) \cdot x \]

      11. lower-*.f64 77.7

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, x \cdot x, 0.16666666666666666\right), \color{blue}{x \cdot x}, 1\right) \cdot x \]

   5. Applied rewrites 77.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, x \cdot x, 0.16666666666666666\right), x \cdot x, 1\right) \cdot x} \]

   6. Taylor expanded in x around inf

      \[\leadsto \left(\frac{1}{120} \cdot {x}^{4}\right) \cdot x \]
   7. Step-by-step derivation

   8. Applied rewrites 77.7%

      \[\leadsto \left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot 0.008333333333333333\right) \cdot x \]

   9. Taylor expanded in x around inf

      \[\leadsto \frac{1}{120} \cdot \color{blue}{{x}^{5}} \]
   10. Step-by-step derivation

   11. Applied rewrites 77.7%

       \[\leadsto \left(0.008333333333333333 \cdot x\right) \cdot \color{blue}{\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 67.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{x} - e^{-x} \leq 0.0002:\\ \;\;\;\;1 \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x \cdot x\right) \cdot x\right) \cdot 0.16666666666666666\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (- (exp x) (exp (- x))) 0.0002)
   (* 1.0 x)
   (* (* (* x x) x) 0.16666666666666666)))

C

double code(double x) {
	double tmp;
	if ((exp(x) - exp(-x)) <= 0.0002) {
		tmp = 1.0 * x;
	} else {
		tmp = ((x * x) * x) * 0.16666666666666666;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((exp(x) - exp(-x)) <= 0.0002d0) then
        tmp = 1.0d0 * x
    else
        tmp = ((x * x) * x) * 0.16666666666666666d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if ((Math.exp(x) - Math.exp(-x)) <= 0.0002) {
		tmp = 1.0 * x;
	} else {
		tmp = ((x * x) * x) * 0.16666666666666666;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (math.exp(x) - math.exp(-x)) <= 0.0002:
		tmp = 1.0 * x
	else:
		tmp = ((x * x) * x) * 0.16666666666666666
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (Float64(exp(x) - exp(Float64(-x))) <= 0.0002)
		tmp = Float64(1.0 * x);
	else
		tmp = Float64(Float64(Float64(x * x) * x) * 0.16666666666666666);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((exp(x) - exp(-x)) <= 0.0002)
		tmp = 1.0 * x;
	else
		tmp = ((x * x) * x) * 0.16666666666666666;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[(N[Exp[x], $MachinePrecision] - N[Exp[(-x)], $MachinePrecision]), $MachinePrecision], 0.0002], N[(1.0 * x), $MachinePrecision], N[(N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision] * 0.16666666666666666), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (exp.f64 x) (exp.f64 (neg.f64 x))) < 2.0000000000000001e-4

   1. Initial program 40.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right)\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right)\right)\right) \cdot x} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right)\right)\right) \cdot x} \]

      3. +-commutative N/A

         \[\leadsto \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right)\right) + 1\right)} \cdot x \]

      4. *-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right)\right) \cdot {x}^{2}} + 1\right) \cdot x \]

      5. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right), {x}^{2}, 1\right)} \cdot x \]

      6. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{{x}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right) + \frac{1}{6}}, {x}^{2}, 1\right) \cdot x \]

      7. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right) \cdot {x}^{2}} + \frac{1}{6}, {x}^{2}, 1\right) \cdot x \]

      8. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}, {x}^{2}, \frac{1}{6}\right)}, {x}^{2}, 1\right) \cdot x \]

      9. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{1}{5040} \cdot {x}^{2} + \frac{1}{120}}, {x}^{2}, \frac{1}{6}\right), {x}^{2}, 1\right) \cdot x \]

      10. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{5040}, {x}^{2}, \frac{1}{120}\right)}, {x}^{2}, \frac{1}{6}\right), {x}^{2}, 1\right) \cdot x \]

      11. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{5040}, \color{blue}{x \cdot x}, \frac{1}{120}\right), {x}^{2}, \frac{1}{6}\right), {x}^{2}, 1\right) \cdot x \]

