Expression, p14

Percentage Accurate: 99.9% → 100.0%

Time: 4.5s

Alternatives: 5

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: 100.0% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

a, b, c, d = sort([a, b, c, d])
function code(a, b, c, d)
	return fma(Float64(b + d), a, Float64(c * a))
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[(b + d), $MachinePrecision] * a + N[(c * a), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

   1. lift-*.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. +-commutative N/A

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

   4. lift-+.f64 N/A

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

   5. associate-+r+ N/A

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

   6. distribute-rgt-in N/A

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

   7. +-commutative N/A

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

   8. lower-fma.f64 N/A

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

   9. +-commutative N/A

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

   10. lower-+.f64 N/A

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

   11. lower-*.f64 100.0

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

4. Applied rewrites 100.0%

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

5. Final simplification 100.0%

   \[\leadsto \mathsf{fma}\left(b + d, a, c \cdot a\right) \]
6. Add Preprocessing

Alternative 2: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

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 = ((c + d) + b) * a
end function

Java

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

Python

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

Julia

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

MATLAB

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

Derivation

1. Initial program 100.0%

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

   1. lift-+.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. associate-+l+ N/A

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

   4. +-commutative N/A

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

   5. lower-+.f64 N/A

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

   6. +-commutative N/A

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

   7. lower-+.f64 100.0

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

4. Applied rewrites 100.0%

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

5. Final simplification 100.0%

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

Alternative 3: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

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 = ((c + b) + d) * a
end function

Java

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

Python

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

Julia

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

MATLAB

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

Derivation

1. Initial program 100.0%

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

3. Final simplification 100.0%

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

Alternative 4: 99.7% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

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 = (c + d) * a
end function

Java

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

Python

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

Julia

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

MATLAB

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in b around 0

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

   1. +-commutative N/A

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

   2. lower-+.f64 65.7

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

5. Applied rewrites 65.7%

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

6. Final simplification 65.7%

   \[\leadsto \left(c + d\right) \cdot a \]
7. Add Preprocessing

Alternative 5: 93.7% accurate, 2.0× 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 100.0%

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

3. Taylor expanded in d around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 34.0

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

5. Applied rewrites 34.0%

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

6. Final simplification 34.0%

   \[\leadsto a \cdot d \]
7. Add Preprocessing

Developer Target 1: 99.9% accurate, 0.7× 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 2024295 
(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)))