Expression, p6

Percentage Accurate: 94.3% → 100.0%

Time: 14.7s

Alternatives: 8

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 94.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.2%

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

   1. +-commutative 94.2%

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

   2. +-commutative 94.2%

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

   3. associate-+r+ 95.2%

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

   4. +-commutative 95.2%

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

   5. associate-+r+ 95.2%

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

   6. +-commutative 95.2%

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

   7. flip-+ 94.2%

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

   8. fma-neg 94.3%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(d + b, d + b, -\left(a + c\right) \cdot \left(a + c\right)\right)}}{\left(d + b\right) - \left(a + c\right)} \cdot 2 \]

   9. +-commutative 94.3%

      \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{b + d}, d + b, -\left(a + c\right) \cdot \left(a + c\right)\right)}{\left(d + b\right) - \left(a + c\right)} \cdot 2 \]

   10. +-commutative 94.3%

       \[\leadsto \frac{\mathsf{fma}\left(b + d, \color{blue}{b + d}, -\left(a + c\right) \cdot \left(a + c\right)\right)}{\left(d + b\right) - \left(a + c\right)} \cdot 2 \]

   11. pow2 94.3%

       \[\leadsto \frac{\mathsf{fma}\left(b + d, b + d, -\color{blue}{{\left(a + c\right)}^{2}}\right)}{\left(d + b\right) - \left(a + c\right)} \cdot 2 \]

   12. +-commutative 94.3%

       \[\leadsto \frac{\mathsf{fma}\left(b + d, b + d, -{\left(a + c\right)}^{2}\right)}{\color{blue}{\left(b + d\right)} - \left(a + c\right)} \cdot 2 \]

3. Applied egg-rr 94.3%

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

   1. +-commutative 94.3%

      \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{d + b}, b + d, -{\left(a + c\right)}^{2}\right)}{\left(b + d\right) - \left(a + c\right)} \cdot 2 \]

   2. +-commutative 94.3%

      \[\leadsto \frac{\mathsf{fma}\left(d + b, \color{blue}{d + b}, -{\left(a + c\right)}^{2}\right)}{\left(b + d\right) - \left(a + c\right)} \cdot 2 \]

   3. +-commutative 94.3%

      \[\leadsto \frac{\mathsf{fma}\left(d + b, d + b, -{\color{blue}{\left(c + a\right)}}^{2}\right)}{\left(b + d\right) - \left(a + c\right)} \cdot 2 \]

   4. associate--r+ 94.3%

      \[\leadsto \frac{\mathsf{fma}\left(d + b, d + b, -{\left(c + a\right)}^{2}\right)}{\color{blue}{\left(\left(b + d\right) - a\right) - c}} \cdot 2 \]

   5. +-commutative 94.3%

      \[\leadsto \frac{\mathsf{fma}\left(d + b, d + b, -{\left(c + a\right)}^{2}\right)}{\left(\color{blue}{\left(d + b\right)} - a\right) - c} \cdot 2 \]

   6. associate--l+ 94.3%

      \[\leadsto \frac{\mathsf{fma}\left(d + b, d + b, -{\left(c + a\right)}^{2}\right)}{\color{blue}{\left(d + \left(b - a\right)\right)} - c} \cdot 2 \]

5. Simplified 94.3%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(d + b, d + b, -{\left(c + a\right)}^{2}\right)}{\left(d + \left(b - a\right)\right) - c}} \cdot 2 \]

6. Taylor expanded in d around inf 98.6%

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

   1. *-un-lft-identity 98.6%

      \[\leadsto \left(\left(c + \color{blue}{1 \cdot \left(a + \left(d + 2 \cdot b\right)\right)}\right) - b\right) \cdot 2 \]

8. Applied egg-rr 98.6%

   \[\leadsto \left(\left(c + \color{blue}{1 \cdot \left(a + \left(d + 2 \cdot b\right)\right)}\right) - b\right) \cdot 2 \]
9. Step-by-step derivation