      12. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{5040}, \color{blue}{x \cdot x}, \frac{1}{120}\right), {x}^{2}, \frac{1}{6}\right), {x}^{2}, 1\right) \cdot x \]

      13. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{5040}, x \cdot x, \frac{1}{120}\right), \color{blue}{x \cdot x}, \frac{1}{6}\right), {x}^{2}, 1\right) \cdot x \]

      14. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{5040}, x \cdot x, \frac{1}{120}\right), \color{blue}{x \cdot x}, \frac{1}{6}\right), {x}^{2}, 1\right) \cdot x \]

      15. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{5040}, x \cdot x, \frac{1}{120}\right), x \cdot x, \frac{1}{6}\right), \color{blue}{x \cdot x}, 1\right) \cdot x \]

      16. lower-*.f64 97.0

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0001984126984126984, x \cdot x, 0.008333333333333333\right), x \cdot x, 0.16666666666666666\right), \color{blue}{x \cdot x}, 1\right) \cdot x \]

   5. Applied rewrites 97.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0001984126984126984, x \cdot x, 0.008333333333333333\right), x \cdot x, 0.16666666666666666\right), x \cdot x, 1\right) \cdot x} \]

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 67.1%

      \[\leadsto 1 \cdot x \]

   if 2.0000000000000001e-4 < (-.f64 (exp.f64 x) (exp.f64 (neg.f64 x)))

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 64.0

         \[\leadsto \mathsf{fma}\left(\color{blue}{x \cdot x}, 0.16666666666666666, 1\right) \cdot x \]

   5. Applied rewrites 64.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot x, 0.16666666666666666, 1\right) \cdot x} \]

   6. Taylor expanded in x around inf

      \[\leadsto \frac{1}{6} \cdot \color{blue}{{x}^{3}} \]
   7. Step-by-step derivation

   8. Applied rewrites 64.0%

      \[\leadsto \left(\left(x \cdot x\right) \cdot x\right) \cdot \color{blue}{0.16666666666666666} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 67.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{x} - e^{-x} \leq 0.0002:\\ \;\;\;\;1 \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(0.16666666666666666 \cdot x\right) \cdot \left(x \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (- (exp x) (exp (- x))) 0.0002)
   (* 1.0 x)
   (* (* 0.16666666666666666 x) (* x x))))

C

double code(double x) {
	double tmp;
	if ((exp(x) - exp(-x)) <= 0.0002) {
		tmp = 1.0 * x;
	} else {
		tmp = (0.16666666666666666 * x) * (x * x);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((exp(x) - exp(-x)) <= 0.0002d0) then
        tmp = 1.0d0 * x
    else
        tmp = (0.16666666666666666d0 * x) * (x * x)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if ((Math.exp(x) - Math.exp(-x)) <= 0.0002) {
		tmp = 1.0 * x;
	} else {
		tmp = (0.16666666666666666 * x) * (x * x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (math.exp(x) - math.exp(-x)) <= 0.0002:
		tmp = 1.0 * x
	else:
		tmp = (0.16666666666666666 * x) * (x * x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (Float64(exp(x) - exp(Float64(-x))) <= 0.0002)
		tmp = Float64(1.0 * x);
	else
		tmp = Float64(Float64(0.16666666666666666 * x) * Float64(x * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((exp(x) - exp(-x)) <= 0.0002)
		tmp = 1.0 * x;
	else
		tmp = (0.16666666666666666 * x) * (x * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[(N[Exp[x], $MachinePrecision] - N[Exp[(-x)], $MachinePrecision]), $MachinePrecision], 0.0002], N[(1.0 * x), $MachinePrecision], N[(N[(0.16666666666666666 * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (exp.f64 x) (exp.f64 (neg.f64 x))) < 2.0000000000000001e-4

   1. Initial program 40.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right)\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right)\right)\right) \cdot x} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right)\right)\right) \cdot x} \]

      3. +-commutative N/A

         \[\leadsto \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right)\right) + 1\right)} \cdot x \]

      4. *-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right)\right) \cdot {x}^{2}} + 1\right) \cdot x \]

      5. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right), {x}^{2}, 1\right)} \cdot x \]

