Hyperbolic sine

Percentage Accurate: 53.9% → 99.4%

Time: 8.5s

Alternatives: 11

Speedup: 9.9×

• 49.0% 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: 53.9% 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: 99.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;e^{x\_m} - e^{-x\_m} \leq 0.0002:\\ \;\;\;\;0.5 \cdot \left(\mathsf{fma}\left(0.3333333333333333, x\_m \cdot x\_m, 2\right) \cdot x\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\left(e^{x\_m} - 1\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m)
 :precision binary64
 (*
  x_s
  (if (<= (- (exp x_m) (exp (- x_m))) 0.0002)
    (* 0.5 (* (fma 0.3333333333333333 (* x_m x_m) 2.0) x_m))
    (* (- (exp x_m) 1.0) 0.5))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	double tmp;
	if ((exp(x_m) - exp(-x_m)) <= 0.0002) {
		tmp = 0.5 * (fma(0.3333333333333333, (x_m * x_m), 2.0) * x_m);
	} else {
		tmp = (exp(x_m) - 1.0) * 0.5;
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m)
	tmp = 0.0
	if (Float64(exp(x_m) - exp(Float64(-x_m))) <= 0.0002)
		tmp = Float64(0.5 * Float64(fma(0.3333333333333333, Float64(x_m * x_m), 2.0) * x_m));
	else
		tmp = Float64(Float64(exp(x_m) - 1.0) * 0.5);
	end
	return Float64(x_s * tmp)
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_] := N[(x$95$s * If[LessEqual[N[(N[Exp[x$95$m], $MachinePrecision] - N[Exp[(-x$95$m)], $MachinePrecision]), $MachinePrecision], 0.0002], N[(0.5 * N[(N[(0.3333333333333333 * N[(x$95$m * x$95$m), $MachinePrecision] + 2.0), $MachinePrecision] * x$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(N[Exp[x$95$m], $MachinePrecision] - 1.0), $MachinePrecision] * 0.5), $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 36.2%

      \[\frac{e^{x} - e^{-x}}{2} \]
   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 93.2

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

   5. Applied rewrites 93.2%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x \cdot x, 0.3333333333333333, 2\right) \cdot x}}{2} \]
   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 93.2%

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

   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

      \[\leadsto \frac{e^{x} - \color{blue}{1}}{2} \]
   4. Step-by-step derivation

   5. Applied rewrites 100.0%

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

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{e^{x} - 1}{2}} \]

      2. div-inv N/A

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

      3. lower-*.f64 N/A

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

      4. metadata-eval 100.0

         \[\leadsto \left(e^{x} - 1\right) \cdot \color{blue}{0.5} \]

   7. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\left(e^{x} - 1\right) \cdot 0.5} \]
3. Recombined 2 regimes into one program.

