exp2 (problem 3.3.7)

Percentage Accurate: 54.1% → 99.1%

Time: 10.1s

Alternatives: 4

Speedup: 68.7×

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: 54.1% 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: 99.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x \cdot x + \mathsf{fma}\left(x \cdot x, 0.002777777777777778, 0.08333333333333333\right) \cdot {x}^{4} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (+
  (* x x)
  (* (fma (* x x) 0.002777777777777778 0.08333333333333333) (pow x 4.0))))

C

double code(double x) {
	return (x * x) + (fma((x * x), 0.002777777777777778, 0.08333333333333333) * pow(x, 4.0));
}

Julia

function code(x)
	return Float64(Float64(x * x) + Float64(fma(Float64(x * x), 0.002777777777777778, 0.08333333333333333) * (x ^ 4.0)))
end

Wolfram

code[x_] := N[(N[(x * x), $MachinePrecision] + N[(N[(N[(x * x), $MachinePrecision] * 0.002777777777777778 + 0.08333333333333333), $MachinePrecision] * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 53.0%

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

   1. associate-+l- 53.0%

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

   2. sub-neg 53.0%

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

   3. sub-neg 53.0%

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

   4. distribute-neg-in 53.0%

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

   5. remove-double-neg 53.0%

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

   6. +-commutative 53.0%

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

   7. metadata-eval 53.0%

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

3. Simplified 53.0%

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

5. Taylor expanded in x around 0 99.6%

   \[\leadsto \color{blue}{{x}^{2} \cdot \left(1 + {x}^{2} \cdot \left(0.08333333333333333 + 0.002777777777777778 \cdot {x}^{2}\right)\right)} \]
6. Step-by-step derivation

   1. distribute-rgt-in 99.6%

      \[\leadsto \color{blue}{1 \cdot {x}^{2} + \left({x}^{2} \cdot \left(0.08333333333333333 + 0.002777777777777778 \cdot {x}^{2}\right)\right) \cdot {x}^{2}} \]

   2. *-lft-identity 99.6%

      \[\leadsto \color{blue}{{x}^{2}} + \left({x}^{2} \cdot \left(0.08333333333333333 + 0.002777777777777778 \cdot {x}^{2}\right)\right) \cdot {x}^{2} \]

   3. *-commutative 99.6%

      \[\leadsto {x}^{2} + \color{blue}{\left(\left(0.08333333333333333 + 0.002777777777777778 \cdot {x}^{2}\right) \cdot {x}^{2}\right)} \cdot {x}^{2} \]

   4. associate-*l* 99.6%

      \[\leadsto {x}^{2} + \color{blue}{\left(0.08333333333333333 + 0.002777777777777778 \cdot {x}^{2}\right) \cdot \left({x}^{2} \cdot {x}^{2}\right)} \]

   5. +-commutative 99.6%

      \[\leadsto {x}^{2} + \color{blue}{\left(0.002777777777777778 \cdot {x}^{2} + 0.08333333333333333\right)} \cdot \left({x}^{2} \cdot {x}^{2}\right) \]

   6. *-commutative 99.6%

      \[\leadsto {x}^{2} + \left(\color{blue}{{x}^{2} \cdot 0.002777777777777778} + 0.08333333333333333\right) \cdot \left({x}^{2} \cdot {x}^{2}\right) \]

   7. fma-define 99.6%

      \[\leadsto {x}^{2} + \color{blue}{\mathsf{fma}\left({x}^{2}, 0.002777777777777778, 0.08333333333333333\right)} \cdot \left({x}^{2} \cdot {x}^{2}\right) \]

   8. pow-sqr 99.6%

      \[\leadsto {x}^{2} + \mathsf{fma}\left({x}^{2}, 0.002777777777777778, 0.08333333333333333\right) \cdot \color{blue}{{x}^{\left(2 \cdot 2\right)}} \]

   9. metadata-eval 99.6%

      \[\leadsto {x}^{2} + \mathsf{fma}\left({x}^{2}, 0.002777777777777778, 0.08333333333333333\right) \cdot {x}^{\color{blue}{4}} \]

