FastMath dist3

Percentage Accurate: 97.8% → 100.0%

Time: 4.7s

Alternatives: 6

Speedup: 1.9×

• 21.0% 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 97.6%

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

   1. +-commutative 97.6%

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

   2. *-commutative 97.6%

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

   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. Add Preprocessing

Alternative 2: 62.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d2 \leq -38 \lor \neg \left(d2 \leq 38\right):\\ \;\;\;\;d1 \cdot d2\\ \mathbf{else}:\\ \;\;\;\;d1 \cdot 37\\ \end{array} \end{array} \]

FPCore

(FPCore (d1 d2 d3)
 :precision binary64
 (if (or (<= d2 -38.0) (not (<= d2 38.0))) (* d1 d2) (* d1 37.0)))

C

double code(double d1, double d2, double d3) {
	double tmp;
	if ((d2 <= -38.0) || !(d2 <= 38.0)) {
		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 <= (-38.0d0)) .or. (.not. (d2 <= 38.0d0))) 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 <= -38.0) || !(d2 <= 38.0)) {
		tmp = d1 * d2;
	} else {
		tmp = d1 * 37.0;
	}
	return tmp;
}

Python

def code(d1, d2, d3):
	tmp = 0
	if (d2 <= -38.0) or not (d2 <= 38.0):
		tmp = d1 * d2
	else:
		tmp = d1 * 37.0
	return tmp

Julia

function code(d1, d2, d3)
	tmp = 0.0
	if ((d2 <= -38.0) || !(d2 <= 38.0))
		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 <= -38.0) || ~((d2 <= 38.0)))
		tmp = d1 * d2;
	else
		tmp = d1 * 37.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[d1_, d2_, d3_] := If[Or[LessEqual[d2, -38.0], N[Not[LessEqual[d2, 38.0]], $MachinePrecision]], N[(d1 * d2), $MachinePrecision], N[(d1 * 37.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d2 < -38 or 38 < d2

   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 77.7%

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

   if -38 < d2 < 38

   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 99.9%

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

      5. remove-double-neg 99.9%

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

      6. distribute-lft-neg-in 99.9%

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

      7. distribute-rgt-neg-in 99.9%

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

      8. distribute-lft-neg-in 99.9%

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

      9. distribute-rgt-neg-in 99.9%

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

      10. remove-double-neg 99.9%

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

      11. associate-+r+ 99.9%

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

      12. +-commutative 99.9%

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

      13. associate-+r+ 99.9%

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

      14. +-commutative 99.9%

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

      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 51.8%

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

   6. Taylor expanded in d2 around 0 50.7%

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

4. Final simplification 64.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d2 \leq -38 \lor \neg \left(d2 \leq 38\right):\\ \;\;\;\;d1 \cdot d2\\ \mathbf{else}:\\ \;\;\;\;d1 \cdot 37\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 52.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d2 \leq -38:\\ \;\;\;\;d1 \cdot d2\\ \mathbf{elif}\;d2 \leq 1.12 \cdot 10^{-268}:\\ \;\;\;\;d1 \cdot 37\\ \mathbf{else}:\\ \;\;\;\;d1 \cdot d3\\ \end{array} \end{array} \]

FPCore

(FPCore (d1 d2 d3)
 :precision binary64
 (if (<= d2 -38.0) (* d1 d2) (if (<= d2 1.12e-268) (* d1 37.0) (* d1 d3))))

C

double code(double d1, double d2, double d3) {
	double tmp;
	if (d2 <= -38.0) {
		tmp = d1 * d2;
	} else if (d2 <= 1.12e-268) {
		tmp = d1 * 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 (d2 <= (-38.0d0)) then
        tmp = d1 * d2
    else if (d2 <= 1.12d-268) then
        tmp = d1 * 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 (d2 <= -38.0) {
		tmp = d1 * d2;
	} else if (d2 <= 1.12e-268) {
		tmp = d1 * 37.0;
	} else {
		tmp = d1 * d3;
	}
	return tmp;
}

Python

def code(d1, d2, d3):
	tmp = 0
	if d2 <= -38.0:
		tmp = d1 * d2
	elif d2 <= 1.12e-268:
		tmp = d1 * 37.0
	else:
		tmp = d1 * d3
	return tmp

Julia

function code(d1, d2, d3)
	tmp = 0.0
	if (d2 <= -38.0)
		tmp = Float64(d1 * d2);
	elseif (d2 <= 1.12e-268)
		tmp = Float64(d1 * 37.0);
	else
		tmp = Float64(d1 * d3);
	end
	return tmp
end

MATLAB

function tmp_2 = code(d1, d2, d3)
	tmp = 0.0;
	if (d2 <= -38.0)
		tmp = d1 * d2;
	elseif (d2 <= 1.12e-268)
		tmp = d1 * 37.0;
	else
		tmp = d1 * d3;
	end
	tmp_2 = tmp;
end

