exp2 (problem 3.3.7)

Percentage Accurate: 53.4% → 99.7%

Time: 12.8s

Alternatives: 9

Speedup: 34.8×

Specification

\[\left|x\right| \leq 710\]

Math

\[\begin{array}{l} \\ \left(e^{x} - 2\right) + e^{-x} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 53.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(e^{x} - 2\right) + e^{-x} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := e^{x\_m} - 2\\ \mathbf{if}\;e^{-x\_m} + t\_0 \leq 2 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(x\_m, x\_m, \left(\left(0.08333333333333333 \cdot \left(x\_m \cdot x\_m\right)\right) \cdot x\_m\right) \cdot x\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{t\_0}^{3} + {\left(e^{x\_m}\right)}^{-3}}{\mathsf{fma}\left(t\_0, \sinh x\_m \cdot 2 + -2, e^{-2 \cdot x\_m}\right)}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0 (- (exp x_m) 2.0)))
   (if (<= (+ (exp (- x_m)) t_0) 2e-7)
     (fma x_m x_m (* (* (* 0.08333333333333333 (* x_m x_m)) x_m) x_m))
     (/
      (+ (pow t_0 3.0) (pow (exp x_m) -3.0))
      (fma t_0 (+ (* (sinh x_m) 2.0) -2.0) (exp (* -2.0 x_m)))))))

C

x_m = fabs(x);
double code(double x_m) {
	double t_0 = exp(x_m) - 2.0;
	double tmp;
	if ((exp(-x_m) + t_0) <= 2e-7) {
		tmp = fma(x_m, x_m, (((0.08333333333333333 * (x_m * x_m)) * x_m) * x_m));
	} else {
		tmp = (pow(t_0, 3.0) + pow(exp(x_m), -3.0)) / fma(t_0, ((sinh(x_m) * 2.0) + -2.0), exp((-2.0 * x_m)));
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	t_0 = Float64(exp(x_m) - 2.0)
	tmp = 0.0
	if (Float64(exp(Float64(-x_m)) + t_0) <= 2e-7)
		tmp = fma(x_m, x_m, Float64(Float64(Float64(0.08333333333333333 * Float64(x_m * x_m)) * x_m) * x_m));
	else
		tmp = Float64(Float64((t_0 ^ 3.0) + (exp(x_m) ^ -3.0)) / fma(t_0, Float64(Float64(sinh(x_m) * 2.0) + -2.0), exp(Float64(-2.0 * x_m))));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := Block[{t$95$0 = N[(N[Exp[x$95$m], $MachinePrecision] - 2.0), $MachinePrecision]}, If[LessEqual[N[(N[Exp[(-x$95$m)], $MachinePrecision] + t$95$0), $MachinePrecision], 2e-7], N[(x$95$m * x$95$m + N[(N[(N[(0.08333333333333333 * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] * x$95$m), $MachinePrecision] * x$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(N[Power[t$95$0, 3.0], $MachinePrecision] + N[Power[N[Exp[x$95$m], $MachinePrecision], -3.0], $MachinePrecision]), $MachinePrecision] / N[(t$95$0 * N[(N[(N[Sinh[x$95$m], $MachinePrecision] * 2.0), $MachinePrecision] + -2.0), $MachinePrecision] + N[Exp[N[(-2.0 * x$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (-.f64 (exp.f64 x) #s(literal 2 binary64)) (exp.f64 (neg.f64 x))) < 1.9999999999999999e-7

   1. Initial program 47.8%

      \[\left(e^{x} - 2\right) + e^{-x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. *-rgt-identity N/A

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

      6. lower-fma.f64 N/A

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

      7. pow-sqr N/A

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

      8. lower-pow.f64 N/A

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

      9. metadata-eval N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 100.0

          \[\leadsto \mathsf{fma}\left({x}^{4}, 0.08333333333333333, \color{blue}{x \cdot x}\right) \]

   5. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{4}, 0.08333333333333333, x \cdot x\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 100.0%

      \[\leadsto \mathsf{fma}\left(x, \color{blue}{x}, 0.08333333333333333 \cdot {x}^{4}\right) \]
   8. Step-by-step derivation

   9. Applied rewrites 100.0%

      \[\leadsto \mathsf{fma}\left(x, x, x \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot 0.08333333333333333\right)\right)\right) \]

   if 1.9999999999999999e-7 < (+.f64 (-.f64 (exp.f64 x) #s(literal 2 binary64)) (exp.f64 (neg.f64 x)))

   1. Initial program 98.8%

      \[\left(e^{x} - 2\right) + e^{-x} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. flip3-+ N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. lift-exp.f64 N/A

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

      7. lift-neg.f64 N/A

         \[\leadsto \frac{{\left(e^{\color{blue}{\mathsf{neg}\left(x\right)}}\right)}^{3} + {\left(e^{x} - 2\right)}^{3}}{\left(e^{x} - 2\right) \cdot \left(e^{x} - 2\right) + \left(e^{-x} \cdot e^{-x} - \left(e^{x} - 2\right) \cdot e^{-x}\right)} \]

