Hyperbolic sine

Percentage Accurate: 54.7% → 99.5%

Time: 12.9s

Alternatives: 19

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

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

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
      (fma
       x_m
       (* x_m (fma x_m (* x_m 0.016666666666666666) 0.3333333333333333))
       2.0))
     2.0)
    (* 0.5 (expm1 x_m)))))

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

      \[\leadsto \frac{\color{blue}{x \cdot \left(2 + {x}^{2} \cdot \left(\frac{1}{3} + {x}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {x}^{2}\right)\right)\right)}}{2} \]
   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 97.0%

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

   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 97.0

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

   8. Simplified 97.0%

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

   9. Taylor expanded in x around 0

      \[\leadsto \frac{x \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot \color{blue}{\frac{1}{60}}, \frac{1}{3}\right), 2\right)}{2} \]
   10. Step-by-step derivation

   11. Simplified 94.6%

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

   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. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-expm1.f64 100.0

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

   8. Simplified 100.0%

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

Alternative 2: 88.0% accurate, 0.9× speedup

Math

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

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 (fma 0.3333333333333333 (* x_m x_m) 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 tmp;
	if ((exp(x_m) - exp(-x_m)) <= 0.002) {
		tmp = (x_m * fma(0.3333333333333333, (x_m * x_m), 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)
	tmp = 0.0
	if (Float64(exp(x_m) - exp(Float64(-x_m))) <= 0.002)
		tmp = Float64(Float64(x_m * fma(0.3333333333333333, Float64(x_m * x_m), 2.0)) / 2.0);
	else
		tmp = Float64(Float64(x_m * Float64(x_m * 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[N[(N[Exp[x$95$m], $MachinePrecision] - N[Exp[(-x$95$m)], $MachinePrecision]), $MachinePrecision], 0.002], N[(N[(x$95$m * N[(0.3333333333333333 * N[(x$95$m * x$95$m), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(x$95$m * N[(x$95$m * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.020833333333333332), $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%

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

   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%

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

   9. Taylor expanded in x around inf

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

       1. *-commutative N/A

          \[\leadsto \color{blue}{{x}^{4} \cdot \frac{1}{48}} \]

       2. lower-*.f64 N/A

          \[\leadsto \color{blue}{{x}^{4} \cdot \frac{1}{48}} \]

       3. metadata-eval N/A

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

       4. pow-sqr N/A

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

       5. unpow2 N/A

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

       6. associate-*l* N/A

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

       7. unpow2 N/A

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

       8. cube-mult N/A

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

       9. lower-*.f64 N/A

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

       10. cube-mult N/A

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

       11. unpow2 N/A

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

       12. lower-*.f64 N/A

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

       13. unpow2 N/A

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

       14. lower-*.f64 79.9

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

   11. Simplified 79.9%

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

Alternative 3: 44.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;e^{x\_m} - e^{-x\_m} \leq 0.002:\\ \;\;\;\;x\_m \cdot \mathsf{fma}\left(x\_m, 0.25, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot \left(\left(x\_m \cdot x\_m\right) \cdot 0.16666666666666666\right)\\ \end{array} \end{array} \]

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 (<= (- (exp x_m) (exp (- x_m))) 0.002)
    (* x_m (fma x_m 0.25 0.5))
    (* 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 ((exp(x_m) - exp(-x_m)) <= 0.002) {
		tmp = x_m * fma(x_m, 0.25, 0.5);
	} 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 (Float64(exp(x_m) - exp(Float64(-x_m))) <= 0.002)
		tmp = Float64(x_m * fma(x_m, 0.25, 0.5));
	else
		tmp = Float64(x_m * Float64(Float64(x_m * x_m) * 0.16666666666666666));
	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[(x$95$m * N[(x$95$m * 0.25 + 0.5), $MachinePrecision]), $MachinePrecision], 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 (-.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

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

   5. Simplified 5.1%

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

   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 12.7

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

   8. Simplified 12.7%

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

   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

      \[\leadsto \frac{\color{blue}{x \cdot \left(2 + \frac{1}{3} \cdot {x}^{2}\right)}}{2} \]
   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 4: 36.4% accurate, 1.0× speedup

Math

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

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 0.5)
    (* x_m (* x_m 0.25)))))

