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

Percentage Accurate: 99.8% → 99.9%

Time: 4.1s

Alternatives: 2

Speedup: 0.0×

• 25.4% 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: 99.9% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(x, x, 2 \cdot \left(x \cdot x\right)\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (fma x x (* 2.0 (* x x))))

C

double code(double x) {
	return fma(x, x, (2.0 * (x * x)));
}

Julia

function code(x)
	return fma(x, x, Float64(2.0 * Float64(x * x)))
end

Wolfram

code[x_] := N[(x * x + N[(2.0 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $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. *-commutative 99.8%

      \[\leadsto \color{blue}{{x}^{2} \cdot 3} \]

   2. rem-square-sqrt 99.3%

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

   3. pow2 99.3%

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

   4. unpow-prod-down 99.4%

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

   5. add-log-exp 52.0%

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

   6. add-cube-cbrt 52.0%

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

   7. log-prod 52.0%

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

5. Applied egg-rr 52.0%

   \[\leadsto \color{blue}{\log \left(\sqrt[3]{{\left(e^{{x}^{2}}\right)}^{3}} \cdot \sqrt[3]{{\left(e^{{x}^{2}}\right)}^{3}}\right) + \log \left(\sqrt[3]{{\left(e^{{x}^{2}}\right)}^{3}}\right)} \]
6. Step-by-step derivation

   1. +-commutative 52.0%

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

   2. rem-cbrt-cube 52.0%

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

   3. rem-log-exp 55.0%

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

   4. unpow2 55.0%

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

   5. fma-define 55.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, x, \log \left(\sqrt[3]{{\left(e^{{x}^{2}}\right)}^{3}} \cdot \sqrt[3]{{\left(e^{{x}^{2}}\right)}^{3}}\right)\right)} \]

   6. rem-cbrt-cube 55.0%

      \[\leadsto \mathsf{fma}\left(x, x, \log \left(\color{blue}{e^{{x}^{2}}} \cdot \sqrt[3]{{\left(e^{{x}^{2}}\right)}^{3}}\right)\right) \]

   7. rem-cbrt-cube 55.0%

      \[\leadsto \mathsf{fma}\left(x, x, \log \left(e^{{x}^{2}} \cdot \color{blue}{e^{{x}^{2}}}\right)\right) \]

   8. prod-exp 55.1%

      \[\leadsto \mathsf{fma}\left(x, x, \log \color{blue}{\left(e^{{x}^{2} + {x}^{2}}\right)}\right) \]

   9. rem-log-exp 99.9%

      \[\leadsto \mathsf{fma}\left(x, x, \color{blue}{{x}^{2} + {x}^{2}}\right) \]

   10. count-2 99.9%

       \[\leadsto \mathsf{fma}\left(x, x, \color{blue}{2 \cdot {x}^{2}}\right) \]

7. Simplified 99.9%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x, x, 2 \cdot {x}^{2}\right)} \]
8. Step-by-step derivation

   1. unpow2 99.9%

      \[\leadsto \mathsf{fma}\left(x, x, 2 \cdot \color{blue}{\left(x \cdot x\right)}\right) \]

9. Applied egg-rr 99.9%

   \[\leadsto \mathsf{fma}\left(x, x, 2 \cdot \color{blue}{\left(x \cdot x\right)}\right) \]
10. Add Preprocessing

Alternative 2: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(N[(x * x), $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 99.9%

      \[\leadsto \mathsf{fma}\left(x, x, 2 \cdot \color{blue}{\left(x \cdot x\right)}\right) \]

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

Reproduce

herbie shell --seed 2024125 
(FPCore (x)
  :name "Diagrams.Tangent:$catParam from diagrams-lib-1.3.0.3, F"
  :precision binary64
  (* (* x 3.0) x))