Expression 1, p15

Percentage Accurate: 99.4% → 99.6%

Time: 9.4s

Alternatives: 8

Speedup: 1.0×

Specification

\[\left(\left(\left(\left(\left(\left(\left(\left(1 \leq a \land a \leq 2\right) \land 2 \leq b\right) \land b \leq 4\right) \land 4 \leq c\right) \land c \leq 8\right) \land 8 \leq d\right) \land d \leq 16\right) \land 16 \leq e\right) \land e \leq 32\]

Math

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

FPCore

(FPCore (a b c d e) :precision binary64 (+ (+ (+ (+ e d) c) b) a))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.4% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (a b c d e) :precision binary64 (+ (+ (+ (+ e d) c) b) a))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (a b c d e) :precision binary64 (+ (+ e d) (+ (+ c b) a)))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

   \[\left(\left(\left(e + d\right) + c\right) + b\right) + a \]
2. Step-by-step derivation

   1. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\left(\left(\left(e + d\right) + c\right) + b\right), \color{blue}{a}\right) \]

   2. associate-+l+ N/A

      \[\leadsto \mathsf{+.f64}\left(\left(\left(e + d\right) + \left(c + b\right)\right), a\right) \]

   3. +-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(\left(\left(c + b\right) + \left(e + d\right)\right), a\right) \]

   4. associate-+l+ N/A

      \[\leadsto \mathsf{+.f64}\left(\left(c + \left(b + \left(e + d\right)\right)\right), a\right) \]

   5. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(c, \left(b + \left(e + d\right)\right)\right), a\right) \]

   6. +-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(c, \left(\left(e + d\right) + b\right)\right), a\right) \]

   7. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(c, \mathsf{+.f64}\left(\left(e + d\right), b\right)\right), a\right) \]

   8. +-lowering-+.f64 99.4%

      \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(c, \mathsf{+.f64}\left(\mathsf{+.f64}\left(e, d\right), b\right)\right), a\right) \]

3. Simplified 99.4%

   \[\leadsto \color{blue}{\left(c + \left(\left(e + d\right) + b\right)\right) + a} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. associate-+r+ N/A

      \[\leadsto \left(\left(c + \left(e + d\right)\right) + b\right) + a \]

   2. +-commutative N/A

      \[\leadsto \left(\left(\left(e + d\right) + c\right) + b\right) + a \]

   3. associate-+l+ N/A

      \[\leadsto \left(\left(e + d\right) + \left(c + b\right)\right) + a \]

   4. associate-+l+ N/A

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

   5. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\left(e + d\right), \color{blue}{\left(\left(c + b\right) + a\right)}\right) \]

   6. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(e, d\right), \left(\color{blue}{\left(c + b\right)} + a\right)\right) \]

   7. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(e, d\right), \mathsf{+.f64}\left(\left(c + b\right), \color{blue}{a}\right)\right) \]

   8. +-lowering-+.f64 99.7%

      \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(e, d\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(c, b\right), a\right)\right) \]

6. Applied egg-rr 99.7%

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

Alternative 2: 99.6% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (a b c d e) :precision binary64 (+ (+ c (+ e b)) (+ d a)))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

   \[\left(\left(\left(e + d\right) + c\right) + b\right) + a \]
2. Step-by-step derivation

   1. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\left(\left(\left(e + d\right) + c\right) + b\right), \color{blue}{a}\right) \]

   2. associate-+l+ N/A

      \[\leadsto \mathsf{+.f64}\left(\left(\left(e + d\right) + \left(c + b\right)\right), a\right) \]

   3. +-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(\left(\left(c + b\right) + \left(e + d\right)\right), a\right) \]

   4. associate-+l+ N/A

      \[\leadsto \mathsf{+.f64}\left(\left(c + \left(b + \left(e + d\right)\right)\right), a\right) \]

   5. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(c, \left(b + \left(e + d\right)\right)\right), a\right) \]

   6. +-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(c, \left(\left(e + d\right) + b\right)\right), a\right) \]

   7. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(c, \mathsf{+.f64}\left(\left(e + d\right), b\right)\right), a\right) \]

   8. +-lowering-+.f64 99.4%

      \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(c, \mathsf{+.f64}\left(\mathsf{+.f64}\left(e, d\right), b\right)\right), a\right) \]