   1. *-lft-identity 98.6%

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

   2. associate-+r+ 100.0%

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

   3. +-commutative 100.0%

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

10. Simplified 100.0%

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

11. Final simplification 100.0%

    \[\leadsto 2 \cdot \left(\left(c + \left(\left(d + a\right) + 2 \cdot b\right)\right) - b\right) \]

Alternative 2: 13.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.605:\\ \;\;\;\;\left(d + a\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(a + \left(c + d\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= b -2.605) (* (+ d a) 2.0) (* 2.0 (+ a (+ c d)))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (b <= -2.605) {
		tmp = (d + a) * 2.0;
	} else {
		tmp = 2.0 * (a + (c + d));
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if (b <= (-2.605d0)) then
        tmp = (d + a) * 2.0d0
    else
        tmp = 2.0d0 * (a + (c + d))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if (b <= -2.605) {
		tmp = (d + a) * 2.0;
	} else {
		tmp = 2.0 * (a + (c + d));
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if b <= -2.605:
		tmp = (d + a) * 2.0
	else:
		tmp = 2.0 * (a + (c + d))
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (b <= -2.605)
		tmp = Float64(Float64(d + a) * 2.0);
	else
		tmp = Float64(2.0 * Float64(a + Float64(c + d)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if (b <= -2.605)
		tmp = (d + a) * 2.0;
	else
		tmp = 2.0 * (a + (c + d));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[b, -2.605], N[(N[(d + a), $MachinePrecision] * 2.0), $MachinePrecision], N[(2.0 * N[(a + N[(c + d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -2.60499999999999998

   1. Initial program 93.8%

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

   2. Taylor expanded in b around 0 7.0%

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

   3. Taylor expanded in c around 0 13.6%

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

      1. +-commutative 13.6%

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

   5. Simplified 13.6%

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

   if -2.60499999999999998 < b

   1. Initial program 94.5%

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

   2. Taylor expanded in b around 0 14.1%

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

4. Final simplification 13.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.605:\\ \;\;\;\;\left(d + a\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(a + \left(c + d\right)\right)\\ \end{array} \]

Alternative 3: 94.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.2%

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

2. Final simplification 94.2%

   \[\leadsto 2 \cdot \left(a + \left(b + \left(c + d\right)\right)\right) \]

Alternative 4: 95.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.2%

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

   1. *-commutative 94.2%

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

   2. associate-+r+ 93.9%

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

   3. distribute-lft-in 93.9%

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

   4. distribute-lft-in 93.9%

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

   5. associate-+r+ 94.7%

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

   6. distribute-lft-in 94.7%

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

   7. +-commutative 94.7%

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

   8. associate-+r+ 95.2%

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

   9. +-commutative 95.2%

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

   10. distribute-lft-in 95.2%

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

   11. +-commutative 95.2%

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

   12. *-commutative 95.2%

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

   13. +-commutative 95.2%

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

3. Simplified 95.2%

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

   1. *-un-lft-identity 95.2%

      \[\leadsto \left(d + \color{blue}{1 \cdot \left(b + \left(a + c\right)\right)}\right) \cdot 2 \]

   2. *-commutative 95.2%

      \[\leadsto \left(d + \color{blue}{\left(b + \left(a + c\right)\right) \cdot 1}\right) \cdot 2 \]

   3. +-commutative 95.2%

      \[\leadsto \left(d + \color{blue}{\left(\left(a + c\right) + b\right)} \cdot 1\right) \cdot 2 \]

   4. associate-+l+ 95.8%

      \[\leadsto \left(d + \color{blue}{\left(a + \left(c + b\right)\right)} \cdot 1\right) \cdot 2 \]

   5. +-commutative 95.8%

      \[\leadsto \left(d + \left(a + \color{blue}{\left(b + c\right)}\right) \cdot 1\right) \cdot 2 \]

5. Applied egg-rr 95.8%

   \[\leadsto \left(d + \color{blue}{\left(a + \left(b + c\right)\right) \cdot 1}\right) \cdot 2 \]
6. Step-by-step derivation

