expax (section 3.5)

Percentage Accurate: 53.2% → 99.9%

Time: 23.3s

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: 53.2% 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: 99.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \frac{1}{1 + {e}^{\left(a \cdot x\right)}} \cdot \mathsf{expm1}\left(a \cdot \left(x \cdot 2\right)\right) \end{array} \]

FPCore

(FPCore (a x)
 :precision binary64
 (* (/ 1.0 (+ 1.0 (pow E (* a x)))) (expm1 (* a (* x 2.0)))))

C

double code(double a, double x) {
	return (1.0 / (1.0 + pow(((double) M_E), (a * x)))) * expm1((a * (x * 2.0)));
}

Java

public static double code(double a, double x) {
	return (1.0 / (1.0 + Math.pow(Math.E, (a * x)))) * Math.expm1((a * (x * 2.0)));
}

Python

def code(a, x):
	return (1.0 / (1.0 + math.pow(math.e, (a * x)))) * math.expm1((a * (x * 2.0)))

Julia

function code(a, x)
	return Float64(Float64(1.0 / Float64(1.0 + (exp(1) ^ Float64(a * x)))) * expm1(Float64(a * Float64(x * 2.0))))
end

Wolfram

code[a_, x_] := N[(N[(1.0 / N[(1.0 + N[Power[E, N[(a * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(Exp[N[(a * N[(x * 2.0), $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.9%

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

   1. flip-- N/A

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

   2. clear-num N/A

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

   3. associate-/r/ N/A

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

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

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

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \left(e^{a \cdot x} + 1\right)\right), \left(\color{blue}{e^{a \cdot x} \cdot e^{a \cdot x}} - 1 \cdot 1\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\left(e^{a \cdot x}\right), 1\right)\right), \left(e^{a \cdot x} \cdot \color{blue}{e^{a \cdot x}} - 1 \cdot 1\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\left(a \cdot x\right)\right), 1\right)\right), \left(e^{a \cdot x} \cdot e^{\color{blue}{a \cdot x}} - 1 \cdot 1\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(a, x\right)\right), 1\right)\right), \left(e^{a \cdot x} \cdot e^{\color{blue}{a} \cdot x} - 1 \cdot 1\right)\right) \]

   9. prod-exp N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(a, x\right)\right), 1\right)\right), \left(e^{a \cdot x + a \cdot x} - \color{blue}{1} \cdot 1\right)\right) \]

   10. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(a, x\right)\right), 1\right)\right), \left(e^{a \cdot x + a \cdot x} - 1\right)\right) \]

   11. expm1-define N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(a, x\right)\right), 1\right)\right), \left(\mathsf{expm1}\left(a \cdot x + a \cdot x\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(a, x\right)\right), 1\right)\right), \mathsf{expm1.f64}\left(\left(a \cdot x + a \cdot x\right)\right)\right) \]

   13. distribute-lft-out N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(a, x\right)\right), 1\right)\right), \mathsf{expm1.f64}\left(\left(a \cdot \left(x + x\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(a, x\right)\right), 1\right)\right), \mathsf{expm1.f64}\left(\mathsf{*.f64}\left(a, \left(x + x\right)\right)\right)\right) \]

   15. count-2 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(a, x\right)\right), 1\right)\right), \mathsf{expm1.f64}\left(\mathsf{*.f64}\left(a, \left(2 \cdot x\right)\right)\right)\right) \]

   16. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(a, x\right)\right), 1\right)\right), \mathsf{expm1.f64}\left(\mathsf{*.f64}\left(a, \left(x \cdot 2\right)\right)\right)\right) \]

   17. *-lowering-*.f64 99.9%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(a, x\right)\right), 1\right)\right), \mathsf{expm1.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(x, 2\right)\right)\right)\right) \]

6. Applied egg-rr 99.9%

   \[\leadsto \color{blue}{\frac{1}{e^{a \cdot x} + 1} \cdot \mathsf{expm1}\left(a \cdot \left(x \cdot 2\right)\right)} \]
7. Step-by-step derivation

   1. *-lft-identity N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\left(e^{1 \cdot \left(a \cdot x\right)}\right), 1\right)\right), \mathsf{expm1.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(x, 2\right)\right)\right)\right) \]