3. Simplified 99.4%

   \[\leadsto \color{blue}{\left(c + \left(\left(e + d\right) + b\right)\right) + a} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. associate-+r+ N/A

      \[\leadsto \left(\left(c + \left(e + d\right)\right) + b\right) + a \]

   2. +-commutative N/A

      \[\leadsto \left(\left(\left(e + d\right) + c\right) + b\right) + a \]

   3. associate-+l+ N/A

      \[\leadsto \left(\left(e + d\right) + \left(c + b\right)\right) + a \]

   4. associate-+l+ N/A

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

   5. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\left(e + d\right), \color{blue}{\left(\left(c + b\right) + a\right)}\right) \]

   6. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(e, d\right), \left(\color{blue}{\left(c + b\right)} + a\right)\right) \]

   7. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(e, d\right), \mathsf{+.f64}\left(\left(c + b\right), \color{blue}{a}\right)\right) \]

   8. +-lowering-+.f64 99.7%

      \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(e, d\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(c, b\right), a\right)\right) \]

6. Applied egg-rr 99.7%

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

   1. associate-+r+ N/A

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

   2. +-commutative N/A

      \[\leadsto \left(\left(e + d\right) + \left(b + c\right)\right) + a \]

   3. +-commutative N/A

      \[\leadsto \left(\left(d + e\right) + \left(b + c\right)\right) + a \]

   4. +-commutative N/A

      \[\leadsto \left(\left(d + e\right) + \left(c + b\right)\right) + a \]

   5. associate-+l+ N/A

      \[\leadsto \left(d + \left(e + \left(c + b\right)\right)\right) + a \]

   6. associate-+l+ N/A

      \[\leadsto \left(d + \left(\left(e + c\right) + b\right)\right) + a \]

   7. +-commutative N/A

      \[\leadsto \left(d + \left(b + \left(e + c\right)\right)\right) + a \]

   8. +-commutative N/A

      \[\leadsto \left(\left(b + \left(e + c\right)\right) + d\right) + a \]

   9. associate-+l+ N/A

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

   10. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(\left(b + \left(e + c\right)\right), \color{blue}{\left(d + a\right)}\right) \]

   11. associate-+r+ N/A

       \[\leadsto \mathsf{+.f64}\left(\left(\left(b + e\right) + c\right), \left(\color{blue}{d} + a\right)\right) \]

   12. +-commutative N/A

       \[\leadsto \mathsf{+.f64}\left(\left(c + \left(b + e\right)\right), \left(\color{blue}{d} + a\right)\right) \]

   13. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(c, \left(b + e\right)\right), \left(\color{blue}{d} + a\right)\right) \]

   14. +-commutative N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(c, \left(e + b\right)\right), \left(d + a\right)\right) \]

   15. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(c, \mathsf{+.f64}\left(e, b\right)\right), \left(d + a\right)\right) \]

   16. +-lowering-+.f64 99.6%

       \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(c, \mathsf{+.f64}\left(e, b\right)\right), \mathsf{+.f64}\left(d, \color{blue}{a}\right)\right) \]

8. Applied egg-rr 99.6%

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

Alternative 3: 99.6% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (a b c d e) :precision binary64 (+ e (+ (+ d c) (+ b a))))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

   \[\left(\left(\left(e + d\right) + c\right) + b\right) + a \]
2. Step-by-step derivation

   1. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\left(\left(\left(e + d\right) + c\right) + b\right), \color{blue}{a}\right) \]

   2. associate-+l+ N/A

      \[\leadsto \mathsf{+.f64}\left(\left(\left(e + d\right) + \left(c + b\right)\right), a\right) \]

   3. +-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(\left(\left(c + b\right) + \left(e + d\right)\right), a\right) \]

   4. associate-+l+ N/A

      \[\leadsto \mathsf{+.f64}\left(\left(c + \left(b + \left(e + d\right)\right)\right), a\right) \]

   5. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(c, \left(b + \left(e + d\right)\right)\right), a\right) \]

   6. +-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(c, \left(\left(e + d\right) + b\right)\right), a\right) \]

   7. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(c, \mathsf{+.f64}\left(\left(e + d\right), b\right)\right), a\right) \]

   8. +-lowering-+.f64 99.4%

      \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(c, \mathsf{+.f64}\left(\mathsf{+.f64}\left(e, d\right), b\right)\right), a\right) \]

