Difference of squares

Percentage Accurate: 93.9% → 100.0%

Time: 1.3s

Alternatives: 2

Speedup: 1.0×

• 55.7% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[a \cdot a - b \cdot b \]

FPCore

(FPCore (a b)
  :precision binary64
  (- (* a a) (* b b)))

C

double code(double a, double b) {
	return (a * a) - (b * b);
}

Fortran

real(8) function code(a, b)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = (a * a) - (b * b)
end function

Java

public static double code(double a, double b) {
	return (a * a) - (b * b);
}

Python

def code(a, b):
	return (a * a) - (b * b)

Julia

function code(a, b)
	return Float64(Float64(a * a) - Float64(b * b))
end

MATLAB

function tmp = code(a, b)
	tmp = (a * a) - (b * b);
end

Wolfram

code[a_, b_] := N[(N[(a * a), $MachinePrecision] - N[(b * b), $MachinePrecision]), $MachinePrecision]

Initial Program: 93.9% accurate, 1.0× speedup

Math

\[a \cdot a - b \cdot b \]

FPCore

(FPCore (a b)
  :precision binary64
  (- (* a a) (* b b)))

C

double code(double a, double b) {
	return (a * a) - (b * b);
}

Fortran

real(8) function code(a, b)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = (a * a) - (b * b)
end function

Java

public static double code(double a, double b) {
	return (a * a) - (b * b);
}

Python

def code(a, b):
	return (a * a) - (b * b)

Julia

function code(a, b)
	return Float64(Float64(a * a) - Float64(b * b))
end

MATLAB

function tmp = code(a, b)
	tmp = (a * a) - (b * b);
end

Wolfram

code[a_, b_] := N[(N[(a * a), $MachinePrecision] - N[(b * b), $MachinePrecision]), $MachinePrecision]

Alternative 1: 100.0% accurate, 1.0× speedup

Math

\[\left(a - b\right) \cdot \left(b + a\right) \]

FPCore

(FPCore (a b)
  :precision binary64
  (* (- a b) (+ b a)))

C

double code(double a, double b) {
	return (a - b) * (b + a);
}

Fortran

real(8) function code(a, b)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = (a - b) * (b + a)
end function

Java

public static double code(double a, double b) {
	return (a - b) * (b + a);
}

Python

def code(a, b):
	return (a - b) * (b + a)

Julia

function code(a, b)
	return Float64(Float64(a - b) * Float64(b + a))
end

MATLAB

function tmp = code(a, b)
	tmp = (a - b) * (b + a);
end

Wolfram

code[a_, b_] := N[(N[(a - b), $MachinePrecision] * N[(b + a), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 93.9%

   \[a \cdot a - b \cdot b \]
2. Step-by-step derivation

   1. lift--.f64 N/A

      \[\leadsto \color{blue}{a \cdot a - b \cdot b} \]

   2. lift-*.f64 N/A

      \[\leadsto \color{blue}{a \cdot a} - b \cdot b \]

   3. lift-*.f64 N/A

      \[\leadsto a \cdot a - \color{blue}{b \cdot b} \]

   4. difference-of-squares N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. lower--.f64 N/A

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

   8. +-commutative N/A

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

   9. lower-+.f64 100.0%

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

3. Applied rewrites 100.0%

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

Alternative 2: 60.7% accurate, 1.0× speedup

Math

\[\left(\left|a\right| - \left|b\right|\right) \cdot \left|b\right| \]

FPCore

(FPCore (a b)
  :precision binary64
  (* (- (fabs a) (fabs b)) (fabs b)))

C

double code(double a, double b) {
	return (fabs(a) - fabs(b)) * fabs(b);
}

Fortran

real(8) function code(a, b)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = (abs(a) - abs(b)) * abs(b)
end function

Java

public static double code(double a, double b) {
	return (Math.abs(a) - Math.abs(b)) * Math.abs(b);
}

Python

def code(a, b):
	return (math.fabs(a) - math.fabs(b)) * math.fabs(b)

Julia

function code(a, b)
	return Float64(Float64(abs(a) - abs(b)) * abs(b))
end

MATLAB

function tmp = code(a, b)
	tmp = (abs(a) - abs(b)) * abs(b);
end

Wolfram

code[a_, b_] := N[(N[(N[Abs[a], $MachinePrecision] - N[Abs[b], $MachinePrecision]), $MachinePrecision] * N[Abs[b], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 93.9%

   \[a \cdot a - b \cdot b \]
2. Step-by-step derivation

   1. lift--.f64 N/A

      \[\leadsto \color{blue}{a \cdot a - b \cdot b} \]

   2. lift-*.f64 N/A

      \[\leadsto \color{blue}{a \cdot a} - b \cdot b \]

   3. lift-*.f64 N/A

      \[\leadsto a \cdot a - \color{blue}{b \cdot b} \]

   4. difference-of-squares N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. lower--.f64 N/A

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

   8. +-commutative N/A

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

   9. lower-+.f64 100.0%

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

3. Applied rewrites 100.0%

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

4. Taylor expanded in a around 0

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

6. Applied rewrites 56.8%

   \[\leadsto \left(a - b\right) \cdot \color{blue}{b} \]
7. Add Preprocessing

Developer Target 1: 100.0% accurate, 1.0× speedup

Math

\[\left(a + b\right) \cdot \left(a - b\right) \]

FPCore

(FPCore (a b)
  :precision binary64
  (* (+ a b) (- a b)))

C

double code(double a, double b) {
	return (a + b) * (a - b);
}

Fortran

real(8) function code(a, b)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = (a + b) * (a - b)
end function

Java

public static double code(double a, double b) {
	return (a + b) * (a - b);
}

Python

def code(a, b):
	return (a + b) * (a - b)

Julia

function code(a, b)
	return Float64(Float64(a + b) * Float64(a - b))
end

MATLAB

function tmp = code(a, b)
	tmp = (a + b) * (a - b);
end

Wolfram

code[a_, b_] := N[(N[(a + b), $MachinePrecision] * N[(a - b), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2025323 
(FPCore (a b)
  :name "Difference of squares"
  :precision binary64

  :alt
  (! :herbie-platform c (* (+ a b) (- a b)))

  (- (* a a) (* b b)))