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 * 0.5;
	} else {
		tmp = x_m * (x_m * 0.25);
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8) :: tmp
    if ((exp(x_m) - exp(-x_m)) <= 0.002d0) then
        tmp = x_m * 0.5d0
    else
        tmp = x_m * (x_m * 0.25d0)
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m) {
	double tmp;
	if ((Math.exp(x_m) - Math.exp(-x_m)) <= 0.002) {
		tmp = x_m * 0.5;
	} else {
		tmp = x_m * (x_m * 0.25);
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m):
	tmp = 0
	if (math.exp(x_m) - math.exp(-x_m)) <= 0.002:
		tmp = x_m * 0.5
	else:
		tmp = x_m * (x_m * 0.25)
	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(x_m * 0.5);
	else
		tmp = Float64(x_m * Float64(x_m * 0.25));
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m)
	tmp = 0.0;
	if ((exp(x_m) - exp(-x_m)) <= 0.002)
		tmp = x_m * 0.5;
	else
		tmp = x_m * (x_m * 0.25);
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_] := N[(x$95$s * If[LessEqual[N[(N[Exp[x$95$m], $MachinePrecision] - N[Exp[(-x$95$m)], $MachinePrecision]), $MachinePrecision], 0.002], N[(x$95$m * 0.5), $MachinePrecision], N[(x$95$m * N[(x$95$m * 0.25), $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

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

   5. Simplified 5.1%

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

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot x} \]
   7. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-*.f64 14.3

         \[\leadsto \color{blue}{x \cdot 0.5} \]

   8. Simplified 14.3%

      \[\leadsto \color{blue}{x \cdot 0.5} \]

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

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

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

   9. Taylor expanded in x around inf

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

       1. unpow2 N/A

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

       2. associate-*r* N/A

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

       3. *-commutative N/A

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

       4. lower-*.f64 N/A

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

       5. *-commutative N/A

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

       6. lower-*.f64 47.1

          \[\leadsto x \cdot \color{blue}{\left(x \cdot 0.25\right)} \]

   11. Simplified 47.1%

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

Alternative 5: 92.9% accurate, 4.1× speedup

Math

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

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 7.6)
    (/
     (*
      x_m
      (fma
       x_m
       (* x_m (fma x_m (* x_m 0.016666666666666666) 0.3333333333333333))
       2.0))
     2.0)
    (/
     (*
      x_m
      (* 0.0003968253968253968 (* x_m (* 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 <= 7.6) {
		tmp = (x_m * fma(x_m, (x_m * fma(x_m, (x_m * 0.016666666666666666), 0.3333333333333333)), 2.0)) / 2.0;
	} else {
		tmp = (x_m * (0.0003968253968253968 * (x_m * (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 <= 7.6)
		tmp = Float64(Float64(x_m * fma(x_m, Float64(x_m * fma(x_m, Float64(x_m * 0.016666666666666666), 0.3333333333333333)), 2.0)) / 2.0);
	else
		tmp = Float64(Float64(x_m * Float64(0.0003968253968253968 * Float64(x_m * Float64(x_m * Float64(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, 7.6], N[(N[(x$95$m * N[(x$95$m * N[(x$95$m * N[(x$95$m * N[(x$95$m * 0.016666666666666666), $MachinePrecision] + 0.3333333333333333), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(x$95$m * N[(0.0003968253968253968 * N[(x$95$m * N[(x$95$m * N[(x$95$m * N[(x$95$m * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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

      \[\leadsto \frac{\color{blue}{x \cdot \left(2 + {x}^{2} \cdot \left(\frac{1}{3} + {x}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {x}^{2}\right)\right)\right)}}{2} \]
   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 97.0%

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

   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 97.0

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

   8. Simplified 97.0%

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

   9. Taylor expanded in x around 0

      \[\leadsto \frac{x \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot \color{blue}{\frac{1}{60}}, \frac{1}{3}\right), 2\right)}{2} \]
   10. Step-by-step derivation

   11. Simplified 94.6%

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

   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. 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 85.8

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

   8. Simplified 85.8%

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

   9. Taylor expanded in x around inf

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

       1. lower-*.f64 N/A

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

       2. unpow2 N/A

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

       3. lower-*.f64 85.8

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

   11. Simplified 85.8%

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

   12. Taylor expanded in x around inf

       \[\leadsto \frac{\color{blue}{\frac{1}{2520} \cdot {x}^{7}}}{2} \]
   13. Step-by-step derivation

