Hyperbolic sine

Percentage Accurate: 54.7% → 99.5%

Time: 12.2s

Alternatives: 22

Speedup: 12.8×

• 50.3% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 54.7% 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: 99.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;e^{x\_m} - e^{-x\_m} \leq 0.002:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(x\_m, x\_m \cdot 0.016666666666666666, 0.3333333333333333\right), x\_m \cdot \left(x\_m \cdot x\_m\right), x\_m \cdot 2\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{expm1}\left(x\_m\right) \cdot 0.5\\ \end{array} \end{array} \]

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))) 0.002)
    (/
     (fma
      (fma x_m (* x_m 0.016666666666666666) 0.3333333333333333)
      (* x_m (* x_m x_m))
      (* x_m 2.0))
     2.0)
    (* (expm1 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)) <= 0.002) {
		tmp = fma(fma(x_m, (x_m * 0.016666666666666666), 0.3333333333333333), (x_m * (x_m * x_m)), (x_m * 2.0)) / 2.0;
	} else {
		tmp = expm1(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))) <= 0.002)
		tmp = Float64(fma(fma(x_m, Float64(x_m * 0.016666666666666666), 0.3333333333333333), Float64(x_m * Float64(x_m * x_m)), Float64(x_m * 2.0)) / 2.0);
	else
		tmp = Float64(expm1(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], 0.002], N[(N[(N[(x$95$m * N[(x$95$m * 0.016666666666666666), $MachinePrecision] + 0.3333333333333333), $MachinePrecision] * N[(x$95$m * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] + N[(x$95$m * 2.0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(Exp[x$95$m] - 1), $MachinePrecision] * 0.5), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 37.7%

      \[\frac{e^{x} - e^{-x}}{2} \]
   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. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 94.6

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

   5. Simplified 94.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. distribute-rgt-in N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. lift-*.f64 N/A

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

      8. pow3 N/A

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

      9. lift-*.f64 N/A

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

      10. lower-fma.f64 N/A

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

      11. cube-unmult N/A

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

      12. lift-*.f64 N/A

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

      13. lower-*.f64 94.6

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 94.6

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

   7. Applied egg-rr 94.6%

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

   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

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

   5. Simplified 100.0%

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

      1. lift-exp.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-inv N/A

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

      4. lower-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-exp.f64 N/A

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

      7. lower-expm1.f64 N/A

         \[\leadsto \color{blue}{\mathsf{expm1}\left(x\right)} \cdot \frac{1}{2} \]

      8. metadata-eval 100.0

         \[\leadsto \mathsf{expm1}\left(x\right) \cdot \color{blue}{0.5} \]

   7. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(x\right) \cdot 0.5} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 92.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := x\_m \cdot \left(x\_m \cdot x\_m\right)\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;e^{x\_m} - e^{-x\_m} \leq 0.002:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(x\_m, x\_m \cdot 0.016666666666666666, 0.3333333333333333\right), t\_0, x\_m \cdot 2\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m \cdot \left(\mathsf{fma}\left(0.0003968253968253968, x\_m \cdot x\_m, 0.016666666666666666\right) \cdot \left(x\_m \cdot t\_0\right)\right)}{2}\\ \end{array} \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m)
 :precision binary64
 (let* ((t_0 (* x_m (* x_m x_m))))
   (*
    x_s
    (if (<= (- (exp x_m) (exp (- x_m))) 0.002)
      (/
       (fma
        (fma x_m (* x_m 0.016666666666666666) 0.3333333333333333)
        t_0
        (* x_m 2.0))
       2.0)
      (/
       (*
        x_m
        (*
         (fma 0.0003968253968253968 (* x_m x_m) 0.016666666666666666)
         (* x_m t_0)))
       2.0)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	double t_0 = x_m * (x_m * x_m);
	double tmp;
	if ((exp(x_m) - exp(-x_m)) <= 0.002) {
		tmp = fma(fma(x_m, (x_m * 0.016666666666666666), 0.3333333333333333), t_0, (x_m * 2.0)) / 2.0;
	} else {
		tmp = (x_m * (fma(0.0003968253968253968, (x_m * x_m), 0.016666666666666666) * (x_m * t_0))) / 2.0;
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m)
	t_0 = Float64(x_m * Float64(x_m * x_m))
	tmp = 0.0
	if (Float64(exp(x_m) - exp(Float64(-x_m))) <= 0.002)
		tmp = Float64(fma(fma(x_m, Float64(x_m * 0.016666666666666666), 0.3333333333333333), t_0, Float64(x_m * 2.0)) / 2.0);
	else
		tmp = Float64(Float64(x_m * Float64(fma(0.0003968253968253968, Float64(x_m * x_m), 0.016666666666666666) * Float64(x_m * t_0))) / 2.0);
	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_] := Block[{t$95$0 = N[(x$95$m * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * If[LessEqual[N[(N[Exp[x$95$m], $MachinePrecision] - N[Exp[(-x$95$m)], $MachinePrecision]), $MachinePrecision], 0.002], N[(N[(N[(x$95$m * N[(x$95$m * 0.016666666666666666), $MachinePrecision] + 0.3333333333333333), $MachinePrecision] * t$95$0 + N[(x$95$m * 2.0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(x$95$m * N[(N[(0.0003968253968253968 * N[(x$95$m * x$95$m), $MachinePrecision] + 0.016666666666666666), $MachinePrecision] * N[(x$95$m * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 37.7%

      \[\frac{e^{x} - e^{-x}}{2} \]
   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. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 94.6

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

   5. Simplified 94.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. distribute-rgt-in N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. lift-*.f64 N/A

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

      8. pow3 N/A

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

      9. lift-*.f64 N/A

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

      10. lower-fma.f64 N/A

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

      11. cube-unmult N/A

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

      12. lift-*.f64 N/A

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

      13. lower-*.f64 94.6

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 94.6

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

   7. Applied egg-rr 94.6%

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

   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

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

   5. Simplified 85.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-fma.f64 N/A

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

      6. lift-fma.f64 N/A

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

      7. distribute-rgt-in N/A

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

      8. associate-+l+ N/A

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

   7. Applied egg-rr 85.8%

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

   8. Taylor expanded in x around 0

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

   10. Simplified 85.8%

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

   11. Taylor expanded in x around inf

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

   13. Simplified 85.8%

       \[\leadsto \frac{x \cdot \color{blue}{\left(\mathsf{fma}\left(0.0003968253968253968, x \cdot x, 0.016666666666666666\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)}}{2} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 88.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;e^{x\_m} - e^{-x\_m} \leq 0.002:\\ \;\;\;\;\mathsf{fma}\left(x\_m, x\_m \cdot 0.3333333333333333, 2\right) \cdot \left(x\_m \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot \mathsf{fma}\left(x\_m, \mathsf{fma}\left(x\_m, \mathsf{fma}\left(x\_m, 0.020833333333333332, 0.08333333333333333\right), 0.25\right), 0.5\right)\\ \end{array} \end{array} \]

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))) 0.002)
    (* (fma x_m (* x_m 0.3333333333333333) 2.0) (* x_m 0.5))
    (*
     x_m
     (fma
      x_m
      (fma x_m (fma x_m 0.020833333333333332 0.08333333333333333) 0.25)
      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)) <= 0.002) {
		tmp = fma(x_m, (x_m * 0.3333333333333333), 2.0) * (x_m * 0.5);
	} else {
		tmp = x_m * fma(x_m, fma(x_m, fma(x_m, 0.020833333333333332, 0.08333333333333333), 0.25), 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))) <= 0.002)
		tmp = Float64(fma(x_m, Float64(x_m * 0.3333333333333333), 2.0) * Float64(x_m * 0.5));
	else
		tmp = Float64(x_m * fma(x_m, fma(x_m, fma(x_m, 0.020833333333333332, 0.08333333333333333), 0.25), 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], 0.002], N[(N[(x$95$m * N[(x$95$m * 0.3333333333333333), $MachinePrecision] + 2.0), $MachinePrecision] * N[(x$95$m * 0.5), $MachinePrecision]), $MachinePrecision], N[(x$95$m * N[(x$95$m * N[(x$95$m * N[(x$95$m * 0.020833333333333332 + 0.08333333333333333), $MachinePrecision] + 0.25), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 37.7%

      \[\frac{e^{x} - e^{-x}}{2} \]
   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. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 87.6

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

   5. Simplified 87.6%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lift-fma.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. div-inv N/A

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

      14. lower-*.f64 N/A

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

      15. metadata-eval 87.6

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

   7. Applied egg-rr 87.6%

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

   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

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{4} + x \cdot \left(\frac{1}{12} + \frac{1}{48} \cdot x\right)\right)\right)} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{4} + x \cdot \left(\frac{1}{12} + \frac{1}{48} \cdot x\right)\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left(x \cdot \left(\frac{1}{4} + x \cdot \left(\frac{1}{12} + \frac{1}{48} \cdot x\right)\right) + \frac{1}{2}\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(x, \frac{1}{4} + x \cdot \left(\frac{1}{12} + \frac{1}{48} \cdot x\right), \frac{1}{2}\right)} \]

      4. +-commutative N/A

         \[\leadsto x \cdot \mathsf{fma}\left(x, \color{blue}{x \cdot \left(\frac{1}{12} + \frac{1}{48} \cdot x\right) + \frac{1}{4}}, \frac{1}{2}\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, \frac{1}{12} + \frac{1}{48} \cdot x, \frac{1}{4}\right)}, \frac{1}{2}\right) \]

