Expression, p14

Percentage Accurate: 99.9% → 99.9%

Time: 7.8s

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

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

FPCore

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

C

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

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

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

   2. associate-+l+ N/A

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

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

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

   4. +-lowering-+.f64 99.9%

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

3. Simplified 99.9%

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

   1. +-commutative N/A

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

   2. associate-+r+ N/A

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

   3. distribute-rgt-in N/A

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

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

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

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

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

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

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

   7. *-commutative N/A

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

   8. *-lowering-*.f64 100.0%

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

6. Applied egg-rr 100.0%

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

Alternative 2: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

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 + (d + c))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

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

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

   2. associate-+l+ N/A

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

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

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

   4. +-lowering-+.f64 99.9%

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

3. Simplified 99.9%

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

5. Final simplification 99.9%

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

Alternative 3: 65.7% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

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 * (d + c)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

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

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

   2. associate-+l+ N/A

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

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

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

   4. +-lowering-+.f64 99.9%

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

3. Simplified 99.9%

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

5. Taylor expanded in b around 0

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

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

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

   2. +-lowering-+.f64 66.4%

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

7. Simplified 66.4%

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

8. Final simplification 66.4%

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

Alternative 4: 34.1% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ d \cdot a \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

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

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

   2. associate-+l+ N/A

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

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

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

   4. +-lowering-+.f64 99.9%

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

3. Simplified 99.9%

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

5. Taylor expanded in d around inf

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

   1. *-lowering-*.f64 36.6%

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

7. Simplified 36.6%

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

8. Final simplification 36.6%

   \[\leadsto d \cdot a \]
9. 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 2024138 
(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)))