4. Final simplification 94.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{x} - e^{-x} \leq 0.0002:\\ \;\;\;\;0.5 \cdot \left(\mathsf{fma}\left(0.3333333333333333, x \cdot x, 2\right) \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(e^{x} - 1\right) \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 92.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;e^{x\_m} - e^{-x\_m} \leq 0.0002:\\ \;\;\;\;0.5 \cdot \left(\mathsf{fma}\left(0.3333333333333333, x\_m \cdot x\_m, 2\right) \cdot x\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(x\_m \cdot x\_m, 0.0003968253968253968, 0.016666666666666666\right) \cdot x\_m, x\_m, 0.3333333333333333\right) \cdot x\_m\right) \cdot x\_m\right) \cdot x\_m\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m)
 :precision binary64
 (*
  x_s
  (if (<= (- (exp x_m) (exp (- x_m))) 0.0002)
    (* 0.5 (* (fma 0.3333333333333333 (* x_m x_m) 2.0) x_m))
    (*
     (*
      (*
       (*
        (fma
         (* (fma (* x_m x_m) 0.0003968253968253968 0.016666666666666666) x_m)
         x_m
         0.3333333333333333)
        x_m)
       x_m)
      x_m)
     0.5))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	double tmp;
	if ((exp(x_m) - exp(-x_m)) <= 0.0002) {
		tmp = 0.5 * (fma(0.3333333333333333, (x_m * x_m), 2.0) * x_m);
	} else {
		tmp = (((fma((fma((x_m * x_m), 0.0003968253968253968, 0.016666666666666666) * x_m), x_m, 0.3333333333333333) * x_m) * x_m) * x_m) * 0.5;
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m)
	tmp = 0.0
	if (Float64(exp(x_m) - exp(Float64(-x_m))) <= 0.0002)
		tmp = Float64(0.5 * Float64(fma(0.3333333333333333, Float64(x_m * x_m), 2.0) * x_m));
	else
		tmp = Float64(Float64(Float64(Float64(fma(Float64(fma(Float64(x_m * x_m), 0.0003968253968253968, 0.016666666666666666) * x_m), x_m, 0.3333333333333333) * x_m) * x_m) * x_m) * 0.5);
	end
	return Float64(x_s * tmp)
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_] := N[(x$95$s * If[LessEqual[N[(N[Exp[x$95$m], $MachinePrecision] - N[Exp[(-x$95$m)], $MachinePrecision]), $MachinePrecision], 0.0002], N[(0.5 * N[(N[(0.3333333333333333 * N[(x$95$m * x$95$m), $MachinePrecision] + 2.0), $MachinePrecision] * x$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.0003968253968253968 + 0.016666666666666666), $MachinePrecision] * x$95$m), $MachinePrecision] * x$95$m + 0.3333333333333333), $MachinePrecision] * x$95$m), $MachinePrecision] * x$95$m), $MachinePrecision] * x$95$m), $MachinePrecision] * 0.5), $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 36.2%

      \[\frac{e^{x} - e^{-x}}{2} \]
   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 93.2

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

   5. Applied rewrites 93.2%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x \cdot x, 0.3333333333333333, 2\right) \cdot x}}{2} \]
   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 93.2%

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

   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

      \[\leadsto \frac{\color{blue}{x \cdot \left(2 + {x}^{2} \cdot \left(\frac{1}{3} + {x}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {x}^{2}\right)\right)\right)}}{2} \]
   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 88.9

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

   5. Applied rewrites 88.9%

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

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. lower-*.f64 N/A

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

      4. metadata-eval N/A

         \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{Rewrite=>}\left(lower-fma.f64, \left(\mathsf{fma}\left(x \cdot x, \frac{1}{2520}, \frac{1}{60}\right)\right)\right), x \cdot x, \frac{1}{3}\right), x \cdot x, 2\right) \cdot x\right) \cdot \color{blue}{\frac{1}{2}} \]

   7. Applied rewrites 88.9%

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

   8. Taylor expanded in x around inf

      \[\leadsto \left(\left({x}^{6} \cdot \left(\frac{1}{2520} + \left(\frac{\frac{1}{3}}{{x}^{4}} + \frac{1}{60} \cdot \frac{1}{{x}^{2}}\right)\right)\right) \cdot x\right) \cdot \frac{1}{2} \]

   10. Applied rewrites 88.9%

       \[\leadsto \left(\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0003968253968253968, x \cdot x, 0.016666666666666666\right), x \cdot x, 0.3333333333333333\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot 0.5 \]
   11. Step-by-step derivation

   12. Applied rewrites 88.9%

       \[\leadsto \left(\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.0003968253968253968, 0.016666666666666666\right) \cdot x, x, 0.3333333333333333\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot 0.5 \]
3. Recombined 2 regimes into one program.

