expm1 (example 3.7)

Percentage Accurate: 8.2% → 100.0%

Time: 7.5s

Alternatives: 10

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 8.1%

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

   1. expm1-define N/A

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

   2. expm1-lowering-expm1.f64 100.0%

      \[\leadsto \mathsf{expm1.f64}\left(x\right) \]

3. Simplified 100.0%

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

Alternative 2: 99.6% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -0.5 + x \cdot 0.08333333333333333\\ t_1 := x \cdot t\_0\\ \frac{x}{1 + t\_1 \cdot \left(\left(x \cdot x\right) \cdot \left(t\_0 \cdot t\_0\right)\right)} \cdot \left(1 + t\_1 \cdot \left(t\_1 + -1\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (+ -0.5 (* x 0.08333333333333333))) (t_1 (* x t_0)))
   (*
    (/ x (+ 1.0 (* t_1 (* (* x x) (* t_0 t_0)))))
    (+ 1.0 (* t_1 (+ t_1 -1.0))))))

C

double code(double x) {
	double t_0 = -0.5 + (x * 0.08333333333333333);
	double t_1 = x * t_0;
	return (x / (1.0 + (t_1 * ((x * x) * (t_0 * t_0))))) * (1.0 + (t_1 * (t_1 + -1.0)));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    t_0 = (-0.5d0) + (x * 0.08333333333333333d0)
    t_1 = x * t_0
    code = (x / (1.0d0 + (t_1 * ((x * x) * (t_0 * t_0))))) * (1.0d0 + (t_1 * (t_1 + (-1.0d0))))
end function

Java

public static double code(double x) {
	double t_0 = -0.5 + (x * 0.08333333333333333);
	double t_1 = x * t_0;
	return (x / (1.0 + (t_1 * ((x * x) * (t_0 * t_0))))) * (1.0 + (t_1 * (t_1 + -1.0)));
}

Python

def code(x):
	t_0 = -0.5 + (x * 0.08333333333333333)
	t_1 = x * t_0
	return (x / (1.0 + (t_1 * ((x * x) * (t_0 * t_0))))) * (1.0 + (t_1 * (t_1 + -1.0)))

Julia

function code(x)
	t_0 = Float64(-0.5 + Float64(x * 0.08333333333333333))
	t_1 = Float64(x * t_0)
	return Float64(Float64(x / Float64(1.0 + Float64(t_1 * Float64(Float64(x * x) * Float64(t_0 * t_0))))) * Float64(1.0 + Float64(t_1 * Float64(t_1 + -1.0))))
end

MATLAB

function tmp = code(x)
	t_0 = -0.5 + (x * 0.08333333333333333);
	t_1 = x * t_0;
	tmp = (x / (1.0 + (t_1 * ((x * x) * (t_0 * t_0))))) * (1.0 + (t_1 * (t_1 + -1.0)));
end

Wolfram

code[x_] := Block[{t$95$0 = N[(-0.5 + N[(x * 0.08333333333333333), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x * t$95$0), $MachinePrecision]}, N[(N[(x / N[(1.0 + N[(t$95$1 * N[(N[(x * x), $MachinePrecision] * N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(t$95$1 * N[(t$95$1 + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 8.1%

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

   1. expm1-define N/A

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

   2. expm1-lowering-expm1.f64 100.0%

      \[\leadsto \mathsf{expm1.f64}\left(x\right) \]

3. Simplified 100.0%

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

5. Taylor expanded in x around 0

   \[\leadsto \color{blue}{x \cdot \left(1 + x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right)\right)\right)} \]
6. Step-by-step derivation

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(1 + x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right)\right)\right)}\right) \]

   2. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \color{blue}{\left(x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right)\right)\right)}\right)\right) \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right)\right)}\right)\right)\right) \]

   4. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left(x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right)\right)}\right)\right)\right)\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{6} + \frac{1}{24} \cdot x\right)}\right)\right)\right)\right)\right) \]

   6. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \color{blue}{\left(\frac{1}{24} \cdot x\right)}\right)\right)\right)\right)\right)\right) \]

   7. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \left(x \cdot \color{blue}{\frac{1}{24}}\right)\right)\right)\right)\right)\right)\right) \]

   8. *-lowering-*.f64 99.4%

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(x, \color{blue}{\frac{1}{24}}\right)\right)\right)\right)\right)\right)\right) \]