      8. exp-neg N/A

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

      9. lift-exp.f64 N/A

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

      10. inv-pow N/A

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

      11. pow-pow N/A

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

      12. lower-pow.f64 N/A

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

      13. metadata-eval N/A

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

      14. lower-pow.f64 N/A

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

      15. associate-+r- N/A

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

   4. Applied rewrites 98.6%

      \[\leadsto \color{blue}{\frac{{\left(e^{x}\right)}^{-3} + {\left(e^{x} - 2\right)}^{3}}{\mathsf{fma}\left(e^{x} - 2, -2 + 2 \cdot \sinh x, {\left(e^{x}\right)}^{-2}\right)}} \]
   5. Step-by-step derivation

      1. lift-pow.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. pow-exp N/A

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

      4. lower-exp.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 98.3

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

   6. Applied rewrites 98.3%

      \[\leadsto \frac{{\left(e^{x}\right)}^{-3} + {\left(e^{x} - 2\right)}^{3}}{\mathsf{fma}\left(e^{x} - 2, -2 + 2 \cdot \sinh x, \color{blue}{e^{-2 \cdot x}}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-x} + \left(e^{x} - 2\right) \leq 2 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(x, x, \left(\left(0.08333333333333333 \cdot \left(x \cdot x\right)\right) \cdot x\right) \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(e^{x} - 2\right)}^{3} + {\left(e^{x}\right)}^{-3}}{\mathsf{fma}\left(e^{x} - 2, \sinh x \cdot 2 + -2, e^{-2 \cdot x}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := e^{x\_m} - 2\\ \mathbf{if}\;e^{-x\_m} + t\_0 \leq 2 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(x\_m, x\_m, \left(\left(0.08333333333333333 \cdot \left(x\_m \cdot x\_m\right)\right) \cdot x\_m\right) \cdot x\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_0, t\_0, -{\left(e^{x\_m}\right)}^{-2}\right)}{\sinh x\_m \cdot 2 + -2}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0 (- (exp x_m) 2.0)))
   (if (<= (+ (exp (- x_m)) t_0) 2e-7)
     (fma x_m x_m (* (* (* 0.08333333333333333 (* x_m x_m)) x_m) x_m))
     (/ (fma t_0 t_0 (- (pow (exp x_m) -2.0))) (+ (* (sinh x_m) 2.0) -2.0)))))

C

x_m = fabs(x);
double code(double x_m) {
	double t_0 = exp(x_m) - 2.0;
	double tmp;
	if ((exp(-x_m) + t_0) <= 2e-7) {
		tmp = fma(x_m, x_m, (((0.08333333333333333 * (x_m * x_m)) * x_m) * x_m));
	} else {
		tmp = fma(t_0, t_0, -pow(exp(x_m), -2.0)) / ((sinh(x_m) * 2.0) + -2.0);
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	t_0 = Float64(exp(x_m) - 2.0)
	tmp = 0.0
	if (Float64(exp(Float64(-x_m)) + t_0) <= 2e-7)
		tmp = fma(x_m, x_m, Float64(Float64(Float64(0.08333333333333333 * Float64(x_m * x_m)) * x_m) * x_m));
	else
		tmp = Float64(fma(t_0, t_0, Float64(-(exp(x_m) ^ -2.0))) / Float64(Float64(sinh(x_m) * 2.0) + -2.0));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := Block[{t$95$0 = N[(N[Exp[x$95$m], $MachinePrecision] - 2.0), $MachinePrecision]}, If[LessEqual[N[(N[Exp[(-x$95$m)], $MachinePrecision] + t$95$0), $MachinePrecision], 2e-7], N[(x$95$m * x$95$m + N[(N[(N[(0.08333333333333333 * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] * x$95$m), $MachinePrecision] * x$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 * t$95$0 + (-N[Power[N[Exp[x$95$m], $MachinePrecision], -2.0], $MachinePrecision])), $MachinePrecision] / N[(N[(N[Sinh[x$95$m], $MachinePrecision] * 2.0), $MachinePrecision] + -2.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (-.f64 (exp.f64 x) #s(literal 2 binary64)) (exp.f64 (neg.f64 x))) < 1.9999999999999999e-7

   1. Initial program 47.8%

      \[\left(e^{x} - 2\right) + e^{-x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. *-rgt-identity N/A

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

      6. lower-fma.f64 N/A

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

      7. pow-sqr N/A

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

      8. lower-pow.f64 N/A

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

      9. metadata-eval N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 100.0

          \[\leadsto \mathsf{fma}\left({x}^{4}, 0.08333333333333333, \color{blue}{x \cdot x}\right) \]

