Expression, p6

Percentage Accurate: 94.3% → 100.0%

Time: 10.1s

Alternatives: 7

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, 1.0× speedup

Math

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

FPCore

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

C

double code(double a, double b, double c, double d) {
	return ((b + c) + (a + 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 = ((b + c) + (a + d)) * 2.0d0
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.3%

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

   1. add-sqr-sqrt 63.7%

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

   2. pow2 63.7%

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

   3. +-commutative 63.7%

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

   4. associate-+r+ 64.2%

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

   5. +-commutative 64.2%

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

   6. associate-+r+ 64.2%

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

   7. +-commutative 64.2%

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

3. Applied egg-rr 64.2%

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

   1. unpow2 64.2%

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

   2. add-sqr-sqrt 94.9%

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

   3. +-commutative 94.9%

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

   4. +-commutative 94.9%

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

5. Applied egg-rr 94.9%

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

6. Taylor expanded in d around 0 94.3%

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

   1. associate-+r+ 95.5%

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

   2. associate-+r+ 95.5%

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

   3. +-commutative 95.5%

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

   4. associate-+l+ 100.0%

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

8. Simplified 100.0%

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

9. Final simplification 100.0%

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

Alternative 2: 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 94.3%

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

2. Final simplification 94.3%

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

Alternative 3: 95.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.3%

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

   1. +-commutative 94.3%

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

   2. associate-+r+ 94.9%

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

   3. +-commutative 94.9%

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

3. Simplified 94.9%

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

4. Final simplification 94.9%

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

Alternative 4: 95.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

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 + (d + (c + a)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.3%

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

   1. +-commutative 94.3%

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

   2. associate-+r+ 94.9%

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

   3. +-commutative 94.9%

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

3. Simplified 94.9%

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

4. Taylor expanded in a around 0 94.3%

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

   1. +-commutative 94.3%

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

   2. associate-+l+ 94.6%

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

   3. +-commutative 94.6%

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

   4. associate-+r+ 95.5%

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

6. Simplified 95.5%

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

7. Final simplification 95.5%

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

Alternative 5: 14.0% accurate, 1.8× speedup

Math

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

FPCore

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

C

double code(double a, double b, double c, double d) {
	return (b + c) * 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 = (b + c) * 2.0d0
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.3%

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

   1. +-commutative 94.3%

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

   2. associate-+r+ 94.9%

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

   3. +-commutative 94.9%

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

3. Simplified 94.9%

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

4. Taylor expanded in a around 0 94.3%

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

   1. +-commutative 94.3%

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

   2. associate-+l+ 94.6%

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

   3. +-commutative 94.6%

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

   4. associate-+r+ 95.5%

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

6. Simplified 95.5%

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

7. Taylor expanded in c around inf 13.7%

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

8. Final simplification 13.7%

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

Alternative 6: 6.1% accurate, 3.0× speedup

Math

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

FPCore

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

C

double code(double a, double b, double c, double d) {
	return b * 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 = b * 2.0d0
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.3%

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

2. Taylor expanded in b around inf 6.2%

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

3. Final simplification 6.2%

   \[\leadsto b \cdot 2 \]

Alternative 7: 11.8% accurate, 3.0× speedup

Math

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

FPCore

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

C

double code(double a, double b, double c, double d) {
	return c * 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 = c * 2.0d0
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.3%

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

2. Taylor expanded in c around inf 11.5%

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

3. Final simplification 11.5%

   \[\leadsto c \cdot 2 \]

Developer target: 94.0% 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 2023331 
(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))