Hyperbolic sine

Percentage Accurate: 53.3% → 100.0%

Time: 7.7s

Alternatives: 12

Speedup: 9.9×

• 48.6% 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: 53.3% 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 57.9%

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

   1. lift--.f64 N/A

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

   2. lift-exp.f64 N/A

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

   3. lift-exp.f64 N/A

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

   4. lift-neg.f64 N/A

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

   5. sinh-undef N/A

      \[\leadsto \frac{\color{blue}{2 \cdot \sinh x}}{2} \]

   6. *-commutative N/A

      \[\leadsto \frac{\color{blue}{\sinh x \cdot 2}}{2} \]

   7. lower-*.f64 N/A

      \[\leadsto \frac{\color{blue}{\sinh x \cdot 2}}{2} \]

   8. lower-sinh.f64 100.0

      \[\leadsto \frac{\color{blue}{\sinh x} \cdot 2}{2} \]

4. Applied rewrites 100.0%

   \[\leadsto \frac{\color{blue}{\sinh x \cdot 2}}{2} \]
5. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\sinh x \cdot 2}{2}} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{\color{blue}{\sinh x \cdot 2}}{2} \]

   3. associate-/l* N/A

      \[\leadsto \color{blue}{\sinh x \cdot \frac{2}{2}} \]

   4. metadata-eval N/A

      \[\leadsto \sinh x \cdot \color{blue}{1} \]

   5. *-rgt-identity 100.0

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

6. Applied rewrites 100.0%

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

Alternative 2: 87.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{x} - e^{-x} \leq 0.002:\\ \;\;\;\;\mathsf{fma}\left(0.3333333333333333 \cdot \left(x \cdot x\right), x, 2 \cdot x\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\mathsf{fma}\left(0.016666666666666666, x \cdot x, 0.3333333333333333\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (- (exp x) (exp (- x))) 0.002)
   (* (fma (* 0.3333333333333333 (* x x)) x (* 2.0 x)) 0.5)
   (*
    (* (* (* (fma 0.016666666666666666 (* x x) 0.3333333333333333) x) x) x)
    0.5)))

C

double code(double x) {
	double tmp;
	if ((exp(x) - exp(-x)) <= 0.002) {
		tmp = fma((0.3333333333333333 * (x * x)), x, (2.0 * x)) * 0.5;
	} else {
		tmp = (((fma(0.016666666666666666, (x * x), 0.3333333333333333) * x) * x) * x) * 0.5;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(exp(x) - exp(Float64(-x))) <= 0.002)
		tmp = Float64(fma(Float64(0.3333333333333333 * Float64(x * x)), x, Float64(2.0 * x)) * 0.5);
	else
		tmp = Float64(Float64(Float64(Float64(fma(0.016666666666666666, Float64(x * x), 0.3333333333333333) * x) * x) * x) * 0.5);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[(N[Exp[x], $MachinePrecision] - N[Exp[(-x)], $MachinePrecision]), $MachinePrecision], 0.002], N[(N[(N[(0.3333333333333333 * N[(x * x), $MachinePrecision]), $MachinePrecision] * x + N[(2.0 * x), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(N[(N[(N[(0.016666666666666666 * N[(x * x), $MachinePrecision] + 0.3333333333333333), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (exp.f64 x) (exp.f64 (neg.f64 x))) < 2e-3

   1. Initial program 44.4%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 85.3

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

   5. Applied rewrites 85.3%

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

   7. Applied rewrites 85.4%

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

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. metadata-eval N/A

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

      4. lower-*.f64 85.4

         \[\leadsto \color{blue}{\mathsf{fma}\left(0.3333333333333333 \cdot \left(x \cdot x\right), x, 2 \cdot x\right) \cdot 0.5} \]

   9. Applied rewrites 85.3%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(x \cdot x, 0.3333333333333333, 2\right) \cdot x\right) \cdot 0.5} \]
   10. Step-by-step derivation

   11. Applied rewrites 85.4%

       \[\leadsto \mathsf{fma}\left(0.3333333333333333 \cdot \left(x \cdot x\right), \color{blue}{x}, 2 \cdot x\right) \cdot 0.5 \]

   if 2e-3 < (-.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

      \[\leadsto \frac{\color{blue}{x \cdot \left(2 + {x}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {x}^{2}\right)\right)}}{2} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(2 + {x}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {x}^{2}\right)\right) \cdot x}}{2} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(2 + {x}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {x}^{2}\right)\right) \cdot x}}{2} \]

      3. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\left({x}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {x}^{2}\right) + 2\right)} \cdot x}{2} \]

      4. *-commutative N/A

         \[\leadsto \frac{\left(\color{blue}{\left(\frac{1}{3} + \frac{1}{60} \cdot {x}^{2}\right) \cdot {x}^{2}} + 2\right) \cdot x}{2} \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{1}{3} + \frac{1}{60} \cdot {x}^{2}, {x}^{2}, 2\right)} \cdot x}{2} \]

      6. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{1}{60} \cdot {x}^{2} + \frac{1}{3}}, {x}^{2}, 2\right) \cdot x}{2} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{60}, {x}^{2}, \frac{1}{3}\right)}, {x}^{2}, 2\right) \cdot x}{2} \]

      8. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{60}, \color{blue}{x \cdot x}, \frac{1}{3}\right), {x}^{2}, 2\right) \cdot x}{2} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{60}, \color{blue}{x \cdot x}, \frac{1}{3}\right), {x}^{2}, 2\right) \cdot x}{2} \]

      10. unpow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{60}, x \cdot x, \frac{1}{3}\right), \color{blue}{x \cdot x}, 2\right) \cdot x}{2} \]

      11. lower-*.f64 84.7

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.016666666666666666, x \cdot x, 0.3333333333333333\right), \color{blue}{x \cdot x}, 2\right) \cdot x}{2} \]

   5. Applied rewrites 84.7%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.016666666666666666, x \cdot x, 0.3333333333333333\right), x \cdot x, 2\right) \cdot x}}{2} \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{60}, x \cdot x, \frac{1}{3}\right), x \cdot x, 2\right) \cdot x}{2}} \]

