Expression 1, p15

Percentage Accurate: 99.4% → 99.6%

Time: 7.5s

Alternatives: 9

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

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. associate-+l+ 99.5%

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

   2. associate-+l+ 99.5%

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

   3. +-commutative 99.5%

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

   4. associate-+l+ 99.6%

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

   5. +-commutative 99.6%

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

3. Simplified 99.6%

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

   1. associate-+l+ 99.7%

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

   2. *-un-lft-identity 99.7%

      \[\leadsto e + \left(\color{blue}{1 \cdot d} + \left(c + \left(b + a\right)\right)\right) \]

   3. fma-def 99.7%

      \[\leadsto e + \color{blue}{\mathsf{fma}\left(1, d, c + \left(b + a\right)\right)} \]

5. Applied egg-rr 99.7%

   \[\leadsto e + \color{blue}{\mathsf{fma}\left(1, d, c + \left(b + a\right)\right)} \]

6. Taylor expanded in e around 0 99.5%

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

   1. associate-+r+ 99.4%

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

   2. +-commutative 99.4%

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

   3. associate-+r+ 99.4%

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

   4. associate-+l+ 99.4%

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

   5. +-commutative 99.4%

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

   6. associate-+l+ 99.5%

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

   7. +-commutative 99.5%

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

   8. associate-+l+ 99.6%

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

   9. +-commutative 99.6%

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

   10. +-commutative 99.6%

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

   11. associate-+l+ 99.7%

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

8. Simplified 99.7%

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

9. Final simplification 99.7%

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

Alternative 2: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. associate-+l+ 99.5%

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

   2. +-commutative 99.5%

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

   3. associate-+l+ 99.4%

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

   4. associate-+l+ 99.5%

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

3. Simplified 99.5%

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

4. Taylor expanded in d around 0 99.5%

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

   1. associate-+r+ 99.5%

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

   2. +-commutative 99.5%

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

6. Simplified 99.5%

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

7. Final simplification 99.5%

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

Alternative 3: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. associate-+l+ 99.5%

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

   2. +-commutative 99.5%

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

   3. associate-+l+ 99.4%

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

   4. associate-+l+ 99.5%

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

3. Simplified 99.5%

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

4. Final simplification 99.5%

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

Alternative 4: 99.6% accurate, 1.0× speedup

Math

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

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(d + Float64(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[(d + N[(c + N[(b + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.5%

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

   1. associate-+l+ 99.5%

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

   2. associate-+l+ 99.5%

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

   3. +-commutative 99.5%

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

   4. associate-+l+ 99.6%

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

   5. +-commutative 99.6%

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

3. Simplified 99.6%

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

   1. associate-+l+ 99.7%

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

   2. *-un-lft-identity 99.7%

      \[\leadsto e + \left(\color{blue}{1 \cdot d} + \left(c + \left(b + a\right)\right)\right) \]

   3. fma-def 99.7%

      \[\leadsto e + \color{blue}{\mathsf{fma}\left(1, d, c + \left(b + a\right)\right)} \]

5. Applied egg-rr 99.7%

   \[\leadsto e + \color{blue}{\mathsf{fma}\left(1, d, c + \left(b + a\right)\right)} \]
6. Step-by-step derivation

   1. fma-udef 99.7%

      \[\leadsto e + \color{blue}{\left(1 \cdot d + \left(c + \left(b + a\right)\right)\right)} \]

   2. *-un-lft-identity 99.7%

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

   3. +-commutative 99.7%

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

7. Applied egg-rr 99.7%

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

8. Final simplification 99.7%

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

Alternative 5: 25.7% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. associate-+l+ 99.5%

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

   2. +-commutative 99.5%

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

   3. associate-+l+ 99.4%

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

   4. associate-+l+ 99.5%

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

3. Simplified 99.5%

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

4. Taylor expanded in a around 0 25.7%

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

5. Final simplification 25.7%

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

Alternative 6: 23.2% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. associate-+l+ 99.5%

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

   2. +-commutative 99.5%

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

   3. associate-+l+ 99.4%

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

   4. associate-+l+ 99.5%

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

3. Simplified 99.5%

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

4. Taylor expanded in a around 0 25.7%

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

5. Taylor expanded in b around 0 23.2%

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

6. Final simplification 23.2%

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

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 99.5%

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

   1. associate-+l+ 99.5%

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

   2. +-commutative 99.5%

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

   3. associate-+l+ 99.4%

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

   4. associate-+l+ 99.5%

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

3. Simplified 99.5%

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

4. Taylor expanded in e around inf 21.2%

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

5. Final simplification 21.2%

   \[\leadsto e + d \]

Alternative 8: 17.3% accurate, 9.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. associate-+l+ 99.5%

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

   2. +-commutative 99.5%

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

   3. associate-+l+ 99.4%

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

   4. associate-+l+ 99.5%

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

3. Simplified 99.5%

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

4. Taylor expanded in d around inf 17.3%

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

5. Final simplification 17.3%

   \[\leadsto d \]

Alternative 9: 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 99.5%

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

   1. associate-+l+ 99.5%

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

   2. associate-+l+ 99.5%

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

   3. +-commutative 99.5%

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

   4. associate-+l+ 99.6%

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

   5. +-commutative 99.6%

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

3. Simplified 99.6%

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

   1. associate-+l+ 99.7%

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

   2. *-un-lft-identity 99.7%

      \[\leadsto e + \left(\color{blue}{1 \cdot d} + \left(c + \left(b + a\right)\right)\right) \]

   3. fma-def 99.7%

      \[\leadsto e + \color{blue}{\mathsf{fma}\left(1, d, c + \left(b + a\right)\right)} \]

5. Applied egg-rr 99.7%

   \[\leadsto e + \color{blue}{\mathsf{fma}\left(1, d, c + \left(b + a\right)\right)} \]

6. Taylor expanded in e around inf 18.9%

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

7. 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 2023334 
(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))