FastMath dist3

Percentage Accurate: 97.5% → 100.0%

Time: 7.8s

Alternatives: 6

Speedup: 1.9×

• 20.7% 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: 97.5% 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 96.1%

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

   1. +-commutative 96.1%

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

   2. +-commutative 96.1%

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

   3. *-commutative 96.1%

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

   4. distribute-lft-out 100.0%

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

   5. distribute-lft-out 100.0%

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

   6. remove-double-neg 100.0%

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

   7. sub-neg 100.0%

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

   8. sub-neg 100.0%

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

   9. remove-double-neg 100.0%

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

   10. associate-+r+ 100.0%

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

   11. +-commutative 100.0%

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

   12. +-commutative 100.0%

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

   13. associate-+r+ 100.0%

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

   14. +-commutative 100.0%

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

   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. Final simplification 100.0%

   \[\leadsto d1 \cdot \left(\left(d2 + 37\right) + d3\right) \]

Alternative 2: 51.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d2 \leq -36:\\ \;\;\;\;d1 \cdot d2\\ \mathbf{elif}\;d2 \leq -1.08 \cdot 10^{-182}:\\ \;\;\;\;d1 \cdot 37\\ \mathbf{else}:\\ \;\;\;\;d1 \cdot d3\\ \end{array} \end{array} \]

FPCore

(FPCore (d1 d2 d3)
 :precision binary64
 (if (<= d2 -36.0) (* d1 d2) (if (<= d2 -1.08e-182) (* d1 37.0) (* d1 d3))))

C

double code(double d1, double d2, double d3) {
	double tmp;
	if (d2 <= -36.0) {
		tmp = d1 * d2;
	} else if (d2 <= -1.08e-182) {
		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 <= (-36.0d0)) then
        tmp = d1 * d2
    else if (d2 <= (-1.08d-182)) 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 <= -36.0) {
		tmp = d1 * d2;
	} else if (d2 <= -1.08e-182) {
		tmp = d1 * 37.0;
	} else {
		tmp = d1 * d3;
	}
	return tmp;
}

Python

def code(d1, d2, d3):
	tmp = 0
	if d2 <= -36.0:
		tmp = d1 * d2
	elif d2 <= -1.08e-182:
		tmp = d1 * 37.0
	else:
		tmp = d1 * d3
	return tmp

Julia

function code(d1, d2, d3)
	tmp = 0.0
	if (d2 <= -36.0)
		tmp = Float64(d1 * d2);
	elseif (d2 <= -1.08e-182)
		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 <= -36.0)
		tmp = d1 * d2;
	elseif (d2 <= -1.08e-182)
		tmp = d1 * 37.0;
	else
		tmp = d1 * d3;
	end
	tmp_2 = tmp;
end

Wolfram

code[d1_, d2_, d3_] := If[LessEqual[d2, -36.0], N[(d1 * d2), $MachinePrecision], If[LessEqual[d2, -1.08e-182], N[(d1 * 37.0), $MachinePrecision], N[(d1 * d3), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if d2 < -36

   1. Initial program 89.5%

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

      1. +-commutative 89.5%

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

      2. +-commutative 89.5%

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

      3. *-commutative 89.5%

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

      4. distribute-lft-out 100.0%

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

      5. distribute-lft-out 100.0%

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

      6. remove-double-neg 100.0%

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

      7. sub-neg 100.0%

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

      8. sub-neg 100.0%

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

      9. remove-double-neg 100.0%

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

      10. associate-+r+ 100.0%

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

      11. +-commutative 100.0%

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

      12. +-commutative 100.0%

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

      13. associate-+r+ 100.0%

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

      14. +-commutative 100.0%

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

      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. Taylor expanded in d2 around inf 73.9%

