expm1 (example 3.7)

Percentage Accurate: 8.4% → 100.0%

Time: 4.6s

Alternatives: 4

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: 8.4% 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 8.8%

   \[e^{x} - 1 \]
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. Final simplification 100.0%

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

Alternative 2: 99.4% accurate, 9.4× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\left(x \cdot 0.08333333333333333 + \frac{1}{x}\right) - 0.5} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/ 1.0 (- (+ (* x 0.08333333333333333) (/ 1.0 x)) 0.5)))

C

double code(double x) {
	return 1.0 / (((x * 0.08333333333333333) + (1.0 / x)) - 0.5);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 1.0d0 / (((x * 0.08333333333333333d0) + (1.0d0 / x)) - 0.5d0)
end function

Java

public static double code(double x) {
	return 1.0 / (((x * 0.08333333333333333) + (1.0 / x)) - 0.5);
}

Python

def code(x):
	return 1.0 / (((x * 0.08333333333333333) + (1.0 / x)) - 0.5)

Julia

function code(x)
	return Float64(1.0 / Float64(Float64(Float64(x * 0.08333333333333333) + Float64(1.0 / x)) - 0.5))
end

MATLAB

function tmp = code(x)
	tmp = 1.0 / (((x * 0.08333333333333333) + (1.0 / x)) - 0.5);
end

Wolfram

code[x_] := N[(1.0 / N[(N[(N[(x * 0.08333333333333333), $MachinePrecision] + N[(1.0 / x), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 8.8%

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

   1. expm1-undefine 8.8%

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

   2. flip3-- 8.8%

      \[\leadsto \color{blue}{\frac{{\left(e^{x}\right)}^{3} - {1}^{3}}{e^{x} \cdot e^{x} + \left(1 \cdot 1 + e^{x} \cdot 1\right)}} \]

   3. clear-num 8.8%

      \[\leadsto \color{blue}{\frac{1}{\frac{e^{x} \cdot e^{x} + \left(1 \cdot 1 + e^{x} \cdot 1\right)}{{\left(e^{x}\right)}^{3} - {1}^{3}}}} \]

   4. pow2 8.8%

      \[\leadsto \frac{1}{\frac{\color{blue}{{\left(e^{x}\right)}^{2}} + \left(1 \cdot 1 + e^{x} \cdot 1\right)}{{\left(e^{x}\right)}^{3} - {1}^{3}}} \]

   5. metadata-eval 8.8%

      \[\leadsto \frac{1}{\frac{{\left(e^{x}\right)}^{2} + \left(\color{blue}{1} + e^{x} \cdot 1\right)}{{\left(e^{x}\right)}^{3} - {1}^{3}}} \]

   6. *-rgt-identity 8.8%

      \[\leadsto \frac{1}{\frac{{\left(e^{x}\right)}^{2} + \left(1 + \color{blue}{e^{x}}\right)}{{\left(e^{x}\right)}^{3} - {1}^{3}}} \]

   7. +-commutative 8.8%

      \[\leadsto \frac{1}{\frac{{\left(e^{x}\right)}^{2} + \color{blue}{\left(e^{x} + 1\right)}}{{\left(e^{x}\right)}^{3} - {1}^{3}}} \]

   8. metadata-eval 8.8%

      \[\leadsto \frac{1}{\frac{{\left(e^{x}\right)}^{2} + \left(e^{x} + 1\right)}{{\left(e^{x}\right)}^{3} - \color{blue}{1}}} \]

   9. pow3 8.8%

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

   10. sub-neg 8.8%

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

   11. pow3 8.8%

       \[\leadsto \frac{1}{\frac{{\left(e^{x}\right)}^{2} + \left(e^{x} + 1\right)}{\color{blue}{{\left(e^{x}\right)}^{3}} + \left(-1\right)}} \]

   12. pow-exp 9.0%

       \[\leadsto \frac{1}{\frac{{\left(e^{x}\right)}^{2} + \left(e^{x} + 1\right)}{\color{blue}{e^{x \cdot 3}} + \left(-1\right)}} \]

   13. metadata-eval 9.0%

       \[\leadsto \frac{1}{\frac{{\left(e^{x}\right)}^{2} + \left(e^{x} + 1\right)}{e^{x \cdot 3} + \color{blue}{-1}}} \]

6. Applied egg-rr 9.0%

   \[\leadsto \color{blue}{\frac{1}{\frac{{\left(e^{x}\right)}^{2} + \left(e^{x} + 1\right)}{e^{x \cdot 3} + -1}}} \]

7. Taylor expanded in x around 0 99.2%

   \[\leadsto \frac{1}{\color{blue}{\left(0.08333333333333333 \cdot x + \frac{1}{x}\right) - 0.5}} \]