7. Simplified 99.4%

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

   1. distribute-lft-in N/A

      \[\leadsto x \cdot 1 + \color{blue}{x \cdot \left(x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + x \cdot \frac{1}{24}\right)\right)\right)} \]

   2. flip3-+ N/A

      \[\leadsto \frac{{\left(x \cdot 1\right)}^{3} + {\left(x \cdot \left(x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + x \cdot \frac{1}{24}\right)\right)\right)\right)}^{3}}{\color{blue}{\left(x \cdot 1\right) \cdot \left(x \cdot 1\right) + \left(\left(x \cdot \left(x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + x \cdot \frac{1}{24}\right)\right)\right)\right) \cdot \left(x \cdot \left(x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + x \cdot \frac{1}{24}\right)\right)\right)\right) - \left(x \cdot 1\right) \cdot \left(x \cdot \left(x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + x \cdot \frac{1}{24}\right)\right)\right)\right)\right)}} \]

   3. clear-num N/A

      \[\leadsto \frac{1}{\color{blue}{\frac{\left(x \cdot 1\right) \cdot \left(x \cdot 1\right) + \left(\left(x \cdot \left(x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + x \cdot \frac{1}{24}\right)\right)\right)\right) \cdot \left(x \cdot \left(x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + x \cdot \frac{1}{24}\right)\right)\right)\right) - \left(x \cdot 1\right) \cdot \left(x \cdot \left(x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + x \cdot \frac{1}{24}\right)\right)\right)\right)\right)}{{\left(x \cdot 1\right)}^{3} + {\left(x \cdot \left(x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + x \cdot \frac{1}{24}\right)\right)\right)\right)}^{3}}}} \]

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{\left(x \cdot 1\right) \cdot \left(x \cdot 1\right) + \left(\left(x \cdot \left(x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + x \cdot \frac{1}{24}\right)\right)\right)\right) \cdot \left(x \cdot \left(x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + x \cdot \frac{1}{24}\right)\right)\right)\right) - \left(x \cdot 1\right) \cdot \left(x \cdot \left(x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + x \cdot \frac{1}{24}\right)\right)\right)\right)\right)}{{\left(x \cdot 1\right)}^{3} + {\left(x \cdot \left(x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + x \cdot \frac{1}{24}\right)\right)\right)\right)}^{3}}\right)}\right) \]

9. Applied egg-rr 32.3%

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

10. Taylor expanded in x around 0

    \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{1 + x \cdot \left(\frac{1}{12} \cdot x - \frac{1}{2}\right)}{x}\right)}\right) \]
11. Step-by-step derivation

    1. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(1 + x \cdot \left(\frac{1}{12} \cdot x - \frac{1}{2}\right)\right), \color{blue}{x}\right)\right) \]

    2. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left(x \cdot \left(\frac{1}{12} \cdot x - \frac{1}{2}\right)\right)\right), x\right)\right) \]

    3. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(\frac{1}{12} \cdot x - \frac{1}{2}\right)\right)\right), x\right)\right) \]

    4. sub-neg N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(\frac{1}{12} \cdot x + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)\right)\right), x\right)\right) \]

    5. metadata-eval N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(\frac{1}{12} \cdot x + \frac{-1}{2}\right)\right)\right), x\right)\right) \]

    6. +-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(\frac{-1}{2} + \frac{1}{12} \cdot x\right)\right)\right), x\right)\right) \]

    7. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \left(\frac{1}{12} \cdot x\right)\right)\right)\right), x\right)\right) \]

    8. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \left(x \cdot \frac{1}{12}\right)\right)\right)\right), x\right)\right) \]

    9. *-lowering-*.f64 99.1%

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(x, \frac{1}{12}\right)\right)\right)\right), x\right)\right) \]

12. Simplified 99.1%

    \[\leadsto \frac{1}{\color{blue}{\frac{1 + x \cdot \left(-0.5 + x \cdot 0.08333333333333333\right)}{x}}} \]
13. Step-by-step derivation

    1. clear-num N/A

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

    2. flip3-+ N/A

       \[\leadsto \frac{x}{\frac{{1}^{3} + {\left(x \cdot \left(\frac{-1}{2} + x \cdot \frac{1}{12}\right)\right)}^{3}}{\color{blue}{1 \cdot 1 + \left(\left(x \cdot \left(\frac{-1}{2} + x \cdot \frac{1}{12}\right)\right) \cdot \left(x \cdot \left(\frac{-1}{2} + x \cdot \frac{1}{12}\right)\right) - 1 \cdot \left(x \cdot \left(\frac{-1}{2} + x \cdot \frac{1}{12}\right)\right)\right)}}} \]