4. Final simplification 92.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{x} - e^{-x} \leq 0.0002:\\ \;\;\;\;0.5 \cdot \left(\mathsf{fma}\left(0.3333333333333333, x \cdot x, 2\right) \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.0003968253968253968, 0.016666666666666666\right) \cdot x, x, 0.3333333333333333\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 92.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;e^{x\_m} - e^{-x\_m} \leq 0.0002:\\ \;\;\;\;0.5 \cdot \left(\mathsf{fma}\left(0.3333333333333333, x\_m \cdot x\_m, 2\right) \cdot x\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\left(\left(\mathsf{fma}\left(x\_m \cdot x\_m, 0.0003968253968253968, 0.016666666666666666\right) \cdot x\_m\right) \cdot x\_m\right) \cdot x\_m\right) \cdot x\_m\right) \cdot x\_m\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m)
 :precision binary64
 (*
  x_s
  (if (<= (- (exp x_m) (exp (- x_m))) 0.0002)
    (* 0.5 (* (fma 0.3333333333333333 (* x_m x_m) 2.0) x_m))
    (*
     (*
      (*
       (*
        (*
         (* (fma (* x_m x_m) 0.0003968253968253968 0.016666666666666666) x_m)
         x_m)
        x_m)
       x_m)
      x_m)
     0.5))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	double tmp;
	if ((exp(x_m) - exp(-x_m)) <= 0.0002) {
		tmp = 0.5 * (fma(0.3333333333333333, (x_m * x_m), 2.0) * x_m);
	} else {
		tmp = (((((fma((x_m * x_m), 0.0003968253968253968, 0.016666666666666666) * x_m) * x_m) * x_m) * x_m) * x_m) * 0.5;
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m)
	tmp = 0.0
	if (Float64(exp(x_m) - exp(Float64(-x_m))) <= 0.0002)
		tmp = Float64(0.5 * Float64(fma(0.3333333333333333, Float64(x_m * x_m), 2.0) * x_m));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(fma(Float64(x_m * x_m), 0.0003968253968253968, 0.016666666666666666) * x_m) * x_m) * x_m) * x_m) * x_m) * 0.5);
	end
	return Float64(x_s * tmp)
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_] := N[(x$95$s * If[LessEqual[N[(N[Exp[x$95$m], $MachinePrecision] - N[Exp[(-x$95$m)], $MachinePrecision]), $MachinePrecision], 0.0002], N[(0.5 * N[(N[(0.3333333333333333 * N[(x$95$m * x$95$m), $MachinePrecision] + 2.0), $MachinePrecision] * x$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.0003968253968253968 + 0.016666666666666666), $MachinePrecision] * x$95$m), $MachinePrecision] * x$95$m), $MachinePrecision] * x$95$m), $MachinePrecision] * x$95$m), $MachinePrecision] * x$95$m), $MachinePrecision] * 0.5), $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 36.2%

      \[\frac{e^{x} - e^{-x}}{2} \]
   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 93.2

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

   5. Applied rewrites 93.2%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x \cdot x, 0.3333333333333333, 2\right) \cdot x}}{2} \]
   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 93.2%

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

   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

      \[\leadsto \frac{\color{blue}{x \cdot \left(2 + {x}^{2} \cdot \left(\frac{1}{3} + {x}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {x}^{2}\right)\right)\right)}}{2} \]
   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 88.9

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

   5. Applied rewrites 88.9%

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

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. lower-*.f64 N/A

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

      4. metadata-eval N/A

         \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{Rewrite=>}\left(lower-fma.f64, \left(\mathsf{fma}\left(x \cdot x, \frac{1}{2520}, \frac{1}{60}\right)\right)\right), x \cdot x, \frac{1}{3}\right), x \cdot x, 2\right) \cdot x\right) \cdot \color{blue}{\frac{1}{2}} \]

   7. Applied rewrites 88.9%

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

   8. Taylor expanded in x around inf

      \[\leadsto \left(\left({x}^{6} \cdot \left(\frac{1}{2520} + \left(\frac{\frac{1}{3}}{{x}^{4}} + \frac{1}{60} \cdot \frac{1}{{x}^{2}}\right)\right)\right) \cdot x\right) \cdot \frac{1}{2} \]

   10. Applied rewrites 88.9%

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

   11. Taylor expanded in x around inf

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

   13. Applied rewrites 88.9%

       \[\leadsto \left(\left(\left(\left(\left(\mathsf{fma}\left(x \cdot x, 0.0003968253968253968, 0.016666666666666666\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot 0.5 \]
3. Recombined 2 regimes into one program.