       1. metadata-eval N/A

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

       2. pow-plus N/A

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

       3. metadata-eval N/A

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

       4. pow-plus N/A

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

       5. metadata-eval N/A

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

       6. pow-plus N/A

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

       7. associate-*r* N/A

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

       8. unpow2 N/A

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

       9. associate-*l* N/A

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

       10. associate-*l* N/A

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

       11. *-commutative N/A

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

       12. *-commutative N/A

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

       13. lower-*.f64 N/A

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

       14. associate-*r* N/A

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

       15. *-commutative N/A

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

       16. associate-*l* N/A

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

       17. lower-*.f64 N/A

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

       18. unpow2 N/A

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

       19. associate-*l* N/A

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

       20. *-commutative N/A

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

       21. pow-plus N/A

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

       22. metadata-eval N/A

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

       23. lower-*.f64 N/A

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

       24. metadata-eval N/A

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

       25. pow-plus N/A

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

       26. *-commutative N/A

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

       27. lower-*.f64 N/A

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

   14. Simplified 85.8%

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

Alternative 6: 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 7: 92.8% accurate, 4.4× speedup

Math

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

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
       (* 0.0003968253968253968 (* x_m (* x_m x_m)))
       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, (0.0003968253968253968 * (x_m * (x_m * x_m))), 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(0.0003968253968253968 * Float64(x_m * Float64(x_m * x_m))), 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[(0.0003968253968253968 * N[(x$95$m * N[(x$95$m * x$95$m), $MachinePrecision]), $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. Taylor expanded in x around inf

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

    1. lower-*.f64 N/A

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

    2. unpow2 N/A

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

    3. lower-*.f64 94.1

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

11. Simplified 94.1%

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

12. Taylor expanded in x around 0

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

    1. lower-*.f64 N/A

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

    2. cube-mult N/A

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

    3. unpow2 N/A

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

    4. lower-*.f64 N/A

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

    5. unpow2 N/A

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

    6. lower-*.f64 94.1

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

14. Simplified 94.1%

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

Alternative 8: 92.6% accurate, 4.5× speedup

Math

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

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
     (* 0.0003968253968253968 (* (* 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(x_m, (0.0003968253968253968 * ((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(x_m, Float64(0.0003968253968253968 * Float64(Float64(x_m * x_m) * 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[(x$95$m * N[(0.0003968253968253968 * 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] / 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. Taylor expanded in x around inf

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

    1. lower-*.f64 N/A

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

    2. metadata-eval N/A

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

    3. pow-plus N/A

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

    4. metadata-eval N/A

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

    5. pow-sqr N/A

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

    6. associate-*r* N/A

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

    7. unpow2 N/A

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

    8. unpow3 N/A

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

    9. lower-*.f64 N/A

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

    10. unpow2 N/A

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

    11. lower-*.f64 N/A

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

    12. cube-mult N/A

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

    13. unpow2 N/A

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

    14. lower-*.f64 N/A

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

    15. unpow2 N/A

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

    16. lower-*.f64 93.8

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

11. Simplified 93.8%

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

Alternative 9: 90.6% accurate, 4.9× speedup

Math

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

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)
    (/ (* x_m (fma 0.3333333333333333 (* x_m x_m) 2.0)) 2.0)
    (/
     (*
      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 = (x_m * fma(0.3333333333333333, (x_m * x_m), 2.0)) / 2.0;
	} 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(Float64(x_m * fma(0.3333333333333333, Float64(x_m * x_m), 2.0)) / 2.0);
	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[(0.3333333333333333 * N[(x$95$m * x$95$m), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] / 2.0), $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%

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

   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 10: 90.7% accurate, 5.0× speedup

Math

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

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)
    (/ (* x_m (fma 0.3333333333333333 (* x_m x_m) 2.0)) 2.0)
    (/ (* 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 = (x_m * fma(0.3333333333333333, (x_m * x_m), 2.0)) / 2.0;
	} 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(Float64(x_m * fma(0.3333333333333333, Float64(x_m * x_m), 2.0)) / 2.0);
	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[(0.3333333333333333 * N[(x$95$m * x$95$m), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] / 2.0), $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%

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

   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. 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%