3. Simplified 99.4%

   \[\leadsto \color{blue}{\left(c + \left(\left(e + d\right) + b\right)\right) + a} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. associate-+r+ N/A

      \[\leadsto \left(\left(c + \left(e + d\right)\right) + b\right) + a \]

   2. +-commutative N/A

      \[\leadsto \left(\left(\left(e + d\right) + c\right) + b\right) + a \]

   3. associate-+l+ N/A

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

   4. associate-+l+ N/A

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

   5. associate-+l+ N/A

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

   6. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(e, \color{blue}{\left(\left(d + c\right) + \left(b + a\right)\right)}\right) \]

   7. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(e, \mathsf{+.f64}\left(\left(d + c\right), \color{blue}{\left(b + a\right)}\right)\right) \]

   8. +-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(e, \mathsf{+.f64}\left(\left(c + d\right), \left(\color{blue}{b} + a\right)\right)\right) \]

   9. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(e, \mathsf{+.f64}\left(\mathsf{+.f64}\left(c, d\right), \left(\color{blue}{b} + a\right)\right)\right) \]

   10. +-lowering-+.f64 99.6%

       \[\leadsto \mathsf{+.f64}\left(e, \mathsf{+.f64}\left(\mathsf{+.f64}\left(c, d\right), \mathsf{+.f64}\left(b, \color{blue}{a}\right)\right)\right) \]

6. Applied egg-rr 99.6%

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

7. Final simplification 99.6%

   \[\leadsto e + \left(\left(d + c\right) + \left(b + a\right)\right) \]
8. Add Preprocessing

Alternative 4: 25.7% accurate, 1.3× speedup

Math

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

FPCore

(FPCore (a b c d e) :precision binary64 (+ d (+ b (+ e c))))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a_, b_, c_, d_, e_] := N[(d + N[(b + N[(e + c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.4%

   \[\left(\left(\left(e + d\right) + c\right) + b\right) + a \]
2. Step-by-step derivation

   1. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\left(\left(\left(e + d\right) + c\right) + b\right), \color{blue}{a}\right) \]

   2. associate-+l+ N/A

      \[\leadsto \mathsf{+.f64}\left(\left(\left(e + d\right) + \left(c + b\right)\right), a\right) \]

   3. +-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(\left(\left(c + b\right) + \left(e + d\right)\right), a\right) \]

   4. associate-+l+ N/A

      \[\leadsto \mathsf{+.f64}\left(\left(c + \left(b + \left(e + d\right)\right)\right), a\right) \]

   5. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(c, \left(b + \left(e + d\right)\right)\right), a\right) \]

   6. +-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(c, \left(\left(e + d\right) + b\right)\right), a\right) \]

   7. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(c, \mathsf{+.f64}\left(\left(e + d\right), b\right)\right), a\right) \]

   8. +-lowering-+.f64 99.4%

      \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(c, \mathsf{+.f64}\left(\mathsf{+.f64}\left(e, d\right), b\right)\right), a\right) \]

3. Simplified 99.4%

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

5. Taylor expanded in a around 0

   \[\leadsto \color{blue}{b + \left(c + \left(d + e\right)\right)} \]
6. Step-by-step derivation

   1. associate-+r+ N/A

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

   2. +-commutative N/A

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

   3. associate-+l+ N/A

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

   4. +-commutative N/A

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

   5. associate-+r+ N/A

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

   6. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(d, \color{blue}{\left(b + \left(c + e\right)\right)}\right) \]

   7. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(d, \mathsf{+.f64}\left(b, \color{blue}{\left(c + e\right)}\right)\right) \]

   8. +-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(d, \mathsf{+.f64}\left(b, \left(e + \color{blue}{c}\right)\right)\right) \]

   9. +-lowering-+.f64 25.7%

      \[\leadsto \mathsf{+.f64}\left(d, \mathsf{+.f64}\left(b, \mathsf{+.f64}\left(e, \color{blue}{c}\right)\right)\right) \]

7. Simplified 25.7%

   \[\leadsto \color{blue}{d + \left(b + \left(e + c\right)\right)} \]
8. Add Preprocessing