4. Final simplification 92.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{x} - e^{-x} \leq 0.0002:\\ \;\;\;\;0.5 \cdot \left(\mathsf{fma}\left(0.3333333333333333, x \cdot x, 2\right) \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\left(\left(\mathsf{fma}\left(x \cdot x, 0.0003968253968253968, 0.016666666666666666\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 83.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;e^{x\_m} - e^{-x\_m} \leq 0.0002:\\ \;\;\;\;\left(2 \cdot x\_m\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(x\_m \cdot x\_m\right) \cdot 0.3333333333333333\right) \cdot x\_m\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m)
 :precision binary64
 (*
  x_s
  (if (<= (- (exp x_m) (exp (- x_m))) 0.0002)
    (* (* 2.0 x_m) 0.5)
    (* (* (* (* x_m x_m) 0.3333333333333333) x_m) 0.5))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	double tmp;
	if ((exp(x_m) - exp(-x_m)) <= 0.0002) {
		tmp = (2.0 * x_m) * 0.5;
	} else {
		tmp = (((x_m * x_m) * 0.3333333333333333) * x_m) * 0.5;
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8) :: tmp
    if ((exp(x_m) - exp(-x_m)) <= 0.0002d0) then
        tmp = (2.0d0 * x_m) * 0.5d0
    else
        tmp = (((x_m * x_m) * 0.3333333333333333d0) * x_m) * 0.5d0
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m) {
	double tmp;
	if ((Math.exp(x_m) - Math.exp(-x_m)) <= 0.0002) {
		tmp = (2.0 * x_m) * 0.5;
	} else {
		tmp = (((x_m * x_m) * 0.3333333333333333) * x_m) * 0.5;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m):
	tmp = 0
	if (math.exp(x_m) - math.exp(-x_m)) <= 0.0002:
		tmp = (2.0 * x_m) * 0.5
	else:
		tmp = (((x_m * x_m) * 0.3333333333333333) * x_m) * 0.5
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m)
	tmp = 0.0
	if (Float64(exp(x_m) - exp(Float64(-x_m))) <= 0.0002)
		tmp = Float64(Float64(2.0 * x_m) * 0.5);
	else
		tmp = Float64(Float64(Float64(Float64(x_m * x_m) * 0.3333333333333333) * x_m) * 0.5);
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m)
	tmp = 0.0;
	if ((exp(x_m) - exp(-x_m)) <= 0.0002)
		tmp = (2.0 * x_m) * 0.5;
	else
		tmp = (((x_m * x_m) * 0.3333333333333333) * x_m) * 0.5;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_] := N[(x$95$s * If[LessEqual[N[(N[Exp[x$95$m], $MachinePrecision] - N[Exp[(-x$95$m)], $MachinePrecision]), $MachinePrecision], 0.0002], N[(N[(2.0 * x$95$m), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.3333333333333333), $MachinePrecision] * x$95$m), $MachinePrecision] * 0.5), $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 36.2%

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

   3. Taylor expanded in x around 0

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

      1. lower-*.f64 71.0

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

   5. Applied rewrites 71.0%

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

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. lower-*.f64 N/A

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

      4. metadata-eval 71.0

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

   7. Applied rewrites 71.0%

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

   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

      \[\leadsto \frac{\color{blue}{x \cdot \left(2 + {x}^{2} \cdot \left(\frac{1}{3} + {x}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {x}^{2}\right)\right)\right)}}{2} \]
   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 88.9

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

   5. Applied rewrites 88.9%

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

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. lower-*.f64 N/A

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

      4. metadata-eval N/A

         \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{Rewrite=>}\left(lower-fma.f64, \left(\mathsf{fma}\left(x \cdot x, \frac{1}{2520}, \frac{1}{60}\right)\right)\right), x \cdot x, \frac{1}{3}\right), x \cdot x, 2\right) \cdot x\right) \cdot \color{blue}{\frac{1}{2}} \]

   7. Applied rewrites 88.9%

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

   8. Taylor expanded in x around inf

      \[\leadsto \left(\left({x}^{6} \cdot \left(\frac{1}{2520} + \left(\frac{\frac{1}{3}}{{x}^{4}} + \frac{1}{60} \cdot \frac{1}{{x}^{2}}\right)\right)\right) \cdot x\right) \cdot \frac{1}{2} \]