      2. div-inv N/A

         \[\leadsto \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{60}, x \cdot x, \frac{1}{3}\right), x \cdot x, 2\right) \cdot x\right) \cdot \frac{1}{2}} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{60}, x \cdot x, \frac{1}{3}\right), x \cdot x, 2\right) \cdot x\right) \cdot \frac{1}{2}} \]

      4. metadata-eval 84.7

         \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(0.016666666666666666, x \cdot x, 0.3333333333333333\right), x \cdot x, 2\right) \cdot x\right) \cdot \color{blue}{0.5} \]

   7. Applied rewrites 84.7%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(0.016666666666666666, x \cdot x, 0.3333333333333333\right), x \cdot x, 2\right) \cdot x\right) \cdot 0.5} \]

   8. Taylor expanded in x around inf

      \[\leadsto \left(\left({x}^{4} \cdot \left(\frac{1}{60} + \frac{1}{3} \cdot \frac{1}{{x}^{2}}\right)\right) \cdot x\right) \cdot \frac{1}{2} \]
   9. Step-by-step derivation

   10. Applied rewrites 84.7%

       \[\leadsto \left(\left(\left(\mathsf{fma}\left(0.016666666666666666, x \cdot x, 0.3333333333333333\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot 0.5 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 93.5% accurate, 3.9× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0003968253968253968, x \cdot x, 0.016666666666666666\right), x \cdot x, 0.3333333333333333\right) \cdot x\right) \cdot x, x, 2 \cdot x\right)}{2} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/
  (fma
   (*
    (*
     (fma
      (fma 0.0003968253968253968 (* x x) 0.016666666666666666)
      (* x x)
      0.3333333333333333)
     x)
    x)
   x
   (* 2.0 x))
  2.0))

C

double code(double x) {
	return fma(((fma(fma(0.0003968253968253968, (x * x), 0.016666666666666666), (x * x), 0.3333333333333333) * x) * x), x, (2.0 * x)) / 2.0;
}

Julia

function code(x)
	return Float64(fma(Float64(Float64(fma(fma(0.0003968253968253968, Float64(x * x), 0.016666666666666666), Float64(x * x), 0.3333333333333333) * x) * x), x, Float64(2.0 * x)) / 2.0)
end

Wolfram

code[x_] := N[(N[(N[(N[(N[(N[(0.0003968253968253968 * N[(x * x), $MachinePrecision] + 0.016666666666666666), $MachinePrecision] * N[(x * x), $MachinePrecision] + 0.3333333333333333), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] * x + N[(2.0 * x), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]

Derivation

1. Initial program 57.9%

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

3. Taylor expanded in x around 0

   \[\leadsto \frac{\color{blue}{x \cdot \left(2 + {x}^{2} \cdot \left(\frac{1}{3} + {x}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {x}^{2}\right)\right)\right)}}{2} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \frac{\color{blue}{\left(2 + {x}^{2} \cdot \left(\frac{1}{3} + {x}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {x}^{2}\right)\right)\right) \cdot x}}{2} \]

   2. lower-*.f64 N/A

      \[\leadsto \frac{\color{blue}{\left(2 + {x}^{2} \cdot \left(\frac{1}{3} + {x}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {x}^{2}\right)\right)\right) \cdot x}}{2} \]

   3. +-commutative N/A

      \[\leadsto \frac{\color{blue}{\left({x}^{2} \cdot \left(\frac{1}{3} + {x}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {x}^{2}\right)\right) + 2\right)} \cdot x}{2} \]

   4. *-commutative N/A

      \[\leadsto \frac{\left(\color{blue}{\left(\frac{1}{3} + {x}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {x}^{2}\right)\right) \cdot {x}^{2}} + 2\right) \cdot x}{2} \]

   5. lower-fma.f64 N/A

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{1}{3} + {x}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {x}^{2}\right), {x}^{2}, 2\right)} \cdot x}{2} \]

   6. +-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{{x}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {x}^{2}\right) + \frac{1}{3}}, {x}^{2}, 2\right) \cdot x}{2} \]

   7. *-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\left(\frac{1}{60} + \frac{1}{2520} \cdot {x}^{2}\right) \cdot {x}^{2}} + \frac{1}{3}, {x}^{2}, 2\right) \cdot x}{2} \]

   8. lower-fma.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{60} + \frac{1}{2520} \cdot {x}^{2}, {x}^{2}, \frac{1}{3}\right)}, {x}^{2}, 2\right) \cdot x}{2} \]

   9. +-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{1}{2520} \cdot {x}^{2} + \frac{1}{60}}, {x}^{2}, \frac{1}{3}\right), {x}^{2}, 2\right) \cdot x}{2} \]

   10. lower-fma.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{2520}, {x}^{2}, \frac{1}{60}\right)}, {x}^{2}, \frac{1}{3}\right), {x}^{2}, 2\right) \cdot x}{2} \]

   11. unpow2 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2520}, \color{blue}{x \cdot x}, \frac{1}{60}\right), {x}^{2}, \frac{1}{3}\right), {x}^{2}, 2\right) \cdot x}{2} \]

   12. lower-*.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2520}, \color{blue}{x \cdot x}, \frac{1}{60}\right), {x}^{2}, \frac{1}{3}\right), {x}^{2}, 2\right) \cdot x}{2} \]

   13. unpow2 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2520}, x \cdot x, \frac{1}{60}\right), \color{blue}{x \cdot x}, \frac{1}{3}\right), {x}^{2}, 2\right) \cdot x}{2} \]

   14. lower-*.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2520}, x \cdot x, \frac{1}{60}\right), \color{blue}{x \cdot x}, \frac{1}{3}\right), {x}^{2}, 2\right) \cdot x}{2} \]

   15. unpow2 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2520}, x \cdot x, \frac{1}{60}\right), x \cdot x, \frac{1}{3}\right), \color{blue}{x \cdot x}, 2\right) \cdot x}{2} \]