   1. *-rgt-identity 95.8%

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

   2. flip-+ 94.6%

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

7. Applied egg-rr 94.6%

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

8. Taylor expanded in d around 0 95.4%

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

   1. associate-+r+ 95.2%

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

   2. +-commutative 95.2%

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

10. Simplified 95.2%

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

11. Taylor expanded in a around 0 95.4%

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

    1. associate-+r+ 95.2%

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

    2. +-commutative 95.2%

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

    3. associate-+r+ 95.2%

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

    4. +-commutative 95.2%

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

    5. associate-+l+ 95.8%

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

13. Simplified 95.8%

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

14. Final simplification 95.8%

    \[\leadsto 2 \cdot \left(a + \left(d + \left(c + b\right)\right)\right) \]

Alternative 5: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.2%

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

   1. *-commutative 94.2%

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

   2. associate-+r+ 93.9%

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

   3. distribute-lft-in 93.9%

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

   4. distribute-lft-in 93.9%

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

   5. associate-+r+ 94.7%

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

   6. distribute-lft-in 94.7%

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

   7. +-commutative 94.7%

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

   8. associate-+r+ 95.2%

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

   9. +-commutative 95.2%

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

   10. distribute-lft-in 95.2%

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

   11. +-commutative 95.2%

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

   12. *-commutative 95.2%

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

   13. +-commutative 95.2%

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

3. Simplified 95.2%

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

   1. *-un-lft-identity 95.2%

      \[\leadsto \left(d + \color{blue}{1 \cdot \left(b + \left(a + c\right)\right)}\right) \cdot 2 \]

   2. *-commutative 95.2%

      \[\leadsto \left(d + \color{blue}{\left(b + \left(a + c\right)\right) \cdot 1}\right) \cdot 2 \]

   3. +-commutative 95.2%

      \[\leadsto \left(d + \color{blue}{\left(\left(a + c\right) + b\right)} \cdot 1\right) \cdot 2 \]

   4. associate-+l+ 95.8%

      \[\leadsto \left(d + \color{blue}{\left(a + \left(c + b\right)\right)} \cdot 1\right) \cdot 2 \]

   5. +-commutative 95.8%

      \[\leadsto \left(d + \left(a + \color{blue}{\left(b + c\right)}\right) \cdot 1\right) \cdot 2 \]

5. Applied egg-rr 95.8%

   \[\leadsto \left(d + \color{blue}{\left(a + \left(b + c\right)\right) \cdot 1}\right) \cdot 2 \]
6. Step-by-step derivation

   1. *-rgt-identity 95.8%

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

   2. flip-+ 94.6%

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

7. Applied egg-rr 94.6%

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

8. Taylor expanded in d around 0 95.4%

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

   1. associate-+r+ 95.2%

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

   2. +-commutative 95.2%

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

10. Simplified 95.2%

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

    1. *-un-lft-identity 95.2%

       \[\leadsto \color{blue}{\left(1 \cdot \left(\left(a + c\right) + \left(d + b\right)\right)\right)} \cdot 2 \]

    2. *-commutative 95.2%

       \[\leadsto \color{blue}{\left(\left(\left(a + c\right) + \left(d + b\right)\right) \cdot 1\right)} \cdot 2 \]

    3. associate-+r+ 95.3%

       \[\leadsto \left(\color{blue}{\left(\left(\left(a + c\right) + d\right) + b\right)} \cdot 1\right) \cdot 2 \]

    4. +-commutative 95.3%

       \[\leadsto \left(\left(\left(\color{blue}{\left(c + a\right)} + d\right) + b\right) \cdot 1\right) \cdot 2 \]

    5. associate-+l+ 100.0%

       \[\leadsto \left(\left(\color{blue}{\left(c + \left(a + d\right)\right)} + b\right) \cdot 1\right) \cdot 2 \]

    6. associate-+l+ 100.0%

       \[\leadsto \left(\color{blue}{\left(c + \left(\left(a + d\right) + b\right)\right)} \cdot 1\right) \cdot 2 \]

12. Applied egg-rr 100.0%

    \[\leadsto \color{blue}{\left(\left(c + \left(\left(a + d\right) + b\right)\right) \cdot 1\right)} \cdot 2 \]

