expax (section 3.5)

Percentage Accurate: 54.1% → 100.0%

Time: 6.1s

Alternatives: 4

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: 54.1% 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 55.7%

   \[e^{a \cdot x} - 1 \]
2. 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. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 100.0

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

3. Applied rewrites 100.0%

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

5. Final simplification 100.0%

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

Alternative 2: 70.1% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot x \leq -10:\\ \;\;\;\;\frac{a \cdot x}{\left(-0.5 \cdot a\right) \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(a \cdot x\right) \cdot a, 0.5, a\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (a x)
 :precision binary64
 (if (<= (* a x) -10.0)
   (/ (* a x) (* (* -0.5 a) x))
   (* (fma (* (* a x) a) 0.5 a) x)))

C

double code(double a, double x) {
	double tmp;
	if ((a * x) <= -10.0) {
		tmp = (a * x) / ((-0.5 * a) * x);
	} else {
		tmp = fma(((a * x) * a), 0.5, a) * x;
	}
	return tmp;
}

Julia

function code(a, x)
	tmp = 0.0
	if (Float64(a * x) <= -10.0)
		tmp = Float64(Float64(a * x) / Float64(Float64(-0.5 * a) * x));
	else
		tmp = Float64(fma(Float64(Float64(a * x) * a), 0.5, a) * x);
	end
	return tmp
end

Wolfram

code[a_, x_] := If[LessEqual[N[(a * x), $MachinePrecision], -10.0], N[(N[(a * x), $MachinePrecision] / N[(N[(-0.5 * a), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(a * x), $MachinePrecision] * a), $MachinePrecision] * 0.5 + a), $MachinePrecision] * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 a x) < -10

   1. Initial program 100.0%

      \[e^{a \cdot x} - 1 \]
   2. 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. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 100.0

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

   3. Applied rewrites 100.0%

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

   5. Taylor expanded in a around 0

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

   7. Applied rewrites 0.8%

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

   9. Applied rewrites 0.8%

      \[\leadsto \frac{a \cdot x}{\color{blue}{\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664 \cdot x, a, 0.16666666666666666\right), a \cdot x, 0.5\right) \cdot a, x, 1\right)}}} \]

   10. Taylor expanded in a around 0

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

   12. Applied rewrites 15.4%

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

   13. Taylor expanded in a around inf

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

   15. Applied rewrites 15.5%

       \[\leadsto \frac{a \cdot x}{\left(-0.5 \cdot a\right) \cdot x} \]

   if -10 < (*.f64 a x)

   1. Initial program 35.9%

      \[e^{a \cdot x} - 1 \]
   2. 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. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 100.0

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

   3. Applied rewrites 100.0%

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

   5. Taylor expanded in a around 0

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

      1. distribute-lft-in N/A

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

      2. associate-*r* N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. distribute-rgt1-in N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. distribute-rgt1-in N/A

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

      12. associate-*r* N/A

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

      13. associate-*l* N/A

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

      14. unpow2 N/A

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

      15. associate-*r* N/A

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

      16. *-commutative N/A

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

      17. distribute-lft-in N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 N/A

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

   7. Applied rewrites 95.0%

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

   9. Applied rewrites 98.2%

      \[\leadsto \mathsf{fma}\left(\left(a \cdot x\right) \cdot a, 0.5, a\right) \cdot x \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 70.4% accurate, 3.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 55.7%

   \[e^{a \cdot x} - 1 \]
2. 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. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 100.0

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

3. Applied rewrites 100.0%

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

5. Taylor expanded in a around 0

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

7. Applied rewrites 68.8%

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

9. Applied rewrites 68.8%

   \[\leadsto \frac{a \cdot x}{\color{blue}{\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664 \cdot x, a, 0.16666666666666666\right), a \cdot x, 0.5\right) \cdot a, x, 1\right)}}} \]

10. Taylor expanded in a around 0

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

12. Applied rewrites 72.7%

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

14. Applied rewrites 72.9%

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

15. Final simplification 72.9%

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

Alternative 4: 66.4% 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 55.7%

   \[e^{a \cdot x} - 1 \]
2. 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. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 100.0

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

3. Applied rewrites 100.0%

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

5. Taylor expanded in a around 0

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

   1. lower-*.f64 69.0

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

7. Applied rewrites 69.0%

   \[\leadsto \color{blue}{a \cdot x} \]
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 2024272 
(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))