   16. lower-*.f64 91.7

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0003968253968253968, x \cdot x, 0.016666666666666666\right), x \cdot x, 0.3333333333333333\right), \color{blue}{x \cdot x}, 2\right) \cdot x}{2} \]

5. Applied rewrites 91.7%

   \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0003968253968253968, x \cdot x, 0.016666666666666666\right), x \cdot x, 0.3333333333333333\right), x \cdot x, 2\right) \cdot x}}{2} \]
6. Step-by-step derivation

7. Applied rewrites 91.7%

   \[\leadsto \frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0003968253968253968, x \cdot x, 0.016666666666666666\right), x \cdot x, 0.3333333333333333\right) \cdot x\right) \cdot x, \color{blue}{x}, 2 \cdot x\right)}{2} \]
8. Add Preprocessing

Alternative 4: 93.5% accurate, 4.9× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.0003968253968253968, 0.016666666666666666\right), x \cdot x, 0.3333333333333333\right) \cdot x, x, 2\right) \cdot x\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  0.5
  (*
   (fma
    (*
     (fma
      (fma (* x x) 0.0003968253968253968 0.016666666666666666)
      (* x x)
      0.3333333333333333)
     x)
    x
    2.0)
   x)))

C

double code(double x) {
	return 0.5 * (fma((fma(fma((x * x), 0.0003968253968253968, 0.016666666666666666), (x * x), 0.3333333333333333) * x), x, 2.0) * x);
}

Julia

function code(x)
	return Float64(0.5 * Float64(fma(Float64(fma(fma(Float64(x * x), 0.0003968253968253968, 0.016666666666666666), Float64(x * x), 0.3333333333333333) * x), x, 2.0) * x))
end

Wolfram

code[x_] := N[(0.5 * N[(N[(N[(N[(N[(N[(x * x), $MachinePrecision] * 0.0003968253968253968 + 0.016666666666666666), $MachinePrecision] * N[(x * x), $MachinePrecision] + 0.3333333333333333), $MachinePrecision] * x), $MachinePrecision] * x + 2.0), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 57.9%

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

3. Taylor expanded in x around 0

   \[\leadsto \frac{\color{blue}{x \cdot \left(2 + {x}^{2} \cdot \left(\frac{1}{3} + {x}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {x}^{2}\right)\right)\right)}}{2} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \frac{\color{blue}{\left(2 + {x}^{2} \cdot \left(\frac{1}{3} + {x}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {x}^{2}\right)\right)\right) \cdot x}}{2} \]

   2. lower-*.f64 N/A

      \[\leadsto \frac{\color{blue}{\left(2 + {x}^{2} \cdot \left(\frac{1}{3} + {x}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {x}^{2}\right)\right)\right) \cdot x}}{2} \]

   3. +-commutative N/A

      \[\leadsto \frac{\color{blue}{\left({x}^{2} \cdot \left(\frac{1}{3} + {x}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {x}^{2}\right)\right) + 2\right)} \cdot x}{2} \]

   4. *-commutative N/A

      \[\leadsto \frac{\left(\color{blue}{\left(\frac{1}{3} + {x}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {x}^{2}\right)\right) \cdot {x}^{2}} + 2\right) \cdot x}{2} \]

   5. lower-fma.f64 N/A

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{1}{3} + {x}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {x}^{2}\right), {x}^{2}, 2\right)} \cdot x}{2} \]

   6. +-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{{x}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {x}^{2}\right) + \frac{1}{3}}, {x}^{2}, 2\right) \cdot x}{2} \]

   7. *-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\left(\frac{1}{60} + \frac{1}{2520} \cdot {x}^{2}\right) \cdot {x}^{2}} + \frac{1}{3}, {x}^{2}, 2\right) \cdot x}{2} \]

   8. lower-fma.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{60} + \frac{1}{2520} \cdot {x}^{2}, {x}^{2}, \frac{1}{3}\right)}, {x}^{2}, 2\right) \cdot x}{2} \]

   9. +-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{1}{2520} \cdot {x}^{2} + \frac{1}{60}}, {x}^{2}, \frac{1}{3}\right), {x}^{2}, 2\right) \cdot x}{2} \]

   10. lower-fma.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{2520}, {x}^{2}, \frac{1}{60}\right)}, {x}^{2}, \frac{1}{3}\right), {x}^{2}, 2\right) \cdot x}{2} \]

   11. unpow2 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2520}, \color{blue}{x \cdot x}, \frac{1}{60}\right), {x}^{2}, \frac{1}{3}\right), {x}^{2}, 2\right) \cdot x}{2} \]

   12. lower-*.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2520}, \color{blue}{x \cdot x}, \frac{1}{60}\right), {x}^{2}, \frac{1}{3}\right), {x}^{2}, 2\right) \cdot x}{2} \]

   13. unpow2 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2520}, x \cdot x, \frac{1}{60}\right), \color{blue}{x \cdot x}, \frac{1}{3}\right), {x}^{2}, 2\right) \cdot x}{2} \]

   14. lower-*.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2520}, x \cdot x, \frac{1}{60}\right), \color{blue}{x \cdot x}, \frac{1}{3}\right), {x}^{2}, 2\right) \cdot x}{2} \]

   15. unpow2 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2520}, x \cdot x, \frac{1}{60}\right), x \cdot x, \frac{1}{3}\right), \color{blue}{x \cdot x}, 2\right) \cdot x}{2} \]

   16. lower-*.f64 91.7

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0003968253968253968, x \cdot x, 0.016666666666666666\right), x \cdot x, 0.3333333333333333\right), \color{blue}{x \cdot x}, 2\right) \cdot x}{2} \]

5. Applied rewrites 91.7%

   \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0003968253968253968, x \cdot x, 0.016666666666666666\right), x \cdot x, 0.3333333333333333\right), x \cdot x, 2\right) \cdot x}}{2} \]
6. Step-by-step derivation