      6. +-commutative N/A

         \[\leadsto x \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{\frac{1}{48} \cdot x + \frac{1}{12}}, \frac{1}{4}\right), \frac{1}{2}\right) \]

      7. *-commutative N/A

         \[\leadsto x \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{x \cdot \frac{1}{48}} + \frac{1}{12}, \frac{1}{4}\right), \frac{1}{2}\right) \]

      8. lower-fma.f64 79.9

         \[\leadsto x \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, 0.020833333333333332, 0.08333333333333333\right)}, 0.25\right), 0.5\right) \]

   8. Simplified 79.9%

      \[\leadsto \color{blue}{x \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.020833333333333332, 0.08333333333333333\right), 0.25\right), 0.5\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 88.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;e^{x\_m} - e^{-x\_m} \leq 0.002:\\ \;\;\;\;\mathsf{fma}\left(x\_m, x\_m \cdot 0.3333333333333333, 2\right) \cdot \left(x\_m \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot \mathsf{fma}\left(x\_m, \left(x\_m \cdot x\_m\right) \cdot 0.020833333333333332, 0.5\right)\\ \end{array} \end{array} \]

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))) 0.002)
    (* (fma x_m (* x_m 0.3333333333333333) 2.0) (* x_m 0.5))
    (* x_m (fma x_m (* (* x_m x_m) 0.020833333333333332) 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)) <= 0.002) {
		tmp = fma(x_m, (x_m * 0.3333333333333333), 2.0) * (x_m * 0.5);
	} else {
		tmp = x_m * fma(x_m, ((x_m * x_m) * 0.020833333333333332), 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))) <= 0.002)
		tmp = Float64(fma(x_m, Float64(x_m * 0.3333333333333333), 2.0) * Float64(x_m * 0.5));
	else
		tmp = Float64(x_m * fma(x_m, Float64(Float64(x_m * x_m) * 0.020833333333333332), 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], 0.002], N[(N[(x$95$m * N[(x$95$m * 0.3333333333333333), $MachinePrecision] + 2.0), $MachinePrecision] * N[(x$95$m * 0.5), $MachinePrecision]), $MachinePrecision], N[(x$95$m * N[(x$95$m * N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.020833333333333332), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 37.7%

      \[\frac{e^{x} - e^{-x}}{2} \]
   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. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 87.6

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

   5. Simplified 87.6%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lift-fma.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. div-inv N/A

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

      14. lower-*.f64 N/A

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

      15. metadata-eval 87.6

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

   7. Applied egg-rr 87.6%

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

   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

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{x \cdot \left(1 + x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right)\right)\right)}}{2} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{x \cdot \color{blue}{\left(x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right)\right) + 1\right)}}{2} \]

      2. distribute-lft-in N/A

         \[\leadsto \frac{\color{blue}{x \cdot \left(x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right)\right)\right) + x \cdot 1}}{2} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(x \cdot x\right) \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right)\right)} + x \cdot 1}{2} \]

      4. unpow2 N/A

         \[\leadsto \frac{\color{blue}{{x}^{2}} \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right)\right) + x \cdot 1}{2} \]

      5. *-rgt-identity N/A

         \[\leadsto \frac{{x}^{2} \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right)\right) + \color{blue}{x}}{2} \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{2} + x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right), x\right)}}{2} \]

      7. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{2} + x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right), x\right)}{2} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{2} + x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right), x\right)}{2} \]

      9. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(x \cdot x, \color{blue}{x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right) + \frac{1}{2}}, x\right)}{2} \]

      10. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left(x, \frac{1}{6} + \frac{1}{24} \cdot x, \frac{1}{2}\right)}, x\right)}{2} \]

      11. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \color{blue}{\frac{1}{24} \cdot x + \frac{1}{6}}, \frac{1}{2}\right), x\right)}{2} \]

      12. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \color{blue}{x \cdot \frac{1}{24}} + \frac{1}{6}, \frac{1}{2}\right), x\right)}{2} \]

      13. lower-fma.f64 79.9

          \[\leadsto \frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, 0.041666666666666664, 0.16666666666666666\right)}, 0.5\right), x\right)}{2} \]

   8. Simplified 79.9%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.041666666666666664, 0.16666666666666666\right), 0.5\right), x\right)}}{2} \]

   9. Taylor expanded in x around inf

      \[\leadsto \frac{\mathsf{fma}\left(x \cdot x, \color{blue}{\frac{1}{24} \cdot {x}^{2}}, x\right)}{2} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \frac{1}{24}}, x\right)}{2} \]

       2. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \frac{1}{24}}, x\right)}{2} \]

       3. unpow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(x \cdot x, \color{blue}{\left(x \cdot x\right)} \cdot \frac{1}{24}, x\right)}{2} \]

       4. lower-*.f64 79.9

          \[\leadsto \frac{\mathsf{fma}\left(x \cdot x, \color{blue}{\left(x \cdot x\right)} \cdot 0.041666666666666664, x\right)}{2} \]

   11. Simplified 79.9%

       \[\leadsto \frac{\mathsf{fma}\left(x \cdot x, \color{blue}{\left(x \cdot x\right) \cdot 0.041666666666666664}, x\right)}{2} \]

   12. Taylor expanded in x around 0

       \[\leadsto \color{blue}{x \cdot \left(\frac{1}{2} + \frac{1}{48} \cdot {x}^{3}\right)} \]
   13. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \color{blue}{x \cdot \left(\frac{1}{2} + \frac{1}{48} \cdot {x}^{3}\right)} \]

       2. +-commutative N/A

          \[\leadsto x \cdot \color{blue}{\left(\frac{1}{48} \cdot {x}^{3} + \frac{1}{2}\right)} \]

       3. unpow3 N/A

          \[\leadsto x \cdot \left(\frac{1}{48} \cdot \color{blue}{\left(\left(x \cdot x\right) \cdot x\right)} + \frac{1}{2}\right) \]

       4. unpow2 N/A

          \[\leadsto x \cdot \left(\frac{1}{48} \cdot \left(\color{blue}{{x}^{2}} \cdot x\right) + \frac{1}{2}\right) \]

       5. associate-*r* N/A

          \[\leadsto x \cdot \left(\color{blue}{\left(\frac{1}{48} \cdot {x}^{2}\right) \cdot x} + \frac{1}{2}\right) \]

       6. *-commutative N/A

          \[\leadsto x \cdot \left(\color{blue}{x \cdot \left(\frac{1}{48} \cdot {x}^{2}\right)} + \frac{1}{2}\right) \]

       7. lower-fma.f64 N/A

          \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(x, \frac{1}{48} \cdot {x}^{2}, \frac{1}{2}\right)} \]

       8. *-commutative N/A

          \[\leadsto x \cdot \mathsf{fma}\left(x, \color{blue}{{x}^{2} \cdot \frac{1}{48}}, \frac{1}{2}\right) \]

       9. lower-*.f64 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(x, \color{blue}{{x}^{2} \cdot \frac{1}{48}}, \frac{1}{2}\right) \]

       10. unpow2 N/A

           \[\leadsto x \cdot \mathsf{fma}\left(x, \color{blue}{\left(x \cdot x\right)} \cdot \frac{1}{48}, \frac{1}{2}\right) \]

       11. lower-*.f64 79.9

           \[\leadsto x \cdot \mathsf{fma}\left(x, \color{blue}{\left(x \cdot x\right)} \cdot 0.020833333333333332, 0.5\right) \]

   14. Simplified 79.9%

       \[\leadsto \color{blue}{x \cdot \mathsf{fma}\left(x, \left(x \cdot x\right) \cdot 0.020833333333333332, 0.5\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 76.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;e^{x\_m} - e^{-x\_m} \leq 0.002:\\ \;\;\;\;x\_m\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot \mathsf{fma}\left(x\_m, 0.25, 0.5\right)\\ \end{array} \end{array} \]

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))) 0.002) x_m (* x_m (fma x_m 0.25 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)) <= 0.002) {
		tmp = x_m;
	} else {
		tmp = x_m * fma(x_m, 0.25, 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))) <= 0.002)
		tmp = x_m;
	else
		tmp = Float64(x_m * fma(x_m, 0.25, 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], 0.002], x$95$m, N[(x$95$m * N[(x$95$m * 0.25 + 0.5), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 37.7%

      \[\frac{e^{x} - e^{-x}}{2} \]
   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 68.6

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

   5. Simplified 68.6%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. metadata-eval N/A

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

      4. lower-*.f64 68.6

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

   7. Applied egg-rr 68.6%

      \[\leadsto \color{blue}{x \cdot 1} \]
   8. Step-by-step derivation

      1. *-rgt-identity 68.6

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

   9. Applied egg-rr 68.6%

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

   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

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot \left(\frac{1}{2} + \frac{1}{4} \cdot x\right)} \]

      2. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left(\frac{1}{4} \cdot x + \frac{1}{2}\right)} \]

      3. *-commutative N/A

         \[\leadsto x \cdot \left(\color{blue}{x \cdot \frac{1}{4}} + \frac{1}{2}\right) \]

      4. lower-fma.f64 47.1

         \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(x, 0.25, 0.5\right)} \]