    3. associate-/r/ N/A

       \[\leadsto \frac{x}{{1}^{3} + {\left(x \cdot \left(\frac{-1}{2} + x \cdot \frac{1}{12}\right)\right)}^{3}} \cdot \color{blue}{\left(1 \cdot 1 + \left(\left(x \cdot \left(\frac{-1}{2} + x \cdot \frac{1}{12}\right)\right) \cdot \left(x \cdot \left(\frac{-1}{2} + x \cdot \frac{1}{12}\right)\right) - 1 \cdot \left(x \cdot \left(\frac{-1}{2} + x \cdot \frac{1}{12}\right)\right)\right)\right)} \]

    4. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\left(\frac{x}{{1}^{3} + {\left(x \cdot \left(\frac{-1}{2} + x \cdot \frac{1}{12}\right)\right)}^{3}}\right), \color{blue}{\left(1 \cdot 1 + \left(\left(x \cdot \left(\frac{-1}{2} + x \cdot \frac{1}{12}\right)\right) \cdot \left(x \cdot \left(\frac{-1}{2} + x \cdot \frac{1}{12}\right)\right) - 1 \cdot \left(x \cdot \left(\frac{-1}{2} + x \cdot \frac{1}{12}\right)\right)\right)\right)}\right) \]

14. Applied egg-rr 99.5%

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

15. Final simplification 99.5%

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

Alternative 3: 99.6% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -0.5 + x \cdot 0.08333333333333333\\ \frac{x}{1 - \left(x \cdot x\right) \cdot \left(t\_0 \cdot t\_0\right)} \cdot \left(1 - x \cdot t\_0\right) \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (+ -0.5 (* x 0.08333333333333333))))
   (* (/ x (- 1.0 (* (* x x) (* t_0 t_0)))) (- 1.0 (* x t_0)))))

C

double code(double x) {
	double t_0 = -0.5 + (x * 0.08333333333333333);
	return (x / (1.0 - ((x * x) * (t_0 * t_0)))) * (1.0 - (x * t_0));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    t_0 = (-0.5d0) + (x * 0.08333333333333333d0)
    code = (x / (1.0d0 - ((x * x) * (t_0 * t_0)))) * (1.0d0 - (x * t_0))
end function

Java

public static double code(double x) {
	double t_0 = -0.5 + (x * 0.08333333333333333);
	return (x / (1.0 - ((x * x) * (t_0 * t_0)))) * (1.0 - (x * t_0));
}

Python

def code(x):
	t_0 = -0.5 + (x * 0.08333333333333333)
	return (x / (1.0 - ((x * x) * (t_0 * t_0)))) * (1.0 - (x * t_0))

Julia

function code(x)
	t_0 = Float64(-0.5 + Float64(x * 0.08333333333333333))
	return Float64(Float64(x / Float64(1.0 - Float64(Float64(x * x) * Float64(t_0 * t_0)))) * Float64(1.0 - Float64(x * t_0)))
end

MATLAB

function tmp = code(x)
	t_0 = -0.5 + (x * 0.08333333333333333);
	tmp = (x / (1.0 - ((x * x) * (t_0 * t_0)))) * (1.0 - (x * t_0));
end

Wolfram

code[x_] := Block[{t$95$0 = N[(-0.5 + N[(x * 0.08333333333333333), $MachinePrecision]), $MachinePrecision]}, N[(N[(x / N[(1.0 - N[(N[(x * x), $MachinePrecision] * N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(x * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 8.1%

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

   1. expm1-define N/A

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

   2. expm1-lowering-expm1.f64 100.0%

      \[\leadsto \mathsf{expm1.f64}\left(x\right) \]

3. Simplified 100.0%

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

5. Taylor expanded in x around 0

   \[\leadsto \color{blue}{x \cdot \left(1 + x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right)\right)\right)} \]
6. Step-by-step derivation

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(1 + x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right)\right)\right)}\right) \]

   2. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \color{blue}{\left(x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right)\right)\right)}\right)\right) \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right)\right)}\right)\right)\right) \]

   4. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left(x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right)\right)}\right)\right)\right)\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{6} + \frac{1}{24} \cdot x\right)}\right)\right)\right)\right)\right) \]

