math.cube on complex, imaginary part

Average Error: 7.9 → 0.3

Time: 9.9s

Precision: binary64

Cost: 832

Math

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

↓

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

FPCore

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

↓

(FPCore (x.re x.im)
 :precision binary64
 (- (* x.re (* 3.0 (* x.re x.im))) (* x.im (* x.im 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_im) + (((x_46_re * x_46_im) + (x_46_im * x_46_re)) * x_46_re);
}

↓

double code(double x_46_re, double x_46_im) {
	return (x_46_re * (3.0 * (x_46_re * x_46_im))) - (x_46_im * (x_46_im * 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_46im) + (((x_46re * x_46im) + (x_46im * x_46re)) * x_46re)
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 * (3.0d0 * (x_46re * x_46im))) - (x_46im * (x_46im * 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_im) + (((x_46_re * x_46_im) + (x_46_im * x_46_re)) * x_46_re);
}

↓

public static double code(double x_46_re, double x_46_im) {
	return (x_46_re * (3.0 * (x_46_re * x_46_im))) - (x_46_im * (x_46_im * 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_im) + (((x_46_re * x_46_im) + (x_46_im * x_46_re)) * x_46_re)

↓

def code(x_46_re, x_46_im):
	return (x_46_re * (3.0 * (x_46_re * x_46_im))) - (x_46_im * (x_46_im * 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_im) + Float64(Float64(Float64(x_46_re * x_46_im) + Float64(x_46_im * x_46_re)) * x_46_re))
end

↓

function code(x_46_re, x_46_im)
	return Float64(Float64(x_46_re * Float64(3.0 * Float64(x_46_re * x_46_im))) - Float64(x_46_im * Float64(x_46_im * 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_im) + (((x_46_re * x_46_im) + (x_46_im * x_46_re)) * x_46_re);
end

↓

function tmp = code(x_46_re, x_46_im)
	tmp = (x_46_re * (3.0 * (x_46_re * x_46_im))) - (x_46_im * (x_46_im * 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$46$im), $MachinePrecision] + N[(N[(N[(x$46$re * x$46$im), $MachinePrecision] + N[(x$46$im * x$46$re), $MachinePrecision]), $MachinePrecision] * x$46$re), $MachinePrecision]), $MachinePrecision]

↓

code[x$46$re_, x$46$im_] := N[(N[(x$46$re * N[(3.0 * N[(x$46$re * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x$46$im * N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Target

Original	7.9
Target	0.2
Herbie	0.3

\[\left(x.re \cdot x.im\right) \cdot \left(2 \cdot x.re\right) + \left(x.im \cdot \left(x.re - x.im\right)\right) \cdot \left(x.re + x.im\right) \]

Derivation

1. Initial program 7.9

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

2. Simplified 0.2

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

   [Start] 7.9	\[ \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
   rational_best_oopsla_all_46_json_45_simplify-35 [=>] 7.9	\[ \color{blue}{\left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re + \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im} \]
   rational_best_oopsla_all_46_json_45_simplify-74 [=>] 7.9	\[ \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re + \color{blue}{x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right)} \]
   rational_best_oopsla_all_46_json_45_simplify-102 [<=] 7.9	\[ \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re + \color{blue}{\left(\left(x.re \cdot x.re\right) \cdot x.im - x.im \cdot \left(x.im \cdot x.im\right)\right)} \]
   rational_best_oopsla_all_46_json_45_simplify-74 [=>] 7.9	\[ \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re + \left(\color{blue}{x.im \cdot \left(x.re \cdot x.re\right)} - x.im \cdot \left(x.im \cdot x.im\right)\right) \]
   rational_best_oopsla_all_46_json_45_simplify-7 [<=] 0.2	\[ \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re + \left(\color{blue}{x.re \cdot \left(x.im \cdot x.re\right)} - x.im \cdot \left(x.im \cdot x.im\right)\right) \]
   rational_best_oopsla_all_46_json_45_simplify-74 [<=] 0.2	\[ \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re + \left(x.re \cdot \color{blue}{\left(x.re \cdot x.im\right)} - x.im \cdot \left(x.im \cdot x.im\right)\right) \]
   rational_best_oopsla_all_46_json_45_simplify-74 [<=] 0.2	\[ \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re + \left(\color{blue}{\left(x.re \cdot x.im\right) \cdot x.re} - x.im \cdot \left(x.im \cdot x.im\right)\right) \]
   rational_best_oopsla_all_46_json_45_simplify-108 [=>] 0.2	\[ \color{blue}{\left(\left(x.re \cdot x.im\right) \cdot x.re + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re\right) - x.im \cdot \left(x.im \cdot x.im\right)} \]
   rational_best_oopsla_all_46_json_45_simplify-74 [=>] 0.2	\[ \left(\left(x.re \cdot x.im\right) \cdot x.re + \color{blue}{x.re \cdot \left(x.re \cdot x.im + x.im \cdot x.re\right)}\right) - x.im \cdot \left(x.im \cdot x.im\right) \]
   rational_best_oopsla_all_46_json_45_simplify-23 [=>] 0.3	\[ \color{blue}{x.re \cdot \left(x.re \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right)\right)} - x.im \cdot \left(x.im \cdot x.im\right) \]
   rational_best_oopsla_all_46_json_45_simplify-35 [<=] 0.3	\[ x.re \cdot \color{blue}{\left(\left(x.re \cdot x.im + x.im \cdot x.re\right) + x.re \cdot x.im\right)} - x.im \cdot \left(x.im \cdot x.im\right) \]
   rational_best_oopsla_all_46_json_45_simplify-35 [=>] 0.3	\[ x.re \cdot \left(\color{blue}{\left(x.im \cdot x.re + x.re \cdot x.im\right)} + x.re \cdot x.im\right) - x.im \cdot \left(x.im \cdot x.im\right) \]
   rational_best_oopsla_all_46_json_45_simplify-23 [=>] 0.3	\[ x.re \cdot \left(\color{blue}{x.re \cdot \left(x.im + x.im\right)} + x.re \cdot x.im\right) - x.im \cdot \left(x.im \cdot x.im\right) \]
   rational_best_oopsla_all_46_json_45_simplify-74 [=>] 0.3	\[ x.re \cdot \left(\color{blue}{\left(x.im + x.im\right) \cdot x.re} + x.re \cdot x.im\right) - x.im \cdot \left(x.im \cdot x.im\right) \]
   rational_best_oopsla_all_46_json_45_simplify-23 [=>] 0.2	\[ x.re \cdot \color{blue}{\left(x.re \cdot \left(\left(x.im + x.im\right) + x.im\right)\right)} - x.im \cdot \left(x.im \cdot x.im\right) \]

3. Taylor expanded in x.im around 0 0.3

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

4. Final simplification 0.3

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

Alternatives

Alternative 1
Error	7.9
Cost	704

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

Reproduce

herbie shell --seed 2023090 
(FPCore (x.re x.im)
  :name "math.cube on complex, imaginary part"
  :precision binary64

  :herbie-target
  (+ (* (* x.re x.im) (* 2.0 x.re)) (* (* x.im (- x.re x.im)) (+ x.re x.im)))

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