7. Simplified 99.6%

   \[\leadsto \color{blue}{{x}^{2} + \mathsf{fma}\left({x}^{2}, 0.002777777777777778, 0.08333333333333333\right) \cdot {x}^{4}} \]
8. Step-by-step derivation

   1. unpow2 99.6%

      \[\leadsto {x}^{2} + \mathsf{fma}\left(\color{blue}{x \cdot x}, 0.002777777777777778, 0.08333333333333333\right) \cdot {x}^{4} \]

9. Applied egg-rr 99.6%

   \[\leadsto {x}^{2} + \mathsf{fma}\left(\color{blue}{x \cdot x}, 0.002777777777777778, 0.08333333333333333\right) \cdot {x}^{4} \]
10. Step-by-step derivation

    1. unpow2 99.6%

       \[\leadsto {x}^{2} + \mathsf{fma}\left(\color{blue}{x \cdot x}, 0.002777777777777778, 0.08333333333333333\right) \cdot {x}^{4} \]

11. Applied egg-rr 99.6%

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

Alternative 2: 99.1% accurate, 12.1× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (x * x) * (1.0d0 + ((x * x) * (0.08333333333333333d0 + ((x * x) * 0.002777777777777778d0))))
end function

Java

public static double code(double x) {
	return (x * x) * (1.0 + ((x * x) * (0.08333333333333333 + ((x * x) * 0.002777777777777778))));
}

Python

def code(x):
	return (x * x) * (1.0 + ((x * x) * (0.08333333333333333 + ((x * x) * 0.002777777777777778))))

Julia

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

MATLAB

function tmp = code(x)
	tmp = (x * x) * (1.0 + ((x * x) * (0.08333333333333333 + ((x * x) * 0.002777777777777778))));
end

Wolfram

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

Derivation

1. Initial program 53.0%

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

   1. associate-+l- 53.0%

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

   2. sub-neg 53.0%

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

   3. sub-neg 53.0%

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

   4. distribute-neg-in 53.0%

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

   5. remove-double-neg 53.0%

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

   6. +-commutative 53.0%

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

   7. metadata-eval 53.0%

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

3. Simplified 53.0%

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

5. Taylor expanded in x around 0 99.6%

   \[\leadsto \color{blue}{{x}^{2} \cdot \left(1 + {x}^{2} \cdot \left(0.08333333333333333 + 0.002777777777777778 \cdot {x}^{2}\right)\right)} \]
6. Step-by-step derivation

   1. *-commutative 99.6%

      \[\leadsto {x}^{2} \cdot \left(1 + {x}^{2} \cdot \left(0.08333333333333333 + \color{blue}{{x}^{2} \cdot 0.002777777777777778}\right)\right) \]

7. Simplified 99.6%

   \[\leadsto \color{blue}{{x}^{2} \cdot \left(1 + {x}^{2} \cdot \left(0.08333333333333333 + {x}^{2} \cdot 0.002777777777777778\right)\right)} \]
8. Step-by-step derivation

   1. unpow2 99.6%

      \[\leadsto {x}^{2} + \mathsf{fma}\left(\color{blue}{x \cdot x}, 0.002777777777777778, 0.08333333333333333\right) \cdot {x}^{4} \]

9. Applied egg-rr 99.6%

   \[\leadsto {x}^{2} \cdot \left(1 + {x}^{2} \cdot \left(0.08333333333333333 + \color{blue}{\left(x \cdot x\right)} \cdot 0.002777777777777778\right)\right) \]
10. Step-by-step derivation

    1. unpow2 99.6%

       \[\leadsto {x}^{2} + \mathsf{fma}\left(\color{blue}{x \cdot x}, 0.002777777777777778, 0.08333333333333333\right) \cdot {x}^{4} \]