   6. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \color{blue}{\left(\frac{1}{24} \cdot x\right)}\right)\right)\right)\right)\right)\right) \]

   7. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \left(x \cdot \color{blue}{\frac{1}{24}}\right)\right)\right)\right)\right)\right)\right) \]

   8. *-lowering-*.f64 99.4%

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(x, \color{blue}{\frac{1}{24}}\right)\right)\right)\right)\right)\right)\right) \]

7. Simplified 99.4%

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

   1. distribute-lft-in N/A

      \[\leadsto x \cdot 1 + \color{blue}{x \cdot \left(x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + x \cdot \frac{1}{24}\right)\right)\right)} \]

   2. flip3-+ N/A

      \[\leadsto \frac{{\left(x \cdot 1\right)}^{3} + {\left(x \cdot \left(x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + x \cdot \frac{1}{24}\right)\right)\right)\right)}^{3}}{\color{blue}{\left(x \cdot 1\right) \cdot \left(x \cdot 1\right) + \left(\left(x \cdot \left(x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + x \cdot \frac{1}{24}\right)\right)\right)\right) \cdot \left(x \cdot \left(x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + x \cdot \frac{1}{24}\right)\right)\right)\right) - \left(x \cdot 1\right) \cdot \left(x \cdot \left(x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + x \cdot \frac{1}{24}\right)\right)\right)\right)\right)}} \]

   3. clear-num N/A

      \[\leadsto \frac{1}{\color{blue}{\frac{\left(x \cdot 1\right) \cdot \left(x \cdot 1\right) + \left(\left(x \cdot \left(x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + x \cdot \frac{1}{24}\right)\right)\right)\right) \cdot \left(x \cdot \left(x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + x \cdot \frac{1}{24}\right)\right)\right)\right) - \left(x \cdot 1\right) \cdot \left(x \cdot \left(x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + x \cdot \frac{1}{24}\right)\right)\right)\right)\right)}{{\left(x \cdot 1\right)}^{3} + {\left(x \cdot \left(x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + x \cdot \frac{1}{24}\right)\right)\right)\right)}^{3}}}} \]

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{\left(x \cdot 1\right) \cdot \left(x \cdot 1\right) + \left(\left(x \cdot \left(x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + x \cdot \frac{1}{24}\right)\right)\right)\right) \cdot \left(x \cdot \left(x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + x \cdot \frac{1}{24}\right)\right)\right)\right) - \left(x \cdot 1\right) \cdot \left(x \cdot \left(x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + x \cdot \frac{1}{24}\right)\right)\right)\right)\right)}{{\left(x \cdot 1\right)}^{3} + {\left(x \cdot \left(x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + x \cdot \frac{1}{24}\right)\right)\right)\right)}^{3}}\right)}\right) \]

9. Applied egg-rr 32.3%

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

10. Taylor expanded in x around 0

    \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{1 + x \cdot \left(\frac{1}{12} \cdot x - \frac{1}{2}\right)}{x}\right)}\right) \]
11. Step-by-step derivation

    1. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(1 + x \cdot \left(\frac{1}{12} \cdot x - \frac{1}{2}\right)\right), \color{blue}{x}\right)\right) \]

    2. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left(x \cdot \left(\frac{1}{12} \cdot x - \frac{1}{2}\right)\right)\right), x\right)\right) \]

    3. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(\frac{1}{12} \cdot x - \frac{1}{2}\right)\right)\right), x\right)\right) \]

    4. sub-neg N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(\frac{1}{12} \cdot x + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)\right)\right), x\right)\right) \]

    5. metadata-eval N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(\frac{1}{12} \cdot x + \frac{-1}{2}\right)\right)\right), x\right)\right) \]

    6. +-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(\frac{-1}{2} + \frac{1}{12} \cdot x\right)\right)\right), x\right)\right) \]

    7. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \left(\frac{1}{12} \cdot x\right)\right)\right)\right), x\right)\right) \]

    8. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \left(x \cdot \frac{1}{12}\right)\right)\right)\right), x\right)\right) \]

    9. *-lowering-*.f64 99.1%

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(x, \frac{1}{12}\right)\right)\right)\right), x\right)\right) \]

12. Simplified 99.1%

    \[\leadsto \frac{1}{\color{blue}{\frac{1 + x \cdot \left(-0.5 + x \cdot 0.08333333333333333\right)}{x}}} \]
13. Step-by-step derivation

