exp2 (problem 3.3.7)

Percentage Accurate: 53.6% → 99.3%

Time: 13.1s

Alternatives: 7

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: 53.6% 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: 99.3% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ \left(x \cdot x\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(4.96031746031746 \cdot 10^{-5}, x \cdot x, 0.002777777777777778\right), 0.08333333333333333\right), x \cdot x, 1\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  (* x x)
  (fma
   (fma
    (* x x)
    (fma 4.96031746031746e-5 (* x x) 0.002777777777777778)
    0.08333333333333333)
   (* x x)
   1.0)))

C

double code(double x) {
	return (x * x) * fma(fma((x * x), fma(4.96031746031746e-5, (x * x), 0.002777777777777778), 0.08333333333333333), (x * x), 1.0);
}

Julia

function code(x)
	return Float64(Float64(x * x) * fma(fma(Float64(x * x), fma(4.96031746031746e-5, Float64(x * x), 0.002777777777777778), 0.08333333333333333), Float64(x * x), 1.0))
end

Wolfram

code[x_] := N[(N[(x * x), $MachinePrecision] * N[(N[(N[(x * x), $MachinePrecision] * N[(4.96031746031746e-5 * N[(x * x), $MachinePrecision] + 0.002777777777777778), $MachinePrecision] + 0.08333333333333333), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 53.6%

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

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{{x}^{2} \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{12} + {x}^{2} \cdot \left(\frac{1}{360} + \frac{1}{20160} \cdot {x}^{2}\right)\right)\right)} \]
4. Step-by-step derivation

   1. unpow2 N/A

      \[\leadsto \color{blue}{\left(x \cdot x\right)} \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{12} + {x}^{2} \cdot \left(\frac{1}{360} + \frac{1}{20160} \cdot {x}^{2}\right)\right)\right) \]

   2. associate-*l* N/A

      \[\leadsto \color{blue}{x \cdot \left(x \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{12} + {x}^{2} \cdot \left(\frac{1}{360} + \frac{1}{20160} \cdot {x}^{2}\right)\right)\right)\right)} \]

   3. lower-*.f64 N/A

      \[\leadsto \color{blue}{x \cdot \left(x \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{12} + {x}^{2} \cdot \left(\frac{1}{360} + \frac{1}{20160} \cdot {x}^{2}\right)\right)\right)\right)} \]

   4. +-commutative N/A

      \[\leadsto x \cdot \left(x \cdot \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{12} + {x}^{2} \cdot \left(\frac{1}{360} + \frac{1}{20160} \cdot {x}^{2}\right)\right) + 1\right)}\right) \]

   5. distribute-lft-in N/A

      \[\leadsto x \cdot \color{blue}{\left(x \cdot \left({x}^{2} \cdot \left(\frac{1}{12} + {x}^{2} \cdot \left(\frac{1}{360} + \frac{1}{20160} \cdot {x}^{2}\right)\right)\right) + x \cdot 1\right)} \]

   6. associate-*r* N/A

      \[\leadsto x \cdot \left(\color{blue}{\left(x \cdot {x}^{2}\right) \cdot \left(\frac{1}{12} + {x}^{2} \cdot \left(\frac{1}{360} + \frac{1}{20160} \cdot {x}^{2}\right)\right)} + x \cdot 1\right) \]

   7. *-commutative N/A

      \[\leadsto x \cdot \left(\color{blue}{\left(\frac{1}{12} + {x}^{2} \cdot \left(\frac{1}{360} + \frac{1}{20160} \cdot {x}^{2}\right)\right) \cdot \left(x \cdot {x}^{2}\right)} + x \cdot 1\right) \]

   8. *-rgt-identity N/A

      \[\leadsto x \cdot \left(\left(\frac{1}{12} + {x}^{2} \cdot \left(\frac{1}{360} + \frac{1}{20160} \cdot {x}^{2}\right)\right) \cdot \left(x \cdot {x}^{2}\right) + \color{blue}{x}\right) \]

   9. lower-fma.f64 N/A

      \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{12} + {x}^{2} \cdot \left(\frac{1}{360} + \frac{1}{20160} \cdot {x}^{2}\right), x \cdot {x}^{2}, x\right)} \]

5. Applied rewrites 99.4%

   \[\leadsto \color{blue}{x \cdot \mathsf{fma}\left(\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, 4.96031746031746 \cdot 10^{-5}, 0.002777777777777778\right), 0.08333333333333333\right), x \cdot \left(x \cdot x\right), x\right)} \]
6. Step-by-step derivation