      6. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{{x}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right) + \frac{1}{6}}, {x}^{2}, 1\right) \cdot x \]

      7. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right) \cdot {x}^{2}} + \frac{1}{6}, {x}^{2}, 1\right) \cdot x \]

      8. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}, {x}^{2}, \frac{1}{6}\right)}, {x}^{2}, 1\right) \cdot x \]

      9. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{1}{5040} \cdot {x}^{2} + \frac{1}{120}}, {x}^{2}, \frac{1}{6}\right), {x}^{2}, 1\right) \cdot x \]

      10. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{5040}, {x}^{2}, \frac{1}{120}\right)}, {x}^{2}, \frac{1}{6}\right), {x}^{2}, 1\right) \cdot x \]

      11. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{5040}, \color{blue}{x \cdot x}, \frac{1}{120}\right), {x}^{2}, \frac{1}{6}\right), {x}^{2}, 1\right) \cdot x \]

      12. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{5040}, \color{blue}{x \cdot x}, \frac{1}{120}\right), {x}^{2}, \frac{1}{6}\right), {x}^{2}, 1\right) \cdot x \]

      13. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{5040}, x \cdot x, \frac{1}{120}\right), \color{blue}{x \cdot x}, \frac{1}{6}\right), {x}^{2}, 1\right) \cdot x \]

      14. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{5040}, x \cdot x, \frac{1}{120}\right), \color{blue}{x \cdot x}, \frac{1}{6}\right), {x}^{2}, 1\right) \cdot x \]

      15. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{5040}, x \cdot x, \frac{1}{120}\right), x \cdot x, \frac{1}{6}\right), \color{blue}{x \cdot x}, 1\right) \cdot x \]

      16. lower-*.f64 97.0

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0001984126984126984, x \cdot x, 0.008333333333333333\right), x \cdot x, 0.16666666666666666\right), \color{blue}{x \cdot x}, 1\right) \cdot x \]

   5. Applied rewrites 97.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0001984126984126984, x \cdot x, 0.008333333333333333\right), x \cdot x, 0.16666666666666666\right), x \cdot x, 1\right) \cdot x} \]

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 67.1%

      \[\leadsto 1 \cdot x \]

   if 2.0000000000000001e-4 < (-.f64 (exp.f64 x) (exp.f64 (neg.f64 x)))

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 64.0

         \[\leadsto \mathsf{fma}\left(\color{blue}{x \cdot x}, 0.16666666666666666, 1\right) \cdot x \]

   5. Applied rewrites 64.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot x, 0.16666666666666666, 1\right) \cdot x} \]

   6. Taylor expanded in x around inf

      \[\leadsto \frac{1}{6} \cdot \color{blue}{{x}^{3}} \]
   7. Step-by-step derivation

   8. Applied rewrites 64.0%

      \[\leadsto \left(\left(x \cdot x\right) \cdot x\right) \cdot \color{blue}{0.16666666666666666} \]
   9. Step-by-step derivation

   10. Applied rewrites 64.0%

       \[\leadsto \left(x \cdot x\right) \cdot \left(0.16666666666666666 \cdot \color{blue}{x}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 66.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{x} - e^{-x} \leq 0.0002:\\ \;\;\;\;1 \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(0.16666666666666666 \cdot x\right) \cdot \left(x \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 93.0% accurate, 5.6× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0001984126984126984, x \cdot x, 0.008333333333333333\right), x \cdot x, 0.16666666666666666\right), x \cdot x, 1\right) \cdot x \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  (fma
   (fma
    (fma 0.0001984126984126984 (* x x) 0.008333333333333333)
    (* x x)
    0.16666666666666666)
   (* x x)
   1.0)
  x))

C

double code(double x) {
	return fma(fma(fma(0.0001984126984126984, (x * x), 0.008333333333333333), (x * x), 0.16666666666666666), (x * x), 1.0) * x;
}

Julia

function code(x)
	return Float64(fma(fma(fma(0.0001984126984126984, Float64(x * x), 0.008333333333333333), Float64(x * x), 0.16666666666666666), Float64(x * x), 1.0) * x)
end