7. Applied rewrites 91.7%

   \[\leadsto \frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0003968253968253968, x \cdot x, 0.016666666666666666\right), x \cdot x, 0.3333333333333333\right) \cdot x\right) \cdot x, \color{blue}{x}, 2 \cdot x\right)}{2} \]
8. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2520}, x \cdot x, \frac{1}{60}\right), x \cdot x, \frac{1}{3}\right) \cdot x\right) \cdot x, x, 2 \cdot x\right)}{2}} \]

   2. div-inv N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2520}, x \cdot x, \frac{1}{60}\right), x \cdot x, \frac{1}{3}\right) \cdot x\right) \cdot x, x, 2 \cdot x\right) \cdot \frac{1}{2}} \]

9. Applied rewrites 91.7%

   \[\leadsto \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.0003968253968253968, 0.016666666666666666\right), x \cdot x, 0.3333333333333333\right) \cdot x, x, 2\right) \cdot x\right) \cdot 0.5} \]

10. Final simplification 91.7%

    \[\leadsto 0.5 \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.0003968253968253968, 0.016666666666666666\right), x \cdot x, 0.3333333333333333\right) \cdot x, x, 2\right) \cdot x\right) \]
11. Add Preprocessing

Alternative 5: 93.0% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \left(\mathsf{fma}\left(\left(\mathsf{fma}\left(x \cdot x, 0.0003968253968253968, 0.016666666666666666\right) \cdot x\right) \cdot x, x \cdot x, 2\right) \cdot x\right) \cdot 0.5 \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  (*
   (fma
    (* (* (fma (* x x) 0.0003968253968253968 0.016666666666666666) x) x)
    (* x x)
    2.0)
   x)
  0.5))

C

double code(double x) {
	return (fma(((fma((x * x), 0.0003968253968253968, 0.016666666666666666) * x) * x), (x * x), 2.0) * x) * 0.5;
}

Julia

function code(x)
	return Float64(Float64(fma(Float64(Float64(fma(Float64(x * x), 0.0003968253968253968, 0.016666666666666666) * x) * x), Float64(x * x), 2.0) * x) * 0.5)
end

Wolfram

code[x_] := N[(N[(N[(N[(N[(N[(N[(x * x), $MachinePrecision] * 0.0003968253968253968 + 0.016666666666666666), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] * N[(x * x), $MachinePrecision] + 2.0), $MachinePrecision] * x), $MachinePrecision] * 0.5), $MachinePrecision]

Derivation

1. Initial program 57.9%

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

3. Taylor expanded in x around 0

   \[\leadsto \frac{\color{blue}{x \cdot \left(2 + {x}^{2} \cdot \left(\frac{1}{3} + {x}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {x}^{2}\right)\right)\right)}}{2} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \frac{\color{blue}{\left(2 + {x}^{2} \cdot \left(\frac{1}{3} + {x}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {x}^{2}\right)\right)\right) \cdot x}}{2} \]

   2. lower-*.f64 N/A

      \[\leadsto \frac{\color{blue}{\left(2 + {x}^{2} \cdot \left(\frac{1}{3} + {x}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {x}^{2}\right)\right)\right) \cdot x}}{2} \]

   3. +-commutative N/A

      \[\leadsto \frac{\color{blue}{\left({x}^{2} \cdot \left(\frac{1}{3} + {x}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {x}^{2}\right)\right) + 2\right)} \cdot x}{2} \]

   4. *-commutative N/A

      \[\leadsto \frac{\left(\color{blue}{\left(\frac{1}{3} + {x}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {x}^{2}\right)\right) \cdot {x}^{2}} + 2\right) \cdot x}{2} \]

   5. lower-fma.f64 N/A

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{1}{3} + {x}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {x}^{2}\right), {x}^{2}, 2\right)} \cdot x}{2} \]

   6. +-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{{x}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {x}^{2}\right) + \frac{1}{3}}, {x}^{2}, 2\right) \cdot x}{2} \]

   7. *-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\left(\frac{1}{60} + \frac{1}{2520} \cdot {x}^{2}\right) \cdot {x}^{2}} + \frac{1}{3}, {x}^{2}, 2\right) \cdot x}{2} \]

   8. lower-fma.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{60} + \frac{1}{2520} \cdot {x}^{2}, {x}^{2}, \frac{1}{3}\right)}, {x}^{2}, 2\right) \cdot x}{2} \]

   9. +-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{1}{2520} \cdot {x}^{2} + \frac{1}{60}}, {x}^{2}, \frac{1}{3}\right), {x}^{2}, 2\right) \cdot x}{2} \]

   10. lower-fma.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{2520}, {x}^{2}, \frac{1}{60}\right)}, {x}^{2}, \frac{1}{3}\right), {x}^{2}, 2\right) \cdot x}{2} \]

   11. unpow2 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2520}, \color{blue}{x \cdot x}, \frac{1}{60}\right), {x}^{2}, \frac{1}{3}\right), {x}^{2}, 2\right) \cdot x}{2} \]

   12. lower-*.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2520}, \color{blue}{x \cdot x}, \frac{1}{60}\right), {x}^{2}, \frac{1}{3}\right), {x}^{2}, 2\right) \cdot x}{2} \]

   13. unpow2 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2520}, x \cdot x, \frac{1}{60}\right), \color{blue}{x \cdot x}, \frac{1}{3}\right), {x}^{2}, 2\right) \cdot x}{2} \]

   14. lower-*.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2520}, x \cdot x, \frac{1}{60}\right), \color{blue}{x \cdot x}, \frac{1}{3}\right), {x}^{2}, 2\right) \cdot x}{2} \]

   15. unpow2 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2520}, x \cdot x, \frac{1}{60}\right), x \cdot x, \frac{1}{3}\right), \color{blue}{x \cdot x}, 2\right) \cdot x}{2} \]

   16. lower-*.f64 91.7

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0003968253968253968, x \cdot x, 0.016666666666666666\right), x \cdot x, 0.3333333333333333\right), \color{blue}{x \cdot x}, 2\right) \cdot x}{2} \]

5. Applied rewrites 91.7%

   \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0003968253968253968, x \cdot x, 0.016666666666666666\right), x \cdot x, 0.3333333333333333\right), x \cdot x, 2\right) \cdot x}}{2} \]