   10. Applied rewrites 88.9%

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

   11. Taylor expanded in x around 0

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

   13. Applied rewrites 70.0%

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

Alternative 5: 93.0% accurate, 4.9× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0003968253968253968, x\_m \cdot x\_m, 0.016666666666666666\right) \cdot x\_m, x\_m, 0.3333333333333333\right), x\_m \cdot x\_m, 2\right) \cdot x\_m\right) \cdot 0.5\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m)
 :precision binary64
 (*
  x_s
  (*
   (*
    (fma
     (fma
      (* (fma 0.0003968253968253968 (* x_m x_m) 0.016666666666666666) x_m)
      x_m
      0.3333333333333333)
     (* x_m x_m)
     2.0)
    x_m)
   0.5)))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	return x_s * ((fma(fma((fma(0.0003968253968253968, (x_m * x_m), 0.016666666666666666) * x_m), x_m, 0.3333333333333333), (x_m * x_m), 2.0) * x_m) * 0.5);
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m)
	return Float64(x_s * Float64(Float64(fma(fma(Float64(fma(0.0003968253968253968, Float64(x_m * x_m), 0.016666666666666666) * x_m), x_m, 0.3333333333333333), Float64(x_m * x_m), 2.0) * x_m) * 0.5))
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_] := N[(x$95$s * N[(N[(N[(N[(N[(N[(0.0003968253968253968 * N[(x$95$m * x$95$m), $MachinePrecision] + 0.016666666666666666), $MachinePrecision] * x$95$m), $MachinePrecision] * x$95$m + 0.3333333333333333), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + 2.0), $MachinePrecision] * x$95$m), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 50.9%

   \[\frac{e^{x} - e^{-x}}{2} \]
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 93.7

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

5. Applied rewrites 93.7%

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

   1. lift-/.f64 N/A

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

   2. div-inv N/A

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

   3. lower-*.f64 N/A

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

   4. metadata-eval N/A

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{Rewrite=>}\left(lower-fma.f64, \left(\mathsf{fma}\left(x \cdot x, \frac{1}{2520}, \frac{1}{60}\right)\right)\right), x \cdot x, \frac{1}{3}\right), x \cdot x, 2\right) \cdot x\right) \cdot \color{blue}{\frac{1}{2}} \]

7. Applied rewrites 93.7%

   \[\leadsto \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.0003968253968253968, 0.016666666666666666\right), x \cdot x, 0.3333333333333333\right), x \cdot x, 2\right) \cdot x\right) \cdot 0.5} \]
8. Step-by-step derivation

9. Applied rewrites 93.7%

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

Alternative 6: 92.5% accurate, 5.0× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(\left(\mathsf{fma}\left(\left(\mathsf{fma}\left(0.0003968253968253968, x\_m \cdot x\_m, 0.016666666666666666\right) \cdot x\_m\right) \cdot x\_m, x\_m \cdot x\_m, 2\right) \cdot x\_m\right) \cdot 0.5\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m)
 :precision binary64
 (*
  x_s
  (*
   (*
    (fma
     (*
      (* (fma 0.0003968253968253968 (* x_m x_m) 0.016666666666666666) x_m)
      x_m)
     (* x_m x_m)
     2.0)
    x_m)
   0.5)))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	return x_s * ((fma(((fma(0.0003968253968253968, (x_m * x_m), 0.016666666666666666) * x_m) * x_m), (x_m * x_m), 2.0) * x_m) * 0.5);
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m)
	return Float64(x_s * Float64(Float64(fma(Float64(Float64(fma(0.0003968253968253968, Float64(x_m * x_m), 0.016666666666666666) * x_m) * x_m), Float64(x_m * x_m), 2.0) * x_m) * 0.5))
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_] := N[(x$95$s * N[(N[(N[(N[(N[(N[(0.0003968253968253968 * N[(x$95$m * x$95$m), $MachinePrecision] + 0.016666666666666666), $MachinePrecision] * x$95$m), $MachinePrecision] * x$95$m), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + 2.0), $MachinePrecision] * x$95$m), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 50.9%

   \[\frac{e^{x} - e^{-x}}{2} \]
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 93.7

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

5. Applied rewrites 93.7%

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

   1. lift-/.f64 N/A

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

   2. div-inv N/A

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

   3. lower-*.f64 N/A

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

   4. metadata-eval N/A

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{Rewrite=>}\left(lower-fma.f64, \left(\mathsf{fma}\left(x \cdot x, \frac{1}{2520}, \frac{1}{60}\right)\right)\right), x \cdot x, \frac{1}{3}\right), x \cdot x, 2\right) \cdot x\right) \cdot \color{blue}{\frac{1}{2}} \]

