expax (section 3.5)

Percentage Accurate: 54.0% → 100.0%

Time: 8.1s

Alternatives: 5

Speedup: 18.2×

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: 54.0% 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 53.0%

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

   1. lift--.f64 N/A

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

   2. lift-exp.f64 N/A

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

   3. lower-expm1.f64 100.0

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

4. Applied rewrites 100.0%

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

Alternative 2: 67.4% accurate, 3.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 53.0%

   \[e^{a \cdot x} - 1 \]
2. Add Preprocessing

3. Taylor expanded in a around 0

   \[\leadsto \color{blue}{a \cdot \left(x + a \cdot \left(\frac{1}{2} \cdot {x}^{2} + a \cdot \left(\frac{1}{24} \cdot \left(a \cdot {x}^{4}\right) + \frac{1}{6} \cdot {x}^{3}\right)\right)\right)} \]

5. Applied rewrites 62.0%

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

7. Applied rewrites 68.3%

   \[\leadsto a \cdot \mathsf{fma}\left(\left(a \cdot x\right) \cdot \mathsf{fma}\left(a, x \cdot \mathsf{fma}\left(a \cdot x, 0.041666666666666664, 0.16666666666666666\right), 0.5\right), \color{blue}{x}, x\right) \]

8. Taylor expanded in a around 0

   \[\leadsto a \cdot \mathsf{fma}\left(\left(a \cdot x\right) \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot \left(a \cdot x\right)\right), x, x\right) \]
9. Step-by-step derivation

10. Applied rewrites 69.1%

    \[\leadsto a \cdot \mathsf{fma}\left(\left(a \cdot x\right) \cdot \mathsf{fma}\left(a, x \cdot 0.16666666666666666, 0.5\right), x, x\right) \]
11. Step-by-step derivation

12. Applied rewrites 69.2%

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

13. Final simplification 69.2%

    \[\leadsto \left(a \cdot x\right) \cdot \mathsf{fma}\left(a \cdot x, \mathsf{fma}\left(a, x \cdot 0.16666666666666666, 0.5\right), 1\right) \]
14. Add Preprocessing

Alternative 3: 67.4% accurate, 3.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 53.0%

   \[e^{a \cdot x} - 1 \]
2. Add Preprocessing

3. Taylor expanded in a around 0

   \[\leadsto \color{blue}{a \cdot \left(x + a \cdot \left(\frac{1}{2} \cdot {x}^{2} + a \cdot \left(\frac{1}{24} \cdot \left(a \cdot {x}^{4}\right) + \frac{1}{6} \cdot {x}^{3}\right)\right)\right)} \]

5. Applied rewrites 62.0%

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

7. Applied rewrites 68.3%

   \[\leadsto a \cdot \mathsf{fma}\left(\left(a \cdot x\right) \cdot \mathsf{fma}\left(a, x \cdot \mathsf{fma}\left(a \cdot x, 0.041666666666666664, 0.16666666666666666\right), 0.5\right), \color{blue}{x}, x\right) \]

8. Taylor expanded in a around 0

   \[\leadsto a \cdot \mathsf{fma}\left(\left(a \cdot x\right) \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot \left(a \cdot x\right)\right), x, x\right) \]
9. Step-by-step derivation

10. Applied rewrites 69.1%

    \[\leadsto a \cdot \mathsf{fma}\left(\left(a \cdot x\right) \cdot \mathsf{fma}\left(a, x \cdot 0.16666666666666666, 0.5\right), x, x\right) \]
11. Add Preprocessing

Alternative 4: 66.7% accurate, 18.2× 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 53.0%

   \[e^{a \cdot x} - 1 \]
2. Add Preprocessing

3. Taylor expanded in a around 0

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

   1. lower-*.f64 68.9

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

5. Applied rewrites 68.9%

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

Alternative 5: 19.3% accurate, 27.3× speedup

Math

\[\begin{array}{l} \\ 1 + -1 \end{array} \]

FPCore

(FPCore (a x) :precision binary64 (+ 1.0 -1.0))

C

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

Fortran

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

Java

public static double code(double a, double x) {
	return 1.0 + -1.0;
}

Python

def code(a, x):
	return 1.0 + -1.0

Julia

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

MATLAB

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

Wolfram

code[a_, x_] := N[(1.0 + -1.0), $MachinePrecision]

Derivation

1. Initial program 53.0%

   \[e^{a \cdot x} - 1 \]
2. Add Preprocessing

3. Taylor expanded in a around 0

   \[\leadsto \color{blue}{1} - 1 \]
4. Step-by-step derivation

5. Applied rewrites 20.5%

   \[\leadsto \color{blue}{1} - 1 \]

6. Final simplification 20.5%

   \[\leadsto 1 + -1 \]
7. 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 2024227 
(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))