7. Applied rewrites 99.4%

   \[\leadsto x \cdot \mathsf{fma}\left(\mathsf{fma}\left(x, \frac{\mathsf{fma}\left(\left(x \cdot \left(x \cdot x\right)\right) \cdot x, 2.460474930713026 \cdot 10^{-9}, -7.71604938271605 \cdot 10^{-6}\right) \cdot x}{\mathsf{fma}\left(x \cdot x, 4.96031746031746 \cdot 10^{-5}, -0.002777777777777778\right)}, 0.08333333333333333\right), x \cdot \left(x \cdot x\right), x\right) \]

9. Applied rewrites 99.4%

   \[\leadsto x \cdot \mathsf{fma}\left(\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, 4.96031746031746 \cdot 10^{-5}, 0.002777777777777778\right), 0.08333333333333333\right) \cdot \left(x \cdot x\right), \color{blue}{x}, x\right) \]
10. Step-by-step derivation

11. Applied rewrites 99.4%

    \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(4.96031746031746 \cdot 10^{-5}, x \cdot x, 0.002777777777777778\right), 0.08333333333333333\right), x \cdot x, 1\right) \cdot \color{blue}{\left(x \cdot x\right)} \]

12. Final simplification 99.4%

    \[\leadsto \left(x \cdot x\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(4.96031746031746 \cdot 10^{-5}, x \cdot x, 0.002777777777777778\right), 0.08333333333333333\right), x \cdot x, 1\right) \]
13. Add Preprocessing

Alternative 2: 99.3% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ x \cdot \mathsf{fma}\left(\left(x \cdot x\right) \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, 4.96031746031746 \cdot 10^{-5}, 0.002777777777777778\right), 0.08333333333333333\right), x, x\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  x
  (fma
   (*
    (* x x)
    (fma
     x
     (* x (fma (* x x) 4.96031746031746e-5 0.002777777777777778))
     0.08333333333333333))
   x
   x)))

C

double code(double x) {
	return x * fma(((x * x) * fma(x, (x * fma((x * x), 4.96031746031746e-5, 0.002777777777777778)), 0.08333333333333333)), x, x);
}

Julia

function code(x)
	return Float64(x * fma(Float64(Float64(x * x) * fma(x, Float64(x * fma(Float64(x * x), 4.96031746031746e-5, 0.002777777777777778)), 0.08333333333333333)), x, x))
end

Wolfram

code[x_] := N[(x * N[(N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * N[(N[(x * x), $MachinePrecision] * 4.96031746031746e-5 + 0.002777777777777778), $MachinePrecision]), $MachinePrecision] + 0.08333333333333333), $MachinePrecision]), $MachinePrecision] * x + x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 53.6%

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

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{{x}^{2} \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{12} + {x}^{2} \cdot \left(\frac{1}{360} + \frac{1}{20160} \cdot {x}^{2}\right)\right)\right)} \]
4. Step-by-step derivation

   1. unpow2 N/A

      \[\leadsto \color{blue}{\left(x \cdot x\right)} \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{12} + {x}^{2} \cdot \left(\frac{1}{360} + \frac{1}{20160} \cdot {x}^{2}\right)\right)\right) \]

   2. associate-*l* N/A

      \[\leadsto \color{blue}{x \cdot \left(x \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{12} + {x}^{2} \cdot \left(\frac{1}{360} + \frac{1}{20160} \cdot {x}^{2}\right)\right)\right)\right)} \]

   3. lower-*.f64 N/A

      \[\leadsto \color{blue}{x \cdot \left(x \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{12} + {x}^{2} \cdot \left(\frac{1}{360} + \frac{1}{20160} \cdot {x}^{2}\right)\right)\right)\right)} \]

   4. +-commutative N/A

      \[\leadsto x \cdot \left(x \cdot \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{12} + {x}^{2} \cdot \left(\frac{1}{360} + \frac{1}{20160} \cdot {x}^{2}\right)\right) + 1\right)}\right) \]

   5. distribute-lft-in N/A

      \[\leadsto x \cdot \color{blue}{\left(x \cdot \left({x}^{2} \cdot \left(\frac{1}{12} + {x}^{2} \cdot \left(\frac{1}{360} + \frac{1}{20160} \cdot {x}^{2}\right)\right)\right) + x \cdot 1\right)} \]

   6. associate-*r* N/A

      \[\leadsto x \cdot \left(\color{blue}{\left(x \cdot {x}^{2}\right) \cdot \left(\frac{1}{12} + {x}^{2} \cdot \left(\frac{1}{360} + \frac{1}{20160} \cdot {x}^{2}\right)\right)} + x \cdot 1\right) \]

