Diagrams.Tangent:$catParam from diagrams-lib-1.3.0.3, F

Percentage Accurate: 99.8% → 99.8%

Time: 7.8s

Alternatives: 3

Speedup: N/A×

• 23.8% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \left(x \cdot 3\right) \cdot x \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* (* x 3.0) x))

C

double code(double x) {
	return (x * 3.0) * x;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (x * 3.0d0) * x
end function

Java

public static double code(double x) {
	return (x * 3.0) * x;
}

Python

def code(x):
	return (x * 3.0) * x

Julia

function code(x)
	return Float64(Float64(x * 3.0) * x)
end

MATLAB

function tmp = code(x)
	tmp = (x * 3.0) * x;
end

Wolfram

code[x_] := N[(N[(x * 3.0), $MachinePrecision] * x), $MachinePrecision]

Initial Program: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(x \cdot 3\right) \cdot x \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* (* x 3.0) x))

C

double code(double x) {
	return (x * 3.0) * x;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (x * 3.0d0) * x
end function

Java

public static double code(double x) {
	return (x * 3.0) * x;
}

Python

def code(x):
	return (x * 3.0) * x

Julia

function code(x)
	return Float64(Float64(x * 3.0) * x)
end

MATLAB

function tmp = code(x)
	tmp = (x * 3.0) * x;
end

Wolfram

code[x_] := N[(N[(x * 3.0), $MachinePrecision] * x), $MachinePrecision]

Alternative 1: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ 3 \cdot \left(x \cdot x\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* 3.0 (* x x)))

C

double code(double x) {
	return 3.0 * (x * x);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 3.0d0 * (x * x)
end function

Java

public static double code(double x) {
	return 3.0 * (x * x);
}

Python

def code(x):
	return 3.0 * (x * x)

Julia

function code(x)
	return Float64(3.0 * Float64(x * x))
end

MATLAB

function tmp = code(x)
	tmp = 3.0 * (x * x);
end

Wolfram

code[x_] := N[(3.0 * N[(x * x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.7%

   \[\left(x \cdot 3\right) \cdot x \]
2. Add Preprocessing

3. Taylor expanded in x around 0 99.8%

   \[\leadsto \color{blue}{3 \cdot {x}^{2}} \]
4. Step-by-step derivation

   1. unpow2 99.8%

      \[\leadsto 3 \cdot \color{blue}{\left(x \cdot x\right)} \]

5. Applied egg-rr 99.8%

   \[\leadsto 3 \cdot \color{blue}{\left(x \cdot x\right)} \]
6. Add Preprocessing

Alternative 2: 4.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ 3 \cdot x \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* 3.0 x))

C

double code(double x) {
	return 3.0 * x;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 3.0d0 * x
end function

Java

public static double code(double x) {
	return 3.0 * x;
}

Python

def code(x):
	return 3.0 * x

Julia

function code(x)
	return Float64(3.0 * x)
end

MATLAB

function tmp = code(x)
	tmp = 3.0 * x;
end

Wolfram

code[x_] := N[(3.0 * x), $MachinePrecision]

Derivation

1. Initial program 99.7%

   \[\left(x \cdot 3\right) \cdot x \]
2. Add Preprocessing
3. Step-by-step derivation

   1. expm1-log1p-u 74.6%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x \cdot 3\right)\right)} \cdot x \]

   2. expm1-undefine 48.2%

      \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(x \cdot 3\right)} - 1\right)} \cdot x \]

   3. flip3-- 30.5%

      \[\leadsto \color{blue}{\frac{{\left(e^{\mathsf{log1p}\left(x \cdot 3\right)}\right)}^{3} - {1}^{3}}{e^{\mathsf{log1p}\left(x \cdot 3\right)} \cdot e^{\mathsf{log1p}\left(x \cdot 3\right)} + \left(1 \cdot 1 + e^{\mathsf{log1p}\left(x \cdot 3\right)} \cdot 1\right)}} \cdot x \]

