math.cube on complex, real part

Average Error: 7.5 → 0.2

Time: 14.3s

Precision: binary64

Cost: 1152

Math

\[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

↓

\[\left(x.re + x.im\right) \cdot \left(\left(x.re - x.im\right) \cdot x.re\right) - \frac{x.im}{-\frac{\frac{-0.5}{x.re}}{x.im}} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (-
  (* (- (* x.re x.re) (* x.im x.im)) x.re)
  (* (+ (* x.re x.im) (* x.im x.re)) x.im)))

↓

(FPCore (x.re x.im)
 :precision binary64
 (-
  (* (+ x.re x.im) (* (- x.re x.im) x.re))
  (/ x.im (- (/ (/ -0.5 x.re) x.im)))))

C

double code(double x_46_re, double x_46_im) {
	return (((x_46_re * x_46_re) - (x_46_im * x_46_im)) * x_46_re) - (((x_46_re * x_46_im) + (x_46_im * x_46_re)) * x_46_im);
}

↓

double code(double x_46_re, double x_46_im) {
	return ((x_46_re + x_46_im) * ((x_46_re - x_46_im) * x_46_re)) - (x_46_im / -((-0.5 / x_46_re) / x_46_im));
}

Fortran

real(8) function code(x_46re, x_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    code = (((x_46re * x_46re) - (x_46im * x_46im)) * x_46re) - (((x_46re * x_46im) + (x_46im * x_46re)) * x_46im)
end function

↓

real(8) function code(x_46re, x_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    code = ((x_46re + x_46im) * ((x_46re - x_46im) * x_46re)) - (x_46im / -(((-0.5d0) / x_46re) / x_46im))
end function

Java

public static double code(double x_46_re, double x_46_im) {
	return (((x_46_re * x_46_re) - (x_46_im * x_46_im)) * x_46_re) - (((x_46_re * x_46_im) + (x_46_im * x_46_re)) * x_46_im);
}

↓

public static double code(double x_46_re, double x_46_im) {
	return ((x_46_re + x_46_im) * ((x_46_re - x_46_im) * x_46_re)) - (x_46_im / -((-0.5 / x_46_re) / x_46_im));
}

Python

def code(x_46_re, x_46_im):
	return (((x_46_re * x_46_re) - (x_46_im * x_46_im)) * x_46_re) - (((x_46_re * x_46_im) + (x_46_im * x_46_re)) * x_46_im)

↓

def code(x_46_re, x_46_im):
	return ((x_46_re + x_46_im) * ((x_46_re - x_46_im) * x_46_re)) - (x_46_im / -((-0.5 / x_46_re) / x_46_im))

Julia

function code(x_46_re, x_46_im)
	return Float64(Float64(Float64(Float64(x_46_re * x_46_re) - Float64(x_46_im * x_46_im)) * x_46_re) - Float64(Float64(Float64(x_46_re * x_46_im) + Float64(x_46_im * x_46_re)) * x_46_im))
end

↓

function code(x_46_re, x_46_im)
	return Float64(Float64(Float64(x_46_re + x_46_im) * Float64(Float64(x_46_re - x_46_im) * x_46_re)) - Float64(x_46_im / Float64(-Float64(Float64(-0.5 / x_46_re) / x_46_im))))
end

MATLAB

function tmp = code(x_46_re, x_46_im)
	tmp = (((x_46_re * x_46_re) - (x_46_im * x_46_im)) * x_46_re) - (((x_46_re * x_46_im) + (x_46_im * x_46_re)) * x_46_im);
end

↓

function tmp = code(x_46_re, x_46_im)
	tmp = ((x_46_re + x_46_im) * ((x_46_re - x_46_im) * x_46_re)) - (x_46_im / -((-0.5 / x_46_re) / x_46_im));
end

Wolfram

code[x$46$re_, x$46$im_] := N[(N[(N[(N[(x$46$re * x$46$re), $MachinePrecision] - N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision] * x$46$re), $MachinePrecision] - N[(N[(N[(x$46$re * x$46$im), $MachinePrecision] + N[(x$46$im * x$46$re), $MachinePrecision]), $MachinePrecision] * x$46$im), $MachinePrecision]), $MachinePrecision]

↓

code[x$46$re_, x$46$im_] := N[(N[(N[(x$46$re + x$46$im), $MachinePrecision] * N[(N[(x$46$re - x$46$im), $MachinePrecision] * x$46$re), $MachinePrecision]), $MachinePrecision] - N[(x$46$im / (-N[(N[(-0.5 / x$46$re), $MachinePrecision] / x$46$im), $MachinePrecision])), $MachinePrecision]), $MachinePrecision]

Target

Original	7.5
Target	0.3
Herbie	0.2

\[\left(x.re \cdot x.re\right) \cdot \left(x.re - x.im\right) + \left(x.re \cdot x.im\right) \cdot \left(x.re - 3 \cdot x.im\right) \]

