FastMath dist3

Percentage Accurate: 97.8% → 100.0%

Time: 6.5s

Alternatives: 7

Speedup: 1.9×

• 21.1% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \left(d1 \cdot d2 + \left(d3 + 5\right) \cdot d1\right) + d1 \cdot 32 \end{array} \]

FPCore

(FPCore (d1 d2 d3)
 :precision binary64
 (+ (+ (* d1 d2) (* (+ d3 5.0) d1)) (* d1 32.0)))

C

double code(double d1, double d2, double d3) {
	return ((d1 * d2) + ((d3 + 5.0) * d1)) + (d1 * 32.0);
}

Fortran

real(8) function code(d1, d2, d3)
    real(8), intent (in) :: d1
    real(8), intent (in) :: d2
    real(8), intent (in) :: d3
    code = ((d1 * d2) + ((d3 + 5.0d0) * d1)) + (d1 * 32.0d0)
end function

Java

public static double code(double d1, double d2, double d3) {
	return ((d1 * d2) + ((d3 + 5.0) * d1)) + (d1 * 32.0);
}

Python

def code(d1, d2, d3):
	return ((d1 * d2) + ((d3 + 5.0) * d1)) + (d1 * 32.0)

Julia

function code(d1, d2, d3)
	return Float64(Float64(Float64(d1 * d2) + Float64(Float64(d3 + 5.0) * d1)) + Float64(d1 * 32.0))
end

MATLAB

function tmp = code(d1, d2, d3)
	tmp = ((d1 * d2) + ((d3 + 5.0) * d1)) + (d1 * 32.0);
end

Wolfram

code[d1_, d2_, d3_] := N[(N[(N[(d1 * d2), $MachinePrecision] + N[(N[(d3 + 5.0), $MachinePrecision] * d1), $MachinePrecision]), $MachinePrecision] + N[(d1 * 32.0), $MachinePrecision]), $MachinePrecision]

Initial Program: 97.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(d1 \cdot d2 + \left(d3 + 5\right) \cdot d1\right) + d1 \cdot 32 \end{array} \]

FPCore

(FPCore (d1 d2 d3)
 :precision binary64
 (+ (+ (* d1 d2) (* (+ d3 5.0) d1)) (* d1 32.0)))

C

double code(double d1, double d2, double d3) {
	return ((d1 * d2) + ((d3 + 5.0) * d1)) + (d1 * 32.0);
}

Fortran

real(8) function code(d1, d2, d3)
    real(8), intent (in) :: d1
    real(8), intent (in) :: d2
    real(8), intent (in) :: d3
    code = ((d1 * d2) + ((d3 + 5.0d0) * d1)) + (d1 * 32.0d0)
end function

Java

public static double code(double d1, double d2, double d3) {
	return ((d1 * d2) + ((d3 + 5.0) * d1)) + (d1 * 32.0);
}

Python

def code(d1, d2, d3):
	return ((d1 * d2) + ((d3 + 5.0) * d1)) + (d1 * 32.0)

Julia

function code(d1, d2, d3)
	return Float64(Float64(Float64(d1 * d2) + Float64(Float64(d3 + 5.0) * d1)) + Float64(d1 * 32.0))
end

MATLAB

function tmp = code(d1, d2, d3)
	tmp = ((d1 * d2) + ((d3 + 5.0) * d1)) + (d1 * 32.0);
end

Wolfram

code[d1_, d2_, d3_] := N[(N[(N[(d1 * d2), $MachinePrecision] + N[(N[(d3 + 5.0), $MachinePrecision] * d1), $MachinePrecision]), $MachinePrecision] + N[(d1 * 32.0), $MachinePrecision]), $MachinePrecision]

Alternative 1: 100.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ d1 \cdot \left(\left(d2 + 37\right) + d3\right) \end{array} \]

FPCore

(FPCore (d1 d2 d3) :precision binary64 (* d1 (+ (+ d2 37.0) d3)))

C

double code(double d1, double d2, double d3) {
	return d1 * ((d2 + 37.0) + d3);
}

Fortran

real(8) function code(d1, d2, d3)
    real(8), intent (in) :: d1
    real(8), intent (in) :: d2
    real(8), intent (in) :: d3
    code = d1 * ((d2 + 37.0d0) + d3)
end function

Java

public static double code(double d1, double d2, double d3) {
	return d1 * ((d2 + 37.0) + d3);
}

Python

def code(d1, d2, d3):
	return d1 * ((d2 + 37.0) + d3)

Julia

function code(d1, d2, d3)
	return Float64(d1 * Float64(Float64(d2 + 37.0) + d3))
end

MATLAB

function tmp = code(d1, d2, d3)
	tmp = d1 * ((d2 + 37.0) + d3);
end

Wolfram

code[d1_, d2_, d3_] := N[(d1 * N[(N[(d2 + 37.0), $MachinePrecision] + d3), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 97.2%

   \[\left(d1 \cdot d2 + \left(d3 + 5\right) \cdot d1\right) + d1 \cdot 32 \]
2. Step-by-step derivation

   1. +-commutative 97.2%

      \[\leadsto \color{blue}{\left(\left(d3 + 5\right) \cdot d1 + d1 \cdot d2\right)} + d1 \cdot 32 \]

   2. *-commutative 97.2%

      \[\leadsto \left(\color{blue}{d1 \cdot \left(d3 + 5\right)} + d1 \cdot d2\right) + d1 \cdot 32 \]

   3. distribute-lft-out 100.0%

      \[\leadsto \color{blue}{d1 \cdot \left(\left(d3 + 5\right) + d2\right)} + d1 \cdot 32 \]

   4. distribute-lft-out 100.0%

      \[\leadsto \color{blue}{d1 \cdot \left(\left(\left(d3 + 5\right) + d2\right) + 32\right)} \]

   5. remove-double-neg 100.0%

      \[\leadsto \color{blue}{\left(-\left(-d1\right)\right)} \cdot \left(\left(\left(d3 + 5\right) + d2\right) + 32\right) \]

   6. distribute-lft-neg-in 100.0%

      \[\leadsto \color{blue}{-\left(-d1\right) \cdot \left(\left(\left(d3 + 5\right) + d2\right) + 32\right)} \]

   7. distribute-rgt-neg-in 100.0%

      \[\leadsto \color{blue}{\left(-d1\right) \cdot \left(-\left(\left(\left(d3 + 5\right) + d2\right) + 32\right)\right)} \]

   8. distribute-lft-neg-in 100.0%

      \[\leadsto \color{blue}{-d1 \cdot \left(-\left(\left(\left(d3 + 5\right) + d2\right) + 32\right)\right)} \]

   9. distribute-rgt-neg-in 100.0%

      \[\leadsto \color{blue}{d1 \cdot \left(-\left(-\left(\left(\left(d3 + 5\right) + d2\right) + 32\right)\right)\right)} \]

   10. remove-double-neg 100.0%

       \[\leadsto d1 \cdot \color{blue}{\left(\left(\left(d3 + 5\right) + d2\right) + 32\right)} \]

   11. associate-+r+ 100.0%

       \[\leadsto d1 \cdot \left(\color{blue}{\left(d3 + \left(5 + d2\right)\right)} + 32\right) \]

   12. +-commutative 100.0%

       \[\leadsto d1 \cdot \left(\left(d3 + \color{blue}{\left(d2 + 5\right)}\right) + 32\right) \]

   13. associate-+r+ 100.0%

       \[\leadsto d1 \cdot \color{blue}{\left(d3 + \left(\left(d2 + 5\right) + 32\right)\right)} \]

   14. +-commutative 100.0%

       \[\leadsto d1 \cdot \color{blue}{\left(\left(\left(d2 + 5\right) + 32\right) + d3\right)} \]

   15. associate-+l+ 100.0%

       \[\leadsto d1 \cdot \left(\color{blue}{\left(d2 + \left(5 + 32\right)\right)} + d3\right) \]

   16. metadata-eval 100.0%

       \[\leadsto d1 \cdot \left(\left(d2 + \color{blue}{37}\right) + d3\right) \]

3. Simplified 100.0%

   \[\leadsto \color{blue}{d1 \cdot \left(\left(d2 + 37\right) + d3\right)} \]
4. Add Preprocessing
5. Add Preprocessing

