math.square on complex, real part

Percentage Accurate: 93.8% → 100.0%
Time: 1.5s
Alternatives: 2
Speedup: 1.0×

Specification

?
\[re \cdot re - im \cdot im \]
(FPCore re_sqr (re im)
  :precision binary64
  (- (* re re) (* im im)))
double re_sqr(double re, double im) {
	return (re * re) - (im * im);
}

real(8) function re_sqr(re, im)
use fmin_fmax_functions
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    re_sqr = (re * re) - (im * im)
end function

public static double re_sqr(double re, double im) {
	return (re * re) - (im * im);
}

def re_sqr(re, im):
	return (re * re) - (im * im)

function re_sqr(re, im)
	return Float64(Float64(re * re) - Float64(im * im))
end

function tmp = re_sqr(re, im)
	tmp = (re * re) - (im * im);
end

re$95$sqr[re_, im_] := N[(N[(re * re), $MachinePrecision] - N[(im * im), $MachinePrecision]), $MachinePrecision]

re \cdot re - im \cdot im

Local Percentage Accuracy vs ?

The average percentage accuracy by input value. Horizontal axis shows value of an input variable; the variable is choosen in the title. Vertical axis is accuracy; higher is better. Red represent the original program, while blue represents Herbie's suggestion. These can be toggled with buttons below the plot. The line is an average while dots represent individual samples.

Accuracy vs Speed?

Herbie found 2 alternatives:

AlternativeAccuracySpeedup
The accuracy (vertical axis) and speed (horizontal axis) of each alternatives. Up and to the right is better. The red square shows the initial program, and each blue circle shows an alternative.The line shows the best available speed-accuracy tradeoffs.

Initial Program: 93.8% accurate, 1.0× speedup?

\[re \cdot re - im \cdot im \]
(FPCore re_sqr (re im)
  :precision binary64
  (- (* re re) (* im im)))
double re_sqr(double re, double im) {
	return (re * re) - (im * im);
}

real(8) function re_sqr(re, im)
use fmin_fmax_functions
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    re_sqr = (re * re) - (im * im)
end function

public static double re_sqr(double re, double im) {
	return (re * re) - (im * im);
}

def re_sqr(re, im):
	return (re * re) - (im * im)

function re_sqr(re, im)
	return Float64(Float64(re * re) - Float64(im * im))
end

function tmp = re_sqr(re, im)
	tmp = (re * re) - (im * im);
end

re$95$sqr[re_, im_] := N[(N[(re * re), $MachinePrecision] - N[(im * im), $MachinePrecision]), $MachinePrecision]

re \cdot re - im \cdot im

Alternative 1: 100.0% accurate, 1.0× speedup?

\[\left(re - im\right) \cdot \left(im + re\right) \]
(FPCore re_sqr (re im)
  :precision binary64
  (* (- re im) (+ im re)))
double re_sqr(double re, double im) {
	return (re - im) * (im + re);
}

real(8) function re_sqr(re, im)
use fmin_fmax_functions
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    re_sqr = (re - im) * (im + re)
end function

public static double re_sqr(double re, double im) {
	return (re - im) * (im + re);
}

def re_sqr(re, im):
	return (re - im) * (im + re)

function re_sqr(re, im)
	return Float64(Float64(re - im) * Float64(im + re))
end

function tmp = re_sqr(re, im)
	tmp = (re - im) * (im + re);
end

re$95$sqr[re_, im_] := N[(N[(re - im), $MachinePrecision] * N[(im + re), $MachinePrecision]), $MachinePrecision]

\left(re - im\right) \cdot \left(im + re\right)
Derivation

  1. Initial program 93.8%

    \[re \cdot re - im \cdot im \]
  2. Step-by-step derivation

    1. lift--.f64N/A

      \[\leadsto \color{blue}{re \cdot re - im \cdot im} \]

    2. lift-*.f64N/A

      \[\leadsto \color{blue}{re \cdot re} - im \cdot im \]

    3. lift-*.f64N/A

      \[\leadsto re \cdot re - \color{blue}{im \cdot im} \]

    4. difference-of-squaresN/A

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

    5. *-commutativeN/A

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

    6. lower-*.f64N/A

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

    7. lower--.f64N/A

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

    8. +-commutativeN/A

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

    9. lower-+.f64100.0%

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

  3. Applied rewrites100.0%

    \[\leadsto \color{blue}{\left(re - im\right) \cdot \left(im + re\right)} \]
  4. Add Preprocessing

Alternative 2: 61.2% accurate, 1.0× speedup?

\[\left(\left|re\right| - \left|im\right|\right) \cdot \left|im\right| \]
(FPCore re_sqr (re im)
  :precision binary64
  (* (- (fabs re) (fabs im)) (fabs im)))
double re_sqr(double re, double im) {
	return (fabs(re) - fabs(im)) * fabs(im);
}

real(8) function re_sqr(re, im)
use fmin_fmax_functions
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    re_sqr = (abs(re) - abs(im)) * abs(im)
end function

public static double re_sqr(double re, double im) {
	return (Math.abs(re) - Math.abs(im)) * Math.abs(im);
}

def re_sqr(re, im):
	return (math.fabs(re) - math.fabs(im)) * math.fabs(im)

function re_sqr(re, im)
	return Float64(Float64(abs(re) - abs(im)) * abs(im))
end

function tmp = re_sqr(re, im)
	tmp = (abs(re) - abs(im)) * abs(im);
end

re$95$sqr[re_, im_] := N[(N[(N[Abs[re], $MachinePrecision] - N[Abs[im], $MachinePrecision]), $MachinePrecision] * N[Abs[im], $MachinePrecision]), $MachinePrecision]

\left(\left|re\right| - \left|im\right|\right) \cdot \left|im\right|
Derivation

  1. Initial program 93.8%

    \[re \cdot re - im \cdot im \]
  2. Step-by-step derivation

    1. lift--.f64N/A

      \[\leadsto \color{blue}{re \cdot re - im \cdot im} \]

    2. lift-*.f64N/A

      \[\leadsto \color{blue}{re \cdot re} - im \cdot im \]

    3. lift-*.f64N/A

      \[\leadsto re \cdot re - \color{blue}{im \cdot im} \]

    4. difference-of-squaresN/A

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

    5. *-commutativeN/A

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

    6. lower-*.f64N/A

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

    7. lower--.f64N/A

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

    8. +-commutativeN/A

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

    9. lower-+.f64100.0%

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

  3. Applied rewrites100.0%

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

  4. Taylor expanded in re around 0

    \[\leadsto \left(re - im\right) \cdot \color{blue}{im} \]
  5. Step-by-step derivation

    1. Applied rewrites57.2%

      \[\leadsto \left(re - im\right) \cdot \color{blue}{im} \]
    2. Add Preprocessing

    Reproduce

    ?
    herbie shell --seed 2025313 -o setup:search
(FPCore re_sqr (re im)
  :name "math.square on complex, real part"
  :precision binary64
  (- (* re re) (* im im)))