   8. Simplified 47.1%

      \[\leadsto \color{blue}{x \cdot \mathsf{fma}\left(x, 0.25, 0.5\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 94.4% accurate, 2.1× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := \mathsf{fma}\left(x\_m, x\_m \cdot 0.016666666666666666, 0.3333333333333333\right)\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 3 \cdot 10^{+77}:\\ \;\;\;\;\frac{\frac{x\_m \cdot \mathsf{fma}\left(t\_0, \left(x\_m \cdot x\_m\right) \cdot \left(t\_0 \cdot \left(x\_m \cdot x\_m\right)\right), -4\right)}{\mathsf{fma}\left(x\_m \cdot x\_m, t\_0, -2\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot \left(x\_m \cdot \left(\left(x\_m \cdot x\_m\right) \cdot 0.020833333333333332\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m)
 :precision binary64
 (let* ((t_0 (fma x_m (* x_m 0.016666666666666666) 0.3333333333333333)))
   (*
    x_s
    (if (<= x_m 3e+77)
      (/
       (/
        (* x_m (fma t_0 (* (* x_m x_m) (* t_0 (* x_m x_m))) -4.0))
        (fma (* x_m x_m) t_0 -2.0))
       2.0)
      (* x_m (* x_m (* (* x_m x_m) 0.020833333333333332)))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	double t_0 = fma(x_m, (x_m * 0.016666666666666666), 0.3333333333333333);
	double tmp;
	if (x_m <= 3e+77) {
		tmp = ((x_m * fma(t_0, ((x_m * x_m) * (t_0 * (x_m * x_m))), -4.0)) / fma((x_m * x_m), t_0, -2.0)) / 2.0;
	} else {
		tmp = x_m * (x_m * ((x_m * x_m) * 0.020833333333333332));
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m)
	t_0 = fma(x_m, Float64(x_m * 0.016666666666666666), 0.3333333333333333)
	tmp = 0.0
	if (x_m <= 3e+77)
		tmp = Float64(Float64(Float64(x_m * fma(t_0, Float64(Float64(x_m * x_m) * Float64(t_0 * Float64(x_m * x_m))), -4.0)) / fma(Float64(x_m * x_m), t_0, -2.0)) / 2.0);
	else
		tmp = Float64(x_m * Float64(x_m * Float64(Float64(x_m * x_m) * 0.020833333333333332)));
	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_] := Block[{t$95$0 = N[(x$95$m * N[(x$95$m * 0.016666666666666666), $MachinePrecision] + 0.3333333333333333), $MachinePrecision]}, N[(x$95$s * If[LessEqual[x$95$m, 3e+77], N[(N[(N[(x$95$m * N[(t$95$0 * N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(t$95$0 * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -4.0), $MachinePrecision]), $MachinePrecision] / N[(N[(x$95$m * x$95$m), $MachinePrecision] * t$95$0 + -2.0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(x$95$m * N[(x$95$m * N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.020833333333333332), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.9999999999999998e77

   1. Initial program 42.0%

      \[\frac{e^{x} - e^{-x}}{2} \]
   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. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 89.8

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

   5. Simplified 89.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-fma.f64 N/A

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

      7. flip-+ N/A

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

      8. associate-*l/ N/A

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

      9. lower-/.f64 N/A

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

   7. Applied egg-rr 72.7%

      \[\leadsto \frac{\color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(x, x \cdot 0.016666666666666666, 0.3333333333333333\right), \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \mathsf{fma}\left(x, x \cdot 0.016666666666666666, 0.3333333333333333\right)\right), -4\right) \cdot x}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.016666666666666666, 0.3333333333333333\right), -2\right)}}}{2} \]

   if 2.9999999999999998e77 < 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. Simplified 100.0%

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

   6. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{x \cdot \left(1 + x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right)\right)\right)}}{2} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{x \cdot \color{blue}{\left(x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right)\right) + 1\right)}}{2} \]

      2. distribute-lft-in N/A

         \[\leadsto \frac{\color{blue}{x \cdot \left(x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right)\right)\right) + x \cdot 1}}{2} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(x \cdot x\right) \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right)\right)} + x \cdot 1}{2} \]

      4. unpow2 N/A

         \[\leadsto \frac{\color{blue}{{x}^{2}} \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right)\right) + x \cdot 1}{2} \]

      5. *-rgt-identity N/A

         \[\leadsto \frac{{x}^{2} \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right)\right) + \color{blue}{x}}{2} \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{2} + x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right), x\right)}}{2} \]

      7. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{2} + x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right), x\right)}{2} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{2} + x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right), x\right)}{2} \]

      9. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(x \cdot x, \color{blue}{x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right) + \frac{1}{2}}, x\right)}{2} \]

      10. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left(x, \frac{1}{6} + \frac{1}{24} \cdot x, \frac{1}{2}\right)}, x\right)}{2} \]

      11. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \color{blue}{\frac{1}{24} \cdot x + \frac{1}{6}}, \frac{1}{2}\right), x\right)}{2} \]

      12. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \color{blue}{x \cdot \frac{1}{24}} + \frac{1}{6}, \frac{1}{2}\right), x\right)}{2} \]

      13. lower-fma.f64 100.0

          \[\leadsto \frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, 0.041666666666666664, 0.16666666666666666\right)}, 0.5\right), x\right)}{2} \]

   8. Simplified 100.0%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.041666666666666664, 0.16666666666666666\right), 0.5\right), x\right)}}{2} \]

   9. Taylor expanded in x around inf

      \[\leadsto \frac{\mathsf{fma}\left(x \cdot x, \color{blue}{\frac{1}{24} \cdot {x}^{2}}, x\right)}{2} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \frac{1}{24}}, x\right)}{2} \]

       2. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \frac{1}{24}}, x\right)}{2} \]

       3. unpow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(x \cdot x, \color{blue}{\left(x \cdot x\right)} \cdot \frac{1}{24}, x\right)}{2} \]

       4. lower-*.f64 100.0

          \[\leadsto \frac{\mathsf{fma}\left(x \cdot x, \color{blue}{\left(x \cdot x\right)} \cdot 0.041666666666666664, x\right)}{2} \]

   11. Simplified 100.0%

       \[\leadsto \frac{\mathsf{fma}\left(x \cdot x, \color{blue}{\left(x \cdot x\right) \cdot 0.041666666666666664}, x\right)}{2} \]

   12. Taylor expanded in x around inf

       \[\leadsto \color{blue}{\frac{1}{48} \cdot {x}^{4}} \]
   13. Step-by-step derivation

       1. metadata-eval N/A

          \[\leadsto \frac{1}{48} \cdot {x}^{\color{blue}{\left(3 + 1\right)}} \]

       2. pow-plus N/A

          \[\leadsto \frac{1}{48} \cdot \color{blue}{\left({x}^{3} \cdot x\right)} \]

       3. associate-*l* N/A

          \[\leadsto \color{blue}{\left(\frac{1}{48} \cdot {x}^{3}\right) \cdot x} \]

       4. *-commutative N/A

          \[\leadsto \color{blue}{x \cdot \left(\frac{1}{48} \cdot {x}^{3}\right)} \]

       5. lower-*.f64 N/A

          \[\leadsto \color{blue}{x \cdot \left(\frac{1}{48} \cdot {x}^{3}\right)} \]

       6. unpow3 N/A

          \[\leadsto x \cdot \left(\frac{1}{48} \cdot \color{blue}{\left(\left(x \cdot x\right) \cdot x\right)}\right) \]

       7. unpow2 N/A

          \[\leadsto x \cdot \left(\frac{1}{48} \cdot \left(\color{blue}{{x}^{2}} \cdot x\right)\right) \]

       8. associate-*r* N/A

          \[\leadsto x \cdot \color{blue}{\left(\left(\frac{1}{48} \cdot {x}^{2}\right) \cdot x\right)} \]

       9. *-commutative N/A

          \[\leadsto x \cdot \color{blue}{\left(x \cdot \left(\frac{1}{48} \cdot {x}^{2}\right)\right)} \]

       10. lower-*.f64 N/A

           \[\leadsto x \cdot \color{blue}{\left(x \cdot \left(\frac{1}{48} \cdot {x}^{2}\right)\right)} \]

       11. *-commutative N/A

           \[\leadsto x \cdot \left(x \cdot \color{blue}{\left({x}^{2} \cdot \frac{1}{48}\right)}\right) \]

       12. lower-*.f64 N/A

           \[\leadsto x \cdot \left(x \cdot \color{blue}{\left({x}^{2} \cdot \frac{1}{48}\right)}\right) \]

       13. unpow2 N/A

           \[\leadsto x \cdot \left(x \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot \frac{1}{48}\right)\right) \]

       14. lower-*.f64 100.0

           \[\leadsto x \cdot \left(x \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot 0.020833333333333332\right)\right) \]

   14. Simplified 100.0%

       \[\leadsto \color{blue}{x \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot 0.020833333333333332\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 78.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3 \cdot 10^{+77}:\\ \;\;\;\;\frac{\frac{x \cdot \mathsf{fma}\left(\mathsf{fma}\left(x, x \cdot 0.016666666666666666, 0.3333333333333333\right), \left(x \cdot x\right) \cdot \left(\mathsf{fma}\left(x, x \cdot 0.016666666666666666, 0.3333333333333333\right) \cdot \left(x \cdot x\right)\right), -4\right)}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.016666666666666666, 0.3333333333333333\right), -2\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot 0.020833333333333332\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 93.0% accurate, 3.9× speedup

Math

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

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
    (*
     x_m
     (fma
      x_m
      (* x_m (fma 0.0003968253968253968 (* x_m x_m) 0.016666666666666666))
      0.3333333333333333))
    (* x_m x_m)
    (* x_m 2.0))
   2.0)))

