expax (section 3.5)

Percentage Accurate: 54.3% → 100.0%

Time: 7.0s

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: 54.3% 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 48.2%

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

   1. expm1-define 100.0%

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

3. Simplified 100.0%

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

5. Final simplification 100.0%

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

Alternative 2: 71.1% accurate, 11.7× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\frac{\frac{1}{a}}{x} + -0.5} \end{array} \]

FPCore

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

C

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

Fortran

real(8) function code(a, x)
    real(8), intent (in) :: a
    real(8), intent (in) :: x
    code = 1.0d0 / (((1.0d0 / a) / x) + (-0.5d0))
end function

Java

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

Python

def code(a, x):
	return 1.0 / (((1.0 / a) / x) + -0.5)

Julia

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

MATLAB

function tmp = code(a, x)
	tmp = 1.0 / (((1.0 / a) / x) + -0.5);
end

Wolfram

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

Derivation

1. Initial program 48.2%

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

   1. expm1-define 100.0%

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

3. Simplified 100.0%

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

   1. add0 100.0%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(a \cdot x\right) + 0} \]

   2. flip3-+ 54.8%

      \[\leadsto \color{blue}{\frac{{\left(\mathsf{expm1}\left(a \cdot x\right)\right)}^{3} + {0}^{3}}{\mathsf{expm1}\left(a \cdot x\right) \cdot \mathsf{expm1}\left(a \cdot x\right) + \left(0 \cdot 0 - \mathsf{expm1}\left(a \cdot x\right) \cdot 0\right)}} \]

   3. pow3 54.9%

      \[\leadsto \frac{\color{blue}{\left(\mathsf{expm1}\left(a \cdot x\right) \cdot \mathsf{expm1}\left(a \cdot x\right)\right) \cdot \mathsf{expm1}\left(a \cdot x\right)} + {0}^{3}}{\mathsf{expm1}\left(a \cdot x\right) \cdot \mathsf{expm1}\left(a \cdot x\right) + \left(0 \cdot 0 - \mathsf{expm1}\left(a \cdot x\right) \cdot 0\right)} \]

   4. metadata-eval 54.9%

      \[\leadsto \frac{\left(\mathsf{expm1}\left(a \cdot x\right) \cdot \mathsf{expm1}\left(a \cdot x\right)\right) \cdot \mathsf{expm1}\left(a \cdot x\right) + \color{blue}{0}}{\mathsf{expm1}\left(a \cdot x\right) \cdot \mathsf{expm1}\left(a \cdot x\right) + \left(0 \cdot 0 - \mathsf{expm1}\left(a \cdot x\right) \cdot 0\right)} \]

   5. add0 54.9%

      \[\leadsto \frac{\color{blue}{\left(\mathsf{expm1}\left(a \cdot x\right) \cdot \mathsf{expm1}\left(a \cdot x\right)\right) \cdot \mathsf{expm1}\left(a \cdot x\right)}}{\mathsf{expm1}\left(a \cdot x\right) \cdot \mathsf{expm1}\left(a \cdot x\right) + \left(0 \cdot 0 - \mathsf{expm1}\left(a \cdot x\right) \cdot 0\right)} \]

   6. pow3 54.8%

      \[\leadsto \frac{\color{blue}{{\left(\mathsf{expm1}\left(a \cdot x\right)\right)}^{3}}}{\mathsf{expm1}\left(a \cdot x\right) \cdot \mathsf{expm1}\left(a \cdot x\right) + \left(0 \cdot 0 - \mathsf{expm1}\left(a \cdot x\right) \cdot 0\right)} \]

   7. pow2 54.8%

      \[\leadsto \frac{{\left(\mathsf{expm1}\left(a \cdot x\right)\right)}^{3}}{\color{blue}{{\left(\mathsf{expm1}\left(a \cdot x\right)\right)}^{2}} + \left(0 \cdot 0 - \mathsf{expm1}\left(a \cdot x\right) \cdot 0\right)} \]