Wolfram

code[d1_, d2_, d3_] := If[LessEqual[d2, -38.0], N[(d1 * d2), $MachinePrecision], If[LessEqual[d2, 1.12e-268], N[(d1 * 37.0), $MachinePrecision], N[(d1 * d3), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if d2 < -38

   1. Initial program 98.5%

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

      1. +-commutative 98.5%

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

      2. *-commutative 98.5%

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

      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 77.8%

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

   if -38 < d2 < 1.11999999999999998e-268

   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 57.0%

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

   6. Taylor expanded in d2 around 0 56.1%

      \[\leadsto d1 \cdot \color{blue}{37} \]

   if 1.11999999999999998e-268 < d2

   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 inf 43.5%

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

Alternative 4: 77.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d2 \leq -0.00036:\\ \;\;\;\;d1 \cdot \left(d2 + 37\right)\\ \mathbf{else}:\\ \;\;\;\;d1 \cdot \left(37 + d3\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (d1 d2 d3)
 :precision binary64
 (if (<= d2 -0.00036) (* d1 (+ d2 37.0)) (* d1 (+ 37.0 d3))))

C

double code(double d1, double d2, double d3) {
	double tmp;
	if (d2 <= -0.00036) {
		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 <= (-0.00036d0)) 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 <= -0.00036) {
		tmp = d1 * (d2 + 37.0);
	} else {
		tmp = d1 * (37.0 + d3);
	}
	return tmp;
}

Python

def code(d1, d2, d3):
	tmp = 0
	if d2 <= -0.00036:
		tmp = d1 * (d2 + 37.0)
	else:
		tmp = d1 * (37.0 + d3)
	return tmp

Julia

function code(d1, d2, d3)
	tmp = 0.0
	if (d2 <= -0.00036)
		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 <= -0.00036)
		tmp = d1 * (d2 + 37.0);
	else
		tmp = d1 * (37.0 + d3);
	end
	tmp_2 = tmp;
end

Wolfram

code[d1_, d2_, d3_] := If[LessEqual[d2, -0.00036], 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 < -3.60000000000000023e-4

   1. Initial program 98.6%

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

      1. +-commutative 98.6%

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

      2. *-commutative 98.6%

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

      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 80.0%

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

   if -3.60000000000000023e-4 < d2

   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 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 77.4%

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

4. Final simplification 78.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d2 \leq -0.00036:\\ \;\;\;\;d1 \cdot \left(d2 + 37\right)\\ \mathbf{else}:\\ \;\;\;\;d1 \cdot \left(37 + d3\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 75.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d3 \leq 9.2 \cdot 10^{+30}:\\ \;\;\;\;d1 \cdot \left(d2 + 37\right)\\ \mathbf{else}:\\ \;\;\;\;d1 \cdot d3\\ \end{array} \end{array} \]

FPCore

(FPCore (d1 d2 d3)
 :precision binary64
 (if (<= d3 9.2e+30) (* d1 (+ d2 37.0)) (* d1 d3)))

C

double code(double d1, double d2, double d3) {
	double tmp;
	if (d3 <= 9.2e+30) {
		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 <= 9.2d+30) 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 <= 9.2e+30) {
		tmp = d1 * (d2 + 37.0);
	} else {
		tmp = d1 * d3;
	}
	return tmp;
}

Python

def code(d1, d2, d3):
	tmp = 0
	if d3 <= 9.2e+30:
		tmp = d1 * (d2 + 37.0)
	else:
		tmp = d1 * d3
	return tmp

Julia

function code(d1, d2, d3)
	tmp = 0.0
	if (d3 <= 9.2e+30)
		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 <= 9.2e+30)
		tmp = d1 * (d2 + 37.0);
	else
		tmp = d1 * d3;
	end
	tmp_2 = tmp;
end

Wolfram

code[d1_, d2_, d3_] := If[LessEqual[d3, 9.2e+30], N[(d1 * N[(d2 + 37.0), $MachinePrecision]), $MachinePrecision], N[(d1 * d3), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d3 < 9.2e30

   1. Initial program 97.4%

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

      1. +-commutative 97.4%

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

      2. *-commutative 97.4%

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

      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 77.7%

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

   if 9.2e30 < d3

   1. Initial program 98.2%

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

      1. +-commutative 98.2%

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

      2. *-commutative 98.2%

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

      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 inf 89.4%

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

4. Final simplification 80.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d3 \leq 9.2 \cdot 10^{+30}:\\ \;\;\;\;d1 \cdot \left(d2 + 37\right)\\ \mathbf{else}:\\ \;\;\;\;d1 \cdot d3\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 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 97.6%

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

   1. +-commutative 97.6%

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

   2. *-commutative 97.6%

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

   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 65.8%

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

6. Taylor expanded in d2 around 0 26.3%

   \[\leadsto d1 \cdot \color{blue}{37} \]
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 2024180 
(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)))