Wolfram

code[x_] := N[(N[(N[(N[(0.0001984126984126984 * N[(x * x), $MachinePrecision] + 0.008333333333333333), $MachinePrecision] * N[(x * x), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * x), $MachinePrecision]

Derivation

1. Initial program 54.7%

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

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right)\right)\right)} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right)\right)\right) \cdot x} \]

   2. lower-*.f64 N/A

      \[\leadsto \color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right)\right)\right) \cdot x} \]

   3. +-commutative N/A

      \[\leadsto \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right)\right) + 1\right)} \cdot x \]

   4. *-commutative N/A

      \[\leadsto \left(\color{blue}{\left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right)\right) \cdot {x}^{2}} + 1\right) \cdot x \]

   5. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right), {x}^{2}, 1\right)} \cdot x \]

   6. +-commutative N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{{x}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right) + \frac{1}{6}}, {x}^{2}, 1\right) \cdot x \]

   7. *-commutative N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right) \cdot {x}^{2}} + \frac{1}{6}, {x}^{2}, 1\right) \cdot x \]

   8. lower-fma.f64 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}, {x}^{2}, \frac{1}{6}\right)}, {x}^{2}, 1\right) \cdot x \]

   9. +-commutative N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{1}{5040} \cdot {x}^{2} + \frac{1}{120}}, {x}^{2}, \frac{1}{6}\right), {x}^{2}, 1\right) \cdot x \]

   10. lower-fma.f64 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{5040}, {x}^{2}, \frac{1}{120}\right)}, {x}^{2}, \frac{1}{6}\right), {x}^{2}, 1\right) \cdot x \]

   11. unpow2 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{5040}, \color{blue}{x \cdot x}, \frac{1}{120}\right), {x}^{2}, \frac{1}{6}\right), {x}^{2}, 1\right) \cdot x \]

   12. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{5040}, \color{blue}{x \cdot x}, \frac{1}{120}\right), {x}^{2}, \frac{1}{6}\right), {x}^{2}, 1\right) \cdot x \]

   13. unpow2 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{5040}, x \cdot x, \frac{1}{120}\right), \color{blue}{x \cdot x}, \frac{1}{6}\right), {x}^{2}, 1\right) \cdot x \]

   14. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{5040}, x \cdot x, \frac{1}{120}\right), \color{blue}{x \cdot x}, \frac{1}{6}\right), {x}^{2}, 1\right) \cdot x \]

   15. unpow2 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{5040}, x \cdot x, \frac{1}{120}\right), x \cdot x, \frac{1}{6}\right), \color{blue}{x \cdot x}, 1\right) \cdot x \]

   16. lower-*.f64 94.4

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0001984126984126984, x \cdot x, 0.008333333333333333\right), x \cdot x, 0.16666666666666666\right), \color{blue}{x \cdot x}, 1\right) \cdot x \]

5. Applied rewrites 94.4%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0001984126984126984, x \cdot x, 0.008333333333333333\right), x \cdot x, 0.16666666666666666\right), x \cdot x, 1\right) \cdot x} \]
6. Add Preprocessing

Alternative 7: 92.5% accurate, 5.9× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot 0.0001984126984126984, x \cdot x, 1\right) \cdot x \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (* (fma (* (* (* (* x x) x) x) 0.0001984126984126984) (* x x) 1.0) x))

C

double code(double x) {
	return fma(((((x * x) * x) * x) * 0.0001984126984126984), (x * x), 1.0) * x;
}

Julia

function code(x)
	return Float64(fma(Float64(Float64(Float64(Float64(x * x) * x) * x) * 0.0001984126984126984), Float64(x * x), 1.0) * x)
end

Wolfram

code[x_] := N[(N[(N[(N[(N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] * 0.0001984126984126984), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * x), $MachinePrecision]

Derivation

1. Initial program 54.7%

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

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right)\right)\right)} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right)\right)\right) \cdot x} \]

   2. lower-*.f64 N/A

      \[\leadsto \color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right)\right)\right) \cdot x} \]