   5. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{4}, 0.08333333333333333, x \cdot x\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 100.0%

      \[\leadsto \mathsf{fma}\left(x, \color{blue}{x}, 0.08333333333333333 \cdot {x}^{4}\right) \]
   8. Step-by-step derivation

   9. Applied rewrites 100.0%

      \[\leadsto \mathsf{fma}\left(x, x, x \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot 0.08333333333333333\right)\right)\right) \]

   if 1.9999999999999999e-7 < (+.f64 (-.f64 (exp.f64 x) #s(literal 2 binary64)) (exp.f64 (neg.f64 x)))

   1. Initial program 98.8%

      \[\left(e^{x} - 2\right) + e^{-x} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. flip-+ N/A

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

      3. lower-/.f64 N/A

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

      4. lower--.f64 N/A

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

      5. pow2 N/A

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

      6. lower-pow.f64 N/A

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

      7. pow2 N/A

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

      8. lift-exp.f64 N/A

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

      9. lift-neg.f64 N/A

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

      10. exp-neg N/A

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

      11. lift-exp.f64 N/A

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

      12. inv-pow N/A

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

      13. pow-pow N/A

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

      14. metadata-eval N/A

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

      15. metadata-eval N/A

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

      16. lower-pow.f64 N/A

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

      17. metadata-eval N/A

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

   4. Applied rewrites 97.9%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-neg.f64 98.2

         \[\leadsto \frac{\mathsf{fma}\left(e^{x} - 2, e^{x} - 2, \color{blue}{-{\left(e^{x}\right)}^{-2}}\right)}{-2 + 2 \cdot \sinh x} \]

   6. Applied rewrites 98.2%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(e^{x} - 2, e^{x} - 2, -{\left(e^{x}\right)}^{-2}\right)}}{-2 + 2 \cdot \sinh x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-x} + \left(e^{x} - 2\right) \leq 2 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(x, x, \left(\left(0.08333333333333333 \cdot \left(x \cdot x\right)\right) \cdot x\right) \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(e^{x} - 2, e^{x} - 2, -{\left(e^{x}\right)}^{-2}\right)}{\sinh x \cdot 2 + -2}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := e^{x\_m} - 2\\ \mathbf{if}\;e^{-x\_m} + t\_0 \leq 2 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(x\_m, x\_m, \left(\left(0.08333333333333333 \cdot \left(x\_m \cdot x\_m\right)\right) \cdot x\_m\right) \cdot x\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{t\_0}^{2} - e^{-2 \cdot x\_m}}{\sinh x\_m \cdot 2 + -2}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0 (- (exp x_m) 2.0)))
   (if (<= (+ (exp (- x_m)) t_0) 2e-7)
     (fma x_m x_m (* (* (* 0.08333333333333333 (* x_m x_m)) x_m) x_m))
     (/ (- (pow t_0 2.0) (exp (* -2.0 x_m))) (+ (* (sinh x_m) 2.0) -2.0)))))

C

x_m = fabs(x);
double code(double x_m) {
	double t_0 = exp(x_m) - 2.0;
	double tmp;
	if ((exp(-x_m) + t_0) <= 2e-7) {
		tmp = fma(x_m, x_m, (((0.08333333333333333 * (x_m * x_m)) * x_m) * x_m));
	} else {
		tmp = (pow(t_0, 2.0) - exp((-2.0 * x_m))) / ((sinh(x_m) * 2.0) + -2.0);
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	t_0 = Float64(exp(x_m) - 2.0)
	tmp = 0.0
	if (Float64(exp(Float64(-x_m)) + t_0) <= 2e-7)
		tmp = fma(x_m, x_m, Float64(Float64(Float64(0.08333333333333333 * Float64(x_m * x_m)) * x_m) * x_m));
	else
		tmp = Float64(Float64((t_0 ^ 2.0) - exp(Float64(-2.0 * x_m))) / Float64(Float64(sinh(x_m) * 2.0) + -2.0));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := Block[{t$95$0 = N[(N[Exp[x$95$m], $MachinePrecision] - 2.0), $MachinePrecision]}, If[LessEqual[N[(N[Exp[(-x$95$m)], $MachinePrecision] + t$95$0), $MachinePrecision], 2e-7], N[(x$95$m * x$95$m + N[(N[(N[(0.08333333333333333 * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] * x$95$m), $MachinePrecision] * x$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(N[Power[t$95$0, 2.0], $MachinePrecision] - N[Exp[N[(-2.0 * x$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(N[(N[Sinh[x$95$m], $MachinePrecision] * 2.0), $MachinePrecision] + -2.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (-.f64 (exp.f64 x) #s(literal 2 binary64)) (exp.f64 (neg.f64 x))) < 1.9999999999999999e-7

   1. Initial program 47.8%

      \[\left(e^{x} - 2\right) + e^{-x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. *-rgt-identity N/A

