Expression 3, p15

Percentage Accurate: 100.0% → 100.0%

Time: 2.0s

Alternatives: 3

Speedup: 1.0×

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. sqr-neg 100.0%

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

   2. +-commutative 100.0%

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

   3. sqr-neg 100.0%

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

   4. fma-define 100.0%

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

   5. cube-unmult 100.0%

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

3. Simplified 100.0%

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

5. Final simplification 100.0%

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

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. Add Preprocessing
3. Step-by-step derivation

   1. distribute-lft-out 100.0%

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

   2. *-commutative 100.0%

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

   3. fma-define 100.0%

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

4. Applied egg-rr 100.0%

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

5. Final simplification 100.0%

   \[\leadsto x \cdot \mathsf{fma}\left(x, x, x\right) \]
6. Add Preprocessing

Alternative 3: 100.0% accurate, 1.0× speedup

Math

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

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 * x) + Float64(x * Float64(x * x)))
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Final simplification 100.0%

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

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 2024041 
(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)))