Expression, p14

Percentage Accurate: 99.9% → 99.9%

Time: 6.8s

Alternatives: 4

Speedup: 1.0×

Specification

\[\left(\left(\left(56789 \leq a \land a \leq 98765\right) \land \left(0 \leq b \land b \leq 1\right)\right) \land \left(0 \leq c \land c \leq 0.0016773\right)\right) \land \left(0 \leq d \land d \leq 0.0016773\right)\]

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} [a, b, c, d] = \mathsf{sort}([a, b, c, d])\\ \\ a \cdot \left(b + c\right) + a \cdot d \end{array} \]

FPCore

NOTE: a, b, c, and d should be sorted in increasing order before calling this function.
(FPCore (a b c d) :precision binary64 (+ (* a (+ b c)) (* a d)))

C

assert(a < b && b < c && c < d);
double code(double a, double b, double c, double d) {
	return (a * (b + c)) + (a * d);
}

Fortran

NOTE: a, b, c, and d should be sorted in increasing order before calling this function.
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)) + (a * d)
end function

Java

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

Python

[a, b, c, d] = sort([a, b, c, d])
def code(a, b, c, d):
	return (a * (b + c)) + (a * d)

Julia

a, b, c, d = sort([a, b, c, d])
function code(a, b, c, d)
	return Float64(Float64(a * Float64(b + c)) + Float64(a * d))
end

MATLAB

a, b, c, d = num2cell(sort([a, b, c, d])){:}
function tmp = code(a, b, c, d)
	tmp = (a * (b + c)) + (a * d);
end

Wolfram

NOTE: a, b, c, and d should be sorted in increasing order before calling this function.
code[a_, b_, c_, d_] := N[(N[(a * N[(b + c), $MachinePrecision]), $MachinePrecision] + N[(a * d), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.9%

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

   1. distribute-rgt-in N/A

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

   2. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\left(\left(b + c\right) \cdot a\right), \color{blue}{\left(d \cdot a\right)}\right) \]

   3. *-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(\left(a \cdot \left(b + c\right)\right), \left(\color{blue}{d} \cdot a\right)\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, \left(b + c\right)\right), \left(\color{blue}{d} \cdot a\right)\right) \]

   5. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, \mathsf{+.f64}\left(b, c\right)\right), \left(d \cdot a\right)\right) \]

   6. *-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, \mathsf{+.f64}\left(b, c\right)\right), \left(a \cdot \color{blue}{d}\right)\right) \]

   7. *-lowering-*.f64 99.9%

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, \mathsf{+.f64}\left(b, c\right)\right), \mathsf{*.f64}\left(a, \color{blue}{d}\right)\right) \]

4. Applied egg-rr 99.9%

   \[\leadsto \color{blue}{a \cdot \left(b + c\right) + a \cdot d} \]
5. Add Preprocessing

Alternative 2: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} [a, b, c, d] = \mathsf{sort}([a, b, c, d])\\ \\ a \cdot \left(\left(b + c\right) + d\right) \end{array} \]

FPCore

NOTE: a, b, c, and d should be sorted in increasing order before calling this function.
(FPCore (a b c d) :precision binary64 (* a (+ (+ b c) d)))

C

assert(a < b && b < c && c < d);
double code(double a, double b, double c, double d) {
	return a * ((b + c) + d);
}

Fortran

NOTE: a, b, c, and d should be sorted in increasing order before calling this function.
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)
end function

Java

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

Python

[a, b, c, d] = sort([a, b, c, d])
def code(a, b, c, d):
	return a * ((b + c) + d)

Julia

a, b, c, d = sort([a, b, c, d])
function code(a, b, c, d)
	return Float64(a * Float64(Float64(b + c) + d))
end

MATLAB

a, b, c, d = num2cell(sort([a, b, c, d])){:}
function tmp = code(a, b, c, d)
	tmp = a * ((b + c) + d);
end

Wolfram

NOTE: a, b, c, and d should be sorted in increasing order before calling this function.
code[a_, b_, c_, d_] := N[(a * N[(N[(b + c), $MachinePrecision] + d), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.9%

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

Alternative 3: 99.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} [a, b, c, d] = \mathsf{sort}([a, b, c, d])\\ \\ a \cdot \left(c + d\right) \end{array} \]

FPCore

NOTE: a, b, c, and d should be sorted in increasing order before calling this function.
(FPCore (a b c d) :precision binary64 (* a (+ c d)))

C

assert(a < b && b < c && c < d);
double code(double a, double b, double c, double d) {
	return a * (c + d);
}

Fortran

NOTE: a, b, c, and d should be sorted in increasing order before calling this function.
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 * (c + d)
end function

Java

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

Python

[a, b, c, d] = sort([a, b, c, d])
def code(a, b, c, d):
	return a * (c + d)

Julia

a, b, c, d = sort([a, b, c, d])
function code(a, b, c, d)
	return Float64(a * Float64(c + d))
end

MATLAB

a, b, c, d = num2cell(sort([a, b, c, d])){:}
function tmp = code(a, b, c, d)
	tmp = a * (c + d);
end

Wolfram

NOTE: a, b, c, and d should be sorted in increasing order before calling this function.
code[a_, b_, c_, d_] := N[(a * N[(c + d), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in b around 0

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

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(a, \color{blue}{\left(c + d\right)}\right) \]

   2. +-lowering-+.f64 68.2%

      \[\leadsto \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(c, \color{blue}{d}\right)\right) \]

5. Simplified 68.2%

   \[\leadsto \color{blue}{a \cdot \left(c + d\right)} \]
6. Add Preprocessing

Alternative 4: 93.7% accurate, 2.3× speedup

Math

\[\begin{array}{l} [a, b, c, d] = \mathsf{sort}([a, b, c, d])\\ \\ a \cdot d \end{array} \]

FPCore

NOTE: a, b, c, and d should be sorted in increasing order before calling this function.
(FPCore (a b c d) :precision binary64 (* a d))

C

assert(a < b && b < c && c < d);
double code(double a, double b, double c, double d) {
	return a * d;
}

Fortran

NOTE: a, b, c, and d should be sorted in increasing order before calling this function.
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 * d
end function

Java

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

Python

[a, b, c, d] = sort([a, b, c, d])
def code(a, b, c, d):
	return a * d

Julia

a, b, c, d = sort([a, b, c, d])
function code(a, b, c, d)
	return Float64(a * d)
end

MATLAB

a, b, c, d = num2cell(sort([a, b, c, d])){:}
function tmp = code(a, b, c, d)
	tmp = a * d;
end

Wolfram

NOTE: a, b, c, and d should be sorted in increasing order before calling this function.
code[a_, b_, c_, d_] := N[(a * d), $MachinePrecision]

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in d around inf

   \[\leadsto \color{blue}{a \cdot d} \]
4. Step-by-step derivation

   1. *-lowering-*.f64 35.4%

      \[\leadsto \mathsf{*.f64}\left(a, \color{blue}{d}\right) \]

5. Simplified 35.4%

   \[\leadsto \color{blue}{a \cdot d} \]
6. Add Preprocessing

Developer Target 1: 99.9% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024139 
(FPCore (a b c d)
  :name "Expression, p14"
  :precision binary64
  :pre (and (and (and (and (<= 56789.0 a) (<= a 98765.0)) (and (<= 0.0 b) (<= b 1.0))) (and (<= 0.0 c) (<= c 0.0016773))) (and (<= 0.0 d) (<= d 0.0016773)))

  :alt
  (! :herbie-platform default (+ (* a b) (* a (+ c d))))

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