Expression 1, p15

Percentage Accurate: 99.4% → 99.6%

Time: 8.4s

Alternatives: 5

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

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

   1. lift-+.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. associate-+l+ N/A

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

   4. +-commutative N/A

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

   5. lift-+.f64 N/A

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

   6. lift-+.f64 N/A

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

   7. +-commutative N/A

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

   8. associate-+l+ N/A

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

   9. associate-+r+ N/A

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

   10. lower-+.f64 N/A

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

   11. lower-+.f64 N/A

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

   12. +-commutative N/A

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

   13. lower-+.f64 N/A

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

   14. lower-+.f64 99.7

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

4. Applied rewrites 99.7%

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

   1. lift-+.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. associate-+r+ N/A

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

   4. lift-+.f64 N/A

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

   5. associate-+r+ N/A

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

   6. lift-+.f64 N/A

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

   7. +-commutative N/A

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

   8. associate-+l+ N/A

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

   9. lift-+.f64 N/A

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

   10. associate-+r+ N/A

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

   11. lift-+.f64 N/A

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

   12. +-commutative N/A

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

   13. lift-+.f64 N/A

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

   14. lift-+.f64 N/A

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

   15. associate-+r+ N/A

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

   16. +-commutative N/A

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

   17. associate-+r+ N/A

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

   18. lower-+.f64 N/A

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

   19. lower-+.f64 99.7

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

   20. lift-+.f64 N/A

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

   21. +-commutative N/A

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

   22. lower-+.f64 99.7

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

6. Applied rewrites 99.7%

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

7. Final simplification 99.7%

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

Alternative 2: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

   1. lift-+.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-+.f64 N/A

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

   4. associate-+l+ N/A

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

   5. associate-+l+ N/A

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

   6. lift-+.f64 N/A

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

   7. +-commutative N/A

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

   8. associate-+l+ N/A

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

   9. lower-+.f64 N/A

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

   10. lower-+.f64 N/A

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

   11. lower-+.f64 N/A

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

   12. +-commutative N/A

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

   13. lower-+.f64 99.6

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

4. Applied rewrites 99.6%

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

5. Final simplification 99.6%

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

Alternative 3: 25.7% accurate, 1.3× speedup

Math

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

FPCore

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

C

double code(double a, double b, double c, double d, double e) {
	return ((b + c) + d) + 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 = ((b + c) + d) + e
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

3. Taylor expanded in a around 0

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

   1. +-commutative N/A

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

   2. lower-+.f64 N/A

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

   3. +-commutative N/A

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

   4. lower-+.f64 N/A

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

   5. lower-+.f64 25.7

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

5. Applied rewrites 25.7%

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

7. Applied rewrites 25.7%

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

8. Final simplification 25.7%

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

Alternative 4: 23.2% accurate, 1.9× speedup

Math

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

FPCore

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

C

double code(double a, double b, double c, double d, double e) {
	return (c + d) + 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 = (c + d) + e
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

3. Taylor expanded in a around 0

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

   1. +-commutative N/A

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

   2. lower-+.f64 N/A

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

   3. +-commutative N/A

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

   4. lower-+.f64 N/A

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

   5. lower-+.f64 25.7

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

5. Applied rewrites 25.7%

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

6. Taylor expanded in b around 0

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

8. Applied rewrites 23.2%

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

10. Applied rewrites 23.2%

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

Alternative 5: 21.2% accurate, 3.3× 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. Add Preprocessing

3. Taylor expanded in a around 0

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

   1. +-commutative N/A

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

   2. lower-+.f64 N/A

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

   3. +-commutative N/A

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

   4. lower-+.f64 N/A

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

   5. lower-+.f64 25.7

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

5. Applied rewrites 25.7%

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

6. Taylor expanded in b around 0

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

8. Applied rewrites 23.2%

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

9. Taylor expanded in c around 0

   \[\leadsto d + e \]
10. Step-by-step derivation

11. Applied rewrites 21.2%

    \[\leadsto e + d \]
12. 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 2024327 
(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))