Expression, p6

Percentage Accurate: 94.2% → 100.0%

Time: 10.3s

Alternatives: 8

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: 94.2% 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, 0.1× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(1, b + c, d + a\right) \cdot 2 \end{array} \]

FPCore

(FPCore (a b c d) :precision binary64 (* (fma 1.0 (+ b c) (+ d a)) 2.0))

C

double code(double a, double b, double c, double d) {
	return fma(1.0, (b + c), (d + a)) * 2.0;
}

Julia

function code(a, b, c, d)
	return Float64(fma(1.0, Float64(b + c), Float64(d + a)) * 2.0)
end

Wolfram

code[a_, b_, c_, d_] := N[(N[(1.0 * N[(b + c), $MachinePrecision] + N[(d + a), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]

Derivation

1. Initial program 94.4%

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

   1. +-commutative 94.4%

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

   2. associate-+r+ 95.9%

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

   3. associate-+l+ 100.0%

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

   4. *-un-lft-identity 100.0%

      \[\leadsto \left(\color{blue}{1 \cdot \left(b + c\right)} + \left(d + a\right)\right) \cdot 2 \]

   5. fma-def 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(1, b + c, d + a\right)} \cdot 2 \]

3. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(1, b + c, d + a\right)} \cdot 2 \]

4. Final simplification 100.0%

   \[\leadsto \mathsf{fma}\left(1, b + c, d + a\right) \cdot 2 \]

Alternative 2: 94.2% 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 94.4%

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

2. Final simplification 94.4%

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

Alternative 3: 95.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.4%

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

2. Taylor expanded in a around 0 95.8%

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

3. Final simplification 95.8%

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

Alternative 4: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.4%

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

2. Taylor expanded in a around 0 95.8%

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

   1. associate-+r+ 100.0%

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

   2. flip-+ 98.2%

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

4. Applied egg-rr 98.2%

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

   1. flip-+ 100.0%

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

6. Applied egg-rr 100.0%

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

7. Final simplification 100.0%

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

Alternative 5: 14.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -13.66:\\ \;\;\;\;\left(d + a\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\left(b + c\right) \cdot 2\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= a -13.66) (* (+ d a) 2.0) (* (+ b c) 2.0)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (a <= -13.66) {
		tmp = (d + a) * 2.0;
	} else {
		tmp = (b + c) * 2.0;
	}
	return tmp;
}

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
    real(8) :: tmp
    if (a <= (-13.66d0)) then
        tmp = (d + a) * 2.0d0
    else
        tmp = (b + c) * 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if (a <= -13.66) {
		tmp = (d + a) * 2.0;
	} else {
		tmp = (b + c) * 2.0;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if a <= -13.66:
		tmp = (d + a) * 2.0
	else:
		tmp = (b + c) * 2.0
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (a <= -13.66)
		tmp = Float64(Float64(d + a) * 2.0);
	else
		tmp = Float64(Float64(b + c) * 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if (a <= -13.66)
		tmp = (d + a) * 2.0;
	else
		tmp = (b + c) * 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[a, -13.66], N[(N[(d + a), $MachinePrecision] * 2.0), $MachinePrecision], N[(N[(b + c), $MachinePrecision] * 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -13.66

   1. Initial program 93.9%

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

   2. Taylor expanded in b around 0 6.7%

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

   3. Taylor expanded in c around 0 12.2%

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

      1. +-commutative 12.2%

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

   5. Simplified 12.2%

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

   if -13.66 < a

   1. Initial program 94.6%

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

   2. Taylor expanded in a around 0 96.1%

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

   3. Taylor expanded in b around inf 17.6%

      \[\leadsto \left(c + \color{blue}{b}\right) \cdot 2 \]
3. Recombined 2 regimes into one program.

4. Final simplification 15.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -13.66:\\ \;\;\;\;\left(d + a\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\left(b + c\right) \cdot 2\\ \end{array} \]

Alternative 6: 13.7% 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 94.4%

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

2. Taylor expanded in a around 0 95.8%

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

3. Taylor expanded in b around inf 14.0%

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

4. Final simplification 14.0%

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

Alternative 7: 6.3% 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 94.4%

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

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 8: 11.5% 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 94.4%

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

2. Taylor expanded in c around inf 11.7%

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

3. Final simplification 11.7%

   \[\leadsto c \cdot 2 \]

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 2023278 
(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))