Alternative 2: 52.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d2 \leq -6.5 \cdot 10^{+29}:\\ \;\;\;\;d1 \cdot d2\\ \mathbf{elif}\;d2 \leq -4 \cdot 10^{-20} \lor \neg \left(d2 \leq -5.5 \cdot 10^{-195}\right):\\ \;\;\;\;d1 \cdot d3\\ \mathbf{else}:\\ \;\;\;\;d1 \cdot 37\\ \end{array} \end{array} \]

FPCore

(FPCore (d1 d2 d3)
 :precision binary64
 (if (<= d2 -6.5e+29)
   (* d1 d2)
   (if (or (<= d2 -4e-20) (not (<= d2 -5.5e-195))) (* d1 d3) (* d1 37.0))))

C

double code(double d1, double d2, double d3) {
	double tmp;
	if (d2 <= -6.5e+29) {
		tmp = d1 * d2;
	} else if ((d2 <= -4e-20) || !(d2 <= -5.5e-195)) {
		tmp = d1 * d3;
	} else {
		tmp = d1 * 37.0;
	}
	return tmp;
}

Fortran

real(8) function code(d1, d2, d3)
    real(8), intent (in) :: d1
    real(8), intent (in) :: d2
    real(8), intent (in) :: d3
    real(8) :: tmp
    if (d2 <= (-6.5d+29)) then
        tmp = d1 * d2
    else if ((d2 <= (-4d-20)) .or. (.not. (d2 <= (-5.5d-195)))) then
        tmp = d1 * d3
    else
        tmp = d1 * 37.0d0
    end if
    code = tmp
end function

Java

public static double code(double d1, double d2, double d3) {
	double tmp;
	if (d2 <= -6.5e+29) {
		tmp = d1 * d2;
	} else if ((d2 <= -4e-20) || !(d2 <= -5.5e-195)) {
		tmp = d1 * d3;
	} else {
		tmp = d1 * 37.0;
	}
	return tmp;
}

Python

def code(d1, d2, d3):
	tmp = 0
	if d2 <= -6.5e+29:
		tmp = d1 * d2
	elif (d2 <= -4e-20) or not (d2 <= -5.5e-195):
		tmp = d1 * d3
	else:
		tmp = d1 * 37.0
	return tmp

Julia

function code(d1, d2, d3)
	tmp = 0.0
	if (d2 <= -6.5e+29)
		tmp = Float64(d1 * d2);
	elseif ((d2 <= -4e-20) || !(d2 <= -5.5e-195))
		tmp = Float64(d1 * d3);
	else
		tmp = Float64(d1 * 37.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(d1, d2, d3)
	tmp = 0.0;
	if (d2 <= -6.5e+29)
		tmp = d1 * d2;
	elseif ((d2 <= -4e-20) || ~((d2 <= -5.5e-195)))
		tmp = d1 * d3;
	else
		tmp = d1 * 37.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[d1_, d2_, d3_] := If[LessEqual[d2, -6.5e+29], N[(d1 * d2), $MachinePrecision], If[Or[LessEqual[d2, -4e-20], N[Not[LessEqual[d2, -5.5e-195]], $MachinePrecision]], N[(d1 * d3), $MachinePrecision], N[(d1 * 37.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if d2 < -6.49999999999999971e29

   1. Initial program 95.4%

      \[\left(d1 \cdot d2 + \left(d3 + 5\right) \cdot d1\right) + d1 \cdot 32 \]
   2. Step-by-step derivation

      1. +-commutative 95.4%

         \[\leadsto \color{blue}{\left(\left(d3 + 5\right) \cdot d1 + d1 \cdot d2\right)} + d1 \cdot 32 \]

      2. *-commutative 95.4%

         \[\leadsto \left(\color{blue}{d1 \cdot \left(d3 + 5\right)} + d1 \cdot d2\right) + d1 \cdot 32 \]

      3. distribute-lft-out 100.0%

         \[\leadsto \color{blue}{d1 \cdot \left(\left(d3 + 5\right) + d2\right)} + d1 \cdot 32 \]

      4. distribute-lft-out 100.0%

         \[\leadsto \color{blue}{d1 \cdot \left(\left(\left(d3 + 5\right) + d2\right) + 32\right)} \]

      5. remove-double-neg 100.0%

         \[\leadsto \color{blue}{\left(-\left(-d1\right)\right)} \cdot \left(\left(\left(d3 + 5\right) + d2\right) + 32\right) \]

      6. distribute-lft-neg-in 100.0%

         \[\leadsto \color{blue}{-\left(-d1\right) \cdot \left(\left(\left(d3 + 5\right) + d2\right) + 32\right)} \]

      7. distribute-rgt-neg-in 100.0%

         \[\leadsto \color{blue}{\left(-d1\right) \cdot \left(-\left(\left(\left(d3 + 5\right) + d2\right) + 32\right)\right)} \]

      8. distribute-lft-neg-in 100.0%

         \[\leadsto \color{blue}{-d1 \cdot \left(-\left(\left(\left(d3 + 5\right) + d2\right) + 32\right)\right)} \]

      9. distribute-rgt-neg-in 100.0%

         \[\leadsto \color{blue}{d1 \cdot \left(-\left(-\left(\left(\left(d3 + 5\right) + d2\right) + 32\right)\right)\right)} \]

      10. remove-double-neg 100.0%

          \[\leadsto d1 \cdot \color{blue}{\left(\left(\left(d3 + 5\right) + d2\right) + 32\right)} \]

      11. associate-+r+ 100.0%

          \[\leadsto d1 \cdot \left(\color{blue}{\left(d3 + \left(5 + d2\right)\right)} + 32\right) \]

      12. +-commutative 100.0%

          \[\leadsto d1 \cdot \left(\left(d3 + \color{blue}{\left(d2 + 5\right)}\right) + 32\right) \]

      13. associate-+r+ 100.0%

          \[\leadsto d1 \cdot \color{blue}{\left(d3 + \left(\left(d2 + 5\right) + 32\right)\right)} \]

      14. +-commutative 100.0%

          \[\leadsto d1 \cdot \color{blue}{\left(\left(\left(d2 + 5\right) + 32\right) + d3\right)} \]

      15. associate-+l+ 100.0%

          \[\leadsto d1 \cdot \left(\color{blue}{\left(d2 + \left(5 + 32\right)\right)} + d3\right) \]

      16. metadata-eval 100.0%

          \[\leadsto d1 \cdot \left(\left(d2 + \color{blue}{37}\right) + d3\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{d1 \cdot \left(\left(d2 + 37\right) + d3\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d2 around inf 84.2%

      \[\leadsto \color{blue}{d1 \cdot d2} \]

   if -6.49999999999999971e29 < d2 < -3.99999999999999978e-20 or -5.5000000000000003e-195 < d2

   1. Initial program 97.1%

      \[\left(d1 \cdot d2 + \left(d3 + 5\right) \cdot d1\right) + d1 \cdot 32 \]
   2. Step-by-step derivation

      1. +-commutative 97.1%

         \[\leadsto \color{blue}{\left(\left(d3 + 5\right) \cdot d1 + d1 \cdot d2\right)} + d1 \cdot 32 \]

      2. *-commutative 97.1%

         \[\leadsto \left(\color{blue}{d1 \cdot \left(d3 + 5\right)} + d1 \cdot d2\right) + d1 \cdot 32 \]

      3. distribute-lft-out 100.0%

         \[\leadsto \color{blue}{d1 \cdot \left(\left(d3 + 5\right) + d2\right)} + d1 \cdot 32 \]

      4. distribute-lft-out 100.0%

         \[\leadsto \color{blue}{d1 \cdot \left(\left(\left(d3 + 5\right) + d2\right) + 32\right)} \]

      5. remove-double-neg 100.0%

         \[\leadsto \color{blue}{\left(-\left(-d1\right)\right)} \cdot \left(\left(\left(d3 + 5\right) + d2\right) + 32\right) \]

      6. distribute-lft-neg-in 100.0%

         \[\leadsto \color{blue}{-\left(-d1\right) \cdot \left(\left(\left(d3 + 5\right) + d2\right) + 32\right)} \]

      7. distribute-rgt-neg-in 100.0%

         \[\leadsto \color{blue}{\left(-d1\right) \cdot \left(-\left(\left(\left(d3 + 5\right) + d2\right) + 32\right)\right)} \]

