exp2 (problem 3.3.7)

Percentage Accurate: 53.3% → 99.3%

Time: 10.1s

Alternatives: 4

Speedup: 68.7×

• Could not converge on a ground truth

  1. x = 109.75744144276558

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: 53.3% 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, 1.7× speedup

Math

\[\begin{array}{l} \\ {x}^{2} \cdot \left(1 + \left(x \cdot x\right) \cdot \left(0.08333333333333333 + \left(x \cdot x\right) \cdot \left(0.002777777777777778 + \left(x \cdot x\right) \cdot 4.96031746031746 \cdot 10^{-5}\right)\right)\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  (pow x 2.0)
  (+
   1.0
   (*
    (* x x)
    (+
     0.08333333333333333
     (* (* x x) (+ 0.002777777777777778 (* (* x x) 4.96031746031746e-5))))))))

C

double code(double x) {
	return pow(x, 2.0) * (1.0 + ((x * x) * (0.08333333333333333 + ((x * x) * (0.002777777777777778 + ((x * x) * 4.96031746031746e-5))))));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (x ** 2.0d0) * (1.0d0 + ((x * x) * (0.08333333333333333d0 + ((x * x) * (0.002777777777777778d0 + ((x * x) * 4.96031746031746d-5))))))
end function

Java

public static double code(double x) {
	return Math.pow(x, 2.0) * (1.0 + ((x * x) * (0.08333333333333333 + ((x * x) * (0.002777777777777778 + ((x * x) * 4.96031746031746e-5))))));
}

Python

def code(x):
	return math.pow(x, 2.0) * (1.0 + ((x * x) * (0.08333333333333333 + ((x * x) * (0.002777777777777778 + ((x * x) * 4.96031746031746e-5))))))

Julia

function code(x)
	return Float64((x ^ 2.0) * Float64(1.0 + Float64(Float64(x * x) * Float64(0.08333333333333333 + Float64(Float64(x * x) * Float64(0.002777777777777778 + Float64(Float64(x * x) * 4.96031746031746e-5)))))))
end

MATLAB

function tmp = code(x)
	tmp = (x ^ 2.0) * (1.0 + ((x * x) * (0.08333333333333333 + ((x * x) * (0.002777777777777778 + ((x * x) * 4.96031746031746e-5))))));
end

Wolfram

code[x_] := N[(N[Power[x, 2.0], $MachinePrecision] * N[(1.0 + N[(N[(x * x), $MachinePrecision] * N[(0.08333333333333333 + N[(N[(x * x), $MachinePrecision] * N[(0.002777777777777778 + N[(N[(x * x), $MachinePrecision] * 4.96031746031746e-5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 52.5%

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

   1. associate-+l- 52.5%

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

   2. sub-neg 52.5%

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

   3. sub-neg 52.5%

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

   4. distribute-neg-in 52.5%

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

   5. remove-double-neg 52.5%

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

   6. +-commutative 52.5%

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

   7. metadata-eval 52.5%

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

3. Simplified 52.5%

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

5. Taylor expanded in x around 0 99.2%

   \[\leadsto \color{blue}{{x}^{2} \cdot \left(1 + {x}^{2} \cdot \left(0.08333333333333333 + {x}^{2} \cdot \left(0.002777777777777778 + 4.96031746031746 \cdot 10^{-5} \cdot {x}^{2}\right)\right)\right)} \]
6. Step-by-step derivation

   1. *-commutative 99.2%

      \[\leadsto {x}^{2} \cdot \left(1 + {x}^{2} \cdot \left(0.08333333333333333 + {x}^{2} \cdot \left(0.002777777777777778 + \color{blue}{{x}^{2} \cdot 4.96031746031746 \cdot 10^{-5}}\right)\right)\right) \]

7. Simplified 99.2%

   \[\leadsto \color{blue}{{x}^{2} \cdot \left(1 + {x}^{2} \cdot \left(0.08333333333333333 + {x}^{2} \cdot \left(0.002777777777777778 + {x}^{2} \cdot 4.96031746031746 \cdot 10^{-5}\right)\right)\right)} \]
8. Step-by-step derivation

   1. unpow2 99.2%

      \[\leadsto {x}^{2} \cdot \left(1 + {x}^{2} \cdot \left(0.08333333333333333 + {x}^{2} \cdot \left(0.002777777777777778 + \color{blue}{\left(x \cdot x\right)} \cdot 4.96031746031746 \cdot 10^{-5}\right)\right)\right) \]

9. Applied egg-rr 99.2%

   \[\leadsto {x}^{2} \cdot \left(1 + {x}^{2} \cdot \left(0.08333333333333333 + {x}^{2} \cdot \left(0.002777777777777778 + \color{blue}{\left(x \cdot x\right)} \cdot 4.96031746031746 \cdot 10^{-5}\right)\right)\right) \]
10. Step-by-step derivation