    1. clear-num N/A

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

    2. flip-+ N/A

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

    3. associate-/r/ N/A

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

    4. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\left(\frac{x}{1 \cdot 1 - \left(x \cdot \left(\frac{-1}{2} + x \cdot \frac{1}{12}\right)\right) \cdot \left(x \cdot \left(\frac{-1}{2} + x \cdot \frac{1}{12}\right)\right)}\right), \color{blue}{\left(1 - x \cdot \left(\frac{-1}{2} + x \cdot \frac{1}{12}\right)\right)}\right) \]

14. Applied egg-rr 99.5%

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

Alternative 4: 99.6% accurate, 9.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 8.1%

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

   1. expm1-define N/A

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

   2. expm1-lowering-expm1.f64 100.0%

      \[\leadsto \mathsf{expm1.f64}\left(x\right) \]

3. Simplified 100.0%

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

5. Taylor expanded in x around 0

   \[\leadsto \color{blue}{x \cdot \left(1 + x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right)\right)\right)} \]
6. Step-by-step derivation

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(1 + x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right)\right)\right)}\right) \]

   2. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \color{blue}{\left(x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right)\right)\right)}\right)\right) \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right)\right)}\right)\right)\right) \]

   4. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left(x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right)\right)}\right)\right)\right)\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{6} + \frac{1}{24} \cdot x\right)}\right)\right)\right)\right)\right) \]

   6. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \color{blue}{\left(\frac{1}{24} \cdot x\right)}\right)\right)\right)\right)\right)\right) \]

   7. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \left(x \cdot \color{blue}{\frac{1}{24}}\right)\right)\right)\right)\right)\right)\right) \]

   8. *-lowering-*.f64 99.4%

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(x, \color{blue}{\frac{1}{24}}\right)\right)\right)\right)\right)\right)\right) \]

7. Simplified 99.4%

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

   1. distribute-lft-in N/A

      \[\leadsto x \cdot 1 + \color{blue}{x \cdot \left(x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + x \cdot \frac{1}{24}\right)\right)\right)} \]

   2. flip3-+ N/A

      \[\leadsto \frac{{\left(x \cdot 1\right)}^{3} + {\left(x \cdot \left(x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + x \cdot \frac{1}{24}\right)\right)\right)\right)}^{3}}{\color{blue}{\left(x \cdot 1\right) \cdot \left(x \cdot 1\right) + \left(\left(x \cdot \left(x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + x \cdot \frac{1}{24}\right)\right)\right)\right) \cdot \left(x \cdot \left(x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + x \cdot \frac{1}{24}\right)\right)\right)\right) - \left(x \cdot 1\right) \cdot \left(x \cdot \left(x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + x \cdot \frac{1}{24}\right)\right)\right)\right)\right)}} \]

   3. clear-num N/A

      \[\leadsto \frac{1}{\color{blue}{\frac{\left(x \cdot 1\right) \cdot \left(x \cdot 1\right) + \left(\left(x \cdot \left(x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + x \cdot \frac{1}{24}\right)\right)\right)\right) \cdot \left(x \cdot \left(x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + x \cdot \frac{1}{24}\right)\right)\right)\right) - \left(x \cdot 1\right) \cdot \left(x \cdot \left(x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + x \cdot \frac{1}{24}\right)\right)\right)\right)\right)}{{\left(x \cdot 1\right)}^{3} + {\left(x \cdot \left(x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + x \cdot \frac{1}{24}\right)\right)\right)\right)}^{3}}}} \]

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{\left(x \cdot 1\right) \cdot \left(x \cdot 1\right) + \left(\left(x \cdot \left(x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + x \cdot \frac{1}{24}\right)\right)\right)\right) \cdot \left(x \cdot \left(x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + x \cdot \frac{1}{24}\right)\right)\right)\right) - \left(x \cdot 1\right) \cdot \left(x \cdot \left(x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + x \cdot \frac{1}{24}\right)\right)\right)\right)\right)}{{\left(x \cdot 1\right)}^{3} + {\left(x \cdot \left(x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + x \cdot \frac{1}{24}\right)\right)\right)\right)}^{3}}\right)}\right) \]

9. Applied egg-rr 32.3%

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

10. Taylor expanded in x around 0

    \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{1 + x \cdot \left(\frac{1}{12} \cdot x - \frac{1}{2}\right)}{x}\right)}\right) \]
11. Step-by-step derivation