      8. distribute-lft-neg-in 100.0%

         \[\leadsto \color{blue}{-d1 \cdot \left(-\left(\left(\left(d3 + 5\right) + d2\right) + 32\right)\right)} \]

      9. distribute-rgt-neg-in 100.0%

         \[\leadsto \color{blue}{d1 \cdot \left(-\left(-\left(\left(\left(d3 + 5\right) + d2\right) + 32\right)\right)\right)} \]

      10. remove-double-neg 100.0%

          \[\leadsto d1 \cdot \color{blue}{\left(\left(\left(d3 + 5\right) + d2\right) + 32\right)} \]

      11. associate-+r+ 100.0%

          \[\leadsto d1 \cdot \left(\color{blue}{\left(d3 + \left(5 + d2\right)\right)} + 32\right) \]

      12. +-commutative 100.0%

          \[\leadsto d1 \cdot \left(\left(d3 + \color{blue}{\left(d2 + 5\right)}\right) + 32\right) \]

      13. associate-+r+ 100.0%

          \[\leadsto d1 \cdot \color{blue}{\left(d3 + \left(\left(d2 + 5\right) + 32\right)\right)} \]

      14. +-commutative 100.0%

          \[\leadsto d1 \cdot \color{blue}{\left(\left(\left(d2 + 5\right) + 32\right) + d3\right)} \]

      15. associate-+l+ 100.0%

          \[\leadsto d1 \cdot \left(\color{blue}{\left(d2 + \left(5 + 32\right)\right)} + d3\right) \]

      16. metadata-eval 100.0%

          \[\leadsto d1 \cdot \left(\left(d2 + \color{blue}{37}\right) + d3\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{d1 \cdot \left(\left(d2 + 37\right) + d3\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d3 around inf 42.5%

      \[\leadsto \color{blue}{d1 \cdot d3} \]

   if -3.99999999999999978e-20 < d2 < -5.5000000000000003e-195

   1. Initial program 99.9%

      \[\left(d1 \cdot d2 + \left(d3 + 5\right) \cdot d1\right) + d1 \cdot 32 \]
   2. Step-by-step derivation

      1. +-commutative 99.9%

         \[\leadsto \color{blue}{\left(\left(d3 + 5\right) \cdot d1 + d1 \cdot d2\right)} + d1 \cdot 32 \]

      2. *-commutative 99.9%

         \[\leadsto \left(\color{blue}{d1 \cdot \left(d3 + 5\right)} + d1 \cdot d2\right) + d1 \cdot 32 \]

      3. distribute-lft-out 99.9%

         \[\leadsto \color{blue}{d1 \cdot \left(\left(d3 + 5\right) + d2\right)} + d1 \cdot 32 \]

      4. distribute-lft-out 100.0%

         \[\leadsto \color{blue}{d1 \cdot \left(\left(\left(d3 + 5\right) + d2\right) + 32\right)} \]

      5. remove-double-neg 100.0%

         \[\leadsto \color{blue}{\left(-\left(-d1\right)\right)} \cdot \left(\left(\left(d3 + 5\right) + d2\right) + 32\right) \]

      6. distribute-lft-neg-in 100.0%

         \[\leadsto \color{blue}{-\left(-d1\right) \cdot \left(\left(\left(d3 + 5\right) + d2\right) + 32\right)} \]

      7. distribute-rgt-neg-in 100.0%

         \[\leadsto \color{blue}{\left(-d1\right) \cdot \left(-\left(\left(\left(d3 + 5\right) + d2\right) + 32\right)\right)} \]

      8. distribute-lft-neg-in 100.0%

         \[\leadsto \color{blue}{-d1 \cdot \left(-\left(\left(\left(d3 + 5\right) + d2\right) + 32\right)\right)} \]

      9. distribute-rgt-neg-in 100.0%

         \[\leadsto \color{blue}{d1 \cdot \left(-\left(-\left(\left(\left(d3 + 5\right) + d2\right) + 32\right)\right)\right)} \]

      10. remove-double-neg 100.0%

          \[\leadsto d1 \cdot \color{blue}{\left(\left(\left(d3 + 5\right) + d2\right) + 32\right)} \]

      11. associate-+r+ 100.0%

          \[\leadsto d1 \cdot \left(\color{blue}{\left(d3 + \left(5 + d2\right)\right)} + 32\right) \]

      12. +-commutative 100.0%

          \[\leadsto d1 \cdot \left(\left(d3 + \color{blue}{\left(d2 + 5\right)}\right) + 32\right) \]

      13. associate-+r+ 100.0%

          \[\leadsto d1 \cdot \color{blue}{\left(d3 + \left(\left(d2 + 5\right) + 32\right)\right)} \]

      14. +-commutative 100.0%

          \[\leadsto d1 \cdot \color{blue}{\left(\left(\left(d2 + 5\right) + 32\right) + d3\right)} \]

      15. associate-+l+ 100.0%

          \[\leadsto d1 \cdot \left(\color{blue}{\left(d2 + \left(5 + 32\right)\right)} + d3\right) \]

      16. metadata-eval 100.0%

          \[\leadsto d1 \cdot \left(\left(d2 + \color{blue}{37}\right) + d3\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{d1 \cdot \left(\left(d2 + 37\right) + d3\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d3 around 0 58.8%

      \[\leadsto \color{blue}{d1 \cdot \left(37 + d2\right)} \]

   6. Taylor expanded in d2 around 0 58.8%

      \[\leadsto d1 \cdot \color{blue}{37} \]
3. Recombined 3 regimes into one program.

4. Final simplification 56.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d2 \leq -6.5 \cdot 10^{+29}:\\ \;\;\;\;d1 \cdot d2\\ \mathbf{elif}\;d2 \leq -4 \cdot 10^{-20} \lor \neg \left(d2 \leq -5.5 \cdot 10^{-195}\right):\\ \;\;\;\;d1 \cdot d3\\ \mathbf{else}:\\ \;\;\;\;d1 \cdot 37\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 81.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d3 \leq 0.00096:\\ \;\;\;\;d1 \cdot \left(d2 + 37\right)\\ \mathbf{else}:\\ \;\;\;\;d1 \cdot \left(d2 + d3\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (d1 d2 d3)
 :precision binary64
 (if (<= d3 0.00096) (* d1 (+ d2 37.0)) (* d1 (+ d2 d3))))

C

double code(double d1, double d2, double d3) {
	double tmp;
	if (d3 <= 0.00096) {
		tmp = d1 * (d2 + 37.0);
	} else {
		tmp = d1 * (d2 + d3);
	}
	return tmp;
}

Fortran

real(8) function code(d1, d2, d3)
    real(8), intent (in) :: d1
    real(8), intent (in) :: d2
    real(8), intent (in) :: d3
    real(8) :: tmp
    if (d3 <= 0.00096d0) then
        tmp = d1 * (d2 + 37.0d0)
    else
        tmp = d1 * (d2 + d3)
    end if
    code = tmp
end function

Java

public static double code(double d1, double d2, double d3) {
	double tmp;
	if (d3 <= 0.00096) {
		tmp = d1 * (d2 + 37.0);
	} else {
		tmp = d1 * (d2 + d3);
	}
	return tmp;
}

Python

def code(d1, d2, d3):
	tmp = 0
	if d3 <= 0.00096:
		tmp = d1 * (d2 + 37.0)
	else:
		tmp = d1 * (d2 + d3)
	return tmp

Julia

function code(d1, d2, d3)
	tmp = 0.0
	if (d3 <= 0.00096)
		tmp = Float64(d1 * Float64(d2 + 37.0));
	else
		tmp = Float64(d1 * Float64(d2 + d3));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d1, d2, d3)
	tmp = 0.0;
	if (d3 <= 0.00096)
		tmp = d1 * (d2 + 37.0);
	else
		tmp = d1 * (d2 + d3);
	end
	tmp_2 = tmp;
end

Wolfram

code[d1_, d2_, d3_] := If[LessEqual[d3, 0.00096], N[(d1 * N[(d2 + 37.0), $MachinePrecision]), $MachinePrecision], N[(d1 * N[(d2 + d3), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d3 < 9.60000000000000024e-4

   1. Initial program 98.0%

      \[\left(d1 \cdot d2 + \left(d3 + 5\right) \cdot d1\right) + d1 \cdot 32 \]
   2. Step-by-step derivation