6. Taylor expanded in x around inf

   \[\leadsto \frac{\mathsf{fma}\left({x}^{4} \cdot \left(\frac{1}{2520} + \frac{1}{60} \cdot \frac{1}{{x}^{2}}\right), x \cdot x, 2\right) \cdot x}{2} \]
7. Step-by-step derivation

8. Applied rewrites 91.2%

   \[\leadsto \frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(0.0003968253968253968, x \cdot x, 0.016666666666666666\right) \cdot x\right) \cdot x, x \cdot x, 2\right) \cdot x}{2} \]
9. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(\frac{1}{2520}, x \cdot x, \frac{1}{60}\right) \cdot x\right) \cdot x, x \cdot x, 2\right) \cdot x}{2}} \]

   2. div-inv N/A

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(\left(\mathsf{fma}\left(\frac{1}{2520}, x \cdot x, \frac{1}{60}\right) \cdot x\right) \cdot x, x \cdot x, 2\right) \cdot x\right) \cdot \frac{1}{2}} \]

   3. metadata-eval N/A

      \[\leadsto \left(\mathsf{fma}\left(\left(\mathsf{fma}\left(\frac{1}{2520}, x \cdot x, \frac{1}{60}\right) \cdot x\right) \cdot x, x \cdot x, 2\right) \cdot x\right) \cdot \color{blue}{\frac{1}{2}} \]

   4. lower-*.f64 91.2

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(\left(\mathsf{fma}\left(0.0003968253968253968, x \cdot x, 0.016666666666666666\right) \cdot x\right) \cdot x, x \cdot x, 2\right) \cdot x\right) \cdot 0.5} \]

10. Applied rewrites 91.2%

    \[\leadsto \color{blue}{\left(\mathsf{fma}\left(\left(\mathsf{fma}\left(x \cdot x, 0.0003968253968253968, 0.016666666666666666\right) \cdot x\right) \cdot x, x \cdot x, 2\right) \cdot x\right) \cdot 0.5} \]
11. Add Preprocessing

Alternative 6: 90.5% accurate, 5.7× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\left(\mathsf{fma}\left(0.016666666666666666, x \cdot x, 0.3333333333333333\right) \cdot x\right) \cdot x, x, 2 \cdot x\right) \cdot 0.5 \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  (fma
   (* (* (fma 0.016666666666666666 (* x x) 0.3333333333333333) x) x)
   x
   (* 2.0 x))
  0.5))

C

double code(double x) {
	return fma(((fma(0.016666666666666666, (x * x), 0.3333333333333333) * x) * x), x, (2.0 * x)) * 0.5;
}

Julia

function code(x)
	return Float64(fma(Float64(Float64(fma(0.016666666666666666, Float64(x * x), 0.3333333333333333) * x) * x), x, Float64(2.0 * x)) * 0.5)
end

Wolfram

code[x_] := N[(N[(N[(N[(N[(0.016666666666666666 * N[(x * x), $MachinePrecision] + 0.3333333333333333), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] * x + N[(2.0 * x), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]

Derivation

1. Initial program 57.9%

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

3. Taylor expanded in x around 0

   \[\leadsto \frac{\color{blue}{x \cdot \left(2 + {x}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {x}^{2}\right)\right)}}{2} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \frac{\color{blue}{\left(2 + {x}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {x}^{2}\right)\right) \cdot x}}{2} \]

   2. lower-*.f64 N/A

      \[\leadsto \frac{\color{blue}{\left(2 + {x}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {x}^{2}\right)\right) \cdot x}}{2} \]

   3. +-commutative N/A

      \[\leadsto \frac{\color{blue}{\left({x}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {x}^{2}\right) + 2\right)} \cdot x}{2} \]

   4. *-commutative N/A

      \[\leadsto \frac{\left(\color{blue}{\left(\frac{1}{3} + \frac{1}{60} \cdot {x}^{2}\right) \cdot {x}^{2}} + 2\right) \cdot x}{2} \]

   5. lower-fma.f64 N/A

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{1}{3} + \frac{1}{60} \cdot {x}^{2}, {x}^{2}, 2\right)} \cdot x}{2} \]

   6. +-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{1}{60} \cdot {x}^{2} + \frac{1}{3}}, {x}^{2}, 2\right) \cdot x}{2} \]

   7. lower-fma.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{60}, {x}^{2}, \frac{1}{3}\right)}, {x}^{2}, 2\right) \cdot x}{2} \]

   8. unpow2 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{60}, \color{blue}{x \cdot x}, \frac{1}{3}\right), {x}^{2}, 2\right) \cdot x}{2} \]

   9. lower-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{60}, \color{blue}{x \cdot x}, \frac{1}{3}\right), {x}^{2}, 2\right) \cdot x}{2} \]

   10. unpow2 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{60}, x \cdot x, \frac{1}{3}\right), \color{blue}{x \cdot x}, 2\right) \cdot x}{2} \]

   11. lower-*.f64 88.6

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.016666666666666666, x \cdot x, 0.3333333333333333\right), \color{blue}{x \cdot x}, 2\right) \cdot x}{2} \]

5. Applied rewrites 88.6%

   \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.016666666666666666, x \cdot x, 0.3333333333333333\right), x \cdot x, 2\right) \cdot x}}{2} \]
6. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{60}, x \cdot x, \frac{1}{3}\right), x \cdot x, 2\right) \cdot x}{2}} \]

   2. div-inv N/A

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{60}, x \cdot x, \frac{1}{3}\right), x \cdot x, 2\right) \cdot x\right) \cdot \frac{1}{2}} \]

   3. lower-*.f64 N/A

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{60}, x \cdot x, \frac{1}{3}\right), x \cdot x, 2\right) \cdot x\right) \cdot \frac{1}{2}} \]

   4. metadata-eval 88.6

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(0.016666666666666666, x \cdot x, 0.3333333333333333\right), x \cdot x, 2\right) \cdot x\right) \cdot \color{blue}{0.5} \]