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

   9. Taylor expanded in x around inf

      \[\leadsto \frac{\color{blue}{\frac{1}{60} \cdot {x}^{5}}}{2} \]
   10. Step-by-step derivation

       1. lower-*.f64 N/A

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

       2. metadata-eval N/A

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

       3. pow-plus N/A

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

       4. metadata-eval N/A

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

       5. pow-sqr N/A

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

       6. associate-*r* N/A

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

       7. unpow2 N/A

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

       8. unpow3 N/A

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

       9. lower-*.f64 N/A

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

       10. unpow2 N/A

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

       11. lower-*.f64 N/A

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

       12. cube-mult N/A

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

       13. unpow2 N/A

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

       14. lower-*.f64 N/A

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

       15. unpow2 N/A

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

       16. lower-*.f64 84.1

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

   11. Simplified 84.1%

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

Alternative 11: 88.2% accurate, 5.3× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 5.5:\\ \;\;\;\;\frac{x\_m \cdot \mathsf{fma}\left(0.3333333333333333, x\_m \cdot x\_m, 2\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\left(x\_m \cdot \left(x\_m \cdot \left(x\_m \cdot x\_m\right)\right)\right) \cdot \left(0.020833333333333332 + \frac{0.08333333333333333}{x\_m}\right)\\ \end{array} \end{array} \]

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 5.5)
    (/ (* x_m (fma 0.3333333333333333 (* x_m x_m) 2.0)) 2.0)
    (*
     (* x_m (* x_m (* x_m x_m)))
     (+ 0.020833333333333332 (/ 0.08333333333333333 x_m))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	double tmp;
	if (x_m <= 5.5) {
		tmp = (x_m * fma(0.3333333333333333, (x_m * x_m), 2.0)) / 2.0;
	} else {
		tmp = (x_m * (x_m * (x_m * x_m))) * (0.020833333333333332 + (0.08333333333333333 / x_m));
	}
	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 <= 5.5)
		tmp = Float64(Float64(x_m * fma(0.3333333333333333, Float64(x_m * x_m), 2.0)) / 2.0);
	else
		tmp = Float64(Float64(x_m * Float64(x_m * Float64(x_m * x_m))) * Float64(0.020833333333333332 + Float64(0.08333333333333333 / x_m)));
	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, 5.5], N[(N[(x$95$m * N[(0.3333333333333333 * N[(x$95$m * x$95$m), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(x$95$m * N[(x$95$m * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(0.020833333333333332 + N[(0.08333333333333333 / x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 5.5

   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%

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

   if 5.5 < 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

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

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

   9. Taylor expanded in x around inf

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

       1. lower-*.f64 N/A

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

       2. metadata-eval N/A

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

       3. pow-sqr N/A

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

       4. unpow2 N/A

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

       5. associate-*l* N/A

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

       6. unpow2 N/A

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

       7. cube-mult N/A

          \[\leadsto \left(x \cdot \color{blue}{{x}^{3}}\right) \cdot \left(\frac{1}{48} + \frac{1}{12} \cdot \frac{1}{x}\right) \]

       8. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(x \cdot {x}^{3}\right)} \cdot \left(\frac{1}{48} + \frac{1}{12} \cdot \frac{1}{x}\right) \]

       9. cube-mult N/A

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

       10. unpow2 N/A

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

       11. lower-*.f64 N/A

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

       12. unpow2 N/A

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

       13. lower-*.f64 N/A

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

       14. lower-+.f64 N/A

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

       15. associate-*r/ N/A

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

       16. metadata-eval N/A

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

       17. lower-/.f64 79.9

           \[\leadsto \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \left(0.020833333333333332 + \color{blue}{\frac{0.08333333333333333}{x}}\right) \]

   11. Simplified 79.9%

       \[\leadsto \color{blue}{\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \left(0.020833333333333332 + \frac{0.08333333333333333}{x}\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 90.7% accurate, 5.6× speedup

Math

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

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(x_m, Float64(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[(x$95$m * N[(x$95$m * N[(x$95$m * N[(x$95$m * 0.016666666666666666), $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. Taylor expanded in x around 0

   \[\leadsto \frac{x \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot \color{blue}{\frac{1}{60}}, \frac{1}{3}\right), 2\right)}{2} \]
10. Step-by-step derivation