C

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

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m)
	return Float64(x_s * Float64(fma(Float64(x_m * fma(x_m, Float64(x_m * fma(0.0003968253968253968, Float64(x_m * x_m), 0.016666666666666666)), 0.3333333333333333)), Float64(x_m * x_m), Float64(x_m * 2.0)) / 2.0))
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[(x$95$m * N[(x$95$m * N[(x$95$m * N[(0.0003968253968253968 * N[(x$95$m * x$95$m), $MachinePrecision] + 0.016666666666666666), $MachinePrecision]), $MachinePrecision] + 0.3333333333333333), $MachinePrecision]), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + N[(x$95$m * 2.0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 53.8%

   \[\frac{e^{x} - e^{-x}}{2} \]
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. lower-*.f64 N/A

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

   2. +-commutative N/A

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

   3. lower-fma.f64 N/A

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

5. Simplified 94.1%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-fma.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-fma.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. associate-*l* N/A

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

   8. *-commutative N/A

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

   9. lower-fma.f64 N/A

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

7. Applied egg-rr 94.1%

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

   1. lift-*.f64 N/A

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

   2. lift-fma.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-fma.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. distribute-rgt-in N/A

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

   7. associate-*l* N/A

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

   8. lift-*.f64 N/A

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

   9. *-commutative N/A

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

   10. lift-*.f64 N/A

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

   11. lower-fma.f64 94.1

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

9. Applied egg-rr 94.1%

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

Alternative 8: 92.9% accurate, 4.1× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := x\_m \cdot \left(x\_m \cdot x\_m\right)\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 7.6:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(x\_m, x\_m \cdot 0.016666666666666666, 0.3333333333333333\right), t\_0, x\_m \cdot 2\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m \cdot \left(0.0003968253968253968 \cdot \left(x\_m \cdot \left(\left(x\_m \cdot x\_m\right) \cdot t\_0\right)\right)\right)}{2}\\ \end{array} \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m)
 :precision binary64
 (let* ((t_0 (* x_m (* x_m x_m))))
   (*
    x_s
    (if (<= x_m 7.6)
      (/
       (fma
        (fma x_m (* x_m 0.016666666666666666) 0.3333333333333333)
        t_0
        (* x_m 2.0))
       2.0)
      (/ (* x_m (* 0.0003968253968253968 (* x_m (* (* x_m x_m) t_0)))) 2.0)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	double t_0 = x_m * (x_m * x_m);
	double tmp;
	if (x_m <= 7.6) {
		tmp = fma(fma(x_m, (x_m * 0.016666666666666666), 0.3333333333333333), t_0, (x_m * 2.0)) / 2.0;
	} else {
		tmp = (x_m * (0.0003968253968253968 * (x_m * ((x_m * x_m) * t_0)))) / 2.0;
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m)
	t_0 = Float64(x_m * Float64(x_m * x_m))
	tmp = 0.0
	if (x_m <= 7.6)
		tmp = Float64(fma(fma(x_m, Float64(x_m * 0.016666666666666666), 0.3333333333333333), t_0, Float64(x_m * 2.0)) / 2.0);
	else
		tmp = Float64(Float64(x_m * Float64(0.0003968253968253968 * Float64(x_m * Float64(Float64(x_m * x_m) * t_0)))) / 2.0);
	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_] := Block[{t$95$0 = N[(x$95$m * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * If[LessEqual[x$95$m, 7.6], N[(N[(N[(x$95$m * N[(x$95$m * 0.016666666666666666), $MachinePrecision] + 0.3333333333333333), $MachinePrecision] * t$95$0 + N[(x$95$m * 2.0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(x$95$m * N[(0.0003968253968253968 * N[(x$95$m * N[(N[(x$95$m * x$95$m), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 7.5999999999999996

   1. Initial program 37.7%

      \[\frac{e^{x} - e^{-x}}{2} \]
   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. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 94.6

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

   5. Simplified 94.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. distribute-rgt-in N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. lift-*.f64 N/A

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

      8. pow3 N/A

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

      9. lift-*.f64 N/A

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

      10. lower-fma.f64 N/A

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

      11. cube-unmult N/A

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

      12. lift-*.f64 N/A

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

      13. lower-*.f64 94.6

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 94.6

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

   7. Applied egg-rr 94.6%

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

   if 7.5999999999999996 < 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. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

   5. Simplified 85.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-fma.f64 N/A

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

      6. lift-fma.f64 N/A

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

      7. distribute-rgt-in N/A

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

      8. associate-+l+ N/A

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

   7. Applied egg-rr 85.8%

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

   8. Taylor expanded in x around 0

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

   10. Simplified 85.8%

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

   11. Taylor expanded in x around inf

       \[\leadsto \frac{x \cdot \color{blue}{\left(\frac{1}{2520} \cdot {x}^{6}\right)}}{2} \]

   13. Simplified 85.8%

       \[\leadsto \frac{x \cdot \color{blue}{\left(0.0003968253968253968 \cdot \left(x \cdot \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right)}}{2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 92.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 7.6:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(x, x \cdot 0.016666666666666666, 0.3333333333333333\right), x \cdot \left(x \cdot x\right), x \cdot 2\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(0.0003968253968253968 \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 93.0% accurate, 4.3× speedup

Math

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

FPCore

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

C

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

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m)
	return Float64(x_s * Float64(Float64(x_m * fma(x_m, Float64(x_m * fma(x_m, Float64(x_m * fma(0.0003968253968253968, Float64(x_m * x_m), 0.016666666666666666)), 0.3333333333333333)), 2.0)) / 2.0))
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[(x$95$m * N[(x$95$m * N[(x$95$m * N[(x$95$m * N[(x$95$m * N[(0.0003968253968253968 * N[(x$95$m * x$95$m), $MachinePrecision] + 0.016666666666666666), $MachinePrecision]), $MachinePrecision] + 0.3333333333333333), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 53.8%

   \[\frac{e^{x} - e^{-x}}{2} \]
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. lower-*.f64 N/A

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

   2. +-commutative N/A

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

   3. lower-fma.f64 N/A

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

5. Simplified 94.1%

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

6. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. unpow2 N/A

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

   3. associate-*l* N/A

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

   4. lower-fma.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. +-commutative N/A

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

   7. unpow2 N/A

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

   8. associate-*l* N/A

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

   9. lower-fma.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. +-commutative N/A

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

   12. lower-fma.f64 N/A

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

   13. unpow2 N/A

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

   14. lower-*.f64 94.1

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

8. Simplified 94.1%

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

Alternative 10: 92.8% accurate, 4.4× speedup

Math

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

FPCore

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

C

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

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m)
	return Float64(x_s * Float64(Float64(x_m * fma(Float64(x_m * x_m), fma(x_m, Float64(Float64(x_m * Float64(x_m * x_m)) * 0.0003968253968253968), 0.3333333333333333), 2.0)) / 2.0))
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[(x$95$m * N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(x$95$m * N[(N[(x$95$m * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] * 0.0003968253968253968), $MachinePrecision] + 0.3333333333333333), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 53.8%

   \[\frac{e^{x} - e^{-x}}{2} \]
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. lower-*.f64 N/A

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

   2. +-commutative N/A

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

   3. lower-fma.f64 N/A

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

5. Simplified 94.1%

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

6. Taylor expanded in x around inf

   \[\leadsto \frac{x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \color{blue}{\frac{1}{2520} \cdot {x}^{3}}, \frac{1}{3}\right), 2\right)}{2} \]
7. Step-by-step derivation

   1. lower-*.f64 N/A

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

   2. cube-mult N/A

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

   3. unpow2 N/A

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

   4. lower-*.f64 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f64 94.1

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

8. Simplified 94.1%

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

9. Final simplification 94.1%

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

Alternative 11: 92.6% accurate, 4.5× speedup

Math

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

FPCore

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

C

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

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m)
	return Float64(x_s * Float64(Float64(x_m * fma(Float64(0.0003968253968253968 * Float64(x_m * Float64(x_m * Float64(x_m * Float64(x_m * x_m))))), x_m, 2.0)) / 2.0))
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[(x$95$m * N[(N[(0.0003968253968253968 * N[(x$95$m * N[(x$95$m * N[(x$95$m * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x$95$m + 2.0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 53.8%

   \[\frac{e^{x} - e^{-x}}{2} \]
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. lower-*.f64 N/A

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

   2. +-commutative N/A

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

   3. lower-fma.f64 N/A

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

5. Simplified 94.1%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-fma.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-fma.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. associate-*l* N/A

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

   8. *-commutative N/A

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

   9. lower-fma.f64 N/A

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

7. Applied egg-rr 94.1%

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

8. Taylor expanded in x around inf

   \[\leadsto \frac{x \cdot \mathsf{fma}\left(\color{blue}{\frac{1}{2520} \cdot {x}^{5}}, x, 2\right)}{2} \]
9. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \frac{x \cdot \mathsf{fma}\left(\color{blue}{\frac{1}{2520} \cdot {x}^{5}}, x, 2\right)}{2} \]

   2. metadata-eval N/A

      \[\leadsto \frac{x \cdot \mathsf{fma}\left(\frac{1}{2520} \cdot {x}^{\color{blue}{\left(4 + 1\right)}}, x, 2\right)}{2} \]

   3. pow-plus N/A

      \[\leadsto \frac{x \cdot \mathsf{fma}\left(\frac{1}{2520} \cdot \color{blue}{\left({x}^{4} \cdot x\right)}, x, 2\right)}{2} \]

   4. *-rgt-identity N/A

      \[\leadsto \frac{x \cdot \mathsf{fma}\left(\frac{1}{2520} \cdot \left({x}^{4} \cdot \color{blue}{\left(x \cdot 1\right)}\right), x, 2\right)}{2} \]