7. Applied rewrites 88.6%

   \[\leadsto \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(0.016666666666666666, x \cdot x, 0.3333333333333333\right), x \cdot x, 2\right) \cdot x\right) \cdot 0.5} \]
8. Step-by-step derivation

9. Applied rewrites 88.6%

   \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left(0.016666666666666666, x \cdot x, 0.3333333333333333\right) \cdot x\right) \cdot x, \color{blue}{x}, 2 \cdot x\right) \cdot 0.5 \]
10. Add Preprocessing

Alternative 7: 90.5% accurate, 6.6× speedup

Math

\[\begin{array}{l} \\ \left(\mathsf{fma}\left(\mathsf{fma}\left(0.016666666666666666, x \cdot x, 0.3333333333333333\right) \cdot x, x, 2\right) \cdot x\right) \cdot 0.5 \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  (* (fma (* (fma 0.016666666666666666 (* x x) 0.3333333333333333) x) x 2.0) x)
  0.5))

C

double code(double x) {
	return (fma((fma(0.016666666666666666, (x * x), 0.3333333333333333) * x), x, 2.0) * x) * 0.5;
}

Julia

function code(x)
	return Float64(Float64(fma(Float64(fma(0.016666666666666666, Float64(x * x), 0.3333333333333333) * x), x, 2.0) * x) * 0.5)
end

Wolfram

code[x_] := N[(N[(N[(N[(N[(0.016666666666666666 * N[(x * x), $MachinePrecision] + 0.3333333333333333), $MachinePrecision] * x), $MachinePrecision] * x + 2.0), $MachinePrecision] * x), $MachinePrecision] * 0.5), $MachinePrecision]

Derivation

1. Initial program 57.9%

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

3. Taylor expanded in x around 0

   \[\leadsto \frac{\color{blue}{x \cdot \left(2 + {x}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {x}^{2}\right)\right)}}{2} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \frac{\color{blue}{\left(2 + {x}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {x}^{2}\right)\right) \cdot x}}{2} \]

   2. lower-*.f64 N/A

      \[\leadsto \frac{\color{blue}{\left(2 + {x}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {x}^{2}\right)\right) \cdot x}}{2} \]

   3. +-commutative N/A

      \[\leadsto \frac{\color{blue}{\left({x}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {x}^{2}\right) + 2\right)} \cdot x}{2} \]

   4. *-commutative N/A

      \[\leadsto \frac{\left(\color{blue}{\left(\frac{1}{3} + \frac{1}{60} \cdot {x}^{2}\right) \cdot {x}^{2}} + 2\right) \cdot x}{2} \]

   5. lower-fma.f64 N/A

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{1}{3} + \frac{1}{60} \cdot {x}^{2}, {x}^{2}, 2\right)} \cdot x}{2} \]

   6. +-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{1}{60} \cdot {x}^{2} + \frac{1}{3}}, {x}^{2}, 2\right) \cdot x}{2} \]

   7. lower-fma.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{60}, {x}^{2}, \frac{1}{3}\right)}, {x}^{2}, 2\right) \cdot x}{2} \]

   8. unpow2 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{60}, \color{blue}{x \cdot x}, \frac{1}{3}\right), {x}^{2}, 2\right) \cdot x}{2} \]

   9. lower-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{60}, \color{blue}{x \cdot x}, \frac{1}{3}\right), {x}^{2}, 2\right) \cdot x}{2} \]

   10. unpow2 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{60}, x \cdot x, \frac{1}{3}\right), \color{blue}{x \cdot x}, 2\right) \cdot x}{2} \]

   11. lower-*.f64 88.6

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.016666666666666666, x \cdot x, 0.3333333333333333\right), \color{blue}{x \cdot x}, 2\right) \cdot x}{2} \]

5. Applied rewrites 88.6%

   \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.016666666666666666, x \cdot x, 0.3333333333333333\right), x \cdot x, 2\right) \cdot x}}{2} \]
6. Step-by-step derivation

7. Applied rewrites 88.6%

   \[\leadsto \frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(0.016666666666666666, x \cdot x, 0.3333333333333333\right) \cdot x\right) \cdot x, \color{blue}{x}, 2 \cdot x\right)}{2} \]
8. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(\frac{1}{60}, x \cdot x, \frac{1}{3}\right) \cdot x\right) \cdot x, x, 2 \cdot x\right)}{2}} \]

   2. div-inv N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\mathsf{fma}\left(\frac{1}{60}, x \cdot x, \frac{1}{3}\right) \cdot x\right) \cdot x, x, 2 \cdot x\right) \cdot \frac{1}{2}} \]

   3. metadata-eval N/A

      \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left(\frac{1}{60}, x \cdot x, \frac{1}{3}\right) \cdot x\right) \cdot x, x, 2 \cdot x\right) \cdot \color{blue}{\frac{1}{2}} \]

   4. lower-*.f64 88.6

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\mathsf{fma}\left(0.016666666666666666, x \cdot x, 0.3333333333333333\right) \cdot x\right) \cdot x, x, 2 \cdot x\right) \cdot 0.5} \]

9. Applied rewrites 88.6%

   \[\leadsto \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(0.016666666666666666, x \cdot x, 0.3333333333333333\right) \cdot x, x, 2\right) \cdot x\right) \cdot 0.5} \]
10. Add Preprocessing

Alternative 8: 90.1% accurate, 6.8× speedup

Math

\[\begin{array}{l} \\ \left(\mathsf{fma}\left(0.016666666666666666 \cdot \left(x \cdot x\right), x \cdot x, 2\right) \cdot x\right) \cdot 0.5 \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (* (* (fma (* 0.016666666666666666 (* x x)) (* x x) 2.0) x) 0.5))

C

double code(double x) {
	return (fma((0.016666666666666666 * (x * x)), (x * x), 2.0) * x) * 0.5;
}

Julia

function code(x)
	return Float64(Float64(fma(Float64(0.016666666666666666 * Float64(x * x)), Float64(x * x), 2.0) * x) * 0.5)
end

