Expression 3, p15

Percentage Accurate: 100.0% → 100.0%

Time: 3.4s

Alternatives: 5

Speedup: 0.0×

Specification

\[0 \leq x \land x \leq 2\]

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(x, x, {x}^{3}\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (fma x x (pow x 3.0)))

C

double code(double x) {
	return fma(x, x, pow(x, 3.0));
}

Julia

function code(x)
	return fma(x, x, (x ^ 3.0))
end

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. distribute-lft-out 100.0%

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

   2. fma-def 100.0%

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

3. Simplified 100.0%

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

4. Taylor expanded in x around 0 100.0%

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

   1. unpow2 100.0%

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

   2. fma-def 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, x, {x}^{3}\right)} \]

6. Simplified 100.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x, x, {x}^{3}\right)} \]

7. Final simplification 100.0%

   \[\leadsto \mathsf{fma}\left(x, x, {x}^{3}\right) \]

Alternative 2: 100.0% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. distribute-lft-out 100.0%

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

   2. fma-def 100.0%

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

3. Simplified 100.0%

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

4. Final simplification 100.0%

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

Alternative 3: 100.0% accurate, 1.3× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 (* (+ x 1.0) (* x x)))

C

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

Fortran

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

Java

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

Python

def code(x):
	return (x + 1.0) * (x * x)

Julia

function code(x)
	return Float64(Float64(x + 1.0) * Float64(x * x))
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. distribute-lft1-in 100.0%

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

3. Applied egg-rr 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \left(x + 1\right) \cdot \left(x \cdot x\right) \]

Alternative 4: 97.5% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x):
	return x * x

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. distribute-lft-out 100.0%

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

   2. fma-def 100.0%

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

3. Simplified 100.0%

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

4. Taylor expanded in x around 0 98.2%

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

   1. unpow2 98.2%

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

6. Simplified 98.2%

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

7. Final simplification 98.2%

   \[\leadsto x \cdot x \]

Alternative 5: 4.9% accurate, 4.5× speedup

Math

\[\begin{array}{l} \\ -x \end{array} \]

FPCore

(FPCore (x) :precision binary64 (- x))

C

double code(double x) {
	return -x;
}

Fortran

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

Java

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

Python

def code(x):
	return -x

Julia

function code(x)
	return Float64(-x)
end

MATLAB

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

Wolfram

code[x_] := (-x)

Derivation

1. Initial program 100.0%

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

   1. distribute-lft1-in 100.0%

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

3. Applied egg-rr 100.0%

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

   1. flip-+ 100.0%

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

   2. associate-*l/ 100.0%

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

   3. metadata-eval 100.0%

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

   4. fma-neg 100.0%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x, x, -1\right)} \cdot \left(x \cdot x\right)}{x - 1} \]

   5. metadata-eval 100.0%

      \[\leadsto \frac{\mathsf{fma}\left(x, x, \color{blue}{-1}\right) \cdot \left(x \cdot x\right)}{x - 1} \]

   6. sub-neg 100.0%

      \[\leadsto \frac{\mathsf{fma}\left(x, x, -1\right) \cdot \left(x \cdot x\right)}{\color{blue}{x + \left(-1\right)}} \]

   7. metadata-eval 100.0%

      \[\leadsto \frac{\mathsf{fma}\left(x, x, -1\right) \cdot \left(x \cdot x\right)}{x + \color{blue}{-1}} \]

5. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x, x, -1\right) \cdot \left(x \cdot x\right)}{x + -1}} \]

6. Taylor expanded in x around 0 99.2%

   \[\leadsto \frac{\color{blue}{-1 \cdot {x}^{2}}}{x + -1} \]
7. Step-by-step derivation

   1. unpow2 99.2%

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

   2. mul-1-neg 99.2%

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

   3. distribute-rgt-neg-in 99.2%

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

8. Simplified 99.2%

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

9. Taylor expanded in x around inf 4.7%

   \[\leadsto \color{blue}{-1 \cdot x} \]
10. Step-by-step derivation

    1. mul-1-neg 4.7%

       \[\leadsto \color{blue}{-x} \]

11. Simplified 4.7%

    \[\leadsto \color{blue}{-x} \]

12. Final simplification 4.7%

    \[\leadsto -x \]

Developer target: 100.0% accurate, 1.3× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 (* (* (+ 1.0 x) x) x))

C

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

Fortran

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

Java

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

Python

def code(x):
	return ((1.0 + x) * x) * x

Julia

function code(x)
	return Float64(Float64(Float64(1.0 + x) * x) * x)
end

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2023208 
(FPCore (x)
  :name "Expression 3, p15"
  :precision binary64
  :pre (and (<= 0.0 x) (<= x 2.0))

  :herbie-target
  (* (* (+ 1.0 x) x) x)

  (+ (* x (* x x)) (* x x)))