11. Applied egg-rr 99.6%

    \[\leadsto {x}^{2} \cdot \left(1 + \color{blue}{\left(x \cdot x\right)} \cdot \left(0.08333333333333333 + \left(x \cdot x\right) \cdot 0.002777777777777778\right)\right) \]
12. Step-by-step derivation

    1. unpow2 99.6%

       \[\leadsto {x}^{2} + \mathsf{fma}\left(\color{blue}{x \cdot x}, 0.002777777777777778, 0.08333333333333333\right) \cdot {x}^{4} \]

13. Applied egg-rr 99.6%

    \[\leadsto \color{blue}{\left(x \cdot x\right)} \cdot \left(1 + \left(x \cdot x\right) \cdot \left(0.08333333333333333 + \left(x \cdot x\right) \cdot 0.002777777777777778\right)\right) \]
14. Add Preprocessing

Alternative 3: 98.9% accurate, 18.7× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (x * x) * (1.0d0 + ((x * x) * 0.08333333333333333d0))
end function

Java

public static double code(double x) {
	return (x * x) * (1.0 + ((x * x) * 0.08333333333333333));
}

Python

def code(x):
	return (x * x) * (1.0 + ((x * x) * 0.08333333333333333))

Julia

function code(x)
	return Float64(Float64(x * x) * Float64(1.0 + Float64(Float64(x * x) * 0.08333333333333333)))
end

MATLAB

function tmp = code(x)
	tmp = (x * x) * (1.0 + ((x * x) * 0.08333333333333333));
end

Wolfram

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

Derivation

1. Initial program 53.0%

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

   1. associate-+l- 53.0%

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

   2. sub-neg 53.0%

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

   3. sub-neg 53.0%

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

   4. distribute-neg-in 53.0%

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

   5. remove-double-neg 53.0%

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

   6. +-commutative 53.0%

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

   7. metadata-eval 53.0%

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

3. Simplified 53.0%

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

5. Taylor expanded in x around 0 99.3%

   \[\leadsto \color{blue}{{x}^{2} \cdot \left(1 + 0.08333333333333333 \cdot {x}^{2}\right)} \]
6. Step-by-step derivation

   1. *-commutative 99.3%

      \[\leadsto {x}^{2} \cdot \left(1 + \color{blue}{{x}^{2} \cdot 0.08333333333333333}\right) \]

7. Simplified 99.3%

   \[\leadsto \color{blue}{{x}^{2} \cdot \left(1 + {x}^{2} \cdot 0.08333333333333333\right)} \]
8. Step-by-step derivation

   1. unpow2 99.6%

      \[\leadsto {x}^{2} + \mathsf{fma}\left(\color{blue}{x \cdot x}, 0.002777777777777778, 0.08333333333333333\right) \cdot {x}^{4} \]

9. Applied egg-rr 99.3%

   \[\leadsto {x}^{2} \cdot \left(1 + \color{blue}{\left(x \cdot x\right)} \cdot 0.08333333333333333\right) \]
10. Step-by-step derivation

    1. unpow2 99.6%

       \[\leadsto {x}^{2} + \mathsf{fma}\left(\color{blue}{x \cdot x}, 0.002777777777777778, 0.08333333333333333\right) \cdot {x}^{4} \]

11. Applied egg-rr 99.3%

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

Alternative 4: 98.4% accurate, 68.7× 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 53.0%

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

   1. associate-+l- 53.0%

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

   2. sub-neg 53.0%

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

   3. sub-neg 53.0%

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

   4. distribute-neg-in 53.0%

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

   5. remove-double-neg 53.0%

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

   6. +-commutative 53.0%

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

   7. metadata-eval 53.0%

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

3. Simplified 53.0%

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

5. Taylor expanded in x around 0 98.5%

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

   1. unpow2 99.6%

      \[\leadsto {x}^{2} + \mathsf{fma}\left(\color{blue}{x \cdot x}, 0.002777777777777778, 0.08333333333333333\right) \cdot {x}^{4} \]

7. Applied egg-rr 98.5%

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

Developer Target 1: 99.9% accurate, 1.0× 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 2024185 
(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))))