Wolfram

code[x_] := N[(N[(N[(N[(0.016666666666666666 * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(x * x), $MachinePrecision] + 2.0), $MachinePrecision] * x), $MachinePrecision] * 0.5), $MachinePrecision]

Derivation

1. Initial program 57.9%

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

3. Taylor expanded in x around 0

   \[\leadsto \frac{\color{blue}{x \cdot \left(2 + {x}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {x}^{2}\right)\right)}}{2} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \frac{\color{blue}{\left(2 + {x}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {x}^{2}\right)\right) \cdot x}}{2} \]

   2. lower-*.f64 N/A

      \[\leadsto \frac{\color{blue}{\left(2 + {x}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {x}^{2}\right)\right) \cdot x}}{2} \]

   3. +-commutative N/A

      \[\leadsto \frac{\color{blue}{\left({x}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {x}^{2}\right) + 2\right)} \cdot x}{2} \]

   4. *-commutative N/A

      \[\leadsto \frac{\left(\color{blue}{\left(\frac{1}{3} + \frac{1}{60} \cdot {x}^{2}\right) \cdot {x}^{2}} + 2\right) \cdot x}{2} \]

   5. lower-fma.f64 N/A

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{1}{3} + \frac{1}{60} \cdot {x}^{2}, {x}^{2}, 2\right)} \cdot x}{2} \]

   6. +-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{1}{60} \cdot {x}^{2} + \frac{1}{3}}, {x}^{2}, 2\right) \cdot x}{2} \]

   7. lower-fma.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{60}, {x}^{2}, \frac{1}{3}\right)}, {x}^{2}, 2\right) \cdot x}{2} \]

   8. unpow2 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{60}, \color{blue}{x \cdot x}, \frac{1}{3}\right), {x}^{2}, 2\right) \cdot x}{2} \]

   9. lower-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{60}, \color{blue}{x \cdot x}, \frac{1}{3}\right), {x}^{2}, 2\right) \cdot x}{2} \]

   10. unpow2 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{60}, x \cdot x, \frac{1}{3}\right), \color{blue}{x \cdot x}, 2\right) \cdot x}{2} \]

   11. lower-*.f64 88.6

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.016666666666666666, x \cdot x, 0.3333333333333333\right), \color{blue}{x \cdot x}, 2\right) \cdot x}{2} \]

5. Applied rewrites 88.6%

   \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.016666666666666666, x \cdot x, 0.3333333333333333\right), x \cdot x, 2\right) \cdot x}}{2} \]
6. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{60}, x \cdot x, \frac{1}{3}\right), x \cdot x, 2\right) \cdot x}{2}} \]

   2. div-inv N/A

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{60}, x \cdot x, \frac{1}{3}\right), x \cdot x, 2\right) \cdot x\right) \cdot \frac{1}{2}} \]

   3. lower-*.f64 N/A

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{60}, x \cdot x, \frac{1}{3}\right), x \cdot x, 2\right) \cdot x\right) \cdot \frac{1}{2}} \]

   4. metadata-eval 88.6

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(0.016666666666666666, x \cdot x, 0.3333333333333333\right), x \cdot x, 2\right) \cdot x\right) \cdot \color{blue}{0.5} \]

7. Applied rewrites 88.6%

   \[\leadsto \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(0.016666666666666666, x \cdot x, 0.3333333333333333\right), x \cdot x, 2\right) \cdot x\right) \cdot 0.5} \]

8. Taylor expanded in x around inf

   \[\leadsto \left(\mathsf{fma}\left(\frac{1}{60} \cdot {x}^{2}, x \cdot x, 2\right) \cdot x\right) \cdot \frac{1}{2} \]
9. Step-by-step derivation

10. Applied rewrites 88.2%

    \[\leadsto \left(\mathsf{fma}\left(0.016666666666666666 \cdot \left(x \cdot x\right), x \cdot x, 2\right) \cdot x\right) \cdot 0.5 \]
11. Add Preprocessing

Alternative 9: 68.9% accurate, 8.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.4:\\ \;\;\;\;\left(2 \cdot x\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\left(\left(0.3333333333333333 \cdot \left(x \cdot x\right)\right) \cdot x\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 2.4)
   (* (* 2.0 x) 0.5)
   (* (* (* 0.3333333333333333 (* x x)) x) 0.5)))

C

double code(double x) {
	double tmp;
	if (x <= 2.4) {
		tmp = (2.0 * x) * 0.5;
	} else {
		tmp = ((0.3333333333333333 * (x * x)) * x) * 0.5;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 2.4d0) then
        tmp = (2.0d0 * x) * 0.5d0
    else
        tmp = ((0.3333333333333333d0 * (x * x)) * x) * 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 2.4) {
		tmp = (2.0 * x) * 0.5;
	} else {
		tmp = ((0.3333333333333333 * (x * x)) * x) * 0.5;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 2.4:
		tmp = (2.0 * x) * 0.5
	else:
		tmp = ((0.3333333333333333 * (x * x)) * x) * 0.5
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 2.4)
		tmp = Float64(Float64(2.0 * x) * 0.5);
	else
		tmp = Float64(Float64(Float64(0.3333333333333333 * Float64(x * x)) * x) * 0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 2.4)
		tmp = (2.0 * x) * 0.5;
	else
		tmp = ((0.3333333333333333 * (x * x)) * x) * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 2.4], N[(N[(2.0 * x), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(N[(0.3333333333333333 * N[(x * x), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.39999999999999991

   1. Initial program 44.4%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 62.2

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

   5. Applied rewrites 62.2%

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

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. lower-*.f64 N/A

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

      4. metadata-eval N/A

         \[\leadsto \mathsf{Rewrite=>}\left(lower-*.f64, \left(2 \cdot x\right)\right) \cdot \color{blue}{\frac{1}{2}} \]

   7. Applied rewrites 62.2%

      \[\leadsto \color{blue}{\left(2 \cdot x\right) \cdot 0.5} \]

   if 2.39999999999999991 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 71.1

