FastMath dist