      1. +-commutative 98.0%

         \[\leadsto \color{blue}{\left(\left(d3 + 5\right) \cdot d1 + d1 \cdot d2\right)} + d1 \cdot 32 \]

      2. *-commutative 98.0%

         \[\leadsto \left(\color{blue}{d1 \cdot \left(d3 + 5\right)} + d1 \cdot d2\right) + d1 \cdot 32 \]

      3. distribute-lft-out 100.0%

         \[\leadsto \color{blue}{d1 \cdot \left(\left(d3 + 5\right) + d2\right)} + d1 \cdot 32 \]

      4. distribute-lft-out 100.0%

         \[\leadsto \color{blue}{d1 \cdot \left(\left(\left(d3 + 5\right) + d2\right) + 32\right)} \]

      5. remove-double-neg 100.0%

         \[\leadsto \color{blue}{\left(-\left(-d1\right)\right)} \cdot \left(\left(\left(d3 + 5\right) + d2\right) + 32\right) \]

      6. distribute-lft-neg-in 100.0%

         \[\leadsto \color{blue}{-\left(-d1\right) \cdot \left(\left(\left(d3 + 5\right) + d2\right) + 32\right)} \]

      7. distribute-rgt-neg-in 100.0%

         \[\leadsto \color{blue}{\left(-d1\right) \cdot \left(-\left(\left(\left(d3 + 5\right) + d2\right) + 32\right)\right)} \]

      8. distribute-lft-neg-in 100.0%

         \[\leadsto \color{blue}{-d1 \cdot \left(-\left(\left(\left(d3 + 5\right) + d2\right) + 32\right)\right)} \]

      9. distribute-rgt-neg-in 100.0%

         \[\leadsto \color{blue}{d1 \cdot \left(-\left(-\left(\left(\left(d3 + 5\right) + d2\right) + 32\right)\right)\right)} \]

      10. remove-double-neg 100.0%

          \[\leadsto d1 \cdot \color{blue}{\left(\left(\left(d3 + 5\right) + d2\right) + 32\right)} \]

      11. associate-+r+ 100.0%

          \[\leadsto d1 \cdot \left(\color{blue}{\left(d3 + \left(5 + d2\right)\right)} + 32\right) \]

      12. +-commutative 100.0%

          \[\leadsto d1 \cdot \left(\left(d3 + \color{blue}{\left(d2 + 5\right)}\right) + 32\right) \]

      13. associate-+r+ 100.0%

          \[\leadsto d1 \cdot \color{blue}{\left(d3 + \left(\left(d2 + 5\right) + 32\right)\right)} \]

      14. +-commutative 100.0%

          \[\leadsto d1 \cdot \color{blue}{\left(\left(\left(d2 + 5\right) + 32\right) + d3\right)} \]

      15. associate-+l+ 100.0%

          \[\leadsto d1 \cdot \left(\color{blue}{\left(d2 + \left(5 + 32\right)\right)} + d3\right) \]

      16. metadata-eval 100.0%

          \[\leadsto d1 \cdot \left(\left(d2 + \color{blue}{37}\right) + d3\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{d1 \cdot \left(\left(d2 + 37\right) + d3\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d3 around 0 75.0%

      \[\leadsto \color{blue}{d1 \cdot \left(37 + d2\right)} \]

   if 9.60000000000000024e-4 < d3

   1. Initial program 94.7%

      \[\left(d1 \cdot d2 + \left(d3 + 5\right) \cdot d1\right) + d1 \cdot 32 \]
   2. Step-by-step derivation

      1. +-commutative 94.7%

         \[\leadsto \color{blue}{\left(\left(d3 + 5\right) \cdot d1 + d1 \cdot d2\right)} + d1 \cdot 32 \]

      2. *-commutative 94.7%

         \[\leadsto \left(\color{blue}{d1 \cdot \left(d3 + 5\right)} + d1 \cdot d2\right) + d1 \cdot 32 \]

      3. distribute-lft-out 100.0%

         \[\leadsto \color{blue}{d1 \cdot \left(\left(d3 + 5\right) + d2\right)} + d1 \cdot 32 \]

      4. distribute-lft-out 100.0%

         \[\leadsto \color{blue}{d1 \cdot \left(\left(\left(d3 + 5\right) + d2\right) + 32\right)} \]

      5. remove-double-neg 100.0%

         \[\leadsto \color{blue}{\left(-\left(-d1\right)\right)} \cdot \left(\left(\left(d3 + 5\right) + d2\right) + 32\right) \]

      6. distribute-lft-neg-in 100.0%

         \[\leadsto \color{blue}{-\left(-d1\right) \cdot \left(\left(\left(d3 + 5\right) + d2\right) + 32\right)} \]

      7. distribute-rgt-neg-in 100.0%

         \[\leadsto \color{blue}{\left(-d1\right) \cdot \left(-\left(\left(\left(d3 + 5\right) + d2\right) + 32\right)\right)} \]

      8. distribute-lft-neg-in 100.0%

         \[\leadsto \color{blue}{-d1 \cdot \left(-\left(\left(\left(d3 + 5\right) + d2\right) + 32\right)\right)} \]

      9. distribute-rgt-neg-in 100.0%

         \[\leadsto \color{blue}{d1 \cdot \left(-\left(-\left(\left(\left(d3 + 5\right) + d2\right) + 32\right)\right)\right)} \]

      10. remove-double-neg 100.0%

          \[\leadsto d1 \cdot \color{blue}{\left(\left(\left(d3 + 5\right) + d2\right) + 32\right)} \]

      11. associate-+r+ 100.0%

          \[\leadsto d1 \cdot \left(\color{blue}{\left(d3 + \left(5 + d2\right)\right)} + 32\right) \]

      12. +-commutative 100.0%

          \[\leadsto d1 \cdot \left(\left(d3 + \color{blue}{\left(d2 + 5\right)}\right) + 32\right) \]

      13. associate-+r+ 100.0%

          \[\leadsto d1 \cdot \color{blue}{\left(d3 + \left(\left(d2 + 5\right) + 32\right)\right)} \]

      14. +-commutative 100.0%

          \[\leadsto d1 \cdot \color{blue}{\left(\left(\left(d2 + 5\right) + 32\right) + d3\right)} \]

      15. associate-+l+ 100.0%

          \[\leadsto d1 \cdot \left(\color{blue}{\left(d2 + \left(5 + 32\right)\right)} + d3\right) \]

      16. metadata-eval 100.0%

          \[\leadsto d1 \cdot \left(\left(d2 + \color{blue}{37}\right) + d3\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{d1 \cdot \left(\left(d2 + 37\right) + d3\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d3 around inf 98.2%

      \[\leadsto \color{blue}{d3 \cdot \left(d1 + \frac{d1 \cdot \left(37 + d2\right)}{d3}\right)} \]

   6. Taylor expanded in d2 around inf 96.0%

      \[\leadsto d3 \cdot \left(d1 + \frac{d1 \cdot \color{blue}{d2}}{d3}\right) \]

   7. Taylor expanded in d3 around 0 92.4%

      \[\leadsto \color{blue}{d1 \cdot d2 + d1 \cdot d3} \]
   8. Step-by-step derivation

      1. distribute-lft-out 97.7%

         \[\leadsto \color{blue}{d1 \cdot \left(d2 + d3\right)} \]

   9. Simplified 97.7%

      \[\leadsto \color{blue}{d1 \cdot \left(d2 + d3\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 80.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d3 \leq 0.00096:\\ \;\;\;\;d1 \cdot \left(d2 + 37\right)\\ \mathbf{else}:\\ \;\;\;\;d1 \cdot \left(d2 + d3\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 76.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d2 \leq -1.6 \cdot 10^{-6}:\\ \;\;\;\;d1 \cdot \left(d2 + 37\right)\\ \mathbf{else}:\\ \;\;\;\;d1 \cdot \left(37 + d3\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (d1 d2 d3)
 :precision binary64
 (if (<= d2 -1.6e-6) (* d1 (+ d2 37.0)) (* d1 (+ 37.0 d3))))

C

double code(double d1, double d2, double d3) {
	double tmp;
	if (d2 <= -1.6e-6) {
		tmp = d1 * (d2 + 37.0);
	} else {
		tmp = d1 * (37.0 + d3);
	}
	return tmp;
}

