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

Percentage Accurate: 99.8% → 99.6%

Time: 8.5s

Alternatives: 5

Speedup: 1.4×

• 25.2% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ 3 \cdot \left(\left(\left(x \cdot 3\right) \cdot x - x \cdot 4\right) + 1\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* 3.0 (+ (- (* (* x 3.0) x) (* x 4.0)) 1.0)))

C

double code(double x) {
	return 3.0 * ((((x * 3.0) * x) - (x * 4.0)) + 1.0);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 3.0d0 * ((((x * 3.0d0) * x) - (x * 4.0d0)) + 1.0d0)
end function

Java

public static double code(double x) {
	return 3.0 * ((((x * 3.0) * x) - (x * 4.0)) + 1.0);
}

Python

def code(x):
	return 3.0 * ((((x * 3.0) * x) - (x * 4.0)) + 1.0)

Julia

function code(x)
	return Float64(3.0 * Float64(Float64(Float64(Float64(x * 3.0) * x) - Float64(x * 4.0)) + 1.0))
end

MATLAB

function tmp = code(x)
	tmp = 3.0 * ((((x * 3.0) * x) - (x * 4.0)) + 1.0);
end

Wolfram

code[x_] := N[(3.0 * N[(N[(N[(N[(x * 3.0), $MachinePrecision] * x), $MachinePrecision] - N[(x * 4.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]

Initial Program: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ 3 \cdot \left(\left(\left(x \cdot 3\right) \cdot x - x \cdot 4\right) + 1\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* 3.0 (+ (- (* (* x 3.0) x) (* x 4.0)) 1.0)))

C

double code(double x) {
	return 3.0 * ((((x * 3.0) * x) - (x * 4.0)) + 1.0);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 3.0d0 * ((((x * 3.0d0) * x) - (x * 4.0d0)) + 1.0d0)
end function

Java

public static double code(double x) {
	return 3.0 * ((((x * 3.0) * x) - (x * 4.0)) + 1.0);
}

Python

def code(x):
	return 3.0 * ((((x * 3.0) * x) - (x * 4.0)) + 1.0)

Julia

function code(x)
	return Float64(3.0 * Float64(Float64(Float64(Float64(x * 3.0) * x) - Float64(x * 4.0)) + 1.0))
end

MATLAB

function tmp = code(x)
	tmp = 3.0 * ((((x * 3.0) * x) - (x * 4.0)) + 1.0);
end

Wolfram

code[x_] := N[(3.0 * N[(N[(N[(N[(x * 3.0), $MachinePrecision] * x), $MachinePrecision] - N[(x * 4.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ 3 + x \cdot \left(x \cdot \left(9 - \frac{12}{x}\right)\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (+ 3.0 (* x (* x (- 9.0 (/ 12.0 x))))))

C

double code(double x) {
	return 3.0 + (x * (x * (9.0 - (12.0 / x))));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 3.0d0 + (x * (x * (9.0d0 - (12.0d0 / x))))
end function

Java

public static double code(double x) {
	return 3.0 + (x * (x * (9.0 - (12.0 / x))));
}

Python

def code(x):
	return 3.0 + (x * (x * (9.0 - (12.0 / x))))

Julia

function code(x)
	return Float64(3.0 + Float64(x * Float64(x * Float64(9.0 - Float64(12.0 / x)))))
end

MATLAB

function tmp = code(x)
	tmp = 3.0 + (x * (x * (9.0 - (12.0 / x))));
end

Wolfram

code[x_] := N[(3.0 + N[(x * N[(x * N[(9.0 - N[(12.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

   \[3 \cdot \left(\left(\left(x \cdot 3\right) \cdot x - x \cdot 4\right) + 1\right) \]
2. Add Preprocessing

3. Taylor expanded in x around 0 99.9%

   \[\leadsto \color{blue}{3 + x \cdot \left(9 \cdot x - 12\right)} \]

4. Taylor expanded in x around inf 99.9%

   \[\leadsto 3 + x \cdot \color{blue}{\left(x \cdot \left(9 - 12 \cdot \frac{1}{x}\right)\right)} \]
5. Step-by-step derivation

   1. associate-*r/ 99.9%

      \[\leadsto 3 + x \cdot \left(x \cdot \left(9 - \color{blue}{\frac{12 \cdot 1}{x}}\right)\right) \]

   2. metadata-eval 99.9%

      \[\leadsto 3 + x \cdot \left(x \cdot \left(9 - \frac{\color{blue}{12}}{x}\right)\right) \]

6. Simplified 99.9%

   \[\leadsto 3 + x \cdot \color{blue}{\left(x \cdot \left(9 - \frac{12}{x}\right)\right)} \]

7. Final simplification 99.9%

   \[\leadsto 3 + x \cdot \left(x \cdot \left(9 - \frac{12}{x}\right)\right) \]
8. Add Preprocessing

Alternative 2: 99.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ 3 + x \cdot \left(x \cdot 9 - 12\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (+ 3.0 (* x (- (* x 9.0) 12.0))))

C

double code(double x) {
	return 3.0 + (x * ((x * 9.0) - 12.0));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 3.0d0 + (x * ((x * 9.0d0) - 12.0d0))
end function

Java

public static double code(double x) {
	return 3.0 + (x * ((x * 9.0) - 12.0));
}

Python

def code(x):
	return 3.0 + (x * ((x * 9.0) - 12.0))

Julia

function code(x)
	return Float64(3.0 + Float64(x * Float64(Float64(x * 9.0) - 12.0)))
end