8. Final simplification 99.2%

   \[\leadsto \frac{1}{\left(x \cdot 0.08333333333333333 + \frac{1}{x}\right) - 0.5} \]
9. Add Preprocessing

Alternative 3: 98.8% accurate, 14.7× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\frac{1}{x} - 0.5} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ 1.0 (- (/ 1.0 x) 0.5)))

C

double code(double x) {
	return 1.0 / ((1.0 / x) - 0.5);
}

Fortran

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

Java

public static double code(double x) {
	return 1.0 / ((1.0 / x) - 0.5);
}

Python

def code(x):
	return 1.0 / ((1.0 / x) - 0.5)

Julia

function code(x)
	return Float64(1.0 / Float64(Float64(1.0 / x) - 0.5))
end

MATLAB

function tmp = code(x)
	tmp = 1.0 / ((1.0 / x) - 0.5);
end

Wolfram

code[x_] := N[(1.0 / N[(N[(1.0 / x), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 8.8%

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

   1. expm1-undefine 8.8%

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

   2. flip3-- 8.8%

      \[\leadsto \color{blue}{\frac{{\left(e^{x}\right)}^{3} - {1}^{3}}{e^{x} \cdot e^{x} + \left(1 \cdot 1 + e^{x} \cdot 1\right)}} \]

   3. clear-num 8.8%

      \[\leadsto \color{blue}{\frac{1}{\frac{e^{x} \cdot e^{x} + \left(1 \cdot 1 + e^{x} \cdot 1\right)}{{\left(e^{x}\right)}^{3} - {1}^{3}}}} \]

   4. pow2 8.8%

      \[\leadsto \frac{1}{\frac{\color{blue}{{\left(e^{x}\right)}^{2}} + \left(1 \cdot 1 + e^{x} \cdot 1\right)}{{\left(e^{x}\right)}^{3} - {1}^{3}}} \]

   5. metadata-eval 8.8%

      \[\leadsto \frac{1}{\frac{{\left(e^{x}\right)}^{2} + \left(\color{blue}{1} + e^{x} \cdot 1\right)}{{\left(e^{x}\right)}^{3} - {1}^{3}}} \]

   6. *-rgt-identity 8.8%

      \[\leadsto \frac{1}{\frac{{\left(e^{x}\right)}^{2} + \left(1 + \color{blue}{e^{x}}\right)}{{\left(e^{x}\right)}^{3} - {1}^{3}}} \]

   7. +-commutative 8.8%

      \[\leadsto \frac{1}{\frac{{\left(e^{x}\right)}^{2} + \color{blue}{\left(e^{x} + 1\right)}}{{\left(e^{x}\right)}^{3} - {1}^{3}}} \]

   8. metadata-eval 8.8%

      \[\leadsto \frac{1}{\frac{{\left(e^{x}\right)}^{2} + \left(e^{x} + 1\right)}{{\left(e^{x}\right)}^{3} - \color{blue}{1}}} \]

   9. pow3 8.8%

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

   10. sub-neg 8.8%

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

   11. pow3 8.8%

       \[\leadsto \frac{1}{\frac{{\left(e^{x}\right)}^{2} + \left(e^{x} + 1\right)}{\color{blue}{{\left(e^{x}\right)}^{3}} + \left(-1\right)}} \]

   12. pow-exp 9.0%

       \[\leadsto \frac{1}{\frac{{\left(e^{x}\right)}^{2} + \left(e^{x} + 1\right)}{\color{blue}{e^{x \cdot 3}} + \left(-1\right)}} \]

   13. metadata-eval 9.0%

       \[\leadsto \frac{1}{\frac{{\left(e^{x}\right)}^{2} + \left(e^{x} + 1\right)}{e^{x \cdot 3} + \color{blue}{-1}}} \]

6. Applied egg-rr 9.0%

   \[\leadsto \color{blue}{\frac{1}{\frac{{\left(e^{x}\right)}^{2} + \left(e^{x} + 1\right)}{e^{x \cdot 3} + -1}}} \]

7. Taylor expanded in x around 0 98.6%

   \[\leadsto \frac{1}{\color{blue}{\frac{1}{x} - 0.5}} \]

8. Final simplification 98.6%

   \[\leadsto \frac{1}{\frac{1}{x} - 0.5} \]
9. Add Preprocessing

Alternative 4: 98.0% 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 8.8%

   \[e^{x} - 1 \]
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 97.6%

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

6. Final simplification 97.6%

   \[\leadsto x \]
7. 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 2024044 
(FPCore (x)
  :name "expm1 (example 3.7)"
  :precision binary64
  :pre (<= (fabs x) 1.0)

  :alt
  (expm1 x)

  (- (exp x) 1.0))