    1. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(1 + x \cdot \left(\frac{1}{12} \cdot x - \frac{1}{2}\right)\right), \color{blue}{x}\right)\right) \]

    2. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left(x \cdot \left(\frac{1}{12} \cdot x - \frac{1}{2}\right)\right)\right), x\right)\right) \]

    3. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(\frac{1}{12} \cdot x - \frac{1}{2}\right)\right)\right), x\right)\right) \]

    4. sub-neg N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(\frac{1}{12} \cdot x + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)\right)\right), x\right)\right) \]

    5. metadata-eval N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(\frac{1}{12} \cdot x + \frac{-1}{2}\right)\right)\right), x\right)\right) \]

    6. +-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(\frac{-1}{2} + \frac{1}{12} \cdot x\right)\right)\right), x\right)\right) \]

    7. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \left(\frac{1}{12} \cdot x\right)\right)\right)\right), x\right)\right) \]

    8. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \left(x \cdot \frac{1}{12}\right)\right)\right)\right), x\right)\right) \]

    9. *-lowering-*.f64 99.1%

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(x, \frac{1}{12}\right)\right)\right)\right), x\right)\right) \]

12. Simplified 99.1%

    \[\leadsto \frac{1}{\color{blue}{\frac{1 + x \cdot \left(-0.5 + x \cdot 0.08333333333333333\right)}{x}}} \]
13. Step-by-step derivation

    1. clear-num N/A

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

    2. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{\left(1 + x \cdot \left(\frac{-1}{2} + x \cdot \frac{1}{12}\right)\right)}\right) \]

    3. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \color{blue}{\left(x \cdot \left(\frac{-1}{2} + x \cdot \frac{1}{12}\right)\right)}\right)\right) \]

    4. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{-1}{2} + x \cdot \frac{1}{12}\right)}\right)\right)\right) \]

    5. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \color{blue}{\left(x \cdot \frac{1}{12}\right)}\right)\right)\right)\right) \]

    6. *-lowering-*.f64 99.5%

       \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(x, \color{blue}{\frac{1}{12}}\right)\right)\right)\right)\right) \]

14. Applied egg-rr 99.5%

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

Alternative 5: 99.4% accurate, 9.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 8.1%

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

   1. expm1-define N/A

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

   2. expm1-lowering-expm1.f64 100.0%

      \[\leadsto \mathsf{expm1.f64}\left(x\right) \]

3. Simplified 100.0%

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

5. Taylor expanded in x around 0

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

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(1 + x \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot x\right)\right)}\right) \]

   2. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \color{blue}{\left(x \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot x\right)\right)}\right)\right) \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{2} + \frac{1}{6} \cdot x\right)}\right)\right)\right) \]

   4. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left(\frac{1}{6} \cdot x\right)}\right)\right)\right)\right) \]

   5. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \left(x \cdot \color{blue}{\frac{1}{6}}\right)\right)\right)\right)\right) \]

   6. *-lowering-*.f64 99.1%

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \color{blue}{\frac{1}{6}}\right)\right)\right)\right)\right) \]

7. Simplified 99.1%

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

   1. +-commutative N/A

      \[\leadsto x \cdot \left(x \cdot \left(\frac{1}{2} + x \cdot \frac{1}{6}\right) + \color{blue}{1}\right) \]

   2. distribute-lft-in N/A

      \[\leadsto x \cdot \left(x \cdot \left(\frac{1}{2} + x \cdot \frac{1}{6}\right)\right) + \color{blue}{x \cdot 1} \]

   3. *-rgt-identity N/A

      \[\leadsto x \cdot \left(x \cdot \left(\frac{1}{2} + x \cdot \frac{1}{6}\right)\right) + x \]

   4. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\left(x \cdot \left(x \cdot \left(\frac{1}{2} + x \cdot \frac{1}{6}\right)\right)\right), \color{blue}{x}\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \left(x \cdot \left(\frac{1}{2} + x \cdot \frac{1}{6}\right)\right)\right), x\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\frac{1}{2} + x \cdot \frac{1}{6}\right)\right)\right), x\right) \]

   7. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \left(x \cdot \frac{1}{6}\right)\right)\right)\right), x\right) \]

   8. *-lowering-*.f64 99.1%

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \frac{1}{6}\right)\right)\right)\right), x\right) \]