   7. *-commutative N/A

      \[\leadsto x \cdot \left(\color{blue}{\left(\frac{1}{12} + {x}^{2} \cdot \left(\frac{1}{360} + \frac{1}{20160} \cdot {x}^{2}\right)\right) \cdot \left(x \cdot {x}^{2}\right)} + x \cdot 1\right) \]

   8. *-rgt-identity N/A

      \[\leadsto x \cdot \left(\left(\frac{1}{12} + {x}^{2} \cdot \left(\frac{1}{360} + \frac{1}{20160} \cdot {x}^{2}\right)\right) \cdot \left(x \cdot {x}^{2}\right) + \color{blue}{x}\right) \]

   9. lower-fma.f64 N/A

      \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{12} + {x}^{2} \cdot \left(\frac{1}{360} + \frac{1}{20160} \cdot {x}^{2}\right), x \cdot {x}^{2}, x\right)} \]

5. Applied rewrites 99.4%

   \[\leadsto \color{blue}{x \cdot \mathsf{fma}\left(\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, 4.96031746031746 \cdot 10^{-5}, 0.002777777777777778\right), 0.08333333333333333\right), x \cdot \left(x \cdot x\right), x\right)} \]
6. Step-by-step derivation

7. Applied rewrites 99.4%

   \[\leadsto x \cdot \mathsf{fma}\left(\mathsf{fma}\left(x, \frac{\mathsf{fma}\left(\left(x \cdot \left(x \cdot x\right)\right) \cdot x, 2.460474930713026 \cdot 10^{-9}, -7.71604938271605 \cdot 10^{-6}\right) \cdot x}{\mathsf{fma}\left(x \cdot x, 4.96031746031746 \cdot 10^{-5}, -0.002777777777777778\right)}, 0.08333333333333333\right), x \cdot \left(x \cdot x\right), x\right) \]

9. Applied rewrites 99.4%

   \[\leadsto x \cdot \mathsf{fma}\left(\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, 4.96031746031746 \cdot 10^{-5}, 0.002777777777777778\right), 0.08333333333333333\right) \cdot \left(x \cdot x\right), \color{blue}{x}, x\right) \]

10. Final simplification 99.4%

    \[\leadsto x \cdot \mathsf{fma}\left(\left(x \cdot x\right) \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, 4.96031746031746 \cdot 10^{-5}, 0.002777777777777778\right), 0.08333333333333333\right), x, x\right) \]
11. Add Preprocessing

Alternative 3: 99.2% accurate, 5.5× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\left(x \cdot x\right) \cdot \mathsf{fma}\left(x \cdot x, 0.002777777777777778, 0.08333333333333333\right), x \cdot x, x \cdot x\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (fma
  (* (* x x) (fma (* x x) 0.002777777777777778 0.08333333333333333))
  (* x x)
  (* x x)))

C

double code(double x) {
	return fma(((x * x) * fma((x * x), 0.002777777777777778, 0.08333333333333333)), (x * x), (x * x));
}

Julia

function code(x)
	return fma(Float64(Float64(x * x) * fma(Float64(x * x), 0.002777777777777778, 0.08333333333333333)), Float64(x * x), Float64(x * x))
end

Wolfram

code[x_] := N[(N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * 0.002777777777777778 + 0.08333333333333333), $MachinePrecision]), $MachinePrecision] * N[(x * x), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 53.6%

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

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{{x}^{2} \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {x}^{2}\right)\right)} \]
4. Step-by-step derivation

   1. unpow2 N/A

      \[\leadsto \color{blue}{\left(x \cdot x\right)} \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {x}^{2}\right)\right) \]

   2. associate-*l* N/A

      \[\leadsto \color{blue}{x \cdot \left(x \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {x}^{2}\right)\right)\right)} \]

   3. *-commutative N/A

      \[\leadsto x \cdot \color{blue}{\left(\left(1 + {x}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {x}^{2}\right)\right) \cdot x\right)} \]

   4. lower-*.f64 N/A

      \[\leadsto \color{blue}{x \cdot \left(\left(1 + {x}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {x}^{2}\right)\right) \cdot x\right)} \]

   5. +-commutative N/A

      \[\leadsto x \cdot \left(\color{blue}{\left({x}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {x}^{2}\right) + 1\right)} \cdot x\right) \]

   6. distribute-lft1-in N/A

      \[\leadsto x \cdot \color{blue}{\left(\left({x}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {x}^{2}\right)\right) \cdot x + x\right)} \]

