expax (section 3.5)

Percentage Accurate: 53.6% → 100.0%

Time: 5.8s

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: 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

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 56.9%

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

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

4. Applied rewrites 100.0%

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

Alternative 2: 68.5% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -4.5 \cdot 10^{+213}:\\ \;\;\;\;\left(\mathsf{fma}\left(0.5 \cdot x, x, \frac{x}{a}\right) \cdot a\right) \cdot a - 1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \left(\mathsf{fma}\left(0.16666666666666666 \cdot a, x, 0.5\right) \cdot x\right) \cdot a, x\right) \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (a x)
 :precision binary64
 (if (<= a -4.5e+213)
   (- (* (* (fma (* 0.5 x) x (/ x a)) a) a) 1.0)
   (* (fma x (* (* (fma (* 0.16666666666666666 a) x 0.5) x) a) x) a)))

C

double code(double a, double x) {
	double tmp;
	if (a <= -4.5e+213) {
		tmp = ((fma((0.5 * x), x, (x / a)) * a) * a) - 1.0;
	} else {
		tmp = fma(x, ((fma((0.16666666666666666 * a), x, 0.5) * x) * a), x) * a;
	}
	return tmp;
}

Julia

function code(a, x)
	tmp = 0.0
	if (a <= -4.5e+213)
		tmp = Float64(Float64(Float64(fma(Float64(0.5 * x), x, Float64(x / a)) * a) * a) - 1.0);
	else
		tmp = Float64(fma(x, Float64(Float64(fma(Float64(0.16666666666666666 * a), x, 0.5) * x) * a), x) * a);
	end
	return tmp
end

Wolfram

code[a_, x_] := If[LessEqual[a, -4.5e+213], N[(N[(N[(N[(N[(0.5 * x), $MachinePrecision] * x + N[(x / a), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision] * a), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(x * N[(N[(N[(N[(0.16666666666666666 * a), $MachinePrecision] * x + 0.5), $MachinePrecision] * x), $MachinePrecision] * a), $MachinePrecision] + x), $MachinePrecision] * a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -4.5000000000000002e213

   1. Initial program 100.0%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

   5. Applied rewrites 0.8%

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

   6. Taylor expanded in a around inf

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

   8. Applied rewrites 20.5%

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

   if -4.5000000000000002e213 < a

   1. Initial program 54.3%

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

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in a around 0

      \[\leadsto \color{blue}{a \cdot \left(x + a \cdot \left(\frac{1}{6} \cdot \left(a \cdot {x}^{3}\right) + \frac{1}{2} \cdot {x}^{2}\right)\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. associate-*r* N/A

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

      7. cube-mult N/A

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

      8. unpow2 N/A

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

      9. associate-*r* N/A

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

      10. distribute-rgt-out N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-*.f64 67.6

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

   7. Applied rewrites 67.6%

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

   9. Applied rewrites 71.4%

      \[\leadsto \mathsf{fma}\left(x, \left(\mathsf{fma}\left(0.16666666666666666 \cdot a, x, 0.5\right) \cdot x\right) \cdot a, x\right) \cdot a \]
3. Recombined 2 regimes into one program.

4. Final simplification 68.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -4.5 \cdot 10^{+213}:\\ \;\;\;\;\left(\mathsf{fma}\left(0.5 \cdot x, x, \frac{x}{a}\right) \cdot a\right) \cdot a - 1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \left(\mathsf{fma}\left(0.16666666666666666 \cdot a, x, 0.5\right) \cdot x\right) \cdot a, x\right) \cdot a\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 68.2% accurate, 3.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 56.9%

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

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

4. Applied rewrites 100.0%

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

5. Taylor expanded in a around 0

   \[\leadsto \color{blue}{a \cdot \left(x + a \cdot \left(\frac{1}{6} \cdot \left(a \cdot {x}^{3}\right) + \frac{1}{2} \cdot {x}^{2}\right)\right)} \]
6. Step-by-step derivation

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. associate-*r* N/A

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

   7. cube-mult N/A

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

   8. unpow2 N/A

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

   9. associate-*r* N/A

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

   10. distribute-rgt-out N/A

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

   11. lower-*.f64 N/A

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

   12. unpow2 N/A

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

   13. lower-*.f64 N/A

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

   14. lower-fma.f64 N/A

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

   15. lower-*.f64 63.9

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

7. Applied rewrites 63.9%

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

9. Applied rewrites 67.4%

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

10. Final simplification 67.4%

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

Alternative 4: 67.7% accurate, 18.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 56.9%

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

3. Taylor expanded in a around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 66.6

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

5. Applied rewrites 66.6%

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

Alternative 5: 20.0% 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 56.9%

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

3. Taylor expanded in a around 0

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

5. Applied rewrites 22.1%

   \[\leadsto \color{blue}{1} - 1 \]
6. 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 2024342 
(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))