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

      6. lower-fma.f64 N/A

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

      7. pow-sqr N/A

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

      8. lower-pow.f64 N/A

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

      9. metadata-eval N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 100.0

          \[\leadsto \mathsf{fma}\left({x}^{4}, 0.08333333333333333, \color{blue}{x \cdot x}\right) \]

   5. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{4}, 0.08333333333333333, x \cdot x\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 100.0%

      \[\leadsto \mathsf{fma}\left(x, \color{blue}{x}, 0.08333333333333333 \cdot {x}^{4}\right) \]
   8. Step-by-step derivation

   9. Applied rewrites 100.0%

      \[\leadsto \mathsf{fma}\left(x, x, x \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot 0.08333333333333333\right)\right)\right) \]

   if 1.9999999999999999e-7 < (+.f64 (-.f64 (exp.f64 x) #s(literal 2 binary64)) (exp.f64 (neg.f64 x)))

   1. Initial program 98.8%

      \[\left(e^{x} - 2\right) + e^{-x} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. flip-+ N/A

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

      3. lower-/.f64 N/A

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

      4. lower--.f64 N/A

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

      5. pow2 N/A

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

      6. lower-pow.f64 N/A

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

      7. pow2 N/A

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

      8. lift-exp.f64 N/A

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

      9. lift-neg.f64 N/A

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

      10. exp-neg N/A

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

      11. lift-exp.f64 N/A

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

      12. inv-pow N/A

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

      13. pow-pow N/A

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

      14. metadata-eval N/A

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

      15. metadata-eval N/A

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

      16. lower-pow.f64 N/A

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

      17. metadata-eval N/A

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

   4. Applied rewrites 97.9%

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

      1. lift-pow.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. pow-exp N/A

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

      4. lower-exp.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 98.9

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

   6. Applied rewrites 98.9%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-x} + \left(e^{x} - 2\right) \leq 2 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(x, x, \left(\left(0.08333333333333333 \cdot \left(x \cdot x\right)\right) \cdot x\right) \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(e^{x} - 2\right)}^{2} - e^{-2 \cdot x}}{\sinh x \cdot 2 + -2}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;e^{-x\_m} + \left(e^{x\_m} - 2\right) \leq 2 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(x\_m, x\_m, \left(\left(0.08333333333333333 \cdot \left(x\_m \cdot x\_m\right)\right) \cdot x\_m\right) \cdot x\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\mathsf{fma}\left(\cosh x\_m, 2, 2\right)}{\mathsf{fma}\left({\cosh x\_m}^{2}, 4, -4\right)}}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= (+ (exp (- x_m)) (- (exp x_m) 2.0)) 2e-7)
   (fma x_m x_m (* (* (* 0.08333333333333333 (* x_m x_m)) x_m) x_m))
   (/ 1.0 (/ (fma (cosh x_m) 2.0 2.0) (fma (pow (cosh x_m) 2.0) 4.0 -4.0)))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if ((exp(-x_m) + (exp(x_m) - 2.0)) <= 2e-7) {
		tmp = fma(x_m, x_m, (((0.08333333333333333 * (x_m * x_m)) * x_m) * x_m));
	} else {
		tmp = 1.0 / (fma(cosh(x_m), 2.0, 2.0) / fma(pow(cosh(x_m), 2.0), 4.0, -4.0));
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (Float64(exp(Float64(-x_m)) + Float64(exp(x_m) - 2.0)) <= 2e-7)
		tmp = fma(x_m, x_m, Float64(Float64(Float64(0.08333333333333333 * Float64(x_m * x_m)) * x_m) * x_m));
	else
		tmp = Float64(1.0 / Float64(fma(cosh(x_m), 2.0, 2.0) / fma((cosh(x_m) ^ 2.0), 4.0, -4.0)));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[N[(N[Exp[(-x$95$m)], $MachinePrecision] + N[(N[Exp[x$95$m], $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision], 2e-7], N[(x$95$m * x$95$m + N[(N[(N[(0.08333333333333333 * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] * x$95$m), $MachinePrecision] * x$95$m), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(N[Cosh[x$95$m], $MachinePrecision] * 2.0 + 2.0), $MachinePrecision] / N[(N[Power[N[Cosh[x$95$m], $MachinePrecision], 2.0], $MachinePrecision] * 4.0 + -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (-.f64 (exp.f64 x) #s(literal 2 binary64)) (exp.f64 (neg.f64 x))) < 1.9999999999999999e-7

   1. Initial program 47.8%

      \[\left(e^{x} - 2\right) + e^{-x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. *-rgt-identity N/A

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

      6. lower-fma.f64 N/A

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

      7. pow-sqr N/A

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

      8. lower-pow.f64 N/A

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

      9. metadata-eval N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 100.0

          \[\leadsto \mathsf{fma}\left({x}^{4}, 0.08333333333333333, \color{blue}{x \cdot x}\right) \]