   4. log1p-undefine 30.5%

      \[\leadsto \frac{{\left(e^{\color{blue}{\log \left(1 + x \cdot 3\right)}}\right)}^{3} - {1}^{3}}{e^{\mathsf{log1p}\left(x \cdot 3\right)} \cdot e^{\mathsf{log1p}\left(x \cdot 3\right)} + \left(1 \cdot 1 + e^{\mathsf{log1p}\left(x \cdot 3\right)} \cdot 1\right)} \cdot x \]

   5. rem-exp-log 30.5%

      \[\leadsto \frac{{\color{blue}{\left(1 + x \cdot 3\right)}}^{3} - {1}^{3}}{e^{\mathsf{log1p}\left(x \cdot 3\right)} \cdot e^{\mathsf{log1p}\left(x \cdot 3\right)} + \left(1 \cdot 1 + e^{\mathsf{log1p}\left(x \cdot 3\right)} \cdot 1\right)} \cdot x \]

   6. metadata-eval 30.5%

      \[\leadsto \frac{{\left(1 + x \cdot 3\right)}^{3} - \color{blue}{1}}{e^{\mathsf{log1p}\left(x \cdot 3\right)} \cdot e^{\mathsf{log1p}\left(x \cdot 3\right)} + \left(1 \cdot 1 + e^{\mathsf{log1p}\left(x \cdot 3\right)} \cdot 1\right)} \cdot x \]

   7. log1p-undefine 30.5%

      \[\leadsto \frac{{\left(1 + x \cdot 3\right)}^{3} - 1}{e^{\color{blue}{\log \left(1 + x \cdot 3\right)}} \cdot e^{\mathsf{log1p}\left(x \cdot 3\right)} + \left(1 \cdot 1 + e^{\mathsf{log1p}\left(x \cdot 3\right)} \cdot 1\right)} \cdot x \]

   8. rem-exp-log 30.5%

      \[\leadsto \frac{{\left(1 + x \cdot 3\right)}^{3} - 1}{\color{blue}{\left(1 + x \cdot 3\right)} \cdot e^{\mathsf{log1p}\left(x \cdot 3\right)} + \left(1 \cdot 1 + e^{\mathsf{log1p}\left(x \cdot 3\right)} \cdot 1\right)} \cdot x \]

   9. log1p-undefine 30.5%

      \[\leadsto \frac{{\left(1 + x \cdot 3\right)}^{3} - 1}{\left(1 + x \cdot 3\right) \cdot e^{\color{blue}{\log \left(1 + x \cdot 3\right)}} + \left(1 \cdot 1 + e^{\mathsf{log1p}\left(x \cdot 3\right)} \cdot 1\right)} \cdot x \]

   10. rem-exp-log 30.9%

       \[\leadsto \frac{{\left(1 + x \cdot 3\right)}^{3} - 1}{\left(1 + x \cdot 3\right) \cdot \color{blue}{\left(1 + x \cdot 3\right)} + \left(1 \cdot 1 + e^{\mathsf{log1p}\left(x \cdot 3\right)} \cdot 1\right)} \cdot x \]

   11. metadata-eval 30.9%

       \[\leadsto \frac{{\left(1 + x \cdot 3\right)}^{3} - 1}{\left(1 + x \cdot 3\right) \cdot \left(1 + x \cdot 3\right) + \left(\color{blue}{1} + e^{\mathsf{log1p}\left(x \cdot 3\right)} \cdot 1\right)} \cdot x \]

   12. log1p-undefine 30.9%

       \[\leadsto \frac{{\left(1 + x \cdot 3\right)}^{3} - 1}{\left(1 + x \cdot 3\right) \cdot \left(1 + x \cdot 3\right) + \left(1 + e^{\color{blue}{\log \left(1 + x \cdot 3\right)}} \cdot 1\right)} \cdot x \]

