Expression 1, p15

Percentage Accurate: 99.4% → 99.6%

Time: 4.6s

Alternatives: 7

Speedup: 1.0×

• Unsound rule application detected in e-graph. Results may not be sound.

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

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.3%

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

   1. associate-+l+ 99.6%

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

   2. associate-+l+ 99.6%

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

   3. +-commutative 99.6%

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

   4. associate-+l+ 99.7%

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

3. Simplified 99.7%

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

4. Final simplification 99.7%

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

Alternative 2: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.3%

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

   1. associate-+l+ 99.6%

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

   2. associate-+l+ 99.6%

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

   3. +-commutative 99.6%

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

   4. associate-+l+ 99.7%

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

3. Simplified 99.7%

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

4. Taylor expanded in d around 0 99.5%

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

5. Final simplification 99.5%

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

Alternative 3: 25.7% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.3%

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

   1. associate-+l+ 99.6%

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

   2. associate-+l+ 99.6%

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

   3. +-commutative 99.6%

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

   4. associate-+l+ 99.7%

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

3. Simplified 99.7%

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

4. Taylor expanded in a around 0 25.7%

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

   1. associate-+r+ 25.7%

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

   2. +-commutative 25.7%

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

   3. associate-+l+ 25.7%

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

   4. +-commutative 25.7%

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

6. Simplified 25.7%

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

7. Final simplification 25.7%

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

Alternative 4: 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 99.3%

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

   1. associate-+l+ 99.6%

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

   2. associate-+l+ 99.6%

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

   3. +-commutative 99.6%

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

   4. associate-+l+ 99.7%

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

3. Simplified 99.7%

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

4. Taylor expanded in a around 0 25.7%

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

   1. associate-+r+ 25.7%

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

   2. +-commutative 25.7%

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

   3. associate-+l+ 25.7%

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

   4. +-commutative 25.7%

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

6. Simplified 25.7%

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

7. Taylor expanded in b around 0 23.2%

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

   1. +-commutative 23.2%

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

   2. associate-+l+ 23.2%

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

9. Simplified 23.2%

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

10. Final simplification 23.2%

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

Alternative 5: 19.9% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.3%

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

   1. associate-+l+ 99.6%

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

   2. associate-+l+ 99.6%

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

   3. +-commutative 99.6%

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

   4. associate-+l+ 99.7%

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

3. Simplified 99.7%

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

4. Taylor expanded in d around 0 99.5%

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

5. Taylor expanded in e around inf 19.9%

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

6. Final simplification 19.9%

   \[\leadsto e + c \]

Alternative 6: 21.2% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.3%

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

   1. associate-+l+ 99.6%

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

   2. associate-+l+ 99.6%

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

   3. +-commutative 99.6%

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

   4. associate-+l+ 99.7%

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

3. Simplified 99.7%

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

4. Taylor expanded in a around 0 25.7%

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

   1. associate-+r+ 25.7%

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

   2. +-commutative 25.7%

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

   3. associate-+l+ 25.7%

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

   4. +-commutative 25.7%

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

6. Simplified 25.7%

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

7. Taylor expanded in b around 0 23.2%

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

   1. +-commutative 23.2%

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

   2. associate-+l+ 23.2%

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

9. Simplified 23.2%

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

10. Taylor expanded in c around 0 21.2%

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

11. Final simplification 21.2%

    \[\leadsto d + e \]

Alternative 7: 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 99.3%

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

   1. associate-+l+ 99.6%

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

   2. associate-+l+ 99.6%

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

   3. +-commutative 99.6%

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

   4. associate-+l+ 99.7%

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

3. Simplified 99.7%

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

4. Taylor expanded in e around inf 18.9%

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

5. Final simplification 18.9%

   \[\leadsto e \]

Developer target: 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 2023178 
(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))

  :herbie-target
  (+ (+ d (+ c (+ a b))) e)

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