Expression, p14

Percentage Accurate: 99.9% → 100.0%

Time: 3.5s

Alternatives: 4

Speedup: 1.0×

• Unsound rule application detected in e-graph. Results may not be sound.

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, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. +-commutative 100.0%

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

   2. distribute-rgt-in 100.0%

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

   3. fma-def 100.0%

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

   4. *-commutative 100.0%

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

3. Applied egg-rr 100.0%

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

4. Final simplification 100.0%

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

Alternative 2: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

NOTE: 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 + (b + c))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

2. Final simplification 100.0%

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

Alternative 3: 99.7% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

NOTE: 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 + c)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

2. Taylor expanded in b around 0 62.6%

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

3. Final simplification 62.6%

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

Alternative 4: 93.7% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Fortran

NOTE: 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 = d * a
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

2. Taylor expanded in d around inf 30.5%

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

   1. *-commutative 30.5%

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

4. Simplified 30.5%

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

5. Final simplification 30.5%

   \[\leadsto d \cdot a \]

Developer target: 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 2023178 
(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)))

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

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