13. Final simplification 100.0%

    \[\leadsto 2 \cdot \left(c + \left(\left(d + a\right) + b\right)\right) \]

Alternative 6: 13.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.605:\\ \;\;\;\;\left(d + a\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;c \cdot 2\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= b -2.605) (* (+ d a) 2.0) (* c 2.0)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (b <= -2.605) {
		tmp = (d + a) * 2.0;
	} else {
		tmp = c * 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if (b <= (-2.605d0)) then
        tmp = (d + a) * 2.0d0
    else
        tmp = c * 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if (b <= -2.605) {
		tmp = (d + a) * 2.0;
	} else {
		tmp = c * 2.0;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if b <= -2.605:
		tmp = (d + a) * 2.0
	else:
		tmp = c * 2.0
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (b <= -2.605)
		tmp = Float64(Float64(d + a) * 2.0);
	else
		tmp = Float64(c * 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if (b <= -2.605)
		tmp = (d + a) * 2.0;
	else
		tmp = c * 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[b, -2.605], N[(N[(d + a), $MachinePrecision] * 2.0), $MachinePrecision], N[(c * 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -2.60499999999999998

   1. Initial program 93.8%

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

   2. Taylor expanded in b around 0 7.0%

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

   3. Taylor expanded in c around 0 13.6%

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

      1. +-commutative 13.6%

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

   5. Simplified 13.6%

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

   if -2.60499999999999998 < b

   1. Initial program 94.5%

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

   2. Taylor expanded in c around inf 14.0%

      \[\leadsto \color{blue}{c} \cdot 2 \]
3. Recombined 2 regimes into one program.

4. Final simplification 13.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.605:\\ \;\;\;\;\left(d + a\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;c \cdot 2\\ \end{array} \]

Alternative 7: 12.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq 12.7:\\ \;\;\;\;2 \cdot b\\ \mathbf{else}:\\ \;\;\;\;c \cdot 2\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d) :precision binary64 (if (<= d 12.7) (* 2.0 b) (* c 2.0)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (d <= 12.7) {
		tmp = 2.0 * b;
	} else {
		tmp = c * 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if (d <= 12.7d0) then
        tmp = 2.0d0 * b
    else
        tmp = c * 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if (d <= 12.7) {
		tmp = 2.0 * b;
	} else {
		tmp = c * 2.0;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if d <= 12.7:
		tmp = 2.0 * b
	else:
		tmp = c * 2.0
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (d <= 12.7)
		tmp = Float64(2.0 * b);
	else
		tmp = Float64(c * 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if (d <= 12.7)
		tmp = 2.0 * b;
	else
		tmp = c * 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[d, 12.7], N[(2.0 * b), $MachinePrecision], N[(c * 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < 12.699999999999999

   1. Initial program 95.4%

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

   2. Taylor expanded in b around inf 12.6%

      \[\leadsto \color{blue}{b} \cdot 2 \]

   if 12.699999999999999 < d

   1. Initial program 93.8%

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

   2. Taylor expanded in c around inf 12.6%

      \[\leadsto \color{blue}{c} \cdot 2 \]
3. Recombined 2 regimes into one program.

4. Final simplification 12.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq 12.7:\\ \;\;\;\;2 \cdot b\\ \mathbf{else}:\\ \;\;\;\;c \cdot 2\\ \end{array} \]

Alternative 8: 6.3% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ 2 \cdot b \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.2%

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

2. Taylor expanded in b around inf 6.9%

   \[\leadsto \color{blue}{b} \cdot 2 \]

3. Final simplification 6.9%

   \[\leadsto 2 \cdot b \]

Developer target: 94.1% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2023274 
(FPCore (a b c d)
  :name "Expression, p6"
  :precision binary64
  :pre (and (and (and (and (<= -14.0 a) (<= a -13.0)) (and (<= -3.0 b) (<= b -2.0))) (and (<= 3.0 c) (<= c 3.5))) (and (<= 12.5 d) (<= d 13.5)))

  :herbie-target
  (+ (* (+ a b) 2.0) (* (+ c d) 2.0))

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