9. Applied egg-rr 99.1%

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

10. Final simplification 99.1%

    \[\leadsto x + x \cdot \left(x \cdot \left(0.5 + x \cdot 0.16666666666666666\right)\right) \]
11. Add Preprocessing

Alternative 6: 99.4% 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 8.1%

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

   1. expm1-define N/A

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

   2. expm1-lowering-expm1.f64 100.0%

      \[\leadsto \mathsf{expm1.f64}\left(x\right) \]

3. Simplified 100.0%

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

5. Taylor expanded in x around 0

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

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(1 + x \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot x\right)\right)}\right) \]

   2. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \color{blue}{\left(x \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot x\right)\right)}\right)\right) \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{2} + \frac{1}{6} \cdot x\right)}\right)\right)\right) \]

   4. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left(\frac{1}{6} \cdot x\right)}\right)\right)\right)\right) \]

   5. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \left(x \cdot \color{blue}{\frac{1}{6}}\right)\right)\right)\right)\right) \]

   6. *-lowering-*.f64 99.1%

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \color{blue}{\frac{1}{6}}\right)\right)\right)\right)\right) \]

7. Simplified 99.1%

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

Alternative 7: 99.1% accurate, 14.7× speedup

Math

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

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 8.1%

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

   1. expm1-define N/A

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

   2. expm1-lowering-expm1.f64 100.0%

      \[\leadsto \mathsf{expm1.f64}\left(x\right) \]

3. Simplified 100.0%

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

5. Taylor expanded in x around 0

   \[\leadsto \color{blue}{x \cdot \left(1 + x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right)\right)\right)} \]
6. Step-by-step derivation

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(1 + x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right)\right)\right)}\right) \]

   2. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \color{blue}{\left(x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right)\right)\right)}\right)\right) \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right)\right)}\right)\right)\right) \]

   4. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left(x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right)\right)}\right)\right)\right)\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{6} + \frac{1}{24} \cdot x\right)}\right)\right)\right)\right)\right) \]

   6. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \color{blue}{\left(\frac{1}{24} \cdot x\right)}\right)\right)\right)\right)\right)\right) \]

   7. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \left(x \cdot \color{blue}{\frac{1}{24}}\right)\right)\right)\right)\right)\right)\right) \]

   8. *-lowering-*.f64 99.4%

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(x, \color{blue}{\frac{1}{24}}\right)\right)\right)\right)\right)\right)\right) \]

7. Simplified 99.4%

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

   1. distribute-lft-in N/A

      \[\leadsto x \cdot 1 + \color{blue}{x \cdot \left(x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + x \cdot \frac{1}{24}\right)\right)\right)} \]

   2. flip3-+ N/A

      \[\leadsto \frac{{\left(x \cdot 1\right)}^{3} + {\left(x \cdot \left(x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + x \cdot \frac{1}{24}\right)\right)\right)\right)}^{3}}{\color{blue}{\left(x \cdot 1\right) \cdot \left(x \cdot 1\right) + \left(\left(x \cdot \left(x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + x \cdot \frac{1}{24}\right)\right)\right)\right) \cdot \left(x \cdot \left(x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + x \cdot \frac{1}{24}\right)\right)\right)\right) - \left(x \cdot 1\right) \cdot \left(x \cdot \left(x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + x \cdot \frac{1}{24}\right)\right)\right)\right)\right)}} \]

   3. clear-num N/A

      \[\leadsto \frac{1}{\color{blue}{\frac{\left(x \cdot 1\right) \cdot \left(x \cdot 1\right) + \left(\left(x \cdot \left(x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + x \cdot \frac{1}{24}\right)\right)\right)\right) \cdot \left(x \cdot \left(x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + x \cdot \frac{1}{24}\right)\right)\right)\right) - \left(x \cdot 1\right) \cdot \left(x \cdot \left(x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + x \cdot \frac{1}{24}\right)\right)\right)\right)\right)}{{\left(x \cdot 1\right)}^{3} + {\left(x \cdot \left(x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + x \cdot \frac{1}{24}\right)\right)\right)\right)}^{3}}}} \]

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{\left(x \cdot 1\right) \cdot \left(x \cdot 1\right) + \left(\left(x \cdot \left(x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + x \cdot \frac{1}{24}\right)\right)\right)\right) \cdot \left(x \cdot \left(x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + x \cdot \frac{1}{24}\right)\right)\right)\right) - \left(x \cdot 1\right) \cdot \left(x \cdot \left(x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + x \cdot \frac{1}{24}\right)\right)\right)\right)\right)}{{\left(x \cdot 1\right)}^{3} + {\left(x \cdot \left(x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + x \cdot \frac{1}{24}\right)\right)\right)\right)}^{3}}\right)}\right) \]