   8. metadata-eval 54.8%

      \[\leadsto \frac{{\left(\mathsf{expm1}\left(a \cdot x\right)\right)}^{3}}{{\left(\mathsf{expm1}\left(a \cdot x\right)\right)}^{2} + \left(\color{blue}{0} - \mathsf{expm1}\left(a \cdot x\right) \cdot 0\right)} \]

6. Applied egg-rr 54.8%

   \[\leadsto \color{blue}{\frac{{\left(\mathsf{expm1}\left(a \cdot x\right)\right)}^{3}}{{\left(\mathsf{expm1}\left(a \cdot x\right)\right)}^{2} + \left(0 - \mathsf{expm1}\left(a \cdot x\right) \cdot 0\right)}} \]
7. Step-by-step derivation

   1. sub0-neg 54.8%

      \[\leadsto \frac{{\left(\mathsf{expm1}\left(a \cdot x\right)\right)}^{3}}{{\left(\mathsf{expm1}\left(a \cdot x\right)\right)}^{2} + \color{blue}{\left(-\mathsf{expm1}\left(a \cdot x\right) \cdot 0\right)}} \]

   2. mul0-rgt 54.8%

      \[\leadsto \frac{{\left(\mathsf{expm1}\left(a \cdot x\right)\right)}^{3}}{{\left(\mathsf{expm1}\left(a \cdot x\right)\right)}^{2} + \left(-\color{blue}{0}\right)} \]

   3. sub-neg 54.8%

      \[\leadsto \frac{{\left(\mathsf{expm1}\left(a \cdot x\right)\right)}^{3}}{\color{blue}{{\left(\mathsf{expm1}\left(a \cdot x\right)\right)}^{2} - 0}} \]

   4. --rgt-identity 54.8%

      \[\leadsto \frac{{\left(\mathsf{expm1}\left(a \cdot x\right)\right)}^{3}}{\color{blue}{{\left(\mathsf{expm1}\left(a \cdot x\right)\right)}^{2}}} \]

   5. div-inv 54.7%

      \[\leadsto \color{blue}{{\left(\mathsf{expm1}\left(a \cdot x\right)\right)}^{3} \cdot \frac{1}{{\left(\mathsf{expm1}\left(a \cdot x\right)\right)}^{2}}} \]

   6. pow-flip 54.8%

      \[\leadsto {\left(\mathsf{expm1}\left(a \cdot x\right)\right)}^{3} \cdot \color{blue}{{\left(\mathsf{expm1}\left(a \cdot x\right)\right)}^{\left(-2\right)}} \]

   7. metadata-eval 54.8%

      \[\leadsto {\left(\mathsf{expm1}\left(a \cdot x\right)\right)}^{3} \cdot {\left(\mathsf{expm1}\left(a \cdot x\right)\right)}^{\color{blue}{-2}} \]

8. Applied egg-rr 54.8%

   \[\leadsto \color{blue}{{\left(\mathsf{expm1}\left(a \cdot x\right)\right)}^{3} \cdot {\left(\mathsf{expm1}\left(a \cdot x\right)\right)}^{-2}} \]
9. Step-by-step derivation

   1. pow-prod-up 100.0%

      \[\leadsto \color{blue}{{\left(\mathsf{expm1}\left(a \cdot x\right)\right)}^{\left(3 + -2\right)}} \]

   2. metadata-eval 100.0%

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

   3. pow1 100.0%

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

   4. remove-double-div 99.0%

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

10. Applied egg-rr 99.0%

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

11. Taylor expanded in a around 0 73.1%

    \[\leadsto \frac{1}{\color{blue}{\frac{1}{a \cdot x} - 0.5}} \]
12. Step-by-step derivation

    1. sub-neg 73.1%

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

    2. associate-/r* 72.9%

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

    3. metadata-eval 72.9%

       \[\leadsto \frac{1}{\frac{\frac{1}{a}}{x} + \color{blue}{-0.5}} \]

13. Simplified 72.9%

    \[\leadsto \frac{1}{\color{blue}{\frac{\frac{1}{a}}{x} + -0.5}} \]