    1. unpow2 99.2%

       \[\leadsto {x}^{2} \cdot \left(1 + {x}^{2} \cdot \left(0.08333333333333333 + {x}^{2} \cdot \left(0.002777777777777778 + \color{blue}{\left(x \cdot x\right)} \cdot 4.96031746031746 \cdot 10^{-5}\right)\right)\right) \]

11. Applied egg-rr 99.2%

    \[\leadsto {x}^{2} \cdot \left(1 + {x}^{2} \cdot \left(0.08333333333333333 + \color{blue}{\left(x \cdot x\right)} \cdot \left(0.002777777777777778 + \left(x \cdot x\right) \cdot 4.96031746031746 \cdot 10^{-5}\right)\right)\right) \]
12. Step-by-step derivation

    1. unpow2 99.2%

       \[\leadsto {x}^{2} \cdot \left(1 + {x}^{2} \cdot \left(0.08333333333333333 + {x}^{2} \cdot \left(0.002777777777777778 + \color{blue}{\left(x \cdot x\right)} \cdot 4.96031746031746 \cdot 10^{-5}\right)\right)\right) \]

13. Applied egg-rr 99.2%

    \[\leadsto {x}^{2} \cdot \left(1 + \color{blue}{\left(x \cdot x\right)} \cdot \left(0.08333333333333333 + \left(x \cdot x\right) \cdot \left(0.002777777777777778 + \left(x \cdot x\right) \cdot 4.96031746031746 \cdot 10^{-5}\right)\right)\right) \]
14. Add Preprocessing

Alternative 2: 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 52.5%

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

   1. associate-+l- 52.5%

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

   2. sub-neg 52.5%

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

   3. sub-neg 52.5%

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

   4. distribute-neg-in 52.5%

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

   5. remove-double-neg 52.5%

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

   6. +-commutative 52.5%

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

   7. metadata-eval 52.5%

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

3. Simplified 52.5%

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

5. Taylor expanded in x around 0 98.7%

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

   1. *-commutative 98.7%

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

7. Simplified 98.7%

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

   1. unpow2 99.2%

      \[\leadsto {x}^{2} \cdot \left(1 + {x}^{2} \cdot \left(0.08333333333333333 + {x}^{2} \cdot \left(0.002777777777777778 + \color{blue}{\left(x \cdot x\right)} \cdot 4.96031746031746 \cdot 10^{-5}\right)\right)\right) \]

9. Applied egg-rr 98.7%

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

    1. unpow2 99.2%

       \[\leadsto {x}^{2} \cdot \left(1 + {x}^{2} \cdot \left(0.08333333333333333 + {x}^{2} \cdot \left(0.002777777777777778 + \color{blue}{\left(x \cdot x\right)} \cdot 4.96031746031746 \cdot 10^{-5}\right)\right)\right) \]

11. Applied egg-rr 98.7%

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

Alternative 3: 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 52.5%

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

   1. associate-+l- 52.5%

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

   2. sub-neg 52.5%

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

   3. sub-neg 52.5%

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

   4. distribute-neg-in 52.5%

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

   5. remove-double-neg 52.5%

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

   6. +-commutative 52.5%

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

   7. metadata-eval 52.5%

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

3. Simplified 52.5%

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

5. Taylor expanded in x around 0 97.8%

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

   1. unpow2 99.2%

      \[\leadsto {x}^{2} \cdot \left(1 + {x}^{2} \cdot \left(0.08333333333333333 + {x}^{2} \cdot \left(0.002777777777777778 + \color{blue}{\left(x \cdot x\right)} \cdot 4.96031746031746 \cdot 10^{-5}\right)\right)\right) \]

7. Applied egg-rr 97.8%

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

Alternative 4: 5.9% accurate, 206.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 x)

C

double code(double x) {
	return x;
}

Fortran

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

Java

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

Python

def code(x):
	return x

Julia

function code(x)
	return x
end

MATLAB

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

Wolfram

code[x_] := x

Derivation

1. Initial program 52.5%

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

   1. associate-+l- 52.5%

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

   2. sub-neg 52.5%

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

   3. sub-neg 52.5%

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

   4. distribute-neg-in 52.5%

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

   5. remove-double-neg 52.5%

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

   6. +-commutative 52.5%

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

   7. metadata-eval 52.5%

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

3. Simplified 52.5%

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

5. Taylor expanded in x around 0 49.4%

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

6. Taylor expanded in x around 0 5.9%

   \[\leadsto \color{blue}{x} \]
7. 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 2024191 
(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))))