math.cube on complex, real part

Average Error: 6.9 → 0.2

Time: 14.8s

Precision: binary64

Cost: 832

Math

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

↓

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

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.re x.re)) (* x.im (* -3.0 (* x.im x.re)))))

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_re * x_46_re)) + (x_46_im * (-3.0 * (x_46_im * x_46_re)));
}

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_46re * x_46re)) + (x_46im * ((-3.0d0) * (x_46im * x_46re)))
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_re * x_46_re)) + (x_46_im * (-3.0 * (x_46_im * x_46_re)));
}

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_re * x_46_re)) + (x_46_im * (-3.0 * (x_46_im * x_46_re)))

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(x_46_re * Float64(x_46_re * x_46_re)) + Float64(x_46_im * Float64(-3.0 * Float64(x_46_im * x_46_re))))
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_re * x_46_re)) + (x_46_im * (-3.0 * (x_46_im * x_46_re)));
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[(x$46$re * N[(x$46$re * x$46$re), $MachinePrecision]), $MachinePrecision] + N[(x$46$im * N[(-3.0 * N[(x$46$im * x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Target

Original	6.9
Target	0.2
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 6.9

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

2. Simplified 7.0

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

   [Start] 6.9	\[ \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]
   rational.json-simplify-39 [=>] 6.9	\[ \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \color{blue}{x.im \cdot \left(x.re \cdot x.im + x.im \cdot x.re\right)} \]
   rational.json-simplify-39 [=>] 6.9	\[ \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - x.im \cdot \left(\color{blue}{x.im \cdot x.re} + x.im \cdot x.re\right) \]
   rational.json-simplify-39 [<=] 6.9	\[ \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - x.im \cdot \left(x.im \cdot x.re + \color{blue}{x.re \cdot x.im}\right) \]
   rational.json-simplify-35 [=>] 6.9	\[ \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - x.im \cdot \color{blue}{\left(x.re \cdot \left(x.im + x.im\right)\right)} \]
   rational.json-simplify-3 [=>] 6.9	\[ \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \color{blue}{x.re \cdot \left(x.im \cdot \left(x.im + x.im\right)\right)} \]
   rational.json-simplify-33 [=>] 7.0	\[ \color{blue}{x.re \cdot \left(\left(x.re \cdot x.re - x.im \cdot x.im\right) - x.im \cdot \left(x.im + x.im\right)\right)} \]

3. Applied egg-rr 7.0

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

4. Applied egg-rr 7.0

   \[\leadsto \color{blue}{0 - x.re \cdot \left(x.im \cdot \left(x.im \cdot 3\right) - x.re \cdot x.re\right)} \]

5. Simplified 7.0

   \[\leadsto \color{blue}{x.re \cdot \left(x.re \cdot x.re - x.im \cdot \left(x.im \cdot 3\right)\right)} \]
   Proof

   [Start] 7.0	\[ 0 - x.re \cdot \left(x.im \cdot \left(x.im \cdot 3\right) - x.re \cdot x.re\right) \]
   rational.json-simplify-67 [<=] 7.0	\[ \color{blue}{-x.re \cdot \left(x.im \cdot \left(x.im \cdot 3\right) - x.re \cdot x.re\right)} \]
   rational.json-simplify-69 [=>] 7.0	\[ \color{blue}{\frac{x.re \cdot \left(x.im \cdot \left(x.im \cdot 3\right) - x.re \cdot x.re\right)}{-1}} \]
   rational.json-simplify-39 [=>] 7.0	\[ \frac{\color{blue}{\left(x.im \cdot \left(x.im \cdot 3\right) - x.re \cdot x.re\right) \cdot x.re}}{-1} \]
   rational.json-simplify-19 [=>] 7.0	\[ \color{blue}{x.re \cdot \frac{x.im \cdot \left(x.im \cdot 3\right) - x.re \cdot x.re}{-1}} \]
   rational.json-simplify-23 [=>] 7.0	\[ x.re \cdot \color{blue}{\frac{x.re \cdot x.re - x.im \cdot \left(x.im \cdot 3\right)}{--1}} \]
   metadata-eval [=>] 7.0	\[ x.re \cdot \frac{x.re \cdot x.re - x.im \cdot \left(x.im \cdot 3\right)}{\color{blue}{1}} \]
   rational.json-simplify-66 [=>] 7.0	\[ x.re \cdot \color{blue}{\left(x.re \cdot x.re - x.im \cdot \left(x.im \cdot 3\right)\right)} \]

6. Applied egg-rr 7.0

   \[\leadsto \color{blue}{x.re \cdot \left(x.re \cdot x.re\right) - x.re \cdot \left(x.im \cdot \left(x.im \cdot 3\right)\right)} \]

7. Simplified 0.2

   \[\leadsto \color{blue}{x.re \cdot \left(x.re \cdot x.re\right) - \left(x.im \cdot 3\right) \cdot \left(x.re \cdot x.im\right)} \]
   Proof

   [Start] 7.0	\[ x.re \cdot \left(x.re \cdot x.re\right) - x.re \cdot \left(x.im \cdot \left(x.im \cdot 3\right)\right) \]
   rational.json-simplify-39 [=>] 7.0	\[ x.re \cdot \left(x.re \cdot x.re\right) - x.re \cdot \color{blue}{\left(\left(x.im \cdot 3\right) \cdot x.im\right)} \]
   rational.json-simplify-3 [=>] 0.2	\[ x.re \cdot \left(x.re \cdot x.re\right) - \color{blue}{\left(x.im \cdot 3\right) \cdot \left(x.re \cdot x.im\right)} \]

8. Applied egg-rr 0.2

   \[\leadsto \color{blue}{x.re \cdot \left(x.re \cdot x.re\right) + x.im \cdot \left(-3 \cdot \left(x.im \cdot x.re\right)\right)} \]

9. Final simplification 0.2

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

Alternatives

Alternative 1
Error	0.3
Cost	1032

\[\begin{array}{l} t_0 := \left(\left(x.im + x.im\right) - \left(x.re - x.im\right)\right) \cdot \left(x.im \cdot \left(-x.re\right)\right)\\ \mathbf{if}\;x.im \leq -6.5 \cdot 10^{+153}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x.im \leq 1.25 \cdot 10^{+91}:\\ \;\;\;\;x.re \cdot \left(\left(x.im \cdot x.im\right) \cdot -3 + x.re \cdot x.re\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	7.0
Cost	704

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

Reproduce

herbie shell --seed 2023066 
(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)))