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

   if -36 < d2 < -1.08000000000000003e-182

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. +-commutative 100.0%

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

      3. *-commutative 100.0%

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

      4. distribute-lft-out 100.0%

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

      5. distribute-lft-out 99.9%

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

      6. remove-double-neg 99.9%

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

      7. sub-neg 99.9%

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

      8. sub-neg 99.9%

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

      9. remove-double-neg 99.9%

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

      10. associate-+r+ 99.9%

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

      11. +-commutative 99.9%

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

      12. +-commutative 99.9%

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

      13. associate-+r+ 99.9%

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

      14. +-commutative 99.9%

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

      15. associate-+l+ 99.9%

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

      16. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in d3 around 0 58.7%

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

   5. Taylor expanded in d2 around 0 55.0%

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

   if -1.08000000000000003e-182 < d2

   1. Initial program 97.9%

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

      1. +-commutative 97.9%

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

      2. +-commutative 97.9%

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

      3. *-commutative 97.9%

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

      4. distribute-lft-out 99.9%

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

      5. distribute-lft-out 100.0%

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

      6. remove-double-neg 100.0%

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

      7. sub-neg 100.0%

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

      8. sub-neg 100.0%

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

      9. remove-double-neg 100.0%

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

      10. associate-+r+ 100.0%

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

      11. +-commutative 100.0%

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

      12. +-commutative 100.0%

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

      13. associate-+r+ 100.0%

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

      14. +-commutative 100.0%

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

      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. Taylor expanded in d3 around inf 38.6%

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

4. Final simplification 50.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d2 \leq -36:\\ \;\;\;\;d1 \cdot d2\\ \mathbf{elif}\;d2 \leq -1.08 \cdot 10^{-182}:\\ \;\;\;\;d1 \cdot 37\\ \mathbf{else}:\\ \;\;\;\;d1 \cdot d3\\ \end{array} \]

Alternative 3: 75.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d3 \leq 1.7 \cdot 10^{+17}:\\ \;\;\;\;d1 \cdot \left(d2 + 37\right)\\ \mathbf{else}:\\ \;\;\;\;d1 \cdot d3\\ \end{array} \end{array} \]

FPCore

(FPCore (d1 d2 d3)
 :precision binary64
 (if (<= d3 1.7e+17) (* d1 (+ d2 37.0)) (* d1 d3)))

C

double code(double d1, double d2, double d3) {
	double tmp;
	if (d3 <= 1.7e+17) {
		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 <= 1.7d+17) 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 <= 1.7e+17) {
		tmp = d1 * (d2 + 37.0);
	} else {
		tmp = d1 * d3;
	}
	return tmp;
}

Python

def code(d1, d2, d3):
	tmp = 0
	if d3 <= 1.7e+17:
		tmp = d1 * (d2 + 37.0)
	else:
		tmp = d1 * d3
	return tmp

Julia

function code(d1, d2, d3)
	tmp = 0.0
	if (d3 <= 1.7e+17)
		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 <= 1.7e+17)
		tmp = d1 * (d2 + 37.0);
	else
		tmp = d1 * d3;
	end
	tmp_2 = tmp;
end

Wolfram

code[d1_, d2_, d3_] := If[LessEqual[d3, 1.7e+17], N[(d1 * N[(d2 + 37.0), $MachinePrecision]), $MachinePrecision], N[(d1 * d3), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d3 < 1.7e17

   1. Initial program 98.4%

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

      1. +-commutative 98.4%

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

      2. +-commutative 98.4%

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

      3. *-commutative 98.4%

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

      4. distribute-lft-out 99.9%

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

      5. distribute-lft-out 100.0%

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

      6. remove-double-neg 100.0%

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

      7. sub-neg 100.0%

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

      8. sub-neg 100.0%

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

      9. remove-double-neg 100.0%

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

      10. associate-+r+ 100.0%

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

      11. +-commutative 100.0%

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

      12. +-commutative 100.0%

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

      13. associate-+r+ 100.0%

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

      14. +-commutative 100.0%

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

      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. Taylor expanded in d3 around 0 78.0%