Fortran

real(8) function code(d1, d2, d3)
    real(8), intent (in) :: d1
    real(8), intent (in) :: d2
    real(8), intent (in) :: d3
    real(8) :: tmp
    if (d2 <= (-1.6d-6)) then
        tmp = d1 * (d2 + 37.0d0)
    else
        tmp = d1 * (37.0d0 + d3)
    end if
    code = tmp
end function

Java

public static double code(double d1, double d2, double d3) {
	double tmp;
	if (d2 <= -1.6e-6) {
		tmp = d1 * (d2 + 37.0);
	} else {
		tmp = d1 * (37.0 + d3);
	}
	return tmp;
}

Python

def code(d1, d2, d3):
	tmp = 0
	if d2 <= -1.6e-6:
		tmp = d1 * (d2 + 37.0)
	else:
		tmp = d1 * (37.0 + d3)
	return tmp

Julia

function code(d1, d2, d3)
	tmp = 0.0
	if (d2 <= -1.6e-6)
		tmp = Float64(d1 * Float64(d2 + 37.0));
	else
		tmp = Float64(d1 * Float64(37.0 + d3));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d1, d2, d3)
	tmp = 0.0;
	if (d2 <= -1.6e-6)
		tmp = d1 * (d2 + 37.0);
	else
		tmp = d1 * (37.0 + d3);
	end
	tmp_2 = tmp;
end

Wolfram

code[d1_, d2_, d3_] := If[LessEqual[d2, -1.6e-6], N[(d1 * N[(d2 + 37.0), $MachinePrecision]), $MachinePrecision], N[(d1 * N[(37.0 + d3), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d2 < -1.5999999999999999e-6

   1. Initial program 95.7%

      \[\left(d1 \cdot d2 + \left(d3 + 5\right) \cdot d1\right) + d1 \cdot 32 \]
   2. Step-by-step derivation

      1. +-commutative 95.7%

         \[\leadsto \color{blue}{\left(\left(d3 + 5\right) \cdot d1 + d1 \cdot d2\right)} + d1 \cdot 32 \]

      2. *-commutative 95.7%

         \[\leadsto \left(\color{blue}{d1 \cdot \left(d3 + 5\right)} + d1 \cdot d2\right) + d1 \cdot 32 \]

      3. distribute-lft-out 100.0%

         \[\leadsto \color{blue}{d1 \cdot \left(\left(d3 + 5\right) + d2\right)} + d1 \cdot 32 \]

      4. distribute-lft-out 100.0%

         \[\leadsto \color{blue}{d1 \cdot \left(\left(\left(d3 + 5\right) + d2\right) + 32\right)} \]

      5. remove-double-neg 100.0%

         \[\leadsto \color{blue}{\left(-\left(-d1\right)\right)} \cdot \left(\left(\left(d3 + 5\right) + d2\right) + 32\right) \]

      6. distribute-lft-neg-in 100.0%

         \[\leadsto \color{blue}{-\left(-d1\right) \cdot \left(\left(\left(d3 + 5\right) + d2\right) + 32\right)} \]

      7. distribute-rgt-neg-in 100.0%

         \[\leadsto \color{blue}{\left(-d1\right) \cdot \left(-\left(\left(\left(d3 + 5\right) + d2\right) + 32\right)\right)} \]

      8. distribute-lft-neg-in 100.0%

         \[\leadsto \color{blue}{-d1 \cdot \left(-\left(\left(\left(d3 + 5\right) + d2\right) + 32\right)\right)} \]

      9. distribute-rgt-neg-in 100.0%

         \[\leadsto \color{blue}{d1 \cdot \left(-\left(-\left(\left(\left(d3 + 5\right) + d2\right) + 32\right)\right)\right)} \]

      10. remove-double-neg 100.0%

          \[\leadsto d1 \cdot \color{blue}{\left(\left(\left(d3 + 5\right) + d2\right) + 32\right)} \]

      11. associate-+r+ 100.0%

          \[\leadsto d1 \cdot \left(\color{blue}{\left(d3 + \left(5 + d2\right)\right)} + 32\right) \]

      12. +-commutative 100.0%

          \[\leadsto d1 \cdot \left(\left(d3 + \color{blue}{\left(d2 + 5\right)}\right) + 32\right) \]

      13. associate-+r+ 100.0%

          \[\leadsto d1 \cdot \color{blue}{\left(d3 + \left(\left(d2 + 5\right) + 32\right)\right)} \]

      14. +-commutative 100.0%

          \[\leadsto d1 \cdot \color{blue}{\left(\left(\left(d2 + 5\right) + 32\right) + d3\right)} \]

      15. associate-+l+ 100.0%

          \[\leadsto d1 \cdot \left(\color{blue}{\left(d2 + \left(5 + 32\right)\right)} + d3\right) \]

      16. metadata-eval 100.0%

          \[\leadsto d1 \cdot \left(\left(d2 + \color{blue}{37}\right) + d3\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{d1 \cdot \left(\left(d2 + 37\right) + d3\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d3 around 0 81.2%

      \[\leadsto \color{blue}{d1 \cdot \left(37 + d2\right)} \]

   if -1.5999999999999999e-6 < d2

   1. Initial program 97.8%

      \[\left(d1 \cdot d2 + \left(d3 + 5\right) \cdot d1\right) + d1 \cdot 32 \]
   2. Step-by-step derivation

      1. +-commutative 97.8%

         \[\leadsto \color{blue}{\left(\left(d3 + 5\right) \cdot d1 + d1 \cdot d2\right)} + d1 \cdot 32 \]

      2. *-commutative 97.8%

         \[\leadsto \left(\color{blue}{d1 \cdot \left(d3 + 5\right)} + d1 \cdot d2\right) + d1 \cdot 32 \]

      3. distribute-lft-out 99.9%

         \[\leadsto \color{blue}{d1 \cdot \left(\left(d3 + 5\right) + d2\right)} + d1 \cdot 32 \]

      4. distribute-lft-out 100.0%

         \[\leadsto \color{blue}{d1 \cdot \left(\left(\left(d3 + 5\right) + d2\right) + 32\right)} \]

      5. remove-double-neg 100.0%

         \[\leadsto \color{blue}{\left(-\left(-d1\right)\right)} \cdot \left(\left(\left(d3 + 5\right) + d2\right) + 32\right) \]

      6. distribute-lft-neg-in 100.0%

         \[\leadsto \color{blue}{-\left(-d1\right) \cdot \left(\left(\left(d3 + 5\right) + d2\right) + 32\right)} \]

      7. distribute-rgt-neg-in 100.0%

         \[\leadsto \color{blue}{\left(-d1\right) \cdot \left(-\left(\left(\left(d3 + 5\right) + d2\right) + 32\right)\right)} \]

      8. distribute-lft-neg-in 100.0%

         \[\leadsto \color{blue}{-d1 \cdot \left(-\left(\left(\left(d3 + 5\right) + d2\right) + 32\right)\right)} \]

      9. distribute-rgt-neg-in 100.0%

         \[\leadsto \color{blue}{d1 \cdot \left(-\left(-\left(\left(\left(d3 + 5\right) + d2\right) + 32\right)\right)\right)} \]

      10. remove-double-neg 100.0%

          \[\leadsto d1 \cdot \color{blue}{\left(\left(\left(d3 + 5\right) + d2\right) + 32\right)} \]

      11. associate-+r+ 100.0%

          \[\leadsto d1 \cdot \left(\color{blue}{\left(d3 + \left(5 + d2\right)\right)} + 32\right) \]

      12. +-commutative 100.0%

          \[\leadsto d1 \cdot \left(\left(d3 + \color{blue}{\left(d2 + 5\right)}\right) + 32\right) \]

      13. associate-+r+ 100.0%

          \[\leadsto d1 \cdot \color{blue}{\left(d3 + \left(\left(d2 + 5\right) + 32\right)\right)} \]

      14. +-commutative 100.0%

          \[\leadsto d1 \cdot \color{blue}{\left(\left(\left(d2 + 5\right) + 32\right) + d3\right)} \]

      15. associate-+l+ 100.0%

          \[\leadsto d1 \cdot \left(\color{blue}{\left(d2 + \left(5 + 32\right)\right)} + d3\right) \]