Alternative 5: 25.7% accurate, 1.3× speedup

Math

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

FPCore

(FPCore (a b c d e) :precision binary64 (+ c (+ d (+ e b))))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a_, b_, c_, d_, e_] := N[(c + N[(d + N[(e + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.4%

   \[\left(\left(\left(e + d\right) + c\right) + b\right) + a \]
2. Step-by-step derivation

   1. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\left(\left(\left(e + d\right) + c\right) + b\right), \color{blue}{a}\right) \]

   2. associate-+l+ N/A

      \[\leadsto \mathsf{+.f64}\left(\left(\left(e + d\right) + \left(c + b\right)\right), a\right) \]

   3. +-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(\left(\left(c + b\right) + \left(e + d\right)\right), a\right) \]

   4. associate-+l+ N/A

      \[\leadsto \mathsf{+.f64}\left(\left(c + \left(b + \left(e + d\right)\right)\right), a\right) \]

   5. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(c, \left(b + \left(e + d\right)\right)\right), a\right) \]

   6. +-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(c, \left(\left(e + d\right) + b\right)\right), a\right) \]

   7. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(c, \mathsf{+.f64}\left(\left(e + d\right), b\right)\right), a\right) \]

   8. +-lowering-+.f64 99.4%

      \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(c, \mathsf{+.f64}\left(\mathsf{+.f64}\left(e, d\right), b\right)\right), a\right) \]

3. Simplified 99.4%

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

5. Taylor expanded in a around 0

   \[\leadsto \color{blue}{b + \left(c + \left(d + e\right)\right)} \]
6. Step-by-step derivation

   1. associate-+r+ N/A

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

   2. +-commutative N/A

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

   3. associate-+l+ N/A

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

   4. +-commutative N/A

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

   5. associate-+r+ N/A

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

   6. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(d, \color{blue}{\left(b + \left(c + e\right)\right)}\right) \]

   7. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(d, \mathsf{+.f64}\left(b, \color{blue}{\left(c + e\right)}\right)\right) \]

   8. +-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(d, \mathsf{+.f64}\left(b, \left(e + \color{blue}{c}\right)\right)\right) \]

   9. +-lowering-+.f64 25.7%

      \[\leadsto \mathsf{+.f64}\left(d, \mathsf{+.f64}\left(b, \mathsf{+.f64}\left(e, \color{blue}{c}\right)\right)\right) \]

7. Simplified 25.7%

   \[\leadsto \color{blue}{d + \left(b + \left(e + c\right)\right)} \]
8. Step-by-step derivation

   1. +-commutative N/A

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

   2. associate-+l+ N/A

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

   3. associate-+l+ N/A

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

   4. +-commutative N/A

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

   5. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\left(e + d\right), \color{blue}{\left(c + b\right)}\right) \]

   6. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(e, d\right), \left(\color{blue}{c} + b\right)\right) \]

   7. +-lowering-+.f64 25.7%

      \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(e, d\right), \mathsf{+.f64}\left(c, \color{blue}{b}\right)\right) \]

9. Applied egg-rr 25.7%

   \[\leadsto \color{blue}{\left(e + d\right) + \left(c + b\right)} \]
10. Step-by-step derivation

    1. +-commutative N/A

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

    2. associate-+r+ N/A

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

    3. associate-+r+ N/A

       \[\leadsto \left(c + \left(b + e\right)\right) + d \]

    4. +-commutative N/A

       \[\leadsto \left(c + \left(e + b\right)\right) + d \]

    5. associate-+l+ N/A

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

    6. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(c, \color{blue}{\left(\left(e + b\right) + d\right)}\right) \]

    7. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(c, \mathsf{+.f64}\left(\left(e + b\right), \color{blue}{d}\right)\right) \]

    8. +-lowering-+.f64 25.7%

       \[\leadsto \mathsf{+.f64}\left(c, \mathsf{+.f64}\left(\mathsf{+.f64}\left(e, b\right), d\right)\right) \]

11. Applied egg-rr 25.7%

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

12. Final simplification 25.7%

    \[\leadsto c + \left(d + \left(e + b\right)\right) \]
13. Add Preprocessing

Alternative 6: 23.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ d + \left(e + c\right) \end{array} \]

FPCore

(FPCore (a b c d e) :precision binary64 (+ d (+ e c)))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a_, b_, c_, d_, e_] := N[(d + N[(e + c), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.4%

   \[\left(\left(\left(e + d\right) + c\right) + b\right) + a \]
2. Step-by-step derivation