   5. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{4}, 0.08333333333333333, x \cdot x\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 100.0%

      \[\leadsto \mathsf{fma}\left(x, \color{blue}{x}, 0.08333333333333333 \cdot {x}^{4}\right) \]
   8. Step-by-step derivation

   9. Applied rewrites 100.0%

      \[\leadsto \mathsf{fma}\left(x, x, x \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot 0.08333333333333333\right)\right)\right) \]

   if 1.9999999999999999e-7 < (+.f64 (-.f64 (exp.f64 x) #s(literal 2 binary64)) (exp.f64 (neg.f64 x)))

   1. Initial program 98.8%

      \[\left(e^{x} - 2\right) + e^{-x} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. associate-+l- N/A

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

      4. sub-neg N/A

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

      5. associate--r+ N/A

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

      6. lift--.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lift-exp.f64 N/A

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

      9. lift-neg.f64 N/A

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

      10. exp-neg N/A

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

      11. lift-exp.f64 N/A

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

      12. distribute-neg-frac N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f64 98.5

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

   4. Applied rewrites 98.5%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. associate--l- N/A

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

      4. lift-/.f64 N/A

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

      5. div-inv N/A

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

      6. lift-exp.f64 N/A

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

      7. exp-neg N/A

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

      8. mul-1-neg N/A

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

      9. sub-neg N/A

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

      10. associate-+l- N/A

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

      11. lift--.f64 N/A

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

      12. +-commutative N/A

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

      13. lift--.f64 N/A

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

      14. sub-neg N/A

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

      15. metadata-eval N/A

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

      16. associate-+r+ N/A

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

      17. +-commutative N/A

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

      18. lift-exp.f64 N/A

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

      19. cosh-undef N/A

          \[\leadsto \color{blue}{2 \cdot \cosh x} + -2 \]

      20. lift-cosh.f64 N/A

          \[\leadsto 2 \cdot \color{blue}{\cosh x} + -2 \]

      21. *-commutative N/A

          \[\leadsto \color{blue}{\cosh x \cdot 2} + -2 \]

      22. lower-fma.f64 98.4

          \[\leadsto \color{blue}{\mathsf{fma}\left(\cosh x, 2, -2\right)} \]

   6. Applied rewrites 98.4%

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

      1. lift-fma.f64 N/A

         \[\leadsto \color{blue}{\cosh x \cdot 2 + -2} \]

      2. flip-+ N/A

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

      3. clear-num N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. sub-neg N/A

         \[\leadsto \frac{1}{\frac{\color{blue}{\cosh x \cdot 2 + \left(\mathsf{neg}\left(-2\right)\right)}}{\left(\cosh x \cdot 2\right) \cdot \left(\cosh x \cdot 2\right) - -2 \cdot -2}} \]

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

      9. metadata-eval N/A

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

      10. sub-neg N/A

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

      11. swap-sqr N/A

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

      12. unpow2 N/A

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

      13. lift-pow.f64 N/A

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

      14. metadata-eval N/A

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

      15. metadata-eval N/A

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

      16. metadata-eval N/A

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

      17. lower-fma.f64 N/A

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

      18. metadata-eval 99.0

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

   8. Applied rewrites 99.0%

      \[\leadsto \color{blue}{\frac{1}{\frac{\mathsf{fma}\left(\cosh x, 2, 2\right)}{\mathsf{fma}\left({\cosh x}^{2}, 4, -4\right)}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-x} + \left(e^{x} - 2\right) \leq 2 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(x, x, \left(\left(0.08333333333333333 \cdot \left(x \cdot x\right)\right) \cdot x\right) \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\mathsf{fma}\left(\cosh x, 2, 2\right)}{\mathsf{fma}\left({\cosh x}^{2}, 4, -4\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := e^{x\_m} - 2\\ \mathbf{if}\;e^{-x\_m} + t\_0 \leq 2 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(x\_m, x\_m, \left(\left(0.08333333333333333 \cdot \left(x\_m \cdot x\_m\right)\right) \cdot x\_m\right) \cdot x\_m\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 - \frac{-1}{e^{x\_m}}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0 (- (exp x_m) 2.0)))
   (if (<= (+ (exp (- x_m)) t_0) 2e-7)
     (fma x_m x_m (* (* (* 0.08333333333333333 (* x_m x_m)) x_m) x_m))
     (- t_0 (/ -1.0 (exp x_m))))))