   13. rem-exp-log 41.6%

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

4. Applied egg-rr 41.6%

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

5. Taylor expanded in x around 0 53.1%

   \[\leadsto \frac{\color{blue}{9 \cdot x}}{\left(1 + x \cdot 3\right) \cdot \left(1 + x \cdot 3\right) + \left(1 + \left(1 + x \cdot 3\right) \cdot 1\right)} \cdot x \]
6. Step-by-step derivation

   1. *-commutative 53.1%

      \[\leadsto \frac{\color{blue}{x \cdot 9}}{\left(1 + x \cdot 3\right) \cdot \left(1 + x \cdot 3\right) + \left(1 + \left(1 + x \cdot 3\right) \cdot 1\right)} \cdot x \]

7. Simplified 53.1%

   \[\leadsto \frac{\color{blue}{x \cdot 9}}{\left(1 + x \cdot 3\right) \cdot \left(1 + x \cdot 3\right) + \left(1 + \left(1 + x \cdot 3\right) \cdot 1\right)} \cdot x \]

8. Taylor expanded in x around 0 53.1%

   \[\leadsto \frac{x \cdot 9}{\color{blue}{1} + \left(1 + \left(1 + x \cdot 3\right) \cdot 1\right)} \cdot x \]

9. Taylor expanded in x around inf 4.5%

   \[\leadsto \color{blue}{3} \cdot x \]
10. Add Preprocessing

Alternative 3: 4.3% accurate, 5.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 1.0)

C

double code(double x) {
	return 1.0;
}

Fortran

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

Java

public static double code(double x) {
	return 1.0;
}

Python

def code(x):
	return 1.0

Julia

function code(x)
	return 1.0
end

MATLAB

function tmp = code(x)
	tmp = 1.0;
end

Wolfram

code[x_] := 1.0

Derivation

1. Initial program 99.7%

   \[\left(x \cdot 3\right) \cdot x \]
2. Add Preprocessing
3. Step-by-step derivation

   1. expm1-log1p-u 74.6%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x \cdot 3\right)\right)} \cdot x \]

   2. expm1-undefine 48.2%

      \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(x \cdot 3\right)} - 1\right)} \cdot x \]

   3. flip3-- 30.5%

      \[\leadsto \color{blue}{\frac{{\left(e^{\mathsf{log1p}\left(x \cdot 3\right)}\right)}^{3} - {1}^{3}}{e^{\mathsf{log1p}\left(x \cdot 3\right)} \cdot e^{\mathsf{log1p}\left(x \cdot 3\right)} + \left(1 \cdot 1 + e^{\mathsf{log1p}\left(x \cdot 3\right)} \cdot 1\right)}} \cdot x \]

   4. log1p-undefine 30.5%

      \[\leadsto \frac{{\left(e^{\color{blue}{\log \left(1 + x \cdot 3\right)}}\right)}^{3} - {1}^{3}}{e^{\mathsf{log1p}\left(x \cdot 3\right)} \cdot e^{\mathsf{log1p}\left(x \cdot 3\right)} + \left(1 \cdot 1 + e^{\mathsf{log1p}\left(x \cdot 3\right)} \cdot 1\right)} \cdot x \]

   5. rem-exp-log 30.5%

      \[\leadsto \frac{{\color{blue}{\left(1 + x \cdot 3\right)}}^{3} - {1}^{3}}{e^{\mathsf{log1p}\left(x \cdot 3\right)} \cdot e^{\mathsf{log1p}\left(x \cdot 3\right)} + \left(1 \cdot 1 + e^{\mathsf{log1p}\left(x \cdot 3\right)} \cdot 1\right)} \cdot x \]

   6. metadata-eval 30.5%

      \[\leadsto \frac{{\left(1 + x \cdot 3\right)}^{3} - \color{blue}{1}}{e^{\mathsf{log1p}\left(x \cdot 3\right)} \cdot e^{\mathsf{log1p}\left(x \cdot 3\right)} + \left(1 \cdot 1 + e^{\mathsf{log1p}\left(x \cdot 3\right)} \cdot 1\right)} \cdot x \]

