Expression, p6

Percentage Accurate: 94.4% → 100.0%

Time: 6.5s

Alternatives: 6

Speedup: 1.0×

Specification

\[\left(\left(\left(-14 \leq a \land a \leq -13\right) \land \left(-3 \leq b \land b \leq -2\right)\right) \land \left(3 \leq c \land c \leq 3.5\right)\right) \land \left(12.5 \leq d \land d \leq 13.5\right)\]

Math

\[\begin{array}{l} \\ \left(a + \left(b + \left(c + d\right)\right)\right) \cdot 2 \end{array} \]

FPCore

(FPCore (a b c d) :precision binary64 (* (+ a (+ b (+ c d))) 2.0))

C

double code(double a, double b, double c, double d) {
	return (a + (b + (c + d))) * 2.0;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    code = (a + (b + (c + d))) * 2.0d0
end function

Java

public static double code(double a, double b, double c, double d) {
	return (a + (b + (c + d))) * 2.0;
}

Python

def code(a, b, c, d):
	return (a + (b + (c + d))) * 2.0

Julia

function code(a, b, c, d)
	return Float64(Float64(a + Float64(b + Float64(c + d))) * 2.0)
end

MATLAB

function tmp = code(a, b, c, d)
	tmp = (a + (b + (c + d))) * 2.0;
end

Wolfram

code[a_, b_, c_, d_] := N[(N[(a + N[(b + N[(c + d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]

Initial Program: 94.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(a + \left(b + \left(c + d\right)\right)\right) \cdot 2 \end{array} \]

FPCore

(FPCore (a b c d) :precision binary64 (* (+ a (+ b (+ c d))) 2.0))

C

double code(double a, double b, double c, double d) {
	return (a + (b + (c + d))) * 2.0;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    code = (a + (b + (c + d))) * 2.0d0
end function

Java

public static double code(double a, double b, double c, double d) {
	return (a + (b + (c + d))) * 2.0;
}

Python

def code(a, b, c, d):
	return (a + (b + (c + d))) * 2.0

Julia

function code(a, b, c, d)
	return Float64(Float64(a + Float64(b + Float64(c + d))) * 2.0)
end

MATLAB

function tmp = code(a, b, c, d)
	tmp = (a + (b + (c + d))) * 2.0;
end

Wolfram

code[a_, b_, c_, d_] := N[(N[(a + N[(b + N[(c + d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]

Alternative 1: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(\left(b + c\right) + \left(a + d\right)\right) \cdot 2 \end{array} \]

FPCore

(FPCore (a b c d) :precision binary64 (* (+ (+ b c) (+ a d)) 2.0))

C

double code(double a, double b, double c, double d) {
	return ((b + c) + (a + d)) * 2.0;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    code = ((b + c) + (a + d)) * 2.0d0
end function

Java

public static double code(double a, double b, double c, double d) {
	return ((b + c) + (a + d)) * 2.0;
}

Python

def code(a, b, c, d):
	return ((b + c) + (a + d)) * 2.0

Julia

function code(a, b, c, d)
	return Float64(Float64(Float64(b + c) + Float64(a + d)) * 2.0)
end

MATLAB

function tmp = code(a, b, c, d)
	tmp = ((b + c) + (a + d)) * 2.0;
end

Wolfram

code[a_, b_, c_, d_] := N[(N[(N[(b + c), $MachinePrecision] + N[(a + d), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]

Derivation

1. Initial program 94.0%

   \[\left(a + \left(b + \left(c + d\right)\right)\right) \cdot 2 \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-+.f64 N/A

      \[\leadsto \color{blue}{\left(a + \left(b + \left(c + d\right)\right)\right)} \cdot 2 \]

   2. +-commutative N/A

      \[\leadsto \color{blue}{\left(\left(b + \left(c + d\right)\right) + a\right)} \cdot 2 \]