   5. rgt-mult-inverse N/A

      \[\leadsto \frac{x \cdot \mathsf{fma}\left(\frac{1}{2520} \cdot \left({x}^{4} \cdot \left(x \cdot \color{blue}{\left(x \cdot \frac{1}{x}\right)}\right)\right), x, 2\right)}{2} \]

   6. associate-*r/ N/A

      \[\leadsto \frac{x \cdot \mathsf{fma}\left(\frac{1}{2520} \cdot \left({x}^{4} \cdot \left(x \cdot \color{blue}{\frac{x \cdot 1}{x}}\right)\right), x, 2\right)}{2} \]

   7. *-rgt-identity N/A

      \[\leadsto \frac{x \cdot \mathsf{fma}\left(\frac{1}{2520} \cdot \left({x}^{4} \cdot \left(x \cdot \frac{\color{blue}{x}}{x}\right)\right), x, 2\right)}{2} \]

   8. associate-/l* N/A

      \[\leadsto \frac{x \cdot \mathsf{fma}\left(\frac{1}{2520} \cdot \left({x}^{4} \cdot \color{blue}{\frac{x \cdot x}{x}}\right), x, 2\right)}{2} \]

   9. unpow2 N/A

      \[\leadsto \frac{x \cdot \mathsf{fma}\left(\frac{1}{2520} \cdot \left({x}^{4} \cdot \frac{\color{blue}{{x}^{2}}}{x}\right), x, 2\right)}{2} \]

   10. associate-*r/ N/A

       \[\leadsto \frac{x \cdot \mathsf{fma}\left(\frac{1}{2520} \cdot \color{blue}{\frac{{x}^{4} \cdot {x}^{2}}{x}}, x, 2\right)}{2} \]

   11. metadata-eval N/A

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

   12. pow-sqr N/A

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

   13. unpow3 N/A

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

   14. cube-unmult N/A

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

   15. pow-sqr N/A

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

   16. metadata-eval N/A

       \[\leadsto \frac{x \cdot \mathsf{fma}\left(\frac{1}{2520} \cdot \frac{{x}^{2} \cdot {x}^{\color{blue}{4}}}{x}, x, 2\right)}{2} \]

   17. associate-*l/ N/A

       \[\leadsto \frac{x \cdot \mathsf{fma}\left(\frac{1}{2520} \cdot \color{blue}{\left(\frac{{x}^{2}}{x} \cdot {x}^{4}\right)}, x, 2\right)}{2} \]

   18. unpow2 N/A

       \[\leadsto \frac{x \cdot \mathsf{fma}\left(\frac{1}{2520} \cdot \left(\frac{\color{blue}{x \cdot x}}{x} \cdot {x}^{4}\right), x, 2\right)}{2} \]

   19. associate-/l* N/A

       \[\leadsto \frac{x \cdot \mathsf{fma}\left(\frac{1}{2520} \cdot \left(\color{blue}{\left(x \cdot \frac{x}{x}\right)} \cdot {x}^{4}\right), x, 2\right)}{2} \]

   20. *-rgt-identity N/A

       \[\leadsto \frac{x \cdot \mathsf{fma}\left(\frac{1}{2520} \cdot \left(\left(x \cdot \frac{\color{blue}{x \cdot 1}}{x}\right) \cdot {x}^{4}\right), x, 2\right)}{2} \]

   21. associate-*r/ N/A

       \[\leadsto \frac{x \cdot \mathsf{fma}\left(\frac{1}{2520} \cdot \left(\left(x \cdot \color{blue}{\left(x \cdot \frac{1}{x}\right)}\right) \cdot {x}^{4}\right), x, 2\right)}{2} \]

   22. rgt-mult-inverse N/A

       \[\leadsto \frac{x \cdot \mathsf{fma}\left(\frac{1}{2520} \cdot \left(\left(x \cdot \color{blue}{1}\right) \cdot {x}^{4}\right), x, 2\right)}{2} \]

   23. *-rgt-identity N/A

       \[\leadsto \frac{x \cdot \mathsf{fma}\left(\frac{1}{2520} \cdot \left(\color{blue}{x} \cdot {x}^{4}\right), x, 2\right)}{2} \]

   24. lower-*.f64 N/A

       \[\leadsto \frac{x \cdot \mathsf{fma}\left(\frac{1}{2520} \cdot \color{blue}{\left(x \cdot {x}^{4}\right)}, x, 2\right)}{2} \]

   25. metadata-eval N/A

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

   26. pow-sqr N/A

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

10. Simplified 93.8%

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

Alternative 12: 90.6% accurate, 4.9× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 3.3:\\ \;\;\;\;\mathsf{fma}\left(x\_m, x\_m \cdot 0.3333333333333333, 2\right) \cdot \left(x\_m \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m \cdot \left(x\_m \cdot \left(x\_m \cdot \mathsf{fma}\left(0.016666666666666666, x\_m \cdot x\_m, 0.3333333333333333\right)\right)\right)}{2}\\ \end{array} \end{array} \]

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 3.3)
    (* (fma x_m (* x_m 0.3333333333333333) 2.0) (* x_m 0.5))
    (/
     (*
      x_m
      (*
       x_m
       (* x_m (fma 0.016666666666666666 (* x_m x_m) 0.3333333333333333))))
     2.0))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	double tmp;
	if (x_m <= 3.3) {
		tmp = fma(x_m, (x_m * 0.3333333333333333), 2.0) * (x_m * 0.5);
	} else {
		tmp = (x_m * (x_m * (x_m * fma(0.016666666666666666, (x_m * x_m), 0.3333333333333333)))) / 2.0;
	}
	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 <= 3.3)
		tmp = Float64(fma(x_m, Float64(x_m * 0.3333333333333333), 2.0) * Float64(x_m * 0.5));
	else
		tmp = Float64(Float64(x_m * Float64(x_m * Float64(x_m * fma(0.016666666666666666, Float64(x_m * x_m), 0.3333333333333333)))) / 2.0);
	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, 3.3], N[(N[(x$95$m * N[(x$95$m * 0.3333333333333333), $MachinePrecision] + 2.0), $MachinePrecision] * N[(x$95$m * 0.5), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m * N[(x$95$m * N[(x$95$m * N[(0.016666666666666666 * N[(x$95$m * x$95$m), $MachinePrecision] + 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 3.2999999999999998

   1. Initial program 37.7%

      \[\frac{e^{x} - e^{-x}}{2} \]
   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. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 87.6

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

   5. Simplified 87.6%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lift-fma.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. div-inv N/A

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

      14. lower-*.f64 N/A

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

      15. metadata-eval 87.6

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

   7. Applied egg-rr 87.6%

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

   if 3.2999999999999998 < 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. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 84.1

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

   5. Simplified 84.1%

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

   6. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

      5. associate-*r/ N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. metadata-eval N/A

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

      9. pow-sqr N/A

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

      10. associate-/l* N/A

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

      11. *-rgt-identity N/A

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

      12. associate-*r/ N/A

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

      13. rgt-mult-inverse N/A

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

      14. *-rgt-identity N/A

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

      15. *-commutative N/A

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

      16. metadata-eval N/A

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

      17. pow-sqr N/A

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

      18. associate-*l* N/A

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

      19. distribute-rgt-in N/A

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

   8. Simplified 84.1%

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

Alternative 13: 90.7% accurate, 4.9× speedup

Math

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

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 x_m (* x_m 0.016666666666666666) 0.3333333333333333)
    (* x_m (* x_m x_m))
    (* x_m 2.0))
   2.0)))

C

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

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m)
	return Float64(x_s * Float64(fma(fma(x_m, Float64(x_m * 0.016666666666666666), 0.3333333333333333), Float64(x_m * Float64(x_m * x_m)), Float64(x_m * 2.0)) / 2.0))
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[(x$95$m * N[(x$95$m * 0.016666666666666666), $MachinePrecision] + 0.3333333333333333), $MachinePrecision] * N[(x$95$m * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] + N[(x$95$m * 2.0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 53.8%

   \[\frac{e^{x} - e^{-x}}{2} \]
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. lower-*.f64 N/A

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

   2. +-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. unpow2 N/A

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

   5. lower-*.f64 N/A

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

   6. +-commutative N/A

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

   7. unpow2 N/A

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

   8. associate-*r* N/A

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

   9. *-commutative N/A

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

   10. lower-fma.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 91.9

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

5. Simplified 91.9%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-fma.f64 N/A

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

   4. distribute-rgt-in N/A

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

   5. *-commutative N/A

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

   6. associate-*l* N/A

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

   7. lift-*.f64 N/A

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

   8. pow3 N/A

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

   9. lift-*.f64 N/A

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

   10. lower-fma.f64 N/A

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

   11. cube-unmult N/A

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

   12. lift-*.f64 N/A

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

   13. lower-*.f64 91.9

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

   14. lift-*.f64 N/A

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

   15. *-commutative N/A

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

   16. lower-*.f64 91.9

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

7. Applied egg-rr 91.9%

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

Alternative 14: 90.7% accurate, 5.0× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 5:\\ \;\;\;\;\mathsf{fma}\left(x\_m, x\_m \cdot 0.3333333333333333, 2\right) \cdot \left(x\_m \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.016666666666666666 \cdot \left(\left(x\_m \cdot x\_m\right) \cdot \left(x\_m \cdot \left(x\_m \cdot x\_m\right)\right)\right)}{2}\\ \end{array} \end{array} \]