   7. log1p-undefine 30.5%

      \[\leadsto \frac{{\left(1 + x \cdot 3\right)}^{3} - 1}{e^{\color{blue}{\log \left(1 + x \cdot 3\right)}} \cdot e^{\mathsf{log1p}\left(x \cdot 3\right)} + \left(1 \cdot 1 + e^{\mathsf{log1p}\left(x \cdot 3\right)} \cdot 1\right)} \cdot x \]

   8. rem-exp-log 30.5%

      \[\leadsto \frac{{\left(1 + x \cdot 3\right)}^{3} - 1}{\color{blue}{\left(1 + x \cdot 3\right)} \cdot e^{\mathsf{log1p}\left(x \cdot 3\right)} + \left(1 \cdot 1 + e^{\mathsf{log1p}\left(x \cdot 3\right)} \cdot 1\right)} \cdot x \]

   9. log1p-undefine 30.5%

      \[\leadsto \frac{{\left(1 + x \cdot 3\right)}^{3} - 1}{\left(1 + x \cdot 3\right) \cdot e^{\color{blue}{\log \left(1 + x \cdot 3\right)}} + \left(1 \cdot 1 + e^{\mathsf{log1p}\left(x \cdot 3\right)} \cdot 1\right)} \cdot x \]

   10. rem-exp-log 30.9%

       \[\leadsto \frac{{\left(1 + x \cdot 3\right)}^{3} - 1}{\left(1 + x \cdot 3\right) \cdot \color{blue}{\left(1 + x \cdot 3\right)} + \left(1 \cdot 1 + e^{\mathsf{log1p}\left(x \cdot 3\right)} \cdot 1\right)} \cdot x \]

   11. metadata-eval 30.9%

       \[\leadsto \frac{{\left(1 + x \cdot 3\right)}^{3} - 1}{\left(1 + x \cdot 3\right) \cdot \left(1 + x \cdot 3\right) + \left(\color{blue}{1} + e^{\mathsf{log1p}\left(x \cdot 3\right)} \cdot 1\right)} \cdot x \]

   12. log1p-undefine 30.9%

       \[\leadsto \frac{{\left(1 + x \cdot 3\right)}^{3} - 1}{\left(1 + x \cdot 3\right) \cdot \left(1 + x \cdot 3\right) + \left(1 + e^{\color{blue}{\log \left(1 + x \cdot 3\right)}} \cdot 1\right)} \cdot x \]

   13. rem-exp-log 41.6%

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

4. Applied egg-rr 41.6%

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

5. Taylor expanded in x around 0 53.1%

   \[\leadsto \frac{\color{blue}{9 \cdot x}}{\left(1 + x \cdot 3\right) \cdot \left(1 + x \cdot 3\right) + \left(1 + \left(1 + x \cdot 3\right) \cdot 1\right)} \cdot x \]
6. Step-by-step derivation

   1. *-commutative 53.1%

      \[\leadsto \frac{\color{blue}{x \cdot 9}}{\left(1 + x \cdot 3\right) \cdot \left(1 + x \cdot 3\right) + \left(1 + \left(1 + x \cdot 3\right) \cdot 1\right)} \cdot x \]

7. Simplified 53.1%

   \[\leadsto \frac{\color{blue}{x \cdot 9}}{\left(1 + x \cdot 3\right) \cdot \left(1 + x \cdot 3\right) + \left(1 + \left(1 + x \cdot 3\right) \cdot 1\right)} \cdot x \]

8. Taylor expanded in x around inf 4.2%

   \[\leadsto \color{blue}{1} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024110 
(FPCore (x)
  :name "Diagrams.Tangent:$catParam from diagrams-lib-1.3.0.3, F"
  :precision binary64
  (* (* x 3.0) x))