      16. metadata-eval 100.0%

          \[\leadsto d1 \cdot \left(\left(d2 + \color{blue}{37}\right) + d3\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{d1 \cdot \left(\left(d2 + 37\right) + d3\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d2 around 0 76.1%

      \[\leadsto \color{blue}{d1 \cdot \left(37 + d3\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 77.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d2 \leq -1.6 \cdot 10^{-6}:\\ \;\;\;\;d1 \cdot \left(d2 + 37\right)\\ \mathbf{else}:\\ \;\;\;\;d1 \cdot \left(37 + d3\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 76.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d3 \leq 6600000000000:\\ \;\;\;\;d1 \cdot \left(d2 + 37\right)\\ \mathbf{else}:\\ \;\;\;\;d1 \cdot d3\\ \end{array} \end{array} \]

FPCore

(FPCore (d1 d2 d3)
 :precision binary64
 (if (<= d3 6600000000000.0) (* d1 (+ d2 37.0)) (* d1 d3)))

C

double code(double d1, double d2, double d3) {
	double tmp;
	if (d3 <= 6600000000000.0) {
		tmp = d1 * (d2 + 37.0);
	} else {
		tmp = d1 * d3;
	}
	return tmp;
}

Fortran

real(8) function code(d1, d2, d3)
    real(8), intent (in) :: d1
    real(8), intent (in) :: d2
    real(8), intent (in) :: d3
    real(8) :: tmp
    if (d3 <= 6600000000000.0d0) then
        tmp = d1 * (d2 + 37.0d0)
    else
        tmp = d1 * d3
    end if
    code = tmp
end function

Java

public static double code(double d1, double d2, double d3) {
	double tmp;
	if (d3 <= 6600000000000.0) {
		tmp = d1 * (d2 + 37.0);
	} else {
		tmp = d1 * d3;
	}
	return tmp;
}

Python

def code(d1, d2, d3):
	tmp = 0
	if d3 <= 6600000000000.0:
		tmp = d1 * (d2 + 37.0)
	else:
		tmp = d1 * d3
	return tmp

Julia

function code(d1, d2, d3)
	tmp = 0.0
	if (d3 <= 6600000000000.0)
		tmp = Float64(d1 * Float64(d2 + 37.0));
	else
		tmp = Float64(d1 * d3);
	end
	return tmp
end

MATLAB

function tmp_2 = code(d1, d2, d3)
	tmp = 0.0;
	if (d3 <= 6600000000000.0)
		tmp = d1 * (d2 + 37.0);
	else
		tmp = d1 * d3;
	end
	tmp_2 = tmp;
end

Wolfram

code[d1_, d2_, d3_] := If[LessEqual[d3, 6600000000000.0], N[(d1 * N[(d2 + 37.0), $MachinePrecision]), $MachinePrecision], N[(d1 * d3), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d3 < 6.6e12

   1. Initial program 98.0%

      \[\left(d1 \cdot d2 + \left(d3 + 5\right) \cdot d1\right) + d1 \cdot 32 \]
   2. Step-by-step derivation

      1. +-commutative 98.0%

         \[\leadsto \color{blue}{\left(\left(d3 + 5\right) \cdot d1 + d1 \cdot d2\right)} + d1 \cdot 32 \]

      2. *-commutative 98.0%

         \[\leadsto \left(\color{blue}{d1 \cdot \left(d3 + 5\right)} + d1 \cdot d2\right) + d1 \cdot 32 \]

      3. distribute-lft-out 100.0%

         \[\leadsto \color{blue}{d1 \cdot \left(\left(d3 + 5\right) + d2\right)} + d1 \cdot 32 \]

      4. distribute-lft-out 100.0%

         \[\leadsto \color{blue}{d1 \cdot \left(\left(\left(d3 + 5\right) + d2\right) + 32\right)} \]

      5. remove-double-neg 100.0%

         \[\leadsto \color{blue}{\left(-\left(-d1\right)\right)} \cdot \left(\left(\left(d3 + 5\right) + d2\right) + 32\right) \]

      6. distribute-lft-neg-in 100.0%

         \[\leadsto \color{blue}{-\left(-d1\right) \cdot \left(\left(\left(d3 + 5\right) + d2\right) + 32\right)} \]

      7. distribute-rgt-neg-in 100.0%

         \[\leadsto \color{blue}{\left(-d1\right) \cdot \left(-\left(\left(\left(d3 + 5\right) + d2\right) + 32\right)\right)} \]

      8. distribute-lft-neg-in 100.0%

         \[\leadsto \color{blue}{-d1 \cdot \left(-\left(\left(\left(d3 + 5\right) + d2\right) + 32\right)\right)} \]

      9. distribute-rgt-neg-in 100.0%

         \[\leadsto \color{blue}{d1 \cdot \left(-\left(-\left(\left(\left(d3 + 5\right) + d2\right) + 32\right)\right)\right)} \]

      10. remove-double-neg 100.0%

          \[\leadsto d1 \cdot \color{blue}{\left(\left(\left(d3 + 5\right) + d2\right) + 32\right)} \]

      11. associate-+r+ 100.0%

          \[\leadsto d1 \cdot \left(\color{blue}{\left(d3 + \left(5 + d2\right)\right)} + 32\right) \]

      12. +-commutative 100.0%

          \[\leadsto d1 \cdot \left(\left(d3 + \color{blue}{\left(d2 + 5\right)}\right) + 32\right) \]

      13. associate-+r+ 100.0%

          \[\leadsto d1 \cdot \color{blue}{\left(d3 + \left(\left(d2 + 5\right) + 32\right)\right)} \]

      14. +-commutative 100.0%

          \[\leadsto d1 \cdot \color{blue}{\left(\left(\left(d2 + 5\right) + 32\right) + d3\right)} \]

      15. associate-+l+ 100.0%

          \[\leadsto d1 \cdot \left(\color{blue}{\left(d2 + \left(5 + 32\right)\right)} + d3\right) \]

      16. metadata-eval 100.0%

          \[\leadsto d1 \cdot \left(\left(d2 + \color{blue}{37}\right) + d3\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{d1 \cdot \left(\left(d2 + 37\right) + d3\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d3 around 0 74.9%

      \[\leadsto \color{blue}{d1 \cdot \left(37 + d2\right)} \]

   if 6.6e12 < d3

   1. Initial program 94.1%

      \[\left(d1 \cdot d2 + \left(d3 + 5\right) \cdot d1\right) + d1 \cdot 32 \]
   2. Step-by-step derivation

      1. +-commutative 94.1%

         \[\leadsto \color{blue}{\left(\left(d3 + 5\right) \cdot d1 + d1 \cdot d2\right)} + d1 \cdot 32 \]

      2. *-commutative 94.1%

         \[\leadsto \left(\color{blue}{d1 \cdot \left(d3 + 5\right)} + d1 \cdot d2\right) + d1 \cdot 32 \]

      3. distribute-lft-out 100.0%

         \[\leadsto \color{blue}{d1 \cdot \left(\left(d3 + 5\right) + d2\right)} + d1 \cdot 32 \]

      4. distribute-lft-out 100.0%

         \[\leadsto \color{blue}{d1 \cdot \left(\left(\left(d3 + 5\right) + d2\right) + 32\right)} \]

      5. remove-double-neg 100.0%

         \[\leadsto \color{blue}{\left(-\left(-d1\right)\right)} \cdot \left(\left(\left(d3 + 5\right) + d2\right) + 32\right) \]

      6. distribute-lft-neg-in 100.0%

         \[\leadsto \color{blue}{-\left(-d1\right) \cdot \left(\left(\left(d3 + 5\right) + d2\right) + 32\right)} \]

      7. distribute-rgt-neg-in 100.0%

         \[\leadsto \color{blue}{\left(-d1\right) \cdot \left(-\left(\left(\left(d3 + 5\right) + d2\right) + 32\right)\right)} \]

      8. distribute-lft-neg-in 100.0%

         \[\leadsto \color{blue}{-d1 \cdot \left(-\left(\left(\left(d3 + 5\right) + d2\right) + 32\right)\right)} \]

      9. distribute-rgt-neg-in 100.0%

         \[\leadsto \color{blue}{d1 \cdot \left(-\left(-\left(\left(\left(d3 + 5\right) + d2\right) + 32\right)\right)\right)} \]