7. Applied rewrites 93.7%

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

8. Taylor expanded in x around inf

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

10. Applied rewrites 93.4%

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

Alternative 7: 90.2% accurate, 5.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 3.35:\\ \;\;\;\;0.5 \cdot \left(\mathsf{fma}\left(0.3333333333333333, x\_m \cdot x\_m, 2\right) \cdot x\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\mathsf{fma}\left(0.016666666666666666, x\_m \cdot x\_m, 0.3333333333333333\right) \cdot x\_m\right) \cdot x\_m\right) \cdot x\_m\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m)
 :precision binary64
 (*
  x_s
  (if (<= x_m 3.35)
    (* 0.5 (* (fma 0.3333333333333333 (* x_m x_m) 2.0) x_m))
    (*
     (*
      (* (* (fma 0.016666666666666666 (* x_m x_m) 0.3333333333333333) x_m) x_m)
      x_m)
     0.5))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	double tmp;
	if (x_m <= 3.35) {
		tmp = 0.5 * (fma(0.3333333333333333, (x_m * x_m), 2.0) * x_m);
	} else {
		tmp = (((fma(0.016666666666666666, (x_m * x_m), 0.3333333333333333) * x_m) * x_m) * x_m) * 0.5;
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m)
	tmp = 0.0
	if (x_m <= 3.35)
		tmp = Float64(0.5 * Float64(fma(0.3333333333333333, Float64(x_m * x_m), 2.0) * x_m));
	else
		tmp = Float64(Float64(Float64(Float64(fma(0.016666666666666666, Float64(x_m * x_m), 0.3333333333333333) * x_m) * x_m) * x_m) * 0.5);
	end
	return Float64(x_s * tmp)
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_] := N[(x$95$s * If[LessEqual[x$95$m, 3.35], N[(0.5 * N[(N[(0.3333333333333333 * N[(x$95$m * x$95$m), $MachinePrecision] + 2.0), $MachinePrecision] * x$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(0.016666666666666666 * N[(x$95$m * x$95$m), $MachinePrecision] + 0.3333333333333333), $MachinePrecision] * x$95$m), $MachinePrecision] * x$95$m), $MachinePrecision] * x$95$m), $MachinePrecision] * 0.5), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 3.35000000000000009

   1. Initial program 36.2%

      \[\frac{e^{x} - e^{-x}}{2} \]
   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 93.2

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

   5. Applied rewrites 93.2%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x \cdot x, 0.3333333333333333, 2\right) \cdot x}}{2} \]
   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 93.2%

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

   if 3.35000000000000009 < 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 84.0

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

   5. Applied rewrites 84.0%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.016666666666666666, x \cdot x, 0.3333333333333333\right), x \cdot x, 2\right) \cdot x}}{2} \]
   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 84.0

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

   7. Applied rewrites 84.0%

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

   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 84.0%

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

4. Final simplification 91.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3.35:\\ \;\;\;\;0.5 \cdot \left(\mathsf{fma}\left(0.3333333333333333, x \cdot x, 2\right) \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\mathsf{fma}\left(0.016666666666666666, x \cdot x, 0.3333333333333333\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 90.3% accurate, 6.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(\left(\mathsf{fma}\left(\mathsf{fma}\left(0.016666666666666666, x\_m \cdot x\_m, 0.3333333333333333\right), x\_m \cdot x\_m, 2\right) \cdot x\_m\right) \cdot 0.5\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m)
 :precision binary64
 (*
  x_s
  (*
   (*
    (fma
     (fma 0.016666666666666666 (* x_m x_m) 0.3333333333333333)
     (* x_m x_m)
     2.0)
    x_m)
   0.5)))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	return x_s * ((fma(fma(0.016666666666666666, (x_m * x_m), 0.3333333333333333), (x_m * x_m), 2.0) * x_m) * 0.5);
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m)
	return Float64(x_s * Float64(Float64(fma(fma(0.016666666666666666, Float64(x_m * x_m), 0.3333333333333333), Float64(x_m * x_m), 2.0) * x_m) * 0.5))
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_] := N[(x$95$s * N[(N[(N[(N[(0.016666666666666666 * N[(x$95$m * x$95$m), $MachinePrecision] + 0.3333333333333333), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + 2.0), $MachinePrecision] * x$95$m), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 50.9%

   \[\frac{e^{x} - e^{-x}}{2} \]
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 91.9