11. Simplified 91.9%

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

Alternative 13: 90.4% accurate, 5.7× speedup

Math

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

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 x_m) (* 0.016666666666666666 (* 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 * x_m), (0.016666666666666666 * (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 * x_m), Float64(0.016666666666666666 * 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 * x$95$m), $MachinePrecision] * N[(0.016666666666666666 * 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

   \[\leadsto \frac{\color{blue}{x \cdot \left(2 + {x}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {x}^{2}\right)\right)}}{2} \]
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. Taylor expanded in x around inf

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

   1. lower-*.f64 N/A

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

   2. unpow2 N/A

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

   3. lower-*.f64 91.6

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

8. Simplified 91.6%

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

Alternative 14: 88.2% accurate, 8.0× speedup

Math

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

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(Float64(x_m * Float64(x_m * 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[(N[(x$95$m * N[(x$95$m * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.020833333333333332), $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

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

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

   9. Taylor expanded in x around inf

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

       1. *-commutative N/A

          \[\leadsto \color{blue}{{x}^{4} \cdot \frac{1}{48}} \]

       2. lower-*.f64 N/A

          \[\leadsto \color{blue}{{x}^{4} \cdot \frac{1}{48}} \]

       3. metadata-eval N/A

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

       4. pow-sqr N/A

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

       5. unpow2 N/A

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

       6. associate-*l* N/A

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

       7. unpow2 N/A

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

       8. cube-mult N/A

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

       9. lower-*.f64 N/A

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

       10. cube-mult N/A

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

       11. unpow2 N/A

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

       12. lower-*.f64 N/A

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

       13. unpow2 N/A

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

       14. lower-*.f64 79.9

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

   11. Simplified 79.9%

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

Alternative 15: 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

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

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

   9. Taylor expanded in x around inf

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

       1. unpow3 N/A

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

       2. unpow2 N/A

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

       3. associate-*r* N/A

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

       4. *-commutative N/A

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

       5. lower-*.f64 N/A

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

       6. *-commutative N/A

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

       7. lower-*.f64 N/A

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

       8. unpow2 N/A

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

       9. lower-*.f64 79.9

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

   11. Simplified 79.9%

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

Alternative 16: 84.1% accurate, 9.4× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 2.4:\\ \;\;\;\;\frac{x\_m \cdot 2}{2}\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot \left(\left(x\_m \cdot x\_m\right) \cdot 0.16666666666666666\right)\\ \end{array} \end{array} \]

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 2.0) 2.0)
    (* 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 * 2.0) / 2.0;
	} 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 * 2.0d0) / 2.0d0
    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 * 2.0) / 2.0;
	} 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 * 2.0) / 2.0
	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 = Float64(Float64(x_m * 2.0) / 2.0);
	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 * 2.0) / 2.0;
	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], N[(N[(x$95$m * 2.0), $MachinePrecision] / 2.0), $MachinePrecision], 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%

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

   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. Final simplification 69.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.4:\\ \;\;\;\;\frac{x \cdot 2}{2}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\left(x \cdot x\right) \cdot 0.16666666666666666\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 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 18: 36.5% accurate, 18.1× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(x\_m \cdot \mathsf{fma}\left(x\_m, 0.25, 0.5\right)\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m) :precision binary64 (* x_s (* 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) {
	return x_s * (x_m * fma(x_m, 0.25, 0.5));
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m)
	return Float64(x_s * Float64(x_m * fma(x_m, 0.25, 0.5)))
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_] := N[(x$95$s * N[(x$95$m * N[(x$95$m * 0.25 + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 53.8%

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

3. Taylor expanded in x around 0

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

5. Simplified 29.6%

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

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 21.6

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

8. Simplified 21.6%

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

Alternative 19: 12.0% accurate, 36.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 53.8%

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

3. Taylor expanded in x around 0

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

5. Simplified 29.6%

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

6. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{1}{2} \cdot x} \]
7. Step-by-step derivation

   1. *-commutative N/A

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

   2. lower-*.f64 11.9

      \[\leadsto \color{blue}{x \cdot 0.5} \]

8. Simplified 11.9%

   \[\leadsto \color{blue}{x \cdot 0.5} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024215 
(FPCore (x)
  :name "Hyperbolic sine"
  :precision binary64
  (/ (- (exp x) (exp (- x))) 2.0))