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 5.0)
    (* (fma x_m (* x_m 0.3333333333333333) 2.0) (* x_m 0.5))
    (/ (* 0.016666666666666666 (* (* x_m x_m) (* x_m (* x_m x_m)))) 2.0))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	double tmp;
	if (x_m <= 5.0) {
		tmp = fma(x_m, (x_m * 0.3333333333333333), 2.0) * (x_m * 0.5);
	} else {
		tmp = (0.016666666666666666 * ((x_m * x_m) * (x_m * (x_m * x_m)))) / 2.0;
	}
	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 <= 5.0)
		tmp = Float64(fma(x_m, Float64(x_m * 0.3333333333333333), 2.0) * Float64(x_m * 0.5));
	else
		tmp = Float64(Float64(0.016666666666666666 * Float64(Float64(x_m * x_m) * Float64(x_m * Float64(x_m * x_m)))) / 2.0);
	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, 5.0], N[(N[(x$95$m * N[(x$95$m * 0.3333333333333333), $MachinePrecision] + 2.0), $MachinePrecision] * N[(x$95$m * 0.5), $MachinePrecision]), $MachinePrecision], N[(N[(0.016666666666666666 * N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(x$95$m * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 5

   1. Initial program 37.7%

      \[\frac{e^{x} - e^{-x}}{2} \]
   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. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 87.6

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

   5. Simplified 87.6%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lift-fma.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. div-inv N/A

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

      14. lower-*.f64 N/A

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

      15. metadata-eval 87.6

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

   7. Applied egg-rr 87.6%

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

   if 5 < 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. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 84.1

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

   5. Simplified 84.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. distribute-rgt-in N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. lift-*.f64 N/A

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

      8. pow3 N/A

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

      9. lift-*.f64 N/A

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

      10. lower-fma.f64 N/A

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

      11. cube-unmult N/A

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

      12. lift-*.f64 N/A

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

      13. lower-*.f64 84.1

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 84.1

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

   7. Applied egg-rr 84.1%

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

      1. associate-*r* N/A

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

      2. lift-*.f64 N/A

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

      3. lower-fma.f64 84.1

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

   9. Applied egg-rr 84.1%

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

   10. Taylor expanded in x around inf

       \[\leadsto \frac{\color{blue}{\frac{1}{60} \cdot {x}^{5}}}{2} \]

   12. Simplified 84.1%

       \[\leadsto \frac{\color{blue}{0.016666666666666666 \cdot \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)}}{2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 86.7%

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

Alternative 15: 90.7% accurate, 5.6× speedup

Math

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

FPCore

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

C

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

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m)
	return Float64(x_s * Float64(Float64(x_m * fma(Float64(x_m * x_m), fma(x_m, Float64(x_m * 0.016666666666666666), 0.3333333333333333), 2.0)) / 2.0))
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[(x$95$m * N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(x$95$m * N[(x$95$m * 0.016666666666666666), $MachinePrecision] + 0.3333333333333333), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 53.8%

   \[\frac{e^{x} - e^{-x}}{2} \]
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. lower-*.f64 N/A

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

   2. +-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. unpow2 N/A

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

   5. lower-*.f64 N/A

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

   6. +-commutative N/A

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

   7. unpow2 N/A

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

   8. associate-*r* N/A

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

   9. *-commutative N/A

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

   10. lower-fma.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 91.9

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

5. Simplified 91.9%

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

Alternative 16: 90.4% accurate, 5.7× speedup

Math

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

FPCore

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

C

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

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m)
	return Float64(x_s * Float64(Float64(x_m * fma(Float64(x_m * 0.016666666666666666), Float64(x_m * Float64(x_m * x_m)), 2.0)) / 2.0))
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[(x$95$m * N[(N[(x$95$m * 0.016666666666666666), $MachinePrecision] * N[(x$95$m * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 53.8%

   \[\frac{e^{x} - e^{-x}}{2} \]
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. lower-*.f64 N/A

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

   2. +-commutative N/A

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

   3. lower-fma.f64 N/A

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

5. Simplified 94.1%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-fma.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-fma.f64 N/A

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

   6. lift-fma.f64 N/A

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

   7. distribute-rgt-in N/A

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

   8. associate-+l+ N/A

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

7. Applied egg-rr 94.1%

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

8. Taylor expanded in x around 0

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

10. Simplified 93.8%

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

11. Taylor expanded in x around 0

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

13. Simplified 91.6%

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

Alternative 17: 88.2% accurate, 6.2× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 8:\\ \;\;\;\;\mathsf{fma}\left(x\_m, x\_m \cdot 0.3333333333333333, 2\right) \cdot \left(x\_m \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x\_m \cdot \left(x\_m \cdot \left(x\_m \cdot x\_m\right)\right), -0.041666666666666664, -x\_m\right) \cdot -0.5\\ \end{array} \end{array} \]

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 8.0)
    (* (fma x_m (* x_m 0.3333333333333333) 2.0) (* x_m 0.5))
    (* (fma (* x_m (* x_m (* x_m x_m))) -0.041666666666666664 (- 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 <= 8.0) {
		tmp = fma(x_m, (x_m * 0.3333333333333333), 2.0) * (x_m * 0.5);
	} else {
		tmp = fma((x_m * (x_m * (x_m * x_m))), -0.041666666666666664, -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 <= 8.0)
		tmp = Float64(fma(x_m, Float64(x_m * 0.3333333333333333), 2.0) * Float64(x_m * 0.5));
	else
		tmp = Float64(fma(Float64(x_m * Float64(x_m * Float64(x_m * x_m))), -0.041666666666666664, Float64(-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, 8.0], N[(N[(x$95$m * N[(x$95$m * 0.3333333333333333), $MachinePrecision] + 2.0), $MachinePrecision] * N[(x$95$m * 0.5), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x$95$m * N[(x$95$m * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.041666666666666664 + (-x$95$m)), $MachinePrecision] * -0.5), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 8

   1. Initial program 37.7%

      \[\frac{e^{x} - e^{-x}}{2} \]
   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. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 87.6

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

   5. Simplified 87.6%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lift-fma.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. div-inv N/A

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

      14. lower-*.f64 N/A

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

      15. metadata-eval 87.6

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

   7. Applied egg-rr 87.6%

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

   if 8 < 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. Simplified 100.0%

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

   6. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{x \cdot \left(1 + x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right)\right)\right)}}{2} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{x \cdot \color{blue}{\left(x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right)\right) + 1\right)}}{2} \]

      2. distribute-lft-in N/A

         \[\leadsto \frac{\color{blue}{x \cdot \left(x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right)\right)\right) + x \cdot 1}}{2} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(x \cdot x\right) \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right)\right)} + x \cdot 1}{2} \]

      4. unpow2 N/A

         \[\leadsto \frac{\color{blue}{{x}^{2}} \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right)\right) + x \cdot 1}{2} \]

      5. *-rgt-identity N/A

         \[\leadsto \frac{{x}^{2} \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right)\right) + \color{blue}{x}}{2} \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{2} + x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right), x\right)}}{2} \]

      7. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{2} + x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right), x\right)}{2} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{2} + x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right), x\right)}{2} \]

      9. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(x \cdot x, \color{blue}{x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right) + \frac{1}{2}}, x\right)}{2} \]

      10. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left(x, \frac{1}{6} + \frac{1}{24} \cdot x, \frac{1}{2}\right)}, x\right)}{2} \]

      11. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \color{blue}{\frac{1}{24} \cdot x + \frac{1}{6}}, \frac{1}{2}\right), x\right)}{2} \]

      12. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \color{blue}{x \cdot \frac{1}{24}} + \frac{1}{6}, \frac{1}{2}\right), x\right)}{2} \]

      13. lower-fma.f64 79.9

          \[\leadsto \frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, 0.041666666666666664, 0.16666666666666666\right)}, 0.5\right), x\right)}{2} \]

   8. Simplified 79.9%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.041666666666666664, 0.16666666666666666\right), 0.5\right), x\right)}}{2} \]

   9. Taylor expanded in x around inf

      \[\leadsto \frac{\mathsf{fma}\left(x \cdot x, \color{blue}{\frac{1}{24} \cdot {x}^{2}}, x\right)}{2} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \frac{1}{24}}, x\right)}{2} \]

       2. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \frac{1}{24}}, x\right)}{2} \]

       3. unpow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(x \cdot x, \color{blue}{\left(x \cdot x\right)} \cdot \frac{1}{24}, x\right)}{2} \]

       4. lower-*.f64 79.9

          \[\leadsto \frac{\mathsf{fma}\left(x \cdot x, \color{blue}{\left(x \cdot x\right)} \cdot 0.041666666666666664, x\right)}{2} \]

   11. Simplified 79.9%

       \[\leadsto \frac{\mathsf{fma}\left(x \cdot x, \color{blue}{\left(x \cdot x\right) \cdot 0.041666666666666664}, x\right)}{2} \]
   12. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \frac{\color{blue}{\left(x \cdot x\right)} \cdot \left(\left(x \cdot x\right) \cdot \frac{1}{24}\right) + x}{2} \]

       2. lift-*.f64 N/A

          \[\leadsto \frac{\left(x \cdot x\right) \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot \frac{1}{24}\right) + x}{2} \]

       3. lift-*.f64 N/A

          \[\leadsto \frac{\left(x \cdot x\right) \cdot \color{blue}{\left(\left(x \cdot x\right) \cdot \frac{1}{24}\right)} + x}{2} \]