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

5. Applied rewrites 91.9%

   \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.016666666666666666, x \cdot x, 0.3333333333333333\right), x \cdot x, 2\right) \cdot x}}{2} \]
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 91.9

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

7. Applied rewrites 91.9%

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

Alternative 9: 89.9% accurate, 6.8× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(\left(\mathsf{fma}\left(0.016666666666666666 \cdot \left(x\_m \cdot x\_m\right), x\_m \cdot x\_m, 2\right) \cdot x\_m\right) \cdot 0.5\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m)
 :precision binary64
 (*
  x_s
  (* (* (fma (* 0.016666666666666666 (* x_m x_m)) (* x_m x_m) 2.0) x_m) 0.5)))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	return x_s * ((fma((0.016666666666666666 * (x_m * x_m)), (x_m * x_m), 2.0) * x_m) * 0.5);
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m)
	return Float64(x_s * Float64(Float64(fma(Float64(0.016666666666666666 * Float64(x_m * x_m)), Float64(x_m * x_m), 2.0) * x_m) * 0.5))
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_] := N[(x$95$s * N[(N[(N[(N[(0.016666666666666666 * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + 2.0), $MachinePrecision] * x$95$m), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 50.9%

   \[\frac{e^{x} - e^{-x}}{2} \]
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 91.9

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

5. Applied rewrites 91.9%

   \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.016666666666666666, x \cdot x, 0.3333333333333333\right), x \cdot x, 2\right) \cdot x}}{2} \]
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 91.9

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

7. Applied rewrites 91.9%

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

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 91.5%

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

Alternative 10: 84.1% accurate, 9.9× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(0.5 \cdot \left(\mathsf{fma}\left(0.3333333333333333, x\_m \cdot x\_m, 2\right) \cdot x\_m\right)\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m)
 :precision binary64
 (* x_s (* 0.5 (* (fma 0.3333333333333333 (* x_m x_m) 2.0) x_m))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	return x_s * (0.5 * (fma(0.3333333333333333, (x_m * x_m), 2.0) * x_m));
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m)
	return Float64(x_s * Float64(0.5 * Float64(fma(0.3333333333333333, Float64(x_m * x_m), 2.0) * x_m)))
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_] := N[(x$95$s * N[(0.5 * N[(N[(0.3333333333333333 * N[(x$95$m * x$95$m), $MachinePrecision] + 2.0), $MachinePrecision] * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 50.9%

   \[\frac{e^{x} - e^{-x}}{2} \]
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 87.9

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

5. Applied rewrites 87.9%

   \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x \cdot x, 0.3333333333333333, 2\right) \cdot x}}{2} \]
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 87.9%

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

8. Final simplification 87.9%

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

Alternative 11: 52.7% accurate, 19.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(\left(2 \cdot x\_m\right) \cdot 0.5\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m) :precision binary64 (* x_s (* (* 2.0 x_m) 0.5)))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	return x_s * ((2.0 * x_m) * 0.5);
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    code = x_s * ((2.0d0 * x_m) * 0.5d0)
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m) {
	return x_s * ((2.0 * x_m) * 0.5);
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m):
	return x_s * ((2.0 * x_m) * 0.5)

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m)
	return Float64(x_s * Float64(Float64(2.0 * x_m) * 0.5))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp = code(x_s, x_m)
	tmp = x_s * ((2.0 * x_m) * 0.5);
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_] := N[(x$95$s * N[(N[(2.0 * x$95$m), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 50.9%

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

3. Taylor expanded in x around 0

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

   1. lower-*.f64 55.8

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

5. Applied rewrites 55.8%

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

   1. lift-/.f64 N/A

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

   2. div-inv N/A

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

   3. lower-*.f64 N/A

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

   4. metadata-eval 55.8

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

7. Applied rewrites 55.8%

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

Reproduce

herbie shell --seed 2024276 
(FPCore (x)
  :name "Hyperbolic sine"
  :precision binary64
  (/ (- (exp x) (exp (- x))) 2.0))