   3. +-commutative N/A

      \[\leadsto \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right)\right) + 1\right)} \cdot x \]

   4. *-commutative N/A

      \[\leadsto \left(\color{blue}{\left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right)\right) \cdot {x}^{2}} + 1\right) \cdot x \]

   5. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right), {x}^{2}, 1\right)} \cdot x \]

   6. +-commutative N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{{x}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right) + \frac{1}{6}}, {x}^{2}, 1\right) \cdot x \]

   7. *-commutative N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right) \cdot {x}^{2}} + \frac{1}{6}, {x}^{2}, 1\right) \cdot x \]

   8. lower-fma.f64 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}, {x}^{2}, \frac{1}{6}\right)}, {x}^{2}, 1\right) \cdot x \]

   9. +-commutative N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{1}{5040} \cdot {x}^{2} + \frac{1}{120}}, {x}^{2}, \frac{1}{6}\right), {x}^{2}, 1\right) \cdot x \]

   10. lower-fma.f64 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{5040}, {x}^{2}, \frac{1}{120}\right)}, {x}^{2}, \frac{1}{6}\right), {x}^{2}, 1\right) \cdot x \]

   11. unpow2 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{5040}, \color{blue}{x \cdot x}, \frac{1}{120}\right), {x}^{2}, \frac{1}{6}\right), {x}^{2}, 1\right) \cdot x \]

   12. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{5040}, \color{blue}{x \cdot x}, \frac{1}{120}\right), {x}^{2}, \frac{1}{6}\right), {x}^{2}, 1\right) \cdot x \]

   13. unpow2 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{5040}, x \cdot x, \frac{1}{120}\right), \color{blue}{x \cdot x}, \frac{1}{6}\right), {x}^{2}, 1\right) \cdot x \]

   14. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{5040}, x \cdot x, \frac{1}{120}\right), \color{blue}{x \cdot x}, \frac{1}{6}\right), {x}^{2}, 1\right) \cdot x \]

   15. unpow2 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{5040}, x \cdot x, \frac{1}{120}\right), x \cdot x, \frac{1}{6}\right), \color{blue}{x \cdot x}, 1\right) \cdot x \]

   16. lower-*.f64 94.4

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0001984126984126984, x \cdot x, 0.008333333333333333\right), x \cdot x, 0.16666666666666666\right), \color{blue}{x \cdot x}, 1\right) \cdot x \]

5. Applied rewrites 94.4%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0001984126984126984, x \cdot x, 0.008333333333333333\right), x \cdot x, 0.16666666666666666\right), x \cdot x, 1\right) \cdot x} \]

6. Taylor expanded in x around inf

   \[\leadsto \mathsf{fma}\left(\frac{1}{5040} \cdot {x}^{4}, x \cdot x, 1\right) \cdot x \]
7. Step-by-step derivation

8. Applied rewrites 94.3%

   \[\leadsto \mathsf{fma}\left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot 0.0001984126984126984, x \cdot x, 1\right) \cdot x \]
9. Add Preprocessing

Alternative 8: 90.2% accurate, 7.8× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, x \cdot x, 0.16666666666666666\right), x \cdot x, 1\right) \cdot x \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  (fma (fma 0.008333333333333333 (* x x) 0.16666666666666666) (* x x) 1.0)
  x))

C

double code(double x) {
	return fma(fma(0.008333333333333333, (x * x), 0.16666666666666666), (x * x), 1.0) * x;
}

Julia

function code(x)
	return Float64(fma(fma(0.008333333333333333, Float64(x * x), 0.16666666666666666), Float64(x * x), 1.0) * x)
end

Wolfram

code[x_] := N[(N[(N[(0.008333333333333333 * N[(x * x), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * x), $MachinePrecision]

Derivation

1. Initial program 54.7%

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

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right)\right)} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right)\right) \cdot x} \]

   2. lower-*.f64 N/A

      \[\leadsto \color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right)\right) \cdot x} \]

   3. +-commutative N/A

      \[\leadsto \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right) + 1\right)} \cdot x \]

   4. *-commutative N/A

      \[\leadsto \left(\color{blue}{\left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right) \cdot {x}^{2}} + 1\right) \cdot x \]

