exp2 (problem 3.3.7)

Percentage Accurate: 53.6% → 99.2%

Time: 9.8s

Alternatives: 6

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: 99.2% accurate, 5.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 51.6%

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

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

   2. *-commutative N/A

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

   3. associate-+r+ N/A

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

   4. distribute-lft-in N/A

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

   5. +-commutative N/A

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

   6. associate-*r* N/A

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

   7. *-commutative N/A

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

   8. associate-*l* N/A

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

   9. associate-*l* N/A

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

   10. associate-*l* N/A

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

   11. *-commutative N/A

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

   12. lower-fma.f64 N/A

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

5. Applied rewrites 98.7%

   \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{6}, 0.002777777777777778, \mathsf{fma}\left({x}^{4}, 0.08333333333333333, x \cdot x\right)\right)} \]

6. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. distribute-lft-in N/A

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

   3. associate-*r* N/A

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

   4. *-rgt-identity N/A

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

   5. lower-fma.f64 N/A

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

   6. pow-sqr N/A

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

   7. metadata-eval N/A

      \[\leadsto \mathsf{fma}\left({x}^{\color{blue}{4}}, \frac{1}{12} + \frac{1}{360} \cdot {x}^{2}, {x}^{2}\right) \]

   8. lower-pow.f64 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{{x}^{4}}, \frac{1}{12} + \frac{1}{360} \cdot {x}^{2}, {x}^{2}\right) \]

   9. +-commutative N/A

      \[\leadsto \mathsf{fma}\left({x}^{4}, \color{blue}{\frac{1}{360} \cdot {x}^{2} + \frac{1}{12}}, {x}^{2}\right) \]

   10. lower-fma.f64 N/A

       \[\leadsto \mathsf{fma}\left({x}^{4}, \color{blue}{\mathsf{fma}\left(\frac{1}{360}, {x}^{2}, \frac{1}{12}\right)}, {x}^{2}\right) \]

   11. unpow2 N/A

       \[\leadsto \mathsf{fma}\left({x}^{4}, \mathsf{fma}\left(\frac{1}{360}, \color{blue}{x \cdot x}, \frac{1}{12}\right), {x}^{2}\right) \]

   12. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left({x}^{4}, \mathsf{fma}\left(\frac{1}{360}, \color{blue}{x \cdot x}, \frac{1}{12}\right), {x}^{2}\right) \]

   13. unpow2 N/A

       \[\leadsto \mathsf{fma}\left({x}^{4}, \mathsf{fma}\left(\frac{1}{360}, x \cdot x, \frac{1}{12}\right), \color{blue}{x \cdot x}\right) \]

   14. lower-*.f64 98.7

       \[\leadsto \mathsf{fma}\left({x}^{4}, \mathsf{fma}\left(0.002777777777777778, x \cdot x, 0.08333333333333333\right), \color{blue}{x \cdot x}\right) \]

8. Applied rewrites 98.7%

   \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{4}, \mathsf{fma}\left(0.002777777777777778, x \cdot x, 0.08333333333333333\right), x \cdot x\right)} \]
9. Step-by-step derivation

10. Applied rewrites 98.7%

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

Alternative 2: 99.2% accurate, 5.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 51.6%

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

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

   2. *-commutative N/A

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

   3. associate-+r+ N/A

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

   4. distribute-lft-in N/A

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

   5. +-commutative N/A

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

   6. associate-*r* N/A

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

   7. *-commutative N/A

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

   8. associate-*l* N/A

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

   9. associate-*l* N/A

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

   10. associate-*l* N/A

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

   11. *-commutative N/A

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

   12. lower-fma.f64 N/A

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

5. Applied rewrites 98.7%

   \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{6}, 0.002777777777777778, \mathsf{fma}\left({x}^{4}, 0.08333333333333333, x \cdot x\right)\right)} \]

6. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. distribute-lft-in N/A

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

   3. associate-*r* N/A

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

   4. *-rgt-identity N/A

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

   5. lower-fma.f64 N/A

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

   6. pow-sqr N/A

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

   7. metadata-eval N/A

      \[\leadsto \mathsf{fma}\left({x}^{\color{blue}{4}}, \frac{1}{12} + \frac{1}{360} \cdot {x}^{2}, {x}^{2}\right) \]

   8. lower-pow.f64 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{{x}^{4}}, \frac{1}{12} + \frac{1}{360} \cdot {x}^{2}, {x}^{2}\right) \]

   9. +-commutative N/A

      \[\leadsto \mathsf{fma}\left({x}^{4}, \color{blue}{\frac{1}{360} \cdot {x}^{2} + \frac{1}{12}}, {x}^{2}\right) \]

   10. lower-fma.f64 N/A

       \[\leadsto \mathsf{fma}\left({x}^{4}, \color{blue}{\mathsf{fma}\left(\frac{1}{360}, {x}^{2}, \frac{1}{12}\right)}, {x}^{2}\right) \]

   11. unpow2 N/A

       \[\leadsto \mathsf{fma}\left({x}^{4}, \mathsf{fma}\left(\frac{1}{360}, \color{blue}{x \cdot x}, \frac{1}{12}\right), {x}^{2}\right) \]

   12. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left({x}^{4}, \mathsf{fma}\left(\frac{1}{360}, \color{blue}{x \cdot x}, \frac{1}{12}\right), {x}^{2}\right) \]

   13. unpow2 N/A

       \[\leadsto \mathsf{fma}\left({x}^{4}, \mathsf{fma}\left(\frac{1}{360}, x \cdot x, \frac{1}{12}\right), \color{blue}{x \cdot x}\right) \]

