expm1 (example 3.7)

Percentage Accurate: 8.2% → 100.0%

Time: 3.0s

Alternatives: 5

Speedup: 103.0×

Specification

\[\left|x\right| \leq 1\]

Math

\[\begin{array}{l} \\ e^{x} - 1 \end{array} \]

FPCore

(FPCore (x) :precision binary64 (- (exp x) 1.0))

C

double code(double x) {
	return exp(x) - 1.0;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = exp(x) - 1.0d0
end function

Java

public static double code(double x) {
	return Math.exp(x) - 1.0;
}

Python

def code(x):
	return math.exp(x) - 1.0

Julia

function code(x)
	return Float64(exp(x) - 1.0)
end

MATLAB

function tmp = code(x)
	tmp = exp(x) - 1.0;
end

Wolfram

code[x_] := N[(N[Exp[x], $MachinePrecision] - 1.0), $MachinePrecision]

Initial Program: 8.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ e^{x} - 1 \end{array} \]

FPCore

(FPCore (x) :precision binary64 (- (exp x) 1.0))

C

double code(double x) {
	return exp(x) - 1.0;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = exp(x) - 1.0d0
end function

Java

public static double code(double x) {
	return Math.exp(x) - 1.0;
}

Python

def code(x):
	return math.exp(x) - 1.0

Julia

function code(x)
	return Float64(exp(x) - 1.0)
end

MATLAB

function tmp = code(x)
	tmp = exp(x) - 1.0;
end

Wolfram

code[x_] := N[(N[Exp[x], $MachinePrecision] - 1.0), $MachinePrecision]

Alternative 1: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{expm1}\left(x\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (expm1 x))

C

double code(double x) {
	return expm1(x);
}

Java

public static double code(double x) {
	return Math.expm1(x);
}

Python

def code(x):
	return math.expm1(x)

Julia

function code(x)
	return expm1(x)
end

Wolfram

code[x_] := N[(Exp[x] - 1), $MachinePrecision]

Derivation

1. Initial program 7.5%

   \[e^{x} - 1 \]
2. Step-by-step derivation

   1. expm1-define 100.0%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(x\right)} \]

3. Simplified 100.0%

   \[\leadsto \color{blue}{\mathsf{expm1}\left(x\right)} \]
4. Add Preprocessing
5. Add Preprocessing

Alternative 2: 99.7% accurate, 6.9× speedup

Math

\[\begin{array}{l} \\ x \cdot \left(1 + x \cdot \left(0.5 + x \cdot \left(0.16666666666666666 + x \cdot 0.041666666666666664\right)\right)\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  x
  (+
   1.0
   (* x (+ 0.5 (* x (+ 0.16666666666666666 (* x 0.041666666666666664))))))))

C

double code(double x) {
	return x * (1.0 + (x * (0.5 + (x * (0.16666666666666666 + (x * 0.041666666666666664))))));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = x * (1.0d0 + (x * (0.5d0 + (x * (0.16666666666666666d0 + (x * 0.041666666666666664d0))))))
end function

Java

public static double code(double x) {
	return x * (1.0 + (x * (0.5 + (x * (0.16666666666666666 + (x * 0.041666666666666664))))));
}

Python

def code(x):
	return x * (1.0 + (x * (0.5 + (x * (0.16666666666666666 + (x * 0.041666666666666664))))))

Julia

function code(x)
	return Float64(x * Float64(1.0 + Float64(x * Float64(0.5 + Float64(x * Float64(0.16666666666666666 + Float64(x * 0.041666666666666664)))))))
end

MATLAB

function tmp = code(x)
	tmp = x * (1.0 + (x * (0.5 + (x * (0.16666666666666666 + (x * 0.041666666666666664))))));
end

Wolfram

code[x_] := N[(x * N[(1.0 + N[(x * N[(0.5 + N[(x * N[(0.16666666666666666 + N[(x * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 7.5%

   \[e^{x} - 1 \]
2. Step-by-step derivation

   1. expm1-define 100.0%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(x\right)} \]

3. Simplified 100.0%

   \[\leadsto \color{blue}{\mathsf{expm1}\left(x\right)} \]
4. Add Preprocessing

5. Taylor expanded in x around 0 100.0%

   \[\leadsto \color{blue}{x \cdot \left(1 + x \cdot \left(0.5 + x \cdot \left(0.16666666666666666 + 0.041666666666666664 \cdot x\right)\right)\right)} \]
6. Step-by-step derivation

   1. *-commutative 100.0%

      \[\leadsto x \cdot \left(1 + x \cdot \left(0.5 + x \cdot \left(0.16666666666666666 + \color{blue}{x \cdot 0.041666666666666664}\right)\right)\right) \]

7. Simplified 100.0%

   \[\leadsto \color{blue}{x \cdot \left(1 + x \cdot \left(0.5 + x \cdot \left(0.16666666666666666 + x \cdot 0.041666666666666664\right)\right)\right)} \]
8. Add Preprocessing