C

x_m = fabs(x);
double code(double x_m) {
	double t_0 = exp(x_m) - 2.0;
	double tmp;
	if ((exp(-x_m) + t_0) <= 2e-7) {
		tmp = fma(x_m, x_m, (((0.08333333333333333 * (x_m * x_m)) * x_m) * x_m));
	} else {
		tmp = t_0 - (-1.0 / exp(x_m));
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	t_0 = Float64(exp(x_m) - 2.0)
	tmp = 0.0
	if (Float64(exp(Float64(-x_m)) + t_0) <= 2e-7)
		tmp = fma(x_m, x_m, Float64(Float64(Float64(0.08333333333333333 * Float64(x_m * x_m)) * x_m) * x_m));
	else
		tmp = Float64(t_0 - Float64(-1.0 / exp(x_m)));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := Block[{t$95$0 = N[(N[Exp[x$95$m], $MachinePrecision] - 2.0), $MachinePrecision]}, If[LessEqual[N[(N[Exp[(-x$95$m)], $MachinePrecision] + t$95$0), $MachinePrecision], 2e-7], N[(x$95$m * x$95$m + N[(N[(N[(0.08333333333333333 * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] * x$95$m), $MachinePrecision] * x$95$m), $MachinePrecision]), $MachinePrecision], N[(t$95$0 - N[(-1.0 / N[Exp[x$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (-.f64 (exp.f64 x) #s(literal 2 binary64)) (exp.f64 (neg.f64 x))) < 1.9999999999999999e-7

   1. Initial program 47.8%

      \[\left(e^{x} - 2\right) + e^{-x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. *-rgt-identity N/A

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

      6. lower-fma.f64 N/A

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

      7. pow-sqr N/A

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

      8. lower-pow.f64 N/A

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

      9. metadata-eval N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 100.0

          \[\leadsto \mathsf{fma}\left({x}^{4}, 0.08333333333333333, \color{blue}{x \cdot x}\right) \]

   5. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{4}, 0.08333333333333333, x \cdot x\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 100.0%

      \[\leadsto \mathsf{fma}\left(x, \color{blue}{x}, 0.08333333333333333 \cdot {x}^{4}\right) \]
   8. Step-by-step derivation

   9. Applied rewrites 100.0%

      \[\leadsto \mathsf{fma}\left(x, x, x \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot 0.08333333333333333\right)\right)\right) \]

   if 1.9999999999999999e-7 < (+.f64 (-.f64 (exp.f64 x) #s(literal 2 binary64)) (exp.f64 (neg.f64 x)))

   1. Initial program 98.8%

      \[\left(e^{x} - 2\right) + e^{-x} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. associate-+l- N/A

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

      4. sub-neg N/A

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

      5. associate--r+ N/A

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

      6. lift--.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lift-exp.f64 N/A

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

      9. lift-neg.f64 N/A

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

      10. exp-neg N/A

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

      11. lift-exp.f64 N/A

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

      12. distribute-neg-frac N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f64 98.5

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

   4. Applied rewrites 98.5%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-x} + \left(e^{x} - 2\right) \leq 2 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(x, x, \left(\left(0.08333333333333333 \cdot \left(x \cdot x\right)\right) \cdot x\right) \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(e^{x} - 2\right) - \frac{-1}{e^{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := e^{-x\_m} + \left(e^{x\_m} - 2\right)\\ \mathbf{if}\;t\_0 \leq 2 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(x\_m, x\_m, \left(\left(0.08333333333333333 \cdot \left(x\_m \cdot x\_m\right)\right) \cdot x\_m\right) \cdot x\_m\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0 (+ (exp (- x_m)) (- (exp x_m) 2.0))))
   (if (<= t_0 2e-7)
     (fma x_m x_m (* (* (* 0.08333333333333333 (* x_m x_m)) x_m) x_m))
     t_0)))

C

x_m = fabs(x);
double code(double x_m) {
	double t_0 = exp(-x_m) + (exp(x_m) - 2.0);
	double tmp;
	if (t_0 <= 2e-7) {
		tmp = fma(x_m, x_m, (((0.08333333333333333 * (x_m * x_m)) * x_m) * x_m));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	t_0 = Float64(exp(Float64(-x_m)) + Float64(exp(x_m) - 2.0))
	tmp = 0.0
	if (t_0 <= 2e-7)
		tmp = fma(x_m, x_m, Float64(Float64(Float64(0.08333333333333333 * Float64(x_m * x_m)) * x_m) * x_m));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := Block[{t$95$0 = N[(N[Exp[(-x$95$m)], $MachinePrecision] + N[(N[Exp[x$95$m], $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 2e-7], N[(x$95$m * x$95$m + N[(N[(N[(0.08333333333333333 * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] * x$95$m), $MachinePrecision] * x$95$m), $MachinePrecision]), $MachinePrecision], t$95$0]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (-.f64 (exp.f64 x) #s(literal 2 binary64)) (exp.f64 (neg.f64 x))) < 1.9999999999999999e-7

   1. Initial program 47.8%

      \[\left(e^{x} - 2\right) + e^{-x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. *-rgt-identity N/A

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

      6. lower-fma.f64 N/A

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

      7. pow-sqr N/A

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

      8. lower-pow.f64 N/A

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

      9. metadata-eval N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 100.0

          \[\leadsto \mathsf{fma}\left({x}^{4}, 0.08333333333333333, \color{blue}{x \cdot x}\right) \]

   5. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{4}, 0.08333333333333333, x \cdot x\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 100.0%

      \[\leadsto \mathsf{fma}\left(x, \color{blue}{x}, 0.08333333333333333 \cdot {x}^{4}\right) \]
   8. Step-by-step derivation

   9. Applied rewrites 100.0%

      \[\leadsto \mathsf{fma}\left(x, x, x \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot 0.08333333333333333\right)\right)\right) \]

   if 1.9999999999999999e-7 < (+.f64 (-.f64 (exp.f64 x) #s(literal 2 binary64)) (exp.f64 (neg.f64 x)))

   1. Initial program 98.8%

      \[\left(e^{x} - 2\right) + e^{-x} \]
   2. Add Preprocessing
3. Recombined 2 regimes into one program.