   5. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}, {x}^{2}, 1\right)} \cdot x \]

   6. +-commutative N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{120} \cdot {x}^{2} + \frac{1}{6}}, {x}^{2}, 1\right) \cdot x \]

   7. lower-fma.f64 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{120}, {x}^{2}, \frac{1}{6}\right)}, {x}^{2}, 1\right) \cdot x \]

   8. unpow2 N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{120}, \color{blue}{x \cdot x}, \frac{1}{6}\right), {x}^{2}, 1\right) \cdot x \]

   9. lower-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{120}, \color{blue}{x \cdot x}, \frac{1}{6}\right), {x}^{2}, 1\right) \cdot x \]

   10. unpow2 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{120}, x \cdot x, \frac{1}{6}\right), \color{blue}{x \cdot x}, 1\right) \cdot x \]

   11. lower-*.f64 90.5

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, x \cdot x, 0.16666666666666666\right), \color{blue}{x \cdot x}, 1\right) \cdot x \]

5. Applied rewrites 90.5%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, x \cdot x, 0.16666666666666666\right), x \cdot x, 1\right) \cdot x} \]
6. Add Preprocessing

Alternative 9: 89.8% accurate, 8.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(0.008333333333333333 \cdot \left(x \cdot x\right), x \cdot x, 1\right) \cdot x \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (* (fma (* 0.008333333333333333 (* x x)) (* x x) 1.0) x))

C

double code(double x) {
	return fma((0.008333333333333333 * (x * x)), (x * x), 1.0) * x;
}

Julia

function code(x)
	return Float64(fma(Float64(0.008333333333333333 * Float64(x * x)), Float64(x * x), 1.0) * x)
end

Wolfram

code[x_] := N[(N[(N[(0.008333333333333333 * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * x), $MachinePrecision]

Derivation

1. Initial program 54.7%

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

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right)\right)} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right)\right) \cdot x} \]

   2. lower-*.f64 N/A

      \[\leadsto \color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right)\right) \cdot x} \]

   3. +-commutative N/A

      \[\leadsto \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right) + 1\right)} \cdot x \]

   4. *-commutative N/A

      \[\leadsto \left(\color{blue}{\left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right) \cdot {x}^{2}} + 1\right) \cdot x \]

   5. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}, {x}^{2}, 1\right)} \cdot x \]

   6. +-commutative N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{120} \cdot {x}^{2} + \frac{1}{6}}, {x}^{2}, 1\right) \cdot x \]

   7. lower-fma.f64 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{120}, {x}^{2}, \frac{1}{6}\right)}, {x}^{2}, 1\right) \cdot x \]

   8. unpow2 N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{120}, \color{blue}{x \cdot x}, \frac{1}{6}\right), {x}^{2}, 1\right) \cdot x \]

   9. lower-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{120}, \color{blue}{x \cdot x}, \frac{1}{6}\right), {x}^{2}, 1\right) \cdot x \]

   10. unpow2 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{120}, x \cdot x, \frac{1}{6}\right), \color{blue}{x \cdot x}, 1\right) \cdot x \]

   11. lower-*.f64 90.5

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, x \cdot x, 0.16666666666666666\right), \color{blue}{x \cdot x}, 1\right) \cdot x \]

5. Applied rewrites 90.5%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, x \cdot x, 0.16666666666666666\right), x \cdot x, 1\right) \cdot x} \]

6. Taylor expanded in x around inf

   \[\leadsto \mathsf{fma}\left(\frac{1}{120} \cdot {x}^{2}, x \cdot x, 1\right) \cdot x \]
7. Step-by-step derivation

8. Applied rewrites 90.3%

   \[\leadsto \mathsf{fma}\left(0.008333333333333333 \cdot \left(x \cdot x\right), x \cdot x, 1\right) \cdot x \]
9. Add Preprocessing

Alternative 10: 83.7% accurate, 12.8× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(0.16666666666666666 \cdot x, x, 1\right) \cdot x \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* (fma (* 0.16666666666666666 x) x 1.0) x))