   1. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\left(\left(\left(e + d\right) + c\right) + b\right), \color{blue}{a}\right) \]

   2. associate-+l+ N/A

      \[\leadsto \mathsf{+.f64}\left(\left(\left(e + d\right) + \left(c + b\right)\right), a\right) \]

   3. +-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(\left(\left(c + b\right) + \left(e + d\right)\right), a\right) \]

   4. associate-+l+ N/A

      \[\leadsto \mathsf{+.f64}\left(\left(c + \left(b + \left(e + d\right)\right)\right), a\right) \]

   5. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(c, \left(b + \left(e + d\right)\right)\right), a\right) \]

   6. +-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(c, \left(\left(e + d\right) + b\right)\right), a\right) \]

   7. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(c, \mathsf{+.f64}\left(\left(e + d\right), b\right)\right), a\right) \]

   8. +-lowering-+.f64 99.4%

      \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(c, \mathsf{+.f64}\left(\mathsf{+.f64}\left(e, d\right), b\right)\right), a\right) \]

3. Simplified 99.4%

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

5. Taylor expanded in a around 0

   \[\leadsto \color{blue}{b + \left(c + \left(d + e\right)\right)} \]
6. Step-by-step derivation

   1. associate-+r+ N/A

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

   2. +-commutative N/A

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

   3. associate-+l+ N/A

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

   4. +-commutative N/A

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

   5. associate-+r+ N/A

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

   6. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(d, \color{blue}{\left(b + \left(c + e\right)\right)}\right) \]

   7. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(d, \mathsf{+.f64}\left(b, \color{blue}{\left(c + e\right)}\right)\right) \]

   8. +-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(d, \mathsf{+.f64}\left(b, \left(e + \color{blue}{c}\right)\right)\right) \]

   9. +-lowering-+.f64 25.7%

      \[\leadsto \mathsf{+.f64}\left(d, \mathsf{+.f64}\left(b, \mathsf{+.f64}\left(e, \color{blue}{c}\right)\right)\right) \]

7. Simplified 25.7%

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

8. Taylor expanded in b around 0

   \[\leadsto \color{blue}{c + \left(d + e\right)} \]
9. Step-by-step derivation

   1. associate-+r+ N/A

      \[\leadsto \left(c + d\right) + \color{blue}{e} \]

   2. +-commutative N/A

      \[\leadsto \left(d + c\right) + e \]

   3. associate-+l+ N/A

      \[\leadsto d + \color{blue}{\left(c + e\right)} \]

   4. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(d, \color{blue}{\left(c + e\right)}\right) \]

   5. +-lowering-+.f64 23.2%

      \[\leadsto \mathsf{+.f64}\left(d, \mathsf{+.f64}\left(c, \color{blue}{e}\right)\right) \]

10. Simplified 23.2%

    \[\leadsto \color{blue}{d + \left(c + e\right)} \]

11. Final simplification 23.2%

    \[\leadsto d + \left(e + c\right) \]
12. Add Preprocessing

Alternative 7: 21.2% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ e + d \end{array} \]

FPCore

(FPCore (a b c d e) :precision binary64 (+ e d))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a_, b_, c_, d_, e_] := N[(e + d), $MachinePrecision]

Derivation

1. Initial program 99.4%

   \[\left(\left(\left(e + d\right) + c\right) + b\right) + a \]
2. Step-by-step derivation

   1. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\left(\left(\left(e + d\right) + c\right) + b\right), \color{blue}{a}\right) \]

   2. associate-+l+ N/A

      \[\leadsto \mathsf{+.f64}\left(\left(\left(e + d\right) + \left(c + b\right)\right), a\right) \]

   3. +-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(\left(\left(c + b\right) + \left(e + d\right)\right), a\right) \]

   4. associate-+l+ N/A

      \[\leadsto \mathsf{+.f64}\left(\left(c + \left(b + \left(e + d\right)\right)\right), a\right) \]

   5. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(c, \left(b + \left(e + d\right)\right)\right), a\right) \]

   6. +-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(c, \left(\left(e + d\right) + b\right)\right), a\right) \]

   7. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(c, \mathsf{+.f64}\left(\left(e + d\right), b\right)\right), a\right) \]