   2. exp-prod N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\left({\left(e^{1}\right)}^{\left(a \cdot x\right)}\right), 1\right)\right), \mathsf{expm1.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\color{blue}{x}, 2\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\left(e^{1}\right), \left(a \cdot x\right)\right), 1\right)\right), \mathsf{expm1.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\color{blue}{x}, 2\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{exp.f64}\left(1\right), \left(a \cdot x\right)\right), 1\right)\right), \mathsf{expm1.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(x, 2\right)\right)\right)\right) \]

   5. *-lowering-*.f64 100.0%

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{exp.f64}\left(1\right), \mathsf{*.f64}\left(a, x\right)\right), 1\right)\right), \mathsf{expm1.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(x, 2\right)\right)\right)\right) \]

8. Applied egg-rr 100.0%

   \[\leadsto \frac{1}{\color{blue}{{\left(e^{1}\right)}^{\left(a \cdot x\right)}} + 1} \cdot \mathsf{expm1}\left(a \cdot \left(x \cdot 2\right)\right) \]
9. Step-by-step derivation

   1. pow-lowering-pow.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\left(e^{1}\right), \left(a \cdot x\right)\right), 1\right)\right), \mathsf{expm1.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\color{blue}{x}, 2\right)\right)\right)\right) \]

   2. exp-1-e N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{E}\left(\right), \left(a \cdot x\right)\right), 1\right)\right), \mathsf{expm1.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(x, 2\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{E.f64}\left(\right), \left(a \cdot x\right)\right), 1\right)\right), \mathsf{expm1.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(x, 2\right)\right)\right)\right) \]

   4. *-lowering-*.f64 100.0%

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{E.f64}\left(\right), \mathsf{*.f64}\left(a, x\right)\right), 1\right)\right), \mathsf{expm1.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(x, 2\right)\right)\right)\right) \]

10. Applied egg-rr 100.0%

    \[\leadsto \frac{1}{\color{blue}{{e}^{\left(a \cdot x\right)}} + 1} \cdot \mathsf{expm1}\left(a \cdot \left(x \cdot 2\right)\right) \]

11. Final simplification 100.0%

    \[\leadsto \frac{1}{1 + {e}^{\left(a \cdot x\right)}} \cdot \mathsf{expm1}\left(a \cdot \left(x \cdot 2\right)\right) \]
12. Add Preprocessing

Alternative 2: 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 54.9%

   \[e^{a \cdot x} - 1 \]
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 3: 67.7% accurate, 6.2× speedup

Math

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

FPCore

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

C

double code(double a, double x) {
	return (a * x) * (1.0 + ((a * x) * (0.5 + (x * (a * 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 + (x * (a * 0.16666666666666666d0)))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 54.9%

   \[e^{a \cdot x} - 1 \]
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 65.6%

   \[\leadsto \color{blue}{\left(a \cdot x\right) \cdot \left(1 + \left(a \cdot x\right) \cdot \left(0.5 + x \cdot \left(a \cdot 0.16666666666666666\right)\right)\right)} \]
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 54.9%

   \[e^{a \cdot x} - 1 \]
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 64.9%

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

7. Simplified 64.9%

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

Alternative 5: 19.0% accurate, 105.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (a x) :precision binary64 0.0)

C

double code(double a, double x) {
	return 0.0;
}

Fortran

real(8) function code(a, x)
    real(8), intent (in) :: a
    real(8), intent (in) :: x
    code = 0.0d0
end function

Java

public static double code(double a, double x) {
	return 0.0;
}

Python

def code(a, x):
	return 0.0

Julia

function code(a, x)
	return 0.0
end

MATLAB

function tmp = code(a, x)
	tmp = 0.0;
end

Wolfram

code[a_, x_] := 0.0

Derivation

1. Initial program 54.9%

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

3. Taylor expanded in a around 0

   \[\leadsto \mathsf{\_.f64}\left(\color{blue}{1}, 1\right) \]
4. Step-by-step derivation

5. Simplified 18.2%

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

   1. metadata-eval 18.2%

      \[\leadsto 0 \]

7. Applied egg-rr 18.2%

   \[\leadsto \color{blue}{0} \]
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 2024141 
(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))