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

Percentage Accurate: 99.8% → 99.7%

Time: 19.7s

Alternatives: 4

Speedup: 1.0×

• 25.1% 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: 99.7% accurate, 0.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x\_m \cdot \sqrt{\left(x\_m \cdot x\_m\right) \cdot 9} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m) :precision binary64 (* x_m (sqrt (* (* x_m x_m) 9.0))))

C

x_m = fabs(x);
double code(double x_m) {
	return x_m * sqrt(((x_m * x_m) * 9.0));
}

Fortran

x_m = abs(x)
real(8) function code(x_m)
    real(8), intent (in) :: x_m
    code = x_m * sqrt(((x_m * x_m) * 9.0d0))
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	return x_m * Math.sqrt(((x_m * x_m) * 9.0));
}

Python

x_m = math.fabs(x)
def code(x_m):
	return x_m * math.sqrt(((x_m * x_m) * 9.0))

Julia

x_m = abs(x)
function code(x_m)
	return Float64(x_m * sqrt(Float64(Float64(x_m * x_m) * 9.0)))
end

MATLAB

x_m = abs(x);
function tmp = code(x_m)
	tmp = x_m * sqrt(((x_m * x_m) * 9.0));
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := N[(x$95$m * N[Sqrt[N[(N[(x$95$m * x$95$m), $MachinePrecision] * 9.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

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

   1. add-sqr-sqrt 50.9%

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

   2. sqrt-unprod 65.4%

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

   3. swap-sqr 65.5%

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

   4. pow2 65.5%

      \[\leadsto \sqrt{\color{blue}{{x}^{2}} \cdot \left(3 \cdot 3\right)} \cdot x \]

   5. metadata-eval 65.5%

      \[\leadsto \sqrt{{x}^{2} \cdot \color{blue}{9}} \cdot x \]

4. Applied egg-rr 65.5%

   \[\leadsto \color{blue}{\sqrt{{x}^{2} \cdot 9}} \cdot x \]
5. Step-by-step derivation

   1. unpow2 65.5%

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

6. Applied egg-rr 65.5%

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

7. Final simplification 65.5%

   \[\leadsto x \cdot \sqrt{\left(x \cdot x\right) \cdot 9} \]
8. Add Preprocessing

Alternative 2: 99.8% accurate, 1.0× speedup

Math

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

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m) :precision binary64 (* x_m (* x_m 3.0)))

C

x_m = fabs(x);
double code(double x_m) {
	return x_m * (x_m * 3.0);
}

Fortran

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

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	return x_m * (x_m * 3.0);
}

Python

x_m = math.fabs(x)
def code(x_m):
	return x_m * (x_m * 3.0)

Julia

x_m = abs(x)
function code(x_m)
	return Float64(x_m * Float64(x_m * 3.0))
end

MATLAB

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

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := N[(x$95$m * N[(x$95$m * 3.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

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

3. Final simplification 99.8%

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

Alternative 3: 99.8% accurate, 1.0× speedup

Math

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

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m) :precision binary64 (* (* x_m x_m) 3.0))

C

x_m = fabs(x);
double code(double x_m) {
	return (x_m * x_m) * 3.0;
}

Fortran

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

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	return (x_m * x_m) * 3.0;
}

Python

x_m = math.fabs(x)
def code(x_m):
	return (x_m * x_m) * 3.0

Julia

x_m = abs(x)
function code(x_m)
	return Float64(Float64(x_m * x_m) * 3.0)
end

MATLAB

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

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := N[(N[(x$95$m * x$95$m), $MachinePrecision] * 3.0), $MachinePrecision]

Derivation

1. Initial program 99.8%

   \[\left(x \cdot 3\right) \cdot x \]
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 65.5%

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

5. Applied egg-rr 99.8%

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

6. Final simplification 99.8%

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

Alternative 4: 4.2% accurate, 5.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ 1 \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m) :precision binary64 1.0)

C

x_m = fabs(x);
double code(double x_m) {
	return 1.0;
}

Fortran

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

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	return 1.0;
}

Python

x_m = math.fabs(x)
def code(x_m):
	return 1.0

Julia

x_m = abs(x)
function code(x_m)
	return 1.0
end

MATLAB

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

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := 1.0

Derivation

1. Initial program 99.8%

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

   1. expm1-log1p-u 76.8%

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

   2. expm1-undefine 53.7%

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

   3. flip3-- 41.1%

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

   4. log1p-undefine 41.1%

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

   5. rem-exp-log 41.0%

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

   6. metadata-eval 41.0%

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

   7. log1p-undefine 41.0%

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

   8. rem-exp-log 41.1%

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

   9. log1p-undefine 41.1%

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

   10. rem-exp-log 41.5%

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

   11. metadata-eval 41.5%

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

   12. log1p-undefine 41.5%

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

   13. rem-exp-log 51.1%

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

4. Applied egg-rr 51.1%

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

5. Taylor expanded in x around 0 56.7%

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

   1. *-commutative 56.7%

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

7. Simplified 56.7%

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

8. Taylor expanded in x around inf 4.4%

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

Reproduce

herbie shell --seed 2024116 
(FPCore (x)
  :name "Diagrams.Tangent:$catParam from diagrams-lib-1.3.0.3, F"
  :precision binary64
  (* (* x 3.0) x))