      10. remove-double-neg 100.0%

          \[\leadsto d1 \cdot \color{blue}{\left(\left(\left(d3 + 5\right) + d2\right) + 32\right)} \]

      11. associate-+r+ 100.0%

          \[\leadsto d1 \cdot \left(\color{blue}{\left(d3 + \left(5 + d2\right)\right)} + 32\right) \]

      12. +-commutative 100.0%

          \[\leadsto d1 \cdot \left(\left(d3 + \color{blue}{\left(d2 + 5\right)}\right) + 32\right) \]

      13. associate-+r+ 100.0%

          \[\leadsto d1 \cdot \color{blue}{\left(d3 + \left(\left(d2 + 5\right) + 32\right)\right)} \]

      14. +-commutative 100.0%

          \[\leadsto d1 \cdot \color{blue}{\left(\left(\left(d2 + 5\right) + 32\right) + d3\right)} \]

      15. associate-+l+ 100.0%

          \[\leadsto d1 \cdot \left(\color{blue}{\left(d2 + \left(5 + 32\right)\right)} + d3\right) \]

      16. metadata-eval 100.0%

          \[\leadsto d1 \cdot \left(\left(d2 + \color{blue}{37}\right) + d3\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{d1 \cdot \left(\left(d2 + 37\right) + d3\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d3 around inf 79.0%

      \[\leadsto \color{blue}{d1 \cdot d3} \]
3. Recombined 2 regimes into one program.

4. Final simplification 75.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d3 \leq 6600000000000:\\ \;\;\;\;d1 \cdot \left(d2 + 37\right)\\ \mathbf{else}:\\ \;\;\;\;d1 \cdot d3\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 45.3% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d2 \leq -36:\\ \;\;\;\;d1 \cdot d2\\ \mathbf{else}:\\ \;\;\;\;d1 \cdot 37\\ \end{array} \end{array} \]

FPCore

(FPCore (d1 d2 d3)
 :precision binary64
 (if (<= d2 -36.0) (* d1 d2) (* d1 37.0)))

C

double code(double d1, double d2, double d3) {
	double tmp;
	if (d2 <= -36.0) {
		tmp = d1 * d2;
	} else {
		tmp = d1 * 37.0;
	}
	return tmp;
}

Fortran

real(8) function code(d1, d2, d3)
    real(8), intent (in) :: d1
    real(8), intent (in) :: d2
    real(8), intent (in) :: d3
    real(8) :: tmp
    if (d2 <= (-36.0d0)) then
        tmp = d1 * d2
    else
        tmp = d1 * 37.0d0
    end if
    code = tmp
end function

Java

public static double code(double d1, double d2, double d3) {
	double tmp;
	if (d2 <= -36.0) {
		tmp = d1 * d2;
	} else {
		tmp = d1 * 37.0;
	}
	return tmp;
}

Python

def code(d1, d2, d3):
	tmp = 0
	if d2 <= -36.0:
		tmp = d1 * d2
	else:
		tmp = d1 * 37.0
	return tmp

Julia

function code(d1, d2, d3)
	tmp = 0.0
	if (d2 <= -36.0)
		tmp = Float64(d1 * d2);
	else
		tmp = Float64(d1 * 37.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(d1, d2, d3)
	tmp = 0.0;
	if (d2 <= -36.0)
		tmp = d1 * d2;
	else
		tmp = d1 * 37.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[d1_, d2_, d3_] := If[LessEqual[d2, -36.0], N[(d1 * d2), $MachinePrecision], N[(d1 * 37.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d2 < -36

   1. Initial program 95.7%

      \[\left(d1 \cdot d2 + \left(d3 + 5\right) \cdot d1\right) + d1 \cdot 32 \]
   2. Step-by-step derivation

      1. +-commutative 95.7%

         \[\leadsto \color{blue}{\left(\left(d3 + 5\right) \cdot d1 + d1 \cdot d2\right)} + d1 \cdot 32 \]

      2. *-commutative 95.7%

         \[\leadsto \left(\color{blue}{d1 \cdot \left(d3 + 5\right)} + d1 \cdot d2\right) + d1 \cdot 32 \]

      3. distribute-lft-out 100.0%

         \[\leadsto \color{blue}{d1 \cdot \left(\left(d3 + 5\right) + d2\right)} + d1 \cdot 32 \]

      4. distribute-lft-out 100.0%

         \[\leadsto \color{blue}{d1 \cdot \left(\left(\left(d3 + 5\right) + d2\right) + 32\right)} \]

      5. remove-double-neg 100.0%

         \[\leadsto \color{blue}{\left(-\left(-d1\right)\right)} \cdot \left(\left(\left(d3 + 5\right) + d2\right) + 32\right) \]

      6. distribute-lft-neg-in 100.0%

         \[\leadsto \color{blue}{-\left(-d1\right) \cdot \left(\left(\left(d3 + 5\right) + d2\right) + 32\right)} \]

      7. distribute-rgt-neg-in 100.0%

         \[\leadsto \color{blue}{\left(-d1\right) \cdot \left(-\left(\left(\left(d3 + 5\right) + d2\right) + 32\right)\right)} \]

      8. distribute-lft-neg-in 100.0%

         \[\leadsto \color{blue}{-d1 \cdot \left(-\left(\left(\left(d3 + 5\right) + d2\right) + 32\right)\right)} \]

      9. distribute-rgt-neg-in 100.0%

         \[\leadsto \color{blue}{d1 \cdot \left(-\left(-\left(\left(\left(d3 + 5\right) + d2\right) + 32\right)\right)\right)} \]

      10. remove-double-neg 100.0%

          \[\leadsto d1 \cdot \color{blue}{\left(\left(\left(d3 + 5\right) + d2\right) + 32\right)} \]

      11. associate-+r+ 100.0%

          \[\leadsto d1 \cdot \left(\color{blue}{\left(d3 + \left(5 + d2\right)\right)} + 32\right) \]

      12. +-commutative 100.0%

          \[\leadsto d1 \cdot \left(\left(d3 + \color{blue}{\left(d2 + 5\right)}\right) + 32\right) \]

      13. associate-+r+ 100.0%

          \[\leadsto d1 \cdot \color{blue}{\left(d3 + \left(\left(d2 + 5\right) + 32\right)\right)} \]

      14. +-commutative 100.0%

          \[\leadsto d1 \cdot \color{blue}{\left(\left(\left(d2 + 5\right) + 32\right) + d3\right)} \]

      15. associate-+l+ 100.0%

          \[\leadsto d1 \cdot \left(\color{blue}{\left(d2 + \left(5 + 32\right)\right)} + d3\right) \]

      16. metadata-eval 100.0%

          \[\leadsto d1 \cdot \left(\left(d2 + \color{blue}{37}\right) + d3\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{d1 \cdot \left(\left(d2 + 37\right) + d3\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d2 around inf 80.1%

      \[\leadsto \color{blue}{d1 \cdot d2} \]

   if -36 < d2

   1. Initial program 97.8%

      \[\left(d1 \cdot d2 + \left(d3 + 5\right) \cdot d1\right) + d1 \cdot 32 \]
   2. Step-by-step derivation

      1. +-commutative 97.8%

         \[\leadsto \color{blue}{\left(\left(d3 + 5\right) \cdot d1 + d1 \cdot d2\right)} + d1 \cdot 32 \]

      2. *-commutative 97.8%

         \[\leadsto \left(\color{blue}{d1 \cdot \left(d3 + 5\right)} + d1 \cdot d2\right) + d1 \cdot 32 \]

      3. distribute-lft-out 99.9%

         \[\leadsto \color{blue}{d1 \cdot \left(\left(d3 + 5\right) + d2\right)} + d1 \cdot 32 \]

      4. distribute-lft-out 100.0%

         \[\leadsto \color{blue}{d1 \cdot \left(\left(\left(d3 + 5\right) + d2\right) + 32\right)} \]

      5. remove-double-neg 100.0%

         \[\leadsto \color{blue}{\left(-\left(-d1\right)\right)} \cdot \left(\left(\left(d3 + 5\right) + d2\right) + 32\right) \]

      6. distribute-lft-neg-in 100.0%

         \[\leadsto \color{blue}{-\left(-d1\right) \cdot \left(\left(\left(d3 + 5\right) + d2\right) + 32\right)} \]