   3. lift-+.f64 N/A

      \[\leadsto \left(\color{blue}{\left(b + \left(c + d\right)\right)} + a\right) \cdot 2 \]

   4. lift-+.f64 N/A

      \[\leadsto \left(\left(b + \color{blue}{\left(c + d\right)}\right) + a\right) \cdot 2 \]

   5. associate-+r+ N/A

      \[\leadsto \left(\color{blue}{\left(\left(b + c\right) + d\right)} + a\right) \cdot 2 \]

   6. associate-+l+ N/A

      \[\leadsto \color{blue}{\left(\left(b + c\right) + \left(d + a\right)\right)} \cdot 2 \]

   7. lower-+.f64 N/A

      \[\leadsto \color{blue}{\left(\left(b + c\right) + \left(d + a\right)\right)} \cdot 2 \]

   8. lower-+.f64 N/A

      \[\leadsto \left(\color{blue}{\left(b + c\right)} + \left(d + a\right)\right) \cdot 2 \]

   9. +-commutative N/A

      \[\leadsto \left(\left(b + c\right) + \color{blue}{\left(a + d\right)}\right) \cdot 2 \]

   10. lower-+.f64 100.0

       \[\leadsto \left(\left(b + c\right) + \color{blue}{\left(a + d\right)}\right) \cdot 2 \]

4. Applied rewrites 100.0%

   \[\leadsto \color{blue}{\left(\left(b + c\right) + \left(a + d\right)\right)} \cdot 2 \]
5. Add Preprocessing

Alternative 2: 17.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a + \left(b + \left(c + d\right)\right) \leq -0.0005:\\ \;\;\;\;2 \cdot \left(a + d\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(c + \left(a + d\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= (+ a (+ b (+ c d))) -0.0005) (* 2.0 (+ a d)) (* 2.0 (+ c (+ a d)))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((a + (b + (c + d))) <= -0.0005) {
		tmp = 2.0 * (a + d);
	} else {
		tmp = 2.0 * (c + (a + d));
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if ((a + (b + (c + d))) <= (-0.0005d0)) then
        tmp = 2.0d0 * (a + d)
    else
        tmp = 2.0d0 * (c + (a + d))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if ((a + (b + (c + d))) <= -0.0005) {
		tmp = 2.0 * (a + d);
	} else {
		tmp = 2.0 * (c + (a + d));
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (a + (b + (c + d))) <= -0.0005:
		tmp = 2.0 * (a + d)
	else:
		tmp = 2.0 * (c + (a + d))
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (Float64(a + Float64(b + Float64(c + d))) <= -0.0005)
		tmp = Float64(2.0 * Float64(a + d));
	else
		tmp = Float64(2.0 * Float64(c + Float64(a + d)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((a + (b + (c + d))) <= -0.0005)
		tmp = 2.0 * (a + d);
	else
		tmp = 2.0 * (c + (a + d));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[N[(a + N[(b + N[(c + d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -0.0005], N[(2.0 * N[(a + d), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(c + N[(a + d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 a (+.f64 b (+.f64 c d))) < -5.0000000000000001e-4

   1. Initial program 94.0%

      \[\left(a + \left(b + \left(c + d\right)\right)\right) \cdot 2 \]
   2. Add Preprocessing

   3. Taylor expanded in b around 0

      \[\leadsto \color{blue}{\left(a + \left(c + d\right)\right)} \cdot 2 \]
   4. Step-by-step derivation

      1. associate-+r+ N/A

         \[\leadsto \color{blue}{\left(\left(a + c\right) + d\right)} \cdot 2 \]

      2. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(c + a\right)} + d\right) \cdot 2 \]

      3. associate-+l+ N/A

         \[\leadsto \color{blue}{\left(c + \left(a + d\right)\right)} \cdot 2 \]

      4. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(c + \left(a + d\right)\right)} \cdot 2 \]

      5. +-commutative N/A

         \[\leadsto \left(c + \color{blue}{\left(d + a\right)}\right) \cdot 2 \]

