expax (section 3.5)

Percentage Accurate: 54.2% → 100.0%

Time: 7.7s

Alternatives: 4

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: 54.2% 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 52.7%

   \[e^{a \cdot x} - 1 \]
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 99.9%

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

3. Simplified 99.9%

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

Alternative 2: 67.7% accurate, 6.2× speedup

Math

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

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(a * Float64(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[(a * N[(x * N[(1.0 + N[(N[(a * x), $MachinePrecision] * N[(0.5 + N[(a * N[(x * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 52.7%

   \[e^{a \cdot x} - 1 \]
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 99.9%

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

3. Simplified 99.9%

   \[\leadsto \color{blue}{\mathsf{expm1}\left(a \cdot x\right)} \]
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 72.7%

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

   1. associate-*l* N/A

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

   2. *-commutative N/A

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

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

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

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

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

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

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

   6. associate-*r* N/A

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

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

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

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

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

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

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

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

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

   11. *-lowering-*.f64 72.7%

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

9. Applied egg-rr 72.7%

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

10. Final simplification 72.7%

    \[\leadsto a \cdot \left(x \cdot \left(1 + \left(a \cdot x\right) \cdot \left(0.5 + a \cdot \left(x \cdot 0.16666666666666666\right)\right)\right)\right) \]
11. Add Preprocessing

Alternative 3: 67.7% accurate, 6.2× speedup

Math

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

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(a * Float64(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[(a * N[(x * N[(0.5 + N[(a * N[(x * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 52.7%

   \[e^{a \cdot x} - 1 \]
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 99.9%

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

3. Simplified 99.9%

   \[\leadsto \color{blue}{\mathsf{expm1}\left(a \cdot x\right)} \]
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 72.7%

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

Alternative 4: 67.1% 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 52.7%

   \[e^{a \cdot x} - 1 \]
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 99.9%

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

3. Simplified 99.9%

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

5. Taylor expanded in a around 0

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

   1. *-lowering-*.f64 72.3%

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

7. Simplified 72.3%

   \[\leadsto \color{blue}{a \cdot x} \]
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 2024164 
(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))