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-x} + \left(e^{x} - 2\right) \leq 2 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(x, x, \left(\left(0.08333333333333333 \cdot \left(x \cdot x\right)\right) \cdot x\right) \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;e^{-x} + \left(e^{x} - 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 99.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;e^{-x\_m} + \left(e^{x\_m} - 2\right) \leq 2 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(x\_m, x\_m, \left(\left(0.08333333333333333 \cdot \left(x\_m \cdot x\_m\right)\right) \cdot x\_m\right) \cdot x\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2, \cosh x\_m, -2\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= (+ (exp (- x_m)) (- (exp x_m) 2.0)) 2e-7)
   (fma x_m x_m (* (* (* 0.08333333333333333 (* x_m x_m)) x_m) x_m))
   (fma 2.0 (cosh x_m) -2.0)))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if ((exp(-x_m) + (exp(x_m) - 2.0)) <= 2e-7) {
		tmp = fma(x_m, x_m, (((0.08333333333333333 * (x_m * x_m)) * x_m) * x_m));
	} else {
		tmp = fma(2.0, cosh(x_m), -2.0);
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (Float64(exp(Float64(-x_m)) + Float64(exp(x_m) - 2.0)) <= 2e-7)
		tmp = fma(x_m, x_m, Float64(Float64(Float64(0.08333333333333333 * Float64(x_m * x_m)) * x_m) * x_m));
	else
		tmp = fma(2.0, cosh(x_m), -2.0);
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[N[(N[Exp[(-x$95$m)], $MachinePrecision] + N[(N[Exp[x$95$m], $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision], 2e-7], N[(x$95$m * x$95$m + N[(N[(N[(0.08333333333333333 * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] * x$95$m), $MachinePrecision] * x$95$m), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[Cosh[x$95$m], $MachinePrecision] + -2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (-.f64 (exp.f64 x) #s(literal 2 binary64)) (exp.f64 (neg.f64 x))) < 1.9999999999999999e-7

   1. Initial program 47.8%

      \[\left(e^{x} - 2\right) + e^{-x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. *-rgt-identity N/A

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

      6. lower-fma.f64 N/A

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

      7. pow-sqr N/A

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

      8. lower-pow.f64 N/A

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

      9. metadata-eval N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 100.0

          \[\leadsto \mathsf{fma}\left({x}^{4}, 0.08333333333333333, \color{blue}{x \cdot x}\right) \]

   5. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{4}, 0.08333333333333333, x \cdot x\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 100.0%

      \[\leadsto \mathsf{fma}\left(x, \color{blue}{x}, 0.08333333333333333 \cdot {x}^{4}\right) \]
   8. Step-by-step derivation

   9. Applied rewrites 100.0%

      \[\leadsto \mathsf{fma}\left(x, x, x \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot 0.08333333333333333\right)\right)\right) \]

   if 1.9999999999999999e-7 < (+.f64 (-.f64 (exp.f64 x) #s(literal 2 binary64)) (exp.f64 (neg.f64 x)))

   1. Initial program 98.8%

      \[\left(e^{x} - 2\right) + e^{-x} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift--.f64 N/A

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

      4. sub-neg N/A

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

      5. associate-+r+ N/A

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

      6. +-commutative N/A

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

      7. lift-exp.f64 N/A

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

      8. lift-exp.f64 N/A

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

      9. lift-neg.f64 N/A

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

      10. cosh-undef N/A

          \[\leadsto \color{blue}{2 \cdot \cosh x} + \left(\mathsf{neg}\left(2\right)\right) \]

      11. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(2, \cosh x, \mathsf{neg}\left(2\right)\right)} \]

      12. lower-cosh.f64 N/A

          \[\leadsto \mathsf{fma}\left(2, \color{blue}{\cosh x}, \mathsf{neg}\left(2\right)\right) \]

      13. metadata-eval 98.4

          \[\leadsto \mathsf{fma}\left(2, \cosh x, \color{blue}{-2}\right) \]