       4. lift-fma.f64 N/A

          \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x \cdot x, \left(x \cdot x\right) \cdot \frac{1}{24}, x\right)}}{2} \]

       5. frac-2neg N/A

          \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(\mathsf{fma}\left(x \cdot x, \left(x \cdot x\right) \cdot \frac{1}{24}, x\right)\right)}{\mathsf{neg}\left(2\right)}} \]

       6. div-inv N/A

          \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\mathsf{fma}\left(x \cdot x, \left(x \cdot x\right) \cdot \frac{1}{24}, x\right)\right)\right) \cdot \frac{1}{\mathsf{neg}\left(2\right)}} \]

       7. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\mathsf{fma}\left(x \cdot x, \left(x \cdot x\right) \cdot \frac{1}{24}, x\right)\right)\right) \cdot \frac{1}{\mathsf{neg}\left(2\right)}} \]

   13. Applied egg-rr 79.9%

       \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right), -0.041666666666666664, -x\right) \cdot -0.5} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 18: 88.2% accurate, 7.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 8:\\ \;\;\;\;\mathsf{fma}\left(x\_m, \left(x\_m \cdot x\_m\right) \cdot 0.16666666666666666, x\_m\right)\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot \mathsf{fma}\left(x\_m, \left(x\_m \cdot x\_m\right) \cdot 0.020833333333333332, 0.5\right)\\ \end{array} \end{array} \]

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 8.0)
    (fma x_m (* (* x_m x_m) 0.16666666666666666) x_m)
    (* x_m (fma x_m (* (* x_m x_m) 0.020833333333333332) 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 <= 8.0) {
		tmp = fma(x_m, ((x_m * x_m) * 0.16666666666666666), x_m);
	} else {
		tmp = x_m * fma(x_m, ((x_m * x_m) * 0.020833333333333332), 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 <= 8.0)
		tmp = fma(x_m, Float64(Float64(x_m * x_m) * 0.16666666666666666), x_m);
	else
		tmp = Float64(x_m * fma(x_m, Float64(Float64(x_m * x_m) * 0.020833333333333332), 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, 8.0], N[(x$95$m * N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.16666666666666666), $MachinePrecision] + x$95$m), $MachinePrecision], N[(x$95$m * N[(x$95$m * N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.020833333333333332), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 8

   1. Initial program 37.7%

      \[\frac{e^{x} - e^{-x}}{2} \]
   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. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 87.6

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

   5. Simplified 87.6%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. distribute-lft-in N/A

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

      5. *-rgt-identity N/A

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

      6. lower-fma.f64 N/A

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

      7. associate-*r* N/A

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

      8. unpow2 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 87.6

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

   8. Simplified 87.6%

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

   if 8 < 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. Simplified 100.0%

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

   6. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{x \cdot \left(1 + x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right)\right)\right)}}{2} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{x \cdot \color{blue}{\left(x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right)\right) + 1\right)}}{2} \]

      2. distribute-lft-in N/A

         \[\leadsto \frac{\color{blue}{x \cdot \left(x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right)\right)\right) + x \cdot 1}}{2} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(x \cdot x\right) \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right)\right)} + x \cdot 1}{2} \]

      4. unpow2 N/A

         \[\leadsto \frac{\color{blue}{{x}^{2}} \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right)\right) + x \cdot 1}{2} \]

      5. *-rgt-identity N/A

         \[\leadsto \frac{{x}^{2} \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right)\right) + \color{blue}{x}}{2} \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{2} + x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right), x\right)}}{2} \]

      7. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{2} + x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right), x\right)}{2} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{2} + x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right), x\right)}{2} \]

      9. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(x \cdot x, \color{blue}{x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right) + \frac{1}{2}}, x\right)}{2} \]

      10. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left(x, \frac{1}{6} + \frac{1}{24} \cdot x, \frac{1}{2}\right)}, x\right)}{2} \]

      11. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \color{blue}{\frac{1}{24} \cdot x + \frac{1}{6}}, \frac{1}{2}\right), x\right)}{2} \]

      12. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \color{blue}{x \cdot \frac{1}{24}} + \frac{1}{6}, \frac{1}{2}\right), x\right)}{2} \]

      13. lower-fma.f64 79.9

          \[\leadsto \frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, 0.041666666666666664, 0.16666666666666666\right)}, 0.5\right), x\right)}{2} \]

   8. Simplified 79.9%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.041666666666666664, 0.16666666666666666\right), 0.5\right), x\right)}}{2} \]

   9. Taylor expanded in x around inf

      \[\leadsto \frac{\mathsf{fma}\left(x \cdot x, \color{blue}{\frac{1}{24} \cdot {x}^{2}}, x\right)}{2} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \frac{1}{24}}, x\right)}{2} \]

       2. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \frac{1}{24}}, x\right)}{2} \]

       3. unpow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(x \cdot x, \color{blue}{\left(x \cdot x\right)} \cdot \frac{1}{24}, x\right)}{2} \]

       4. lower-*.f64 79.9

          \[\leadsto \frac{\mathsf{fma}\left(x \cdot x, \color{blue}{\left(x \cdot x\right)} \cdot 0.041666666666666664, x\right)}{2} \]

   11. Simplified 79.9%

       \[\leadsto \frac{\mathsf{fma}\left(x \cdot x, \color{blue}{\left(x \cdot x\right) \cdot 0.041666666666666664}, x\right)}{2} \]

   12. Taylor expanded in x around 0

       \[\leadsto \color{blue}{x \cdot \left(\frac{1}{2} + \frac{1}{48} \cdot {x}^{3}\right)} \]
   13. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \color{blue}{x \cdot \left(\frac{1}{2} + \frac{1}{48} \cdot {x}^{3}\right)} \]

       2. +-commutative N/A

          \[\leadsto x \cdot \color{blue}{\left(\frac{1}{48} \cdot {x}^{3} + \frac{1}{2}\right)} \]

       3. unpow3 N/A

          \[\leadsto x \cdot \left(\frac{1}{48} \cdot \color{blue}{\left(\left(x \cdot x\right) \cdot x\right)} + \frac{1}{2}\right) \]

       4. unpow2 N/A

          \[\leadsto x \cdot \left(\frac{1}{48} \cdot \left(\color{blue}{{x}^{2}} \cdot x\right) + \frac{1}{2}\right) \]

       5. associate-*r* N/A

          \[\leadsto x \cdot \left(\color{blue}{\left(\frac{1}{48} \cdot {x}^{2}\right) \cdot x} + \frac{1}{2}\right) \]

       6. *-commutative N/A

          \[\leadsto x \cdot \left(\color{blue}{x \cdot \left(\frac{1}{48} \cdot {x}^{2}\right)} + \frac{1}{2}\right) \]

       7. lower-fma.f64 N/A

          \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(x, \frac{1}{48} \cdot {x}^{2}, \frac{1}{2}\right)} \]

       8. *-commutative N/A

          \[\leadsto x \cdot \mathsf{fma}\left(x, \color{blue}{{x}^{2} \cdot \frac{1}{48}}, \frac{1}{2}\right) \]

       9. lower-*.f64 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(x, \color{blue}{{x}^{2} \cdot \frac{1}{48}}, \frac{1}{2}\right) \]

       10. unpow2 N/A

           \[\leadsto x \cdot \mathsf{fma}\left(x, \color{blue}{\left(x \cdot x\right)} \cdot \frac{1}{48}, \frac{1}{2}\right) \]

       11. lower-*.f64 79.9

           \[\leadsto x \cdot \mathsf{fma}\left(x, \color{blue}{\left(x \cdot x\right)} \cdot 0.020833333333333332, 0.5\right) \]

   14. Simplified 79.9%

       \[\leadsto \color{blue}{x \cdot \mathsf{fma}\left(x, \left(x \cdot x\right) \cdot 0.020833333333333332, 0.5\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 19: 88.2% accurate, 8.0× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 8.5:\\ \;\;\;\;\mathsf{fma}\left(x\_m, \left(x\_m \cdot x\_m\right) \cdot 0.16666666666666666, x\_m\right)\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot \left(x\_m \cdot \left(\left(x\_m \cdot x\_m\right) \cdot 0.020833333333333332\right)\right)\\ \end{array} \end{array} \]

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 8.5)
    (fma x_m (* (* x_m x_m) 0.16666666666666666) x_m)
    (* x_m (* x_m (* (* x_m x_m) 0.020833333333333332))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	double tmp;
	if (x_m <= 8.5) {
		tmp = fma(x_m, ((x_m * x_m) * 0.16666666666666666), x_m);
	} else {
		tmp = x_m * (x_m * ((x_m * x_m) * 0.020833333333333332));
	}
	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 <= 8.5)
		tmp = fma(x_m, Float64(Float64(x_m * x_m) * 0.16666666666666666), x_m);
	else
		tmp = Float64(x_m * Float64(x_m * Float64(Float64(x_m * x_m) * 0.020833333333333332)));
	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, 8.5], N[(x$95$m * N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.16666666666666666), $MachinePrecision] + x$95$m), $MachinePrecision], N[(x$95$m * N[(x$95$m * N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.020833333333333332), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 8.5

   1. Initial program 37.7%

      \[\frac{e^{x} - e^{-x}}{2} \]
   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. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 87.6

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

   5. Simplified 87.6%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. distribute-lft-in N/A

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

      5. *-rgt-identity N/A

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

      6. lower-fma.f64 N/A

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

      7. associate-*r* N/A

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

      8. unpow2 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 87.6

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

   8. Simplified 87.6%

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

   if 8.5 < 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. Simplified 100.0%