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

   if 1.7e17 < d3

   1. Initial program 87.7%

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

      1. +-commutative 87.7%

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

      2. +-commutative 87.7%

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

      3. *-commutative 87.7%

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

      4. distribute-lft-out 100.0%

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

      5. distribute-lft-out 100.0%

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

      6. remove-double-neg 100.0%

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

      7. sub-neg 100.0%

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

      8. sub-neg 100.0%

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

      9. remove-double-neg 100.0%

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

      10. associate-+r+ 100.0%

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

      11. +-commutative 100.0%

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

      12. +-commutative 100.0%

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

      13. associate-+r+ 100.0%

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

      14. +-commutative 100.0%

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

      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. Taylor expanded in d3 around inf 79.5%

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

4. Final simplification 78.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d3 \leq 1.7 \cdot 10^{+17}:\\ \;\;\;\;d1 \cdot \left(d2 + 37\right)\\ \mathbf{else}:\\ \;\;\;\;d1 \cdot d3\\ \end{array} \]

Alternative 4: 75.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d3 \leq 1.65 \cdot 10^{+16}:\\ \;\;\;\;d1 \cdot \left(d2 + 37\right)\\ \mathbf{else}:\\ \;\;\;\;d1 \cdot \left(37 + d3\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (d1 d2 d3)
 :precision binary64
 (if (<= d3 1.65e+16) (* d1 (+ d2 37.0)) (* d1 (+ 37.0 d3))))

C

double code(double d1, double d2, double d3) {
	double tmp;
	if (d3 <= 1.65e+16) {
		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 (d3 <= 1.65d+16) 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 (d3 <= 1.65e+16) {
		tmp = d1 * (d2 + 37.0);
	} else {
		tmp = d1 * (37.0 + d3);
	}
	return tmp;
}

Python

def code(d1, d2, d3):
	tmp = 0
	if d3 <= 1.65e+16:
		tmp = d1 * (d2 + 37.0)
	else:
		tmp = d1 * (37.0 + d3)
	return tmp

Julia

function code(d1, d2, d3)
	tmp = 0.0
	if (d3 <= 1.65e+16)
		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 (d3 <= 1.65e+16)
		tmp = d1 * (d2 + 37.0);
	else
		tmp = d1 * (37.0 + d3);
	end
	tmp_2 = tmp;
end

Wolfram

code[d1_, d2_, d3_] := If[LessEqual[d3, 1.65e+16], 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 d3 < 1.65e16

   1. Initial program 98.4%

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

      1. +-commutative 98.4%

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

      2. +-commutative 98.4%

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

      3. *-commutative 98.4%

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

      4. distribute-lft-out 99.9%

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

      5. distribute-lft-out 100.0%

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

      6. remove-double-neg 100.0%

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

      7. sub-neg 100.0%

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

      8. sub-neg 100.0%

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

      9. remove-double-neg 100.0%

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

      10. associate-+r+ 100.0%

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

      11. +-commutative 100.0%

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

      12. +-commutative 100.0%

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

      13. associate-+r+ 100.0%

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

      14. +-commutative 100.0%

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

      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. Taylor expanded in d3 around 0 78.0%

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

   if 1.65e16 < d3

   1. Initial program 87.7%

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

      1. +-commutative 87.7%

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

      2. +-commutative 87.7%

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

      3. *-commutative 87.7%

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

      4. distribute-lft-out 100.0%

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

      5. distribute-lft-out 100.0%

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

      6. remove-double-neg 100.0%

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

      7. sub-neg 100.0%

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

      8. sub-neg 100.0%

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

      9. remove-double-neg 100.0%

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

      10. associate-+r+ 100.0%

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

      11. +-commutative 100.0%

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

      12. +-commutative 100.0%

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

      13. associate-+r+ 100.0%

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

      14. +-commutative 100.0%

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

      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. Taylor expanded in d2 around 0 79.5%

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

4. Final simplification 78.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d3 \leq 1.65 \cdot 10^{+16}:\\ \;\;\;\;d1 \cdot \left(d2 + 37\right)\\ \mathbf{else}:\\ \;\;\;\;d1 \cdot \left(37 + d3\right)\\ \end{array} \]

Alternative 5: 44.6% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d3 \leq 4.2 \cdot 10^{+15}:\\ \;\;\;\;d1 \cdot 37\\ \mathbf{else}:\\ \;\;\;\;d1 \cdot d3\\ \end{array} \end{array} \]