14. Final simplification 72.9%

    \[\leadsto \frac{1}{\frac{\frac{1}{a}}{x} + -0.5} \]
15. Add Preprocessing

Alternative 3: 71.2% accurate, 11.7× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\frac{1}{a \cdot x} - 0.5} \end{array} \]

FPCore

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

C

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

Fortran

real(8) function code(a, x)
    real(8), intent (in) :: a
    real(8), intent (in) :: x
    code = 1.0d0 / ((1.0d0 / (a * x)) - 0.5d0)
end function

Java

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

Python

def code(a, x):
	return 1.0 / ((1.0 / (a * x)) - 0.5)

Julia

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

MATLAB

function tmp = code(a, x)
	tmp = 1.0 / ((1.0 / (a * x)) - 0.5);
end

Wolfram

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

Derivation

1. Initial program 48.2%

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

   1. expm1-define 100.0%

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

3. Simplified 100.0%

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

   1. add0 100.0%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(a \cdot x\right) + 0} \]

   2. flip3-+ 54.8%

      \[\leadsto \color{blue}{\frac{{\left(\mathsf{expm1}\left(a \cdot x\right)\right)}^{3} + {0}^{3}}{\mathsf{expm1}\left(a \cdot x\right) \cdot \mathsf{expm1}\left(a \cdot x\right) + \left(0 \cdot 0 - \mathsf{expm1}\left(a \cdot x\right) \cdot 0\right)}} \]

   3. pow3 54.9%

      \[\leadsto \frac{\color{blue}{\left(\mathsf{expm1}\left(a \cdot x\right) \cdot \mathsf{expm1}\left(a \cdot x\right)\right) \cdot \mathsf{expm1}\left(a \cdot x\right)} + {0}^{3}}{\mathsf{expm1}\left(a \cdot x\right) \cdot \mathsf{expm1}\left(a \cdot x\right) + \left(0 \cdot 0 - \mathsf{expm1}\left(a \cdot x\right) \cdot 0\right)} \]

   4. metadata-eval 54.9%

      \[\leadsto \frac{\left(\mathsf{expm1}\left(a \cdot x\right) \cdot \mathsf{expm1}\left(a \cdot x\right)\right) \cdot \mathsf{expm1}\left(a \cdot x\right) + \color{blue}{0}}{\mathsf{expm1}\left(a \cdot x\right) \cdot \mathsf{expm1}\left(a \cdot x\right) + \left(0 \cdot 0 - \mathsf{expm1}\left(a \cdot x\right) \cdot 0\right)} \]

   5. add0 54.9%

      \[\leadsto \frac{\color{blue}{\left(\mathsf{expm1}\left(a \cdot x\right) \cdot \mathsf{expm1}\left(a \cdot x\right)\right) \cdot \mathsf{expm1}\left(a \cdot x\right)}}{\mathsf{expm1}\left(a \cdot x\right) \cdot \mathsf{expm1}\left(a \cdot x\right) + \left(0 \cdot 0 - \mathsf{expm1}\left(a \cdot x\right) \cdot 0\right)} \]

   6. pow3 54.8%

      \[\leadsto \frac{\color{blue}{{\left(\mathsf{expm1}\left(a \cdot x\right)\right)}^{3}}}{\mathsf{expm1}\left(a \cdot x\right) \cdot \mathsf{expm1}\left(a \cdot x\right) + \left(0 \cdot 0 - \mathsf{expm1}\left(a \cdot x\right) \cdot 0\right)} \]

   7. pow2 54.8%

      \[\leadsto \frac{{\left(\mathsf{expm1}\left(a \cdot x\right)\right)}^{3}}{\color{blue}{{\left(\mathsf{expm1}\left(a \cdot x\right)\right)}^{2}} + \left(0 \cdot 0 - \mathsf{expm1}\left(a \cdot x\right) \cdot 0\right)} \]

   8. metadata-eval 54.8%

      \[\leadsto \frac{{\left(\mathsf{expm1}\left(a \cdot x\right)\right)}^{3}}{{\left(\mathsf{expm1}\left(a \cdot x\right)\right)}^{2} + \left(\color{blue}{0} - \mathsf{expm1}\left(a \cdot x\right) \cdot 0\right)} \]