   14. lower-*.f64 98.7

       \[\leadsto \mathsf{fma}\left({x}^{4}, \mathsf{fma}\left(0.002777777777777778, x \cdot x, 0.08333333333333333\right), \color{blue}{x \cdot x}\right) \]

8. Applied rewrites 98.7%

   \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{4}, \mathsf{fma}\left(0.002777777777777778, x \cdot x, 0.08333333333333333\right), x \cdot x\right)} \]
9. Step-by-step derivation

10. Applied rewrites 98.7%

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

Alternative 3: 99.2% accurate, 6.3× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (*
  (fma (fma 0.002777777777777778 (* x x) 0.08333333333333333) (* x x) 1.0)
  (* x x)))

C

double code(double x) {
	return fma(fma(0.002777777777777778, (x * x), 0.08333333333333333), (x * x), 1.0) * (x * x);
}

Julia

function code(x)
	return Float64(fma(fma(0.002777777777777778, Float64(x * x), 0.08333333333333333), Float64(x * x), 1.0) * Float64(x * x))
end

Wolfram

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

Derivation

1. Initial program 51.6%

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

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

   2. *-commutative N/A

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

   3. associate-+r+ N/A

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

   4. distribute-lft-in N/A

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

   5. +-commutative N/A

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

   6. associate-*r* N/A

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

   7. *-commutative N/A

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

   8. associate-*l* N/A

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

   9. associate-*l* N/A

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

   10. associate-*l* N/A

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

   11. *-commutative N/A

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

   12. lower-fma.f64 N/A

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

5. Applied rewrites 98.7%

   \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{6}, 0.002777777777777778, \mathsf{fma}\left({x}^{4}, 0.08333333333333333, x \cdot x\right)\right)} \]

6. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. distribute-lft-in N/A

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

   3. associate-*r* N/A

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

   4. *-rgt-identity N/A

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

   5. lower-fma.f64 N/A

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

   6. pow-sqr N/A

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

   7. metadata-eval N/A

      \[\leadsto \mathsf{fma}\left({x}^{\color{blue}{4}}, \frac{1}{12} + \frac{1}{360} \cdot {x}^{2}, {x}^{2}\right) \]

   8. lower-pow.f64 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{{x}^{4}}, \frac{1}{12} + \frac{1}{360} \cdot {x}^{2}, {x}^{2}\right) \]

   9. +-commutative N/A

      \[\leadsto \mathsf{fma}\left({x}^{4}, \color{blue}{\frac{1}{360} \cdot {x}^{2} + \frac{1}{12}}, {x}^{2}\right) \]

   10. lower-fma.f64 N/A

       \[\leadsto \mathsf{fma}\left({x}^{4}, \color{blue}{\mathsf{fma}\left(\frac{1}{360}, {x}^{2}, \frac{1}{12}\right)}, {x}^{2}\right) \]

   11. unpow2 N/A

       \[\leadsto \mathsf{fma}\left({x}^{4}, \mathsf{fma}\left(\frac{1}{360}, \color{blue}{x \cdot x}, \frac{1}{12}\right), {x}^{2}\right) \]

   12. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left({x}^{4}, \mathsf{fma}\left(\frac{1}{360}, \color{blue}{x \cdot x}, \frac{1}{12}\right), {x}^{2}\right) \]

   13. unpow2 N/A

       \[\leadsto \mathsf{fma}\left({x}^{4}, \mathsf{fma}\left(\frac{1}{360}, x \cdot x, \frac{1}{12}\right), \color{blue}{x \cdot x}\right) \]

   14. lower-*.f64 98.7

       \[\leadsto \mathsf{fma}\left({x}^{4}, \mathsf{fma}\left(0.002777777777777778, x \cdot x, 0.08333333333333333\right), \color{blue}{x \cdot x}\right) \]

8. Applied rewrites 98.7%

   \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{4}, \mathsf{fma}\left(0.002777777777777778, x \cdot x, 0.08333333333333333\right), x \cdot x\right)} \]
9. Step-by-step derivation

10. Applied rewrites 98.7%

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

12. Applied rewrites 98.6%

    \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, x \cdot x, 0.08333333333333333\right), x \cdot x, 1\right) \cdot \color{blue}{\left(x \cdot x\right)} \]
13. Add Preprocessing

Alternative 4: 99.0% accurate, 7.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 51.6%

   \[\left(e^{x} - 2\right) + e^{-x} \]
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. +-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 98.5

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

5. Applied rewrites 98.5%

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

7. Applied rewrites 98.5%

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

9. Applied rewrites 98.5%

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

Alternative 5: 99.0% accurate, 9.5× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(0.08333333333333333, x \cdot x, 1\right) \cdot \left(x \cdot x\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (* (fma 0.08333333333333333 (* x x) 1.0) (* x x)))

C

double code(double x) {
	return fma(0.08333333333333333, (x * x), 1.0) * (x * x);
}

Julia

function code(x)
	return Float64(fma(0.08333333333333333, Float64(x * x), 1.0) * Float64(x * x))
end

Wolfram

code[x_] := N[(N[(0.08333333333333333 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 51.6%

   \[\left(e^{x} - 2\right) + e^{-x} \]
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. +-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 98.5

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

5. Applied rewrites 98.5%

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

7. Applied rewrites 98.5%

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

9. Applied rewrites 98.5%

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

11. Applied rewrites 98.5%

    \[\leadsto \mathsf{fma}\left(0.08333333333333333, x \cdot x, 1\right) \cdot \color{blue}{\left(x \cdot x\right)} \]
12. Add Preprocessing

Alternative 6: 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 51.6%

   \[\left(e^{x} - 2\right) + e^{-x} \]
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 97.9

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

5. Applied rewrites 97.9%

   \[\leadsto \color{blue}{x \cdot x} \]
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 2024311 
(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))))