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

   5. Applied rewrites 71.1%

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

   7. Applied rewrites 71.1%

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

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. metadata-eval N/A

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

      4. lower-*.f64 71.1

         \[\leadsto \color{blue}{\mathsf{fma}\left(0.3333333333333333 \cdot \left(x \cdot x\right), x, 2 \cdot x\right) \cdot 0.5} \]

   9. Applied rewrites 71.1%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(x \cdot x, 0.3333333333333333, 2\right) \cdot x\right) \cdot 0.5} \]

   10. Taylor expanded in x around inf

       \[\leadsto \left(\left(\frac{1}{3} \cdot {x}^{2}\right) \cdot x\right) \cdot \frac{1}{2} \]
   11. Step-by-step derivation

   12. Applied rewrites 71.1%

       \[\leadsto \left(\left(\left(x \cdot x\right) \cdot 0.3333333333333333\right) \cdot x\right) \cdot 0.5 \]
3. Recombined 2 regimes into one program.

4. Final simplification 64.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.4:\\ \;\;\;\;\left(2 \cdot x\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\left(\left(0.3333333333333333 \cdot \left(x \cdot x\right)\right) \cdot x\right) \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 84.3% accurate, 8.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(0.3333333333333333 \cdot \left(x \cdot x\right), x, 2 \cdot x\right) \cdot 0.5 \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (* (fma (* 0.3333333333333333 (* x x)) x (* 2.0 x)) 0.5))

C

double code(double x) {
	return fma((0.3333333333333333 * (x * x)), x, (2.0 * x)) * 0.5;
}

Julia

function code(x)
	return Float64(fma(Float64(0.3333333333333333 * Float64(x * x)), x, Float64(2.0 * x)) * 0.5)
end

Wolfram

code[x_] := N[(N[(N[(0.3333333333333333 * N[(x * x), $MachinePrecision]), $MachinePrecision] * x + N[(2.0 * x), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]

Derivation

1. Initial program 57.9%

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

3. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. unpow2 N/A

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

   7. lower-*.f64 81.9

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

5. Applied rewrites 81.9%

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

7. Applied rewrites 81.9%

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

   1. lift-/.f64 N/A

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

   2. div-inv N/A

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

   3. metadata-eval N/A

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

   4. lower-*.f64 81.9

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.3333333333333333 \cdot \left(x \cdot x\right), x, 2 \cdot x\right) \cdot 0.5} \]

9. Applied rewrites 81.9%

   \[\leadsto \color{blue}{\left(\mathsf{fma}\left(x \cdot x, 0.3333333333333333, 2\right) \cdot x\right) \cdot 0.5} \]
10. Step-by-step derivation

11. Applied rewrites 81.9%

    \[\leadsto \mathsf{fma}\left(0.3333333333333333 \cdot \left(x \cdot x\right), \color{blue}{x}, 2 \cdot x\right) \cdot 0.5 \]
12. Add Preprocessing

Alternative 11: 84.3% accurate, 9.9× speedup

Math

\[\begin{array}{l} \\ \left(\mathsf{fma}\left(0.3333333333333333, x \cdot x, 2\right) \cdot x\right) \cdot 0.5 \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (* (* (fma 0.3333333333333333 (* x x) 2.0) x) 0.5))

C

double code(double x) {
	return (fma(0.3333333333333333, (x * x), 2.0) * x) * 0.5;
}

Julia

function code(x)
	return Float64(Float64(fma(0.3333333333333333, Float64(x * x), 2.0) * x) * 0.5)
end

Wolfram

code[x_] := N[(N[(N[(0.3333333333333333 * N[(x * x), $MachinePrecision] + 2.0), $MachinePrecision] * x), $MachinePrecision] * 0.5), $MachinePrecision]

Derivation

1. Initial program 57.9%

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

3. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. unpow2 N/A

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

   7. lower-*.f64 81.9

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

5. Applied rewrites 81.9%

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

   1. lift-/.f64 N/A

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

   2. div-inv N/A

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

   3. lower-*.f64 N/A

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

   4. metadata-eval N/A

      \[\leadsto \left(\mathsf{Rewrite=>}\left(lower-fma.f64, \left(\mathsf{fma}\left(\frac{1}{3}, x \cdot x, 2\right)\right)\right) \cdot x\right) \cdot \color{blue}{\frac{1}{2}} \]

7. Applied rewrites 81.9%

   \[\leadsto \color{blue}{\left(\mathsf{fma}\left(0.3333333333333333, x \cdot x, 2\right) \cdot x\right) \cdot 0.5} \]
8. Add Preprocessing

Alternative 12: 53.2% accurate, 19.7× speedup

Math

\[\begin{array}{l} \\ \left(2 \cdot x\right) \cdot 0.5 \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* (* 2.0 x) 0.5))

C

double code(double x) {
	return (2.0 * x) * 0.5;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (2.0d0 * x) * 0.5d0
end function

Java

public static double code(double x) {
	return (2.0 * x) * 0.5;
}

Python

def code(x):
	return (2.0 * x) * 0.5

Julia

function code(x)
	return Float64(Float64(2.0 * x) * 0.5)
end

MATLAB

function tmp = code(x)
	tmp = (2.0 * x) * 0.5;
end

Wolfram

code[x_] := N[(N[(2.0 * x), $MachinePrecision] * 0.5), $MachinePrecision]

Derivation

1. Initial program 57.9%

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

3. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 48.6

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

5. Applied rewrites 48.6%

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

   1. lift-/.f64 N/A

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

   2. div-inv N/A

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

   3. lower-*.f64 N/A

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

   4. metadata-eval N/A

      \[\leadsto \mathsf{Rewrite=>}\left(lower-*.f64, \left(2 \cdot x\right)\right) \cdot \color{blue}{\frac{1}{2}} \]

7. Applied rewrites 48.6%

   \[\leadsto \color{blue}{\left(2 \cdot x\right) \cdot 0.5} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024270 
(FPCore (x)
  :name "Hyperbolic sine"
  :precision binary64
  (/ (- (exp x) (exp (- x))) 2.0))