6. Applied egg-rr 54.8%

   \[\leadsto \color{blue}{\frac{{\left(\mathsf{expm1}\left(a \cdot x\right)\right)}^{3}}{{\left(\mathsf{expm1}\left(a \cdot x\right)\right)}^{2} + \left(0 - \mathsf{expm1}\left(a \cdot x\right) \cdot 0\right)}} \]
7. Step-by-step derivation

   1. sub0-neg 54.8%

      \[\leadsto \frac{{\left(\mathsf{expm1}\left(a \cdot x\right)\right)}^{3}}{{\left(\mathsf{expm1}\left(a \cdot x\right)\right)}^{2} + \color{blue}{\left(-\mathsf{expm1}\left(a \cdot x\right) \cdot 0\right)}} \]

   2. mul0-rgt 54.8%

      \[\leadsto \frac{{\left(\mathsf{expm1}\left(a \cdot x\right)\right)}^{3}}{{\left(\mathsf{expm1}\left(a \cdot x\right)\right)}^{2} + \left(-\color{blue}{0}\right)} \]

   3. sub-neg 54.8%

      \[\leadsto \frac{{\left(\mathsf{expm1}\left(a \cdot x\right)\right)}^{3}}{\color{blue}{{\left(\mathsf{expm1}\left(a \cdot x\right)\right)}^{2} - 0}} \]

   4. --rgt-identity 54.8%

      \[\leadsto \frac{{\left(\mathsf{expm1}\left(a \cdot x\right)\right)}^{3}}{\color{blue}{{\left(\mathsf{expm1}\left(a \cdot x\right)\right)}^{2}}} \]

   5. div-inv 54.7%

      \[\leadsto \color{blue}{{\left(\mathsf{expm1}\left(a \cdot x\right)\right)}^{3} \cdot \frac{1}{{\left(\mathsf{expm1}\left(a \cdot x\right)\right)}^{2}}} \]

   6. pow-flip 54.8%

      \[\leadsto {\left(\mathsf{expm1}\left(a \cdot x\right)\right)}^{3} \cdot \color{blue}{{\left(\mathsf{expm1}\left(a \cdot x\right)\right)}^{\left(-2\right)}} \]

   7. metadata-eval 54.8%

      \[\leadsto {\left(\mathsf{expm1}\left(a \cdot x\right)\right)}^{3} \cdot {\left(\mathsf{expm1}\left(a \cdot x\right)\right)}^{\color{blue}{-2}} \]

8. Applied egg-rr 54.8%

   \[\leadsto \color{blue}{{\left(\mathsf{expm1}\left(a \cdot x\right)\right)}^{3} \cdot {\left(\mathsf{expm1}\left(a \cdot x\right)\right)}^{-2}} \]
9. Step-by-step derivation

   1. pow-prod-up 100.0%

      \[\leadsto \color{blue}{{\left(\mathsf{expm1}\left(a \cdot x\right)\right)}^{\left(3 + -2\right)}} \]

   2. metadata-eval 100.0%

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

   3. pow1 100.0%

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

   4. remove-double-div 99.0%

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

10. Applied egg-rr 99.0%

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

11. Taylor expanded in a around 0 73.1%

    \[\leadsto \frac{1}{\color{blue}{\frac{1}{a \cdot x} - 0.5}} \]

12. Final simplification 73.1%

    \[\leadsto \frac{1}{\frac{1}{a \cdot x} - 0.5} \]
13. Add Preprocessing

Alternative 4: 66.8% 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 48.2%

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

   1. expm1-define 100.0%

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

3. Simplified 100.0%

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

5. Taylor expanded in a around 0 69.0%

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

6. Final simplification 69.0%

   \[\leadsto a \cdot x \]
7. Add Preprocessing

Developer target: 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 2024046 
(FPCore (a x)
  :name "expax (section 3.5)"
  :precision binary64
  :pre (> 710.0 (* a x))

  :alt
  (expm1 (* a x))

  (- (exp (* a x)) 1.0))