MATLAB

function tmp = code(x)
	tmp = 3.0 + (x * ((x * 9.0) - 12.0));
end

Wolfram

code[x_] := N[(3.0 + N[(x * N[(N[(x * 9.0), $MachinePrecision] - 12.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

   \[3 \cdot \left(\left(\left(x \cdot 3\right) \cdot x - x \cdot 4\right) + 1\right) \]
2. Add Preprocessing

3. Taylor expanded in x around 0 99.9%

   \[\leadsto \color{blue}{3 + x \cdot \left(9 \cdot x - 12\right)} \]

4. Final simplification 99.9%

   \[\leadsto 3 + x \cdot \left(x \cdot 9 - 12\right) \]
5. Add Preprocessing

Alternative 3: 97.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ 3 + x \cdot \left(x \cdot 9\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (+ 3.0 (* x (* x 9.0))))

C

double code(double x) {
	return 3.0 + (x * (x * 9.0));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 3.0d0 + (x * (x * 9.0d0))
end function

Java

public static double code(double x) {
	return 3.0 + (x * (x * 9.0));
}

Python

def code(x):
	return 3.0 + (x * (x * 9.0))

Julia

function code(x)
	return Float64(3.0 + Float64(x * Float64(x * 9.0)))
end

MATLAB

function tmp = code(x)
	tmp = 3.0 + (x * (x * 9.0));
end

Wolfram

code[x_] := N[(3.0 + N[(x * N[(x * 9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

   \[3 \cdot \left(\left(\left(x \cdot 3\right) \cdot x - x \cdot 4\right) + 1\right) \]
2. Add Preprocessing

3. Taylor expanded in x around 0 99.9%

   \[\leadsto \color{blue}{3 + x \cdot \left(9 \cdot x - 12\right)} \]

4. Taylor expanded in x around inf 97.9%

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

   1. *-commutative 97.9%

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

6. Simplified 97.9%

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

7. Final simplification 97.9%

   \[\leadsto 3 + x \cdot \left(x \cdot 9\right) \]
8. Add Preprocessing

Alternative 4: 51.2% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ 3 + x \cdot -12 \end{array} \]

FPCore

(FPCore (x) :precision binary64 (+ 3.0 (* x -12.0)))

C

double code(double x) {
	return 3.0 + (x * -12.0);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 3.0d0 + (x * (-12.0d0))
end function

Java

public static double code(double x) {
	return 3.0 + (x * -12.0);
}

Python

def code(x):
	return 3.0 + (x * -12.0)

Julia

function code(x)
	return Float64(3.0 + Float64(x * -12.0))
end

MATLAB

function tmp = code(x)
	tmp = 3.0 + (x * -12.0);
end

Wolfram

code[x_] := N[(3.0 + N[(x * -12.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

   \[3 \cdot \left(\left(\left(x \cdot 3\right) \cdot x - x \cdot 4\right) + 1\right) \]
2. Add Preprocessing

3. Taylor expanded in x around 0 55.7%

   \[\leadsto \color{blue}{3 + -12 \cdot x} \]
4. Step-by-step derivation

   1. *-commutative 55.7%

      \[\leadsto 3 + \color{blue}{x \cdot -12} \]

5. Simplified 55.7%

   \[\leadsto \color{blue}{3 + x \cdot -12} \]

6. Final simplification 55.7%

   \[\leadsto 3 + x \cdot -12 \]
7. Add Preprocessing

Alternative 5: 50.9% accurate, 13.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 3.0)

C

double code(double x) {
	return 3.0;
}

Fortran

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

Java

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

Python

def code(x):
	return 3.0

Julia

function code(x)
	return 3.0
end

MATLAB

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

Wolfram

code[x_] := 3.0

Derivation

1. Initial program 99.8%

   \[3 \cdot \left(\left(\left(x \cdot 3\right) \cdot x - x \cdot 4\right) + 1\right) \]
2. Add Preprocessing

3. Taylor expanded in x around 0 54.6%

   \[\leadsto \color{blue}{3} \]

4. Final simplification 54.6%

   \[\leadsto 3 \]
5. Add Preprocessing

Developer target: 99.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ 3 + \left(\left(9 \cdot x\right) \cdot x - 12 \cdot x\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (+ 3.0 (- (* (* 9.0 x) x) (* 12.0 x))))

C

double code(double x) {
	return 3.0 + (((9.0 * x) * x) - (12.0 * x));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 3.0d0 + (((9.0d0 * x) * x) - (12.0d0 * x))
end function

Java

public static double code(double x) {
	return 3.0 + (((9.0 * x) * x) - (12.0 * x));
}

Python

def code(x):
	return 3.0 + (((9.0 * x) * x) - (12.0 * x))

Julia

function code(x)
	return Float64(3.0 + Float64(Float64(Float64(9.0 * x) * x) - Float64(12.0 * x)))
end

MATLAB

function tmp = code(x)
	tmp = 3.0 + (((9.0 * x) * x) - (12.0 * x));
end

Wolfram

code[x_] := N[(3.0 + N[(N[(N[(9.0 * x), $MachinePrecision] * x), $MachinePrecision] - N[(12.0 * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2024071 
(FPCore (x)
  :name "Diagrams.Tangent:$catParam from diagrams-lib-1.3.0.3, D"
  :precision binary64

  :alt
  (+ 3.0 (- (* (* 9.0 x) x) (* 12.0 x)))

  (* 3.0 (+ (- (* (* x 3.0) x) (* x 4.0)) 1.0)))