expax (section 3.5)

Percentage Accurate: 53.6% → 100.0%

Time: 8.4s

Alternatives: 5

Speedup: 35.0×

Specification

\[710 > a \cdot x\]

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

function code(a, x)
	return Float64(exp(Float64(a * x)) - 1.0)
end

MATLAB

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

Wolfram

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

Initial Program: 53.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

function code(a, x)
	return Float64(exp(Float64(a * x)) - 1.0)
end

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double a, double x) {
	return expm1((a * x));
}

Java

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

Python

def code(a, x):
	return math.expm1((a * x))

Julia

function code(a, x)
	return expm1(Float64(a * x))
end

Wolfram

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

Derivation

1. Initial program 54.3%

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

   1. expm1-define N/A

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

   2. expm1-lowering-expm1.f64 N/A

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

   3. *-lowering-*.f64 100.0%

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

3. Simplified 100.0%

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

Alternative 2: 67.1% accurate, 5.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 54.3%

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

   1. expm1-define N/A

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

   2. expm1-lowering-expm1.f64 N/A

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

   3. *-lowering-*.f64 100.0%

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

3. Simplified 100.0%

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

5. Taylor expanded in a around 0

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

7. Simplified 65.1%

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

   1. +-commutative N/A

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

   2. distribute-lft-in N/A

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

   3. *-rgt-identity N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

   10. associate-*r* N/A

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

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

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

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

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

   13. *-lowering-*.f64 65.1%

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

9. Applied egg-rr 65.1%

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

10. Final simplification 65.1%

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

Alternative 3: 67.1% accurate, 6.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 54.3%

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

   1. expm1-define N/A

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

   2. expm1-lowering-expm1.f64 N/A

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

   3. *-lowering-*.f64 100.0%

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

3. Simplified 100.0%

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

5. Taylor expanded in a around 0

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

7. Simplified 65.1%

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

Alternative 4: 67.1% accurate, 6.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 54.3%

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

   1. expm1-define N/A

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

   2. expm1-lowering-expm1.f64 N/A

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

   3. *-lowering-*.f64 100.0%

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

3. Simplified 100.0%

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

5. Taylor expanded in a around 0

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

7. Simplified 65.1%

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

   1. +-commutative N/A

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

   2. distribute-lft-in N/A

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

   3. *-rgt-identity N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

   10. associate-*r* N/A

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

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

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

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

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

   13. *-lowering-*.f64 65.1%

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

9. Applied egg-rr 65.1%

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

    1. associate-*l* N/A

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

    2. distribute-lft-out N/A

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

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

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

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

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

    5. associate-*l* N/A

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

    6. associate-*r* N/A

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

    7. *-commutative N/A

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

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

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

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

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

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

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

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

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

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

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

    13. *-lowering-*.f64 65.0%

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

11. Applied egg-rr 65.0%

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

12. Final simplification 65.0%

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

Alternative 5: 66.6% accurate, 35.0× speedup

Math

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

FPCore

(FPCore (a x) :precision binary64 (* a x))

C

double code(double a, double x) {
	return a * x;
}

Fortran

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

Java

public static double code(double a, double x) {
	return a * x;
}

Python

def code(a, x):
	return a * x

Julia

function code(a, x)
	return Float64(a * x)
end

MATLAB

function tmp = code(a, x)
	tmp = a * x;
end

Wolfram

code[a_, x_] := N[(a * x), $MachinePrecision]

Derivation

1. Initial program 54.3%

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

   1. expm1-define N/A

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

   2. expm1-lowering-expm1.f64 N/A

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

   3. *-lowering-*.f64 100.0%

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

3. Simplified 100.0%

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

5. Taylor expanded in a around 0

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

   1. *-lowering-*.f64 63.9%

      \[\leadsto \mathsf{*.f64}\left(a, \color{blue}{x}\right) \]

7. Simplified 63.9%

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

Developer Target 1: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double a, double x) {
	return expm1((a * x));
}

Java

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

Python

def code(a, x):
	return math.expm1((a * x))

Julia

function code(a, x)
	return expm1(Float64(a * x))
end

Wolfram

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

Reproduce

herbie shell --seed 2024139 
(FPCore (a x)
  :name "expax (section 3.5)"
  :precision binary64
  :pre (> 710.0 (* a x))

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

  (- (exp (* a x)) 1.0))