      7. distribute-rgt-neg-in 100.0%

         \[\leadsto \color{blue}{\left(-d1\right) \cdot \left(-\left(\left(\left(d3 + 5\right) + d2\right) + 32\right)\right)} \]

      8. distribute-lft-neg-in 100.0%

         \[\leadsto \color{blue}{-d1 \cdot \left(-\left(\left(\left(d3 + 5\right) + d2\right) + 32\right)\right)} \]

      9. distribute-rgt-neg-in 100.0%

         \[\leadsto \color{blue}{d1 \cdot \left(-\left(-\left(\left(\left(d3 + 5\right) + d2\right) + 32\right)\right)\right)} \]

      10. remove-double-neg 100.0%

          \[\leadsto d1 \cdot \color{blue}{\left(\left(\left(d3 + 5\right) + d2\right) + 32\right)} \]

      11. associate-+r+ 100.0%

          \[\leadsto d1 \cdot \left(\color{blue}{\left(d3 + \left(5 + d2\right)\right)} + 32\right) \]

      12. +-commutative 100.0%

          \[\leadsto d1 \cdot \left(\left(d3 + \color{blue}{\left(d2 + 5\right)}\right) + 32\right) \]

      13. associate-+r+ 100.0%

          \[\leadsto d1 \cdot \color{blue}{\left(d3 + \left(\left(d2 + 5\right) + 32\right)\right)} \]

      14. +-commutative 100.0%

          \[\leadsto d1 \cdot \color{blue}{\left(\left(\left(d2 + 5\right) + 32\right) + d3\right)} \]

      15. associate-+l+ 100.0%

          \[\leadsto d1 \cdot \left(\color{blue}{\left(d2 + \left(5 + 32\right)\right)} + d3\right) \]

      16. metadata-eval 100.0%

          \[\leadsto d1 \cdot \left(\left(d2 + \color{blue}{37}\right) + d3\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{d1 \cdot \left(\left(d2 + 37\right) + d3\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d3 around 0 59.8%

      \[\leadsto \color{blue}{d1 \cdot \left(37 + d2\right)} \]

   6. Taylor expanded in d2 around 0 35.7%

      \[\leadsto d1 \cdot \color{blue}{37} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 26.9% accurate, 4.3× speedup

Math

\[\begin{array}{l} \\ d1 \cdot 37 \end{array} \]

FPCore

(FPCore (d1 d2 d3) :precision binary64 (* d1 37.0))

C

double code(double d1, double d2, double d3) {
	return d1 * 37.0;
}

Fortran

real(8) function code(d1, d2, d3)
    real(8), intent (in) :: d1
    real(8), intent (in) :: d2
    real(8), intent (in) :: d3
    code = d1 * 37.0d0
end function

Java

public static double code(double d1, double d2, double d3) {
	return d1 * 37.0;
}

Python

def code(d1, d2, d3):
	return d1 * 37.0

Julia

function code(d1, d2, d3)
	return Float64(d1 * 37.0)
end

MATLAB

function tmp = code(d1, d2, d3)
	tmp = d1 * 37.0;
end

Wolfram

code[d1_, d2_, d3_] := N[(d1 * 37.0), $MachinePrecision]

Derivation

1. Initial program 97.2%

   \[\left(d1 \cdot d2 + \left(d3 + 5\right) \cdot d1\right) + d1 \cdot 32 \]
2. Step-by-step derivation

   1. +-commutative 97.2%

      \[\leadsto \color{blue}{\left(\left(d3 + 5\right) \cdot d1 + d1 \cdot d2\right)} + d1 \cdot 32 \]

   2. *-commutative 97.2%

      \[\leadsto \left(\color{blue}{d1 \cdot \left(d3 + 5\right)} + d1 \cdot d2\right) + d1 \cdot 32 \]

   3. distribute-lft-out 100.0%

      \[\leadsto \color{blue}{d1 \cdot \left(\left(d3 + 5\right) + d2\right)} + d1 \cdot 32 \]

   4. distribute-lft-out 100.0%

      \[\leadsto \color{blue}{d1 \cdot \left(\left(\left(d3 + 5\right) + d2\right) + 32\right)} \]

   5. remove-double-neg 100.0%

      \[\leadsto \color{blue}{\left(-\left(-d1\right)\right)} \cdot \left(\left(\left(d3 + 5\right) + d2\right) + 32\right) \]

   6. distribute-lft-neg-in 100.0%

      \[\leadsto \color{blue}{-\left(-d1\right) \cdot \left(\left(\left(d3 + 5\right) + d2\right) + 32\right)} \]

   7. distribute-rgt-neg-in 100.0%

      \[\leadsto \color{blue}{\left(-d1\right) \cdot \left(-\left(\left(\left(d3 + 5\right) + d2\right) + 32\right)\right)} \]

   8. distribute-lft-neg-in 100.0%

      \[\leadsto \color{blue}{-d1 \cdot \left(-\left(\left(\left(d3 + 5\right) + d2\right) + 32\right)\right)} \]

   9. distribute-rgt-neg-in 100.0%

      \[\leadsto \color{blue}{d1 \cdot \left(-\left(-\left(\left(\left(d3 + 5\right) + d2\right) + 32\right)\right)\right)} \]

   10. remove-double-neg 100.0%

       \[\leadsto d1 \cdot \color{blue}{\left(\left(\left(d3 + 5\right) + d2\right) + 32\right)} \]

   11. associate-+r+ 100.0%

       \[\leadsto d1 \cdot \left(\color{blue}{\left(d3 + \left(5 + d2\right)\right)} + 32\right) \]

   12. +-commutative 100.0%

       \[\leadsto d1 \cdot \left(\left(d3 + \color{blue}{\left(d2 + 5\right)}\right) + 32\right) \]

   13. associate-+r+ 100.0%

       \[\leadsto d1 \cdot \color{blue}{\left(d3 + \left(\left(d2 + 5\right) + 32\right)\right)} \]

   14. +-commutative 100.0%

       \[\leadsto d1 \cdot \color{blue}{\left(\left(\left(d2 + 5\right) + 32\right) + d3\right)} \]

   15. associate-+l+ 100.0%

       \[\leadsto d1 \cdot \left(\color{blue}{\left(d2 + \left(5 + 32\right)\right)} + d3\right) \]

   16. metadata-eval 100.0%

       \[\leadsto d1 \cdot \left(\left(d2 + \color{blue}{37}\right) + d3\right) \]

3. Simplified 100.0%

   \[\leadsto \color{blue}{d1 \cdot \left(\left(d2 + 37\right) + d3\right)} \]
4. Add Preprocessing

5. Taylor expanded in d3 around 0 65.5%

   \[\leadsto \color{blue}{d1 \cdot \left(37 + d2\right)} \]

6. Taylor expanded in d2 around 0 26.5%

   \[\leadsto d1 \cdot \color{blue}{37} \]
7. Add Preprocessing

Developer Target 1: 100.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ d1 \cdot \left(\left(37 + d3\right) + d2\right) \end{array} \]

FPCore

(FPCore (d1 d2 d3) :precision binary64 (* d1 (+ (+ 37.0 d3) d2)))

C

double code(double d1, double d2, double d3) {
	return d1 * ((37.0 + d3) + d2);
}

Fortran

real(8) function code(d1, d2, d3)
    real(8), intent (in) :: d1
    real(8), intent (in) :: d2
    real(8), intent (in) :: d3
    code = d1 * ((37.0d0 + d3) + d2)
end function

Java

public static double code(double d1, double d2, double d3) {
	return d1 * ((37.0 + d3) + d2);
}

Python

def code(d1, d2, d3):
	return d1 * ((37.0 + d3) + d2)

Julia

function code(d1, d2, d3)
	return Float64(d1 * Float64(Float64(37.0 + d3) + d2))
end

MATLAB

function tmp = code(d1, d2, d3)
	tmp = d1 * ((37.0 + d3) + d2);
end

Wolfram

code[d1_, d2_, d3_] := N[(d1 * N[(N[(37.0 + d3), $MachinePrecision] + d2), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2024176 
(FPCore (d1 d2 d3)
  :name "FastMath dist3"
  :precision binary64

  :alt
  (! :herbie-platform default (* d1 (+ 37 d3 d2)))

  (+ (+ (* d1 d2) (* (+ d3 5.0) d1)) (* d1 32.0)))