   8. +-lowering-+.f64 99.4%

      \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(c, \mathsf{+.f64}\left(\mathsf{+.f64}\left(e, d\right), b\right)\right), a\right) \]

3. Simplified 99.4%

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

5. Taylor expanded in a around 0

   \[\leadsto \color{blue}{b + \left(c + \left(d + e\right)\right)} \]
6. Step-by-step derivation

   1. associate-+r+ N/A

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

   2. +-commutative N/A

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

   3. associate-+l+ N/A

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

   4. +-commutative N/A

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

   5. associate-+r+ N/A

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

   6. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(d, \color{blue}{\left(b + \left(c + e\right)\right)}\right) \]

   7. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(d, \mathsf{+.f64}\left(b, \color{blue}{\left(c + e\right)}\right)\right) \]

   8. +-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(d, \mathsf{+.f64}\left(b, \left(e + \color{blue}{c}\right)\right)\right) \]

   9. +-lowering-+.f64 25.7%

      \[\leadsto \mathsf{+.f64}\left(d, \mathsf{+.f64}\left(b, \mathsf{+.f64}\left(e, \color{blue}{c}\right)\right)\right) \]

7. Simplified 25.7%

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

8. Taylor expanded in e around inf

   \[\leadsto \mathsf{+.f64}\left(d, \color{blue}{e}\right) \]
9. Step-by-step derivation

10. Simplified 21.2%

    \[\leadsto d + \color{blue}{e} \]

11. Final simplification 21.2%

    \[\leadsto e + d \]
12. Add Preprocessing

Alternative 8: 18.9% accurate, 9.0× speedup

Math

\[\begin{array}{l} \\ e \end{array} \]

FPCore

(FPCore (a b c d e) :precision binary64 e)

C

double code(double a, double b, double c, double d, double e) {
	return e;
}

Fortran

real(8) function code(a, b, c, d, e)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8), intent (in) :: e
    code = e
end function

Java

public static double code(double a, double b, double c, double d, double e) {
	return e;
}

Python

def code(a, b, c, d, e):
	return e

Julia

function code(a, b, c, d, e)
	return e
end

MATLAB

function tmp = code(a, b, c, d, e)
	tmp = e;
end

Wolfram

code[a_, b_, c_, d_, e_] := e

Derivation

1. Initial program 99.4%

   \[\left(\left(\left(e + d\right) + c\right) + b\right) + a \]
2. Step-by-step derivation

   1. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\left(\left(\left(e + d\right) + c\right) + b\right), \color{blue}{a}\right) \]

   2. associate-+l+ N/A

      \[\leadsto \mathsf{+.f64}\left(\left(\left(e + d\right) + \left(c + b\right)\right), a\right) \]

   3. +-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(\left(\left(c + b\right) + \left(e + d\right)\right), a\right) \]

   4. associate-+l+ N/A

      \[\leadsto \mathsf{+.f64}\left(\left(c + \left(b + \left(e + d\right)\right)\right), a\right) \]

   5. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(c, \left(b + \left(e + d\right)\right)\right), a\right) \]

   6. +-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(c, \left(\left(e + d\right) + b\right)\right), a\right) \]

   7. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(c, \mathsf{+.f64}\left(\left(e + d\right), b\right)\right), a\right) \]

   8. +-lowering-+.f64 99.4%

      \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(c, \mathsf{+.f64}\left(\mathsf{+.f64}\left(e, d\right), b\right)\right), a\right) \]

3. Simplified 99.4%

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

5. Taylor expanded in e around inf

   \[\leadsto \color{blue}{e} \]
6. Step-by-step derivation

7. Simplified 18.9%

   \[\leadsto \color{blue}{e} \]
8. Add Preprocessing

Developer Target 1: 99.6% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (a b c d e) :precision binary64 (+ (+ d (+ c (+ a b))) e))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024163 
(FPCore (a b c d e)
  :name "Expression 1, p15"
  :precision binary64
  :pre (and (and (and (and (and (and (and (and (and (<= 1.0 a) (<= a 2.0)) (<= 2.0 b)) (<= b 4.0)) (<= 4.0 c)) (<= c 8.0)) (<= 8.0 d)) (<= d 16.0)) (<= 16.0 e)) (<= e 32.0))

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

  (+ (+ (+ (+ e d) c) b) a))