   4. Applied rewrites 98.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(2, \cosh x, -2\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-x} + \left(e^{x} - 2\right) \leq 2 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(x, x, \left(\left(0.08333333333333333 \cdot \left(x \cdot x\right)\right) \cdot x\right) \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2, \cosh x, -2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 99.0% accurate, 7.7× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \mathsf{fma}\left(x\_m, x\_m, \left(\left(0.08333333333333333 \cdot \left(x\_m \cdot x\_m\right)\right) \cdot x\_m\right) \cdot x\_m\right) \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (fma x_m x_m (* (* (* 0.08333333333333333 (* x_m x_m)) x_m) x_m)))

C

x_m = fabs(x);
double code(double x_m) {
	return fma(x_m, x_m, (((0.08333333333333333 * (x_m * x_m)) * x_m) * x_m));
}

Julia

x_m = abs(x)
function code(x_m)
	return fma(x_m, x_m, Float64(Float64(Float64(0.08333333333333333 * Float64(x_m * x_m)) * x_m) * x_m))
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := N[(x$95$m * x$95$m + N[(N[(N[(0.08333333333333333 * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] * x$95$m), $MachinePrecision] * x$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 49.2%

   \[\left(e^{x} - 2\right) + e^{-x} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. distribute-lft-in N/A

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

   3. *-commutative N/A

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

   4. associate-*r* N/A

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

   5. *-rgt-identity N/A

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

   6. lower-fma.f64 N/A

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

   7. pow-sqr N/A

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

   8. lower-pow.f64 N/A

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

   9. metadata-eval N/A

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

   10. unpow2 N/A

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

   11. lower-*.f64 97.8

       \[\leadsto \mathsf{fma}\left({x}^{4}, 0.08333333333333333, \color{blue}{x \cdot x}\right) \]

5. Applied rewrites 97.8%

   \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{4}, 0.08333333333333333, x \cdot x\right)} \]
6. Step-by-step derivation

7. Applied rewrites 97.8%

   \[\leadsto \mathsf{fma}\left(x, \color{blue}{x}, 0.08333333333333333 \cdot {x}^{4}\right) \]
8. Step-by-step derivation

9. Applied rewrites 97.8%

   \[\leadsto \mathsf{fma}\left(x, x, x \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot 0.08333333333333333\right)\right)\right) \]

10. Final simplification 97.8%

    \[\leadsto \mathsf{fma}\left(x, x, \left(\left(0.08333333333333333 \cdot \left(x \cdot x\right)\right) \cdot x\right) \cdot x\right) \]
11. Add Preprocessing

Alternative 9: 98.4% accurate, 34.8× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x\_m \cdot x\_m \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m) :precision binary64 (* x_m x_m))

C

x_m = fabs(x);
double code(double x_m) {
	return x_m * x_m;
}

Fortran

x_m = abs(x)
real(8) function code(x_m)
    real(8), intent (in) :: x_m
    code = x_m * x_m
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	return x_m * x_m;
}

Python

x_m = math.fabs(x)
def code(x_m):
	return x_m * x_m

Julia

x_m = abs(x)
function code(x_m)
	return Float64(x_m * x_m)
end

MATLAB

x_m = abs(x);
function tmp = code(x_m)
	tmp = x_m * x_m;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := N[(x$95$m * x$95$m), $MachinePrecision]

Derivation

1. Initial program 49.2%

   \[\left(e^{x} - 2\right) + e^{-x} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

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

   1. unpow2 N/A

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

   2. lower-*.f64 97.3

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

5. Applied rewrites 97.3%

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

Developer Target 1: 99.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sinh \left(\frac{x}{2}\right)\\ 4 \cdot \left(t\_0 \cdot t\_0\right) \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sinh (/ x 2.0)))) (* 4.0 (* t_0 t_0))))

C

double code(double x) {
	double t_0 = sinh((x / 2.0));
	return 4.0 * (t_0 * t_0);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    t_0 = sinh((x / 2.0d0))
    code = 4.0d0 * (t_0 * t_0)
end function

Java

public static double code(double x) {
	double t_0 = Math.sinh((x / 2.0));
	return 4.0 * (t_0 * t_0);
}

Python

def code(x):
	t_0 = math.sinh((x / 2.0))
	return 4.0 * (t_0 * t_0)

Julia

function code(x)
	t_0 = sinh(Float64(x / 2.0))
	return Float64(4.0 * Float64(t_0 * t_0))
end

MATLAB

function tmp = code(x)
	t_0 = sinh((x / 2.0));
	tmp = 4.0 * (t_0 * t_0);
end

Wolfram

code[x_] := Block[{t$95$0 = N[Sinh[N[(x / 2.0), $MachinePrecision]], $MachinePrecision]}, N[(4.0 * N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]]

Reproduce

herbie shell --seed 2024277 
(FPCore (x)
  :name "exp2 (problem 3.3.7)"
  :precision binary64
  :pre (<= (fabs x) 710.0)

  :alt
  (! :herbie-platform default (* 4 (* (sinh (/ x 2)) (sinh (/ x 2)))))

  (+ (- (exp x) 2.0) (exp (- x))))