9. Applied egg-rr 32.3%

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

10. Taylor expanded in x around 0

    \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{1 + \frac{-1}{2} \cdot x}{x}\right)}\right) \]
11. Step-by-step derivation

    1. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(1 + \frac{-1}{2} \cdot x\right), \color{blue}{x}\right)\right) \]

    2. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{-1}{2} \cdot x\right)\right), x\right)\right) \]

    3. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left(x \cdot \frac{-1}{2}\right)\right), x\right)\right) \]

    4. *-lowering-*.f64 98.5%

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \frac{-1}{2}\right)\right), x\right)\right) \]

12. Simplified 98.5%

    \[\leadsto \frac{1}{\color{blue}{\frac{1 + x \cdot -0.5}{x}}} \]
13. Step-by-step derivation

    1. clear-num N/A

       \[\leadsto \frac{x}{\color{blue}{1 + x \cdot \frac{-1}{2}}} \]

    2. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{\left(1 + x \cdot \frac{-1}{2}\right)}\right) \]

    3. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \color{blue}{\left(x \cdot \frac{-1}{2}\right)}\right)\right) \]

    4. *-lowering-*.f64 98.9%

       \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\frac{-1}{2}}\right)\right)\right) \]

14. Applied egg-rr 98.9%

    \[\leadsto \color{blue}{\frac{x}{1 + x \cdot -0.5}} \]
15. Add Preprocessing

Alternative 8: 99.1% accurate, 14.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 8.1%

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

   1. expm1-define N/A

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

   2. expm1-lowering-expm1.f64 100.0%

      \[\leadsto \mathsf{expm1.f64}\left(x\right) \]

3. Simplified 100.0%

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

5. Taylor expanded in x around 0

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

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(1 + \frac{1}{2} \cdot x\right)}\right) \]

   2. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{2} \cdot x\right)}\right)\right) \]

   3. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(x \cdot \color{blue}{\frac{1}{2}}\right)\right)\right) \]

   4. *-lowering-*.f64 98.9%

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\frac{1}{2}}\right)\right)\right) \]

7. Simplified 98.9%

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

   1. +-commutative N/A

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

   2. distribute-lft-in N/A

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

   3. *-rgt-identity N/A

      \[\leadsto x \cdot \left(x \cdot \frac{1}{2}\right) + x \]

   4. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\left(x \cdot \left(x \cdot \frac{1}{2}\right)\right), \color{blue}{x}\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \left(x \cdot \frac{1}{2}\right)\right), x\right) \]

   6. *-lowering-*.f64 98.9%

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{2}\right)\right), x\right) \]

9. Applied egg-rr 98.9%

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

10. Final simplification 98.9%

    \[\leadsto x + x \cdot \left(x \cdot 0.5\right) \]
11. Add Preprocessing

Alternative 9: 99.1% 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 8.1%

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

   1. expm1-define N/A

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

   2. expm1-lowering-expm1.f64 100.0%

      \[\leadsto \mathsf{expm1.f64}\left(x\right) \]

3. Simplified 100.0%

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

5. Taylor expanded in x around 0

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

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(1 + \frac{1}{2} \cdot x\right)}\right) \]

   2. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{2} \cdot x\right)}\right)\right) \]

   3. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(x \cdot \color{blue}{\frac{1}{2}}\right)\right)\right) \]

   4. *-lowering-*.f64 98.9%

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\frac{1}{2}}\right)\right)\right) \]

7. Simplified 98.9%

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

Alternative 10: 98.1% 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 8.1%

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

   1. expm1-define N/A

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

   2. expm1-lowering-expm1.f64 100.0%

      \[\leadsto \mathsf{expm1.f64}\left(x\right) \]

3. Simplified 100.0%

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

5. Taylor expanded in x around 0

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

7. Simplified 98.0%

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

Developer Target 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]

Reproduce

herbie shell --seed 2024155 
(FPCore (x)
  :name "expm1 (example 3.7)"
  :precision binary64
  :pre (<= (fabs x) 1.0)

  :alt
  (! :herbie-platform default (expm1 x))

  (- (exp x) 1.0))