   7. associate-*l* N/A

      \[\leadsto x \cdot \left(\color{blue}{{x}^{2} \cdot \left(\left(\frac{1}{12} + \frac{1}{360} \cdot {x}^{2}\right) \cdot x\right)} + x\right) \]

   8. lower-fma.f64 N/A

      \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left({x}^{2}, \left(\frac{1}{12} + \frac{1}{360} \cdot {x}^{2}\right) \cdot x, x\right)} \]

   9. unpow2 N/A

      \[\leadsto x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, \left(\frac{1}{12} + \frac{1}{360} \cdot {x}^{2}\right) \cdot x, x\right) \]

   10. lower-*.f64 N/A

       \[\leadsto x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, \left(\frac{1}{12} + \frac{1}{360} \cdot {x}^{2}\right) \cdot x, x\right) \]

   11. *-commutative N/A

       \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{x \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {x}^{2}\right)}, x\right) \]

   12. lower-*.f64 N/A

       \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{x \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {x}^{2}\right)}, x\right) \]

   13. +-commutative N/A

       \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, x \cdot \color{blue}{\left(\frac{1}{360} \cdot {x}^{2} + \frac{1}{12}\right)}, x\right) \]

   14. *-commutative N/A

       \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, x \cdot \left(\color{blue}{{x}^{2} \cdot \frac{1}{360}} + \frac{1}{12}\right), x\right) \]

   15. lower-fma.f64 N/A

       \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, x \cdot \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{360}, \frac{1}{12}\right)}, x\right) \]

   16. unpow2 N/A

       \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{360}, \frac{1}{12}\right), x\right) \]

   17. lower-*.f64 99.2

       \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, 0.002777777777777778, 0.08333333333333333\right), x\right) \]

5. Applied rewrites 99.2%

   \[\leadsto \color{blue}{x \cdot \mathsf{fma}\left(x \cdot x, x \cdot \mathsf{fma}\left(x \cdot x, 0.002777777777777778, 0.08333333333333333\right), x\right)} \]
6. Step-by-step derivation

7. Applied rewrites 99.2%

   \[\leadsto \mathsf{fma}\left(\left(x \cdot x\right) \cdot \mathsf{fma}\left(x \cdot x, 0.002777777777777778, 0.08333333333333333\right), \color{blue}{x \cdot x}, x \cdot x\right) \]
8. Add Preprocessing

Alternative 4: 99.2% accurate, 6.3× speedup

Math

\[\begin{array}{l} \\ x \cdot \mathsf{fma}\left(x \cdot x, x \cdot \mathsf{fma}\left(x \cdot x, 0.002777777777777778, 0.08333333333333333\right), x\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  x
  (fma
   (* x x)
   (* x (fma (* x x) 0.002777777777777778 0.08333333333333333))
   x)))

C

double code(double x) {
	return x * fma((x * x), (x * fma((x * x), 0.002777777777777778, 0.08333333333333333)), x);
}

Julia

function code(x)
	return Float64(x * fma(Float64(x * x), Float64(x * fma(Float64(x * x), 0.002777777777777778, 0.08333333333333333)), x))
end

Wolfram

code[x_] := N[(x * N[(N[(x * x), $MachinePrecision] * N[(x * N[(N[(x * x), $MachinePrecision] * 0.002777777777777778 + 0.08333333333333333), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 53.6%

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

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{{x}^{2} \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {x}^{2}\right)\right)} \]
4. Step-by-step derivation

   1. unpow2 N/A

      \[\leadsto \color{blue}{\left(x \cdot x\right)} \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {x}^{2}\right)\right) \]

   2. associate-*l* N/A

      \[\leadsto \color{blue}{x \cdot \left(x \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {x}^{2}\right)\right)\right)} \]

   3. *-commutative N/A

      \[\leadsto x \cdot \color{blue}{\left(\left(1 + {x}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {x}^{2}\right)\right) \cdot x\right)} \]

   4. lower-*.f64 N/A

      \[\leadsto \color{blue}{x \cdot \left(\left(1 + {x}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {x}^{2}\right)\right) \cdot x\right)} \]

   5. +-commutative N/A

      \[\leadsto x \cdot \left(\color{blue}{\left({x}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {x}^{2}\right) + 1\right)} \cdot x\right) \]

   6. distribute-lft1-in N/A

      \[\leadsto x \cdot \color{blue}{\left(\left({x}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {x}^{2}\right)\right) \cdot x + x\right)} \]

   7. associate-*l* N/A

      \[\leadsto x \cdot \left(\color{blue}{{x}^{2} \cdot \left(\left(\frac{1}{12} + \frac{1}{360} \cdot {x}^{2}\right) \cdot x\right)} + x\right) \]