Derivation

1. Initial program 7.5

   \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

2. Simplified 7.6

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

   [Start] 7.5	\[ \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]
   rational.json-simplify-2 [=>] 7.5	\[ \color{blue}{x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right)} - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]
   rational.json-simplify-34 [=>] 7.5	\[ x.re \cdot \color{blue}{\left(\left(x.im + x.re\right) \cdot \left(x.re - x.im\right)\right)} - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]
   rational.json-simplify-43 [=>] 0.2	\[ \color{blue}{\left(x.im + x.re\right) \cdot \left(\left(x.re - x.im\right) \cdot x.re\right)} - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]
   rational.json-simplify-1 [=>] 0.2	\[ \color{blue}{\left(x.re + x.im\right)} \cdot \left(\left(x.re - x.im\right) \cdot x.re\right) - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]
   rational.json-simplify-2 [=>] 0.2	\[ \left(x.re + x.im\right) \cdot \left(\left(x.re - x.im\right) \cdot x.re\right) - \color{blue}{x.im \cdot \left(x.re \cdot x.im + x.im \cdot x.re\right)} \]
   rational.json-simplify-51 [=>] 0.2	\[ \left(x.re + x.im\right) \cdot \left(\left(x.re - x.im\right) \cdot x.re\right) - x.im \cdot \color{blue}{\left(x.re \cdot \left(x.im + x.im\right)\right)} \]
   rational.json-simplify-43 [=>] 7.6	\[ \left(x.re + x.im\right) \cdot \left(\left(x.re - x.im\right) \cdot x.re\right) - \color{blue}{x.re \cdot \left(\left(x.im + x.im\right) \cdot x.im\right)} \]

3. Applied egg-rr 7.6

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

4. Simplified 0.2

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

   [Start] 7.6	\[ \left(x.re + x.im\right) \cdot \left(\left(x.re - x.im\right) \cdot x.re\right) - \frac{x.re + x.re}{\frac{1}{x.im \cdot x.im}} \]
   rational.json-simplify-46 [=>] 7.1	\[ \left(x.re + x.im\right) \cdot \left(\left(x.re - x.im\right) \cdot x.re\right) - \frac{x.re + x.re}{\color{blue}{\frac{\frac{1}{x.im}}{x.im}}} \]
   rational.json-simplify-61 [=>] 0.2	\[ \left(x.re + x.im\right) \cdot \left(\left(x.re - x.im\right) \cdot x.re\right) - \color{blue}{\frac{x.im}{\frac{\frac{1}{x.im}}{x.re + x.re}}} \]

5. Applied egg-rr 0.2

   \[\leadsto \left(x.re + x.im\right) \cdot \left(\left(x.re - x.im\right) \cdot x.re\right) - \frac{x.im}{\color{blue}{-\frac{\frac{-0.5}{x.re}}{x.im}}} \]

6. Final simplification 0.2

   \[\leadsto \left(x.re + x.im\right) \cdot \left(\left(x.re - x.im\right) \cdot x.re\right) - \frac{x.im}{-\frac{\frac{-0.5}{x.re}}{x.im}} \]

Alternatives

Alternative 1
Error	0.2
Cost	1088

\[x.im \cdot \frac{x.re}{\frac{-0.5}{x.im}} + \left(x.re - x.im\right) \cdot \left(\left(x.re + x.im\right) \cdot x.re\right) \]

Alternative 2
Error	0.3
Cost	968

\[\begin{array}{l} \mathbf{if}\;x.im \leq -3 \cdot 10^{+90}:\\ \;\;\;\;\left(x.re \cdot x.im\right) \cdot \left(x.im \cdot -3\right)\\ \mathbf{elif}\;x.im \leq 1.5 \cdot 10^{+98}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re - 3 \cdot \left(x.im \cdot x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-3 \cdot \left(x.re \cdot x.im\right)\right) \cdot x.im\\ \end{array} \]

Alternative 3
Error	0.2
Cost	968

\[\begin{array}{l} \mathbf{if}\;x.im \leq -3.8 \cdot 10^{+134}:\\ \;\;\;\;\left(x.im \cdot \left(x.re \cdot -3\right)\right) \cdot x.im\\ \mathbf{elif}\;x.im \leq 2 \cdot 10^{+98}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re - x.im \cdot \left(x.im \cdot 3\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-3 \cdot \left(x.re \cdot x.im\right)\right) \cdot x.im\\ \end{array} \]

Alternative 4
Error	19.4
Cost	448

\[-3 \cdot \left(x.im \cdot \left(x.re \cdot x.im\right)\right) \]

Alternative 5
Error	19.4
Cost	448

\[\left(-3 \cdot \left(x.re \cdot x.im\right)\right) \cdot x.im \]

Alternative 6
Error	47.0
Cost	384

\[x.re \cdot \left(x.re \cdot \left(-x.im\right)\right) \]

Reproduce

herbie shell --seed 2023073 
(FPCore (x.re x.im)
  :name "math.cube on complex, real part"
  :precision binary64

  :herbie-target
  (+ (* (* x.re x.re) (- x.re x.im)) (* (* x.re x.im) (- x.re (* 3.0 x.im))))

  (- (* (- (* x.re x.re) (* x.im x.im)) x.re) (* (+ (* x.re x.im) (* x.im x.re)) x.im)))