      6. lower-+.f64 1.6

         \[\leadsto \left(c + \color{blue}{\left(d + a\right)}\right) \cdot 2 \]

   5. Applied rewrites 1.6%

      \[\leadsto \color{blue}{\left(c + \left(d + a\right)\right)} \cdot 2 \]

   6. Taylor expanded in c around 0

      \[\leadsto \left(a + \color{blue}{d}\right) \cdot 2 \]
   7. Step-by-step derivation

   8. Applied rewrites 18.0%

      \[\leadsto \left(a + \color{blue}{d}\right) \cdot 2 \]

   if -5.0000000000000001e-4 < (+.f64 a (+.f64 b (+.f64 c d)))

   1. Initial program 94.5%

      \[\left(a + \left(b + \left(c + d\right)\right)\right) \cdot 2 \]
   2. Add Preprocessing

   3. Taylor expanded in b around 0

      \[\leadsto \color{blue}{\left(a + \left(c + d\right)\right)} \cdot 2 \]
   4. Step-by-step derivation

      1. associate-+r+ N/A

         \[\leadsto \color{blue}{\left(\left(a + c\right) + d\right)} \cdot 2 \]

      2. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(c + a\right)} + d\right) \cdot 2 \]

      3. associate-+l+ N/A

         \[\leadsto \color{blue}{\left(c + \left(a + d\right)\right)} \cdot 2 \]

      4. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(c + \left(a + d\right)\right)} \cdot 2 \]

      5. +-commutative N/A

         \[\leadsto \left(c + \color{blue}{\left(d + a\right)}\right) \cdot 2 \]

      6. lower-+.f64 16.5

         \[\leadsto \left(c + \color{blue}{\left(d + a\right)}\right) \cdot 2 \]

   5. Applied rewrites 16.5%

      \[\leadsto \color{blue}{\left(c + \left(d + a\right)\right)} \cdot 2 \]
3. Recombined 2 regimes into one program.

4. Final simplification 17.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a + \left(b + \left(c + d\right)\right) \leq -0.0005:\\ \;\;\;\;2 \cdot \left(a + d\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(c + \left(a + d\right)\right)\\ \end{array} \]
5. Add Preprocessing

Developer Target 1: 94.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \left(a + b\right) \cdot 2 + \left(c + d\right) \cdot 2 \end{array} \]

FPCore

(FPCore (a b c d) :precision binary64 (+ (* (+ a b) 2.0) (* (+ c d) 2.0)))

C

double code(double a, double b, double c, double d) {
	return ((a + b) * 2.0) + ((c + d) * 2.0);
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    code = ((a + b) * 2.0d0) + ((c + d) * 2.0d0)
end function

Java

public static double code(double a, double b, double c, double d) {
	return ((a + b) * 2.0) + ((c + d) * 2.0);
}

Python

def code(a, b, c, d):
	return ((a + b) * 2.0) + ((c + d) * 2.0)

Julia

function code(a, b, c, d)
	return Float64(Float64(Float64(a + b) * 2.0) + Float64(Float64(c + d) * 2.0))
end

MATLAB

function tmp = code(a, b, c, d)
	tmp = ((a + b) * 2.0) + ((c + d) * 2.0);
end

Wolfram

code[a_, b_, c_, d_] := N[(N[(N[(a + b), $MachinePrecision] * 2.0), $MachinePrecision] + N[(N[(c + d), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2024223 
(FPCore (a b c d)
  :name "Expression, p6"
  :precision binary64
  :pre (and (and (and (and (<= -14.0 a) (<= a -13.0)) (and (<= -3.0 b) (<= b -2.0))) (and (<= 3.0 c) (<= c 3.5))) (and (<= 12.5 d) (<= d 13.5)))

  :alt
  (! :herbie-platform default (let ((e 2)) (+ (* (+ a b) e) (* (+ c d) e))))

  (* (+ a (+ b (+ c d))) 2.0))