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

   6. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{x \cdot \left(1 + x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right)\right)\right)}}{2} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{x \cdot \color{blue}{\left(x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right)\right) + 1\right)}}{2} \]

      2. distribute-lft-in N/A

         \[\leadsto \frac{\color{blue}{x \cdot \left(x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right)\right)\right) + x \cdot 1}}{2} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(x \cdot x\right) \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right)\right)} + x \cdot 1}{2} \]

      4. unpow2 N/A

         \[\leadsto \frac{\color{blue}{{x}^{2}} \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right)\right) + x \cdot 1}{2} \]

      5. *-rgt-identity N/A

         \[\leadsto \frac{{x}^{2} \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right)\right) + \color{blue}{x}}{2} \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{2} + x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right), x\right)}}{2} \]

      7. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{2} + x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right), x\right)}{2} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{2} + x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right), x\right)}{2} \]

      9. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(x \cdot x, \color{blue}{x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right) + \frac{1}{2}}, x\right)}{2} \]

      10. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left(x, \frac{1}{6} + \frac{1}{24} \cdot x, \frac{1}{2}\right)}, x\right)}{2} \]

      11. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \color{blue}{\frac{1}{24} \cdot x + \frac{1}{6}}, \frac{1}{2}\right), x\right)}{2} \]

      12. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \color{blue}{x \cdot \frac{1}{24}} + \frac{1}{6}, \frac{1}{2}\right), x\right)}{2} \]

      13. lower-fma.f64 79.9

          \[\leadsto \frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, 0.041666666666666664, 0.16666666666666666\right)}, 0.5\right), x\right)}{2} \]

   8. Simplified 79.9%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.041666666666666664, 0.16666666666666666\right), 0.5\right), x\right)}}{2} \]

   9. Taylor expanded in x around inf

      \[\leadsto \frac{\mathsf{fma}\left(x \cdot x, \color{blue}{\frac{1}{24} \cdot {x}^{2}}, x\right)}{2} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \frac{1}{24}}, x\right)}{2} \]

       2. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \frac{1}{24}}, x\right)}{2} \]

       3. unpow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(x \cdot x, \color{blue}{\left(x \cdot x\right)} \cdot \frac{1}{24}, x\right)}{2} \]

       4. lower-*.f64 79.9

          \[\leadsto \frac{\mathsf{fma}\left(x \cdot x, \color{blue}{\left(x \cdot x\right)} \cdot 0.041666666666666664, x\right)}{2} \]

   11. Simplified 79.9%

       \[\leadsto \frac{\mathsf{fma}\left(x \cdot x, \color{blue}{\left(x \cdot x\right) \cdot 0.041666666666666664}, x\right)}{2} \]

   12. Taylor expanded in x around inf

       \[\leadsto \color{blue}{\frac{1}{48} \cdot {x}^{4}} \]
   13. Step-by-step derivation

       1. metadata-eval N/A

          \[\leadsto \frac{1}{48} \cdot {x}^{\color{blue}{\left(3 + 1\right)}} \]

       2. pow-plus N/A

          \[\leadsto \frac{1}{48} \cdot \color{blue}{\left({x}^{3} \cdot x\right)} \]

       3. associate-*l* N/A

          \[\leadsto \color{blue}{\left(\frac{1}{48} \cdot {x}^{3}\right) \cdot x} \]

       4. *-commutative N/A

          \[\leadsto \color{blue}{x \cdot \left(\frac{1}{48} \cdot {x}^{3}\right)} \]

       5. lower-*.f64 N/A

          \[\leadsto \color{blue}{x \cdot \left(\frac{1}{48} \cdot {x}^{3}\right)} \]

       6. unpow3 N/A

          \[\leadsto x \cdot \left(\frac{1}{48} \cdot \color{blue}{\left(\left(x \cdot x\right) \cdot x\right)}\right) \]

       7. unpow2 N/A

          \[\leadsto x \cdot \left(\frac{1}{48} \cdot \left(\color{blue}{{x}^{2}} \cdot x\right)\right) \]

       8. associate-*r* N/A

          \[\leadsto x \cdot \color{blue}{\left(\left(\frac{1}{48} \cdot {x}^{2}\right) \cdot x\right)} \]

       9. *-commutative N/A

          \[\leadsto x \cdot \color{blue}{\left(x \cdot \left(\frac{1}{48} \cdot {x}^{2}\right)\right)} \]

       10. lower-*.f64 N/A

           \[\leadsto x \cdot \color{blue}{\left(x \cdot \left(\frac{1}{48} \cdot {x}^{2}\right)\right)} \]

       11. *-commutative N/A

           \[\leadsto x \cdot \left(x \cdot \color{blue}{\left({x}^{2} \cdot \frac{1}{48}\right)}\right) \]

       12. lower-*.f64 N/A

           \[\leadsto x \cdot \left(x \cdot \color{blue}{\left({x}^{2} \cdot \frac{1}{48}\right)}\right) \]

       13. unpow2 N/A

           \[\leadsto x \cdot \left(x \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot \frac{1}{48}\right)\right) \]

       14. lower-*.f64 79.9

           \[\leadsto x \cdot \left(x \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot 0.020833333333333332\right)\right) \]

   14. Simplified 79.9%

       \[\leadsto \color{blue}{x \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot 0.020833333333333332\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 20: 84.1% accurate, 9.9× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 2.4:\\ \;\;\;\;x\_m\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot \left(\left(x\_m \cdot x\_m\right) \cdot 0.16666666666666666\right)\\ \end{array} \end{array} \]

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 2.4) x_m (* x_m (* (* x_m x_m) 0.16666666666666666)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	double tmp;
	if (x_m <= 2.4) {
		tmp = x_m;
	} else {
		tmp = x_m * ((x_m * x_m) * 0.16666666666666666);
	}
	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 (x_m <= 2.4d0) then
        tmp = x_m
    else
        tmp = x_m * ((x_m * x_m) * 0.16666666666666666d0)
    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 (x_m <= 2.4) {
		tmp = x_m;
	} else {
		tmp = x_m * ((x_m * x_m) * 0.16666666666666666);
	}
	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 x_m <= 2.4:
		tmp = x_m
	else:
		tmp = x_m * ((x_m * x_m) * 0.16666666666666666)
	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 <= 2.4)
		tmp = x_m;
	else
		tmp = Float64(x_m * Float64(Float64(x_m * x_m) * 0.16666666666666666));
	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 (x_m <= 2.4)
		tmp = x_m;
	else
		tmp = x_m * ((x_m * x_m) * 0.16666666666666666);
	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[x$95$m, 2.4], x$95$m, N[(x$95$m * N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 2.39999999999999991

   1. Initial program 37.7%

      \[\frac{e^{x} - e^{-x}}{2} \]
   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 68.6

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

   5. Simplified 68.6%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. metadata-eval N/A

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

      4. lower-*.f64 68.6

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

   7. Applied egg-rr 68.6%

      \[\leadsto \color{blue}{x \cdot 1} \]
   8. Step-by-step derivation

      1. *-rgt-identity 68.6

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

   9. Applied egg-rr 68.6%

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

   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. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 71.4

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

   5. Simplified 71.4%

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

   6. Taylor expanded in x around inf

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

      1. cube-mult N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. associate-*r* N/A

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

      9. unpow2 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 71.4

          \[\leadsto x \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot 0.16666666666666666\right) \]

   8. Simplified 71.4%

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

Alternative 21: 84.3% accurate, 12.8× speedup

Math

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

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 x_m (* (* x_m x_m) 0.16666666666666666) x_m)))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	return x_s * fma(x_m, ((x_m * x_m) * 0.16666666666666666), x_m);
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m)
	return Float64(x_s * fma(x_m, Float64(Float64(x_m * x_m) * 0.16666666666666666), 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[(x$95$m * N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.16666666666666666), $MachinePrecision] + x$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 53.8%

   \[\frac{e^{x} - e^{-x}}{2} \]
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. lower-*.f64 N/A

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

   2. +-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. unpow2 N/A

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

   5. lower-*.f64 83.4

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

5. Simplified 83.4%

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

6. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. unpow2 N/A

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

   3. associate-*r* N/A

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

   4. distribute-lft-in N/A

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

   5. *-rgt-identity N/A

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

   6. lower-fma.f64 N/A

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

   7. associate-*r* N/A

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

   8. unpow2 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 N/A

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

   11. unpow2 N/A

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

   12. lower-*.f64 83.4

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

8. Simplified 83.4%

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

Alternative 22: 51.6% accurate, 217.0× speedup

Math

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

FPCore

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

C

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

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 * x_m
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 * x_m;
}

Python

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

Julia

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

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp = code(x_s, x_m)
	tmp = x_s * 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 * x$95$m), $MachinePrecision]

Derivation

1. Initial program 53.8%

   \[\frac{e^{x} - e^{-x}}{2} \]
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 52.3

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

5. Simplified 52.3%

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

   1. *-commutative N/A

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

   2. associate-/l* N/A

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

   3. metadata-eval N/A

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

   4. lower-*.f64 52.3

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

7. Applied egg-rr 52.3%

   \[\leadsto \color{blue}{x \cdot 1} \]
8. Step-by-step derivation

   1. *-rgt-identity 52.3

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

9. Applied egg-rr 52.3%

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

Reproduce

herbie shell --seed 2024215 
(FPCore (x)
  :name "Hyperbolic sine"
  :precision binary64
  (/ (- (exp x) (exp (- x))) 2.0))