C

double code(double x) {
	return fma((0.16666666666666666 * x), x, 1.0) * x;
}

Julia

function code(x)
	return Float64(fma(Float64(0.16666666666666666 * x), x, 1.0) * x)
end

Wolfram

code[x_] := N[(N[(N[(0.16666666666666666 * x), $MachinePrecision] * x + 1.0), $MachinePrecision] * x), $MachinePrecision]

Derivation

1. Initial program 54.7%

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

3. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. unpow2 N/A

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

   7. lower-*.f64 82.7

      \[\leadsto \mathsf{fma}\left(\color{blue}{x \cdot x}, 0.16666666666666666, 1\right) \cdot x \]

5. Applied rewrites 82.7%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot x, 0.16666666666666666, 1\right) \cdot x} \]
6. Step-by-step derivation

7. Applied rewrites 82.7%

   \[\leadsto \mathsf{fma}\left(0.16666666666666666 \cdot x, x, 1\right) \cdot x \]
8. Add Preprocessing

Alternative 11: 51.8% accurate, 36.2× 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 54.7%

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

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right)\right)\right)} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right)\right)\right) \cdot x} \]

   2. lower-*.f64 N/A

      \[\leadsto \color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right)\right)\right) \cdot x} \]

   3. +-commutative N/A

      \[\leadsto \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right)\right) + 1\right)} \cdot x \]

   4. *-commutative N/A

      \[\leadsto \left(\color{blue}{\left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right)\right) \cdot {x}^{2}} + 1\right) \cdot x \]

   5. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right), {x}^{2}, 1\right)} \cdot x \]

   6. +-commutative N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{{x}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right) + \frac{1}{6}}, {x}^{2}, 1\right) \cdot x \]

   7. *-commutative N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right) \cdot {x}^{2}} + \frac{1}{6}, {x}^{2}, 1\right) \cdot x \]

   8. lower-fma.f64 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}, {x}^{2}, \frac{1}{6}\right)}, {x}^{2}, 1\right) \cdot x \]

   9. +-commutative N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{1}{5040} \cdot {x}^{2} + \frac{1}{120}}, {x}^{2}, \frac{1}{6}\right), {x}^{2}, 1\right) \cdot x \]

   10. lower-fma.f64 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{5040}, {x}^{2}, \frac{1}{120}\right)}, {x}^{2}, \frac{1}{6}\right), {x}^{2}, 1\right) \cdot x \]

   11. unpow2 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{5040}, \color{blue}{x \cdot x}, \frac{1}{120}\right), {x}^{2}, \frac{1}{6}\right), {x}^{2}, 1\right) \cdot x \]

   12. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{5040}, \color{blue}{x \cdot x}, \frac{1}{120}\right), {x}^{2}, \frac{1}{6}\right), {x}^{2}, 1\right) \cdot x \]

   13. unpow2 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{5040}, x \cdot x, \frac{1}{120}\right), \color{blue}{x \cdot x}, \frac{1}{6}\right), {x}^{2}, 1\right) \cdot x \]

   14. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{5040}, x \cdot x, \frac{1}{120}\right), \color{blue}{x \cdot x}, \frac{1}{6}\right), {x}^{2}, 1\right) \cdot x \]

   15. unpow2 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{5040}, x \cdot x, \frac{1}{120}\right), x \cdot x, \frac{1}{6}\right), \color{blue}{x \cdot x}, 1\right) \cdot x \]

   16. lower-*.f64 94.4

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0001984126984126984, x \cdot x, 0.008333333333333333\right), x \cdot x, 0.16666666666666666\right), \color{blue}{x \cdot x}, 1\right) \cdot x \]

5. Applied rewrites 94.4%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0001984126984126984, x \cdot x, 0.008333333333333333\right), x \cdot x, 0.16666666666666666\right), x \cdot x, 1\right) \cdot x} \]

6. Taylor expanded in x around 0

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

8. Applied rewrites 51.8%

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

Reproduce

herbie shell --seed 2024235 
(FPCore (x)
  :name "Hyperbolic sine"
  :precision binary64
  (/ (- (exp x) (exp (- x))) 2.0))