Alternative 3: 99.5% accurate, 9.4× speedup

Math

\[\begin{array}{l} \\ x \cdot \left(1 + x \cdot \left(0.5 + x \cdot 0.16666666666666666\right)\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (* x (+ 1.0 (* x (+ 0.5 (* x 0.16666666666666666))))))

C

double code(double x) {
	return x * (1.0 + (x * (0.5 + (x * 0.16666666666666666))));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = x * (1.0d0 + (x * (0.5d0 + (x * 0.16666666666666666d0))))
end function

Java

public static double code(double x) {
	return x * (1.0 + (x * (0.5 + (x * 0.16666666666666666))));
}

Python

def code(x):
	return x * (1.0 + (x * (0.5 + (x * 0.16666666666666666))))

Julia

function code(x)
	return Float64(x * Float64(1.0 + Float64(x * Float64(0.5 + Float64(x * 0.16666666666666666)))))
end

MATLAB

function tmp = code(x)
	tmp = x * (1.0 + (x * (0.5 + (x * 0.16666666666666666))));
end

Wolfram

code[x_] := N[(x * N[(1.0 + N[(x * N[(0.5 + N[(x * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 7.5%

   \[e^{x} - 1 \]
2. Step-by-step derivation

   1. expm1-define 100.0%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(x\right)} \]

3. Simplified 100.0%

   \[\leadsto \color{blue}{\mathsf{expm1}\left(x\right)} \]
4. Add Preprocessing

5. Taylor expanded in x around 0 99.9%

   \[\leadsto \color{blue}{x \cdot \left(1 + x \cdot \left(0.5 + 0.16666666666666666 \cdot x\right)\right)} \]
6. Step-by-step derivation

   1. *-commutative 99.9%

      \[\leadsto x \cdot \left(1 + x \cdot \left(0.5 + \color{blue}{x \cdot 0.16666666666666666}\right)\right) \]

7. Simplified 99.9%

   \[\leadsto \color{blue}{x \cdot \left(1 + x \cdot \left(0.5 + x \cdot 0.16666666666666666\right)\right)} \]
8. Add Preprocessing

Alternative 4: 99.2% accurate, 14.7× speedup

Math

\[\begin{array}{l} \\ x \cdot \left(1 + x \cdot 0.5\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* x (+ 1.0 (* x 0.5))))

C

double code(double x) {
	return x * (1.0 + (x * 0.5));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = x * (1.0d0 + (x * 0.5d0))
end function

Java

public static double code(double x) {
	return x * (1.0 + (x * 0.5));
}

Python

def code(x):
	return x * (1.0 + (x * 0.5))

Julia

function code(x)
	return Float64(x * Float64(1.0 + Float64(x * 0.5)))
end

MATLAB

function tmp = code(x)
	tmp = x * (1.0 + (x * 0.5));
end

Wolfram

code[x_] := N[(x * N[(1.0 + N[(x * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 7.5%

   \[e^{x} - 1 \]
2. Step-by-step derivation

   1. expm1-define 100.0%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(x\right)} \]

3. Simplified 100.0%

   \[\leadsto \color{blue}{\mathsf{expm1}\left(x\right)} \]
4. Add Preprocessing

5. Taylor expanded in x around 0 99.7%

   \[\leadsto \color{blue}{x \cdot \left(1 + 0.5 \cdot x\right)} \]
6. Step-by-step derivation

   1. *-commutative 99.7%

      \[\leadsto x \cdot \left(1 + \color{blue}{x \cdot 0.5}\right) \]

7. Simplified 99.7%

   \[\leadsto \color{blue}{x \cdot \left(1 + x \cdot 0.5\right)} \]
8. Add Preprocessing

Alternative 5: 98.2% accurate, 103.0× 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 7.5%

   \[e^{x} - 1 \]
2. Step-by-step derivation

   1. expm1-define 100.0%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(x\right)} \]

3. Simplified 100.0%

   \[\leadsto \color{blue}{\mathsf{expm1}\left(x\right)} \]
4. Add Preprocessing

5. Taylor expanded in x around 0 98.7%

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

Developer target: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{expm1}\left(x\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (expm1 x))

C

double code(double x) {
	return expm1(x);
}

Java

public static double code(double x) {
	return Math.expm1(x);
}

Python

def code(x):
	return math.expm1(x)

Julia

function code(x)
	return expm1(x)
end

Wolfram

code[x_] := N[(Exp[x] - 1), $MachinePrecision]

Reproduce

herbie shell --seed 2024110 
(FPCore (x)
  :name "expm1 (example 3.7)"
  :precision binary64
  :pre (<= (fabs x) 1.0)

  :alt
  (expm1 x)

  (- (exp x) 1.0))