FPCore

(FPCore (d1 d2 d3)
 :precision binary64
 (if (<= d3 4.2e+15) (* d1 37.0) (* d1 d3)))

C

double code(double d1, double d2, double d3) {
	double tmp;
	if (d3 <= 4.2e+15) {
		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 (d3 <= 4.2d+15) 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 (d3 <= 4.2e+15) {
		tmp = d1 * 37.0;
	} else {
		tmp = d1 * d3;
	}
	return tmp;
}

Python

def code(d1, d2, d3):
	tmp = 0
	if d3 <= 4.2e+15:
		tmp = d1 * 37.0
	else:
		tmp = d1 * d3
	return tmp

Julia

function code(d1, d2, d3)
	tmp = 0.0
	if (d3 <= 4.2e+15)
		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 (d3 <= 4.2e+15)
		tmp = d1 * 37.0;
	else
		tmp = d1 * d3;
	end
	tmp_2 = tmp;
end

Wolfram

code[d1_, d2_, d3_] := If[LessEqual[d3, 4.2e+15], N[(d1 * 37.0), $MachinePrecision], N[(d1 * d3), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d3 < 4.2e15

   1. Initial program 98.4%

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

      1. +-commutative 98.4%

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

      2. +-commutative 98.4%

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

      3. *-commutative 98.4%

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

      4. distribute-lft-out 99.9%

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

      5. distribute-lft-out 100.0%

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

      6. remove-double-neg 100.0%

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

      7. sub-neg 100.0%

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

      8. sub-neg 100.0%

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

      9. remove-double-neg 100.0%

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

      10. associate-+r+ 100.0%

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

      11. +-commutative 100.0%

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

      12. +-commutative 100.0%

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

      13. associate-+r+ 100.0%

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

      14. +-commutative 100.0%

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

      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. Taylor expanded in d3 around 0 78.0%

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

   5. Taylor expanded in d2 around 0 36.2%

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

   if 4.2e15 < d3

   1. Initial program 87.7%

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

      1. +-commutative 87.7%

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

      2. +-commutative 87.7%

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

      3. *-commutative 87.7%

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

      4. distribute-lft-out 100.0%

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

      5. distribute-lft-out 100.0%

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

      6. remove-double-neg 100.0%

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

      7. sub-neg 100.0%

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

      8. sub-neg 100.0%

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

      9. remove-double-neg 100.0%

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

      10. associate-+r+ 100.0%

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

      11. +-commutative 100.0%

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

      12. +-commutative 100.0%

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

      13. associate-+r+ 100.0%

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

      14. +-commutative 100.0%

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

      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. Taylor expanded in d3 around inf 79.5%

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

4. Final simplification 45.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d3 \leq 4.2 \cdot 10^{+15}:\\ \;\;\;\;d1 \cdot 37\\ \mathbf{else}:\\ \;\;\;\;d1 \cdot d3\\ \end{array} \]

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 96.1%

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

   1. +-commutative 96.1%

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

   2. +-commutative 96.1%

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

   3. *-commutative 96.1%

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

   4. distribute-lft-out 100.0%

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

   5. distribute-lft-out 100.0%

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

   6. remove-double-neg 100.0%

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

   7. sub-neg 100.0%

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

   8. sub-neg 100.0%

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

   9. remove-double-neg 100.0%

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

   10. associate-+r+ 100.0%

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

   11. +-commutative 100.0%

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

   12. +-commutative 100.0%

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

   13. associate-+r+ 100.0%

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

   14. +-commutative 100.0%

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

   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. Taylor expanded in d3 around 0 66.0%

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

5. Taylor expanded in d2 around 0 29.2%

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

6. Final simplification 29.2%

   \[\leadsto d1 \cdot 37 \]

Developer target: 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 2023279 
(FPCore (d1 d2 d3)
  :name "FastMath dist3"
  :precision binary64

  :herbie-target
  (* d1 (+ (+ 37.0 d3) d2))

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