   8. lower-fma.f64 N/A

      \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left({x}^{2}, \left(\frac{1}{12} + \frac{1}{360} \cdot {x}^{2}\right) \cdot x, x\right)} \]

   9. unpow2 N/A

      \[\leadsto x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, \left(\frac{1}{12} + \frac{1}{360} \cdot {x}^{2}\right) \cdot x, x\right) \]

   10. lower-*.f64 N/A

       \[\leadsto x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, \left(\frac{1}{12} + \frac{1}{360} \cdot {x}^{2}\right) \cdot x, x\right) \]

   11. *-commutative N/A

       \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{x \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {x}^{2}\right)}, x\right) \]

   12. lower-*.f64 N/A

       \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{x \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {x}^{2}\right)}, x\right) \]

   13. +-commutative N/A

       \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, x \cdot \color{blue}{\left(\frac{1}{360} \cdot {x}^{2} + \frac{1}{12}\right)}, x\right) \]

   14. *-commutative N/A

       \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, x \cdot \left(\color{blue}{{x}^{2} \cdot \frac{1}{360}} + \frac{1}{12}\right), x\right) \]

   15. lower-fma.f64 N/A

       \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, x \cdot \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{360}, \frac{1}{12}\right)}, x\right) \]

   16. unpow2 N/A

       \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{360}, \frac{1}{12}\right), x\right) \]

   17. lower-*.f64 99.2

       \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, 0.002777777777777778, 0.08333333333333333\right), x\right) \]

5. Applied rewrites 99.2%

   \[\leadsto \color{blue}{x \cdot \mathsf{fma}\left(x \cdot x, x \cdot \mathsf{fma}\left(x \cdot x, 0.002777777777777778, 0.08333333333333333\right), x\right)} \]
6. Add Preprocessing

Alternative 5: 99.0% accurate, 7.7× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (fma x x (* x (* 0.08333333333333333 (* x (* x x))))))

C

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

Julia

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

Wolfram

code[x_] := N[(x * x + N[(x * N[(0.08333333333333333 * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 53.6%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift--.f64 N/A

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

   4. associate-+r- N/A

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

   5. flip-- N/A

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

   6. metadata-eval N/A

      \[\leadsto \frac{\left(e^{\mathsf{neg}\left(x\right)} + e^{x}\right) \cdot \left(e^{\mathsf{neg}\left(x\right)} + e^{x}\right) - \color{blue}{4}}{\left(e^{\mathsf{neg}\left(x\right)} + e^{x}\right) + 2} \]

   7. metadata-eval N/A

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

   8. metadata-eval N/A

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

   9. metadata-eval N/A

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

   10. lower-/.f64 N/A

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

4. Applied rewrites 53.6%

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

5. Taylor expanded in x around 0

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

   1. unpow2 N/A

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

   2. associate-*l* N/A

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

   3. lower-*.f64 N/A

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

   4. +-commutative N/A

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

   5. distribute-rgt-in N/A

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

   6. *-commutative N/A

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

   7. associate-*l* N/A

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

   8. *-lft-identity N/A

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

   9. lower-fma.f64 N/A

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

   10. unpow2 N/A

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

   11. lower-*.f64 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f64 98.8

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

7. Applied rewrites 98.8%

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

9. Applied rewrites 98.8%

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

Alternative 6: 99.0% accurate, 9.5× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (* x (fma x (* (* x x) 0.08333333333333333) x)))

C

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

Julia

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

Wolfram

code[x_] := N[(x * N[(x * N[(N[(x * x), $MachinePrecision] * 0.08333333333333333), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 53.6%

   \[\left(e^{x} - 2\right) + e^{-x} \]
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. unpow2 N/A

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

   2. associate-*l* N/A

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

   3. lower-*.f64 N/A

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

   4. +-commutative N/A

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

   5. distribute-lft-in N/A

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

   6. *-rgt-identity N/A

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

   7. lower-fma.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 N/A

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

   10. unpow2 N/A

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

   11. lower-*.f64 98.8

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

5. Applied rewrites 98.8%

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

Alternative 7: 98.4% accurate, 34.8× speedup

Math

\[\begin{array}{l} \\ x \cdot x \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x):
	return x * x

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 53.6%

   \[\left(e^{x} - 2\right) + e^{-x} \]
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 98.1

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

5. Applied rewrites 98.1%

   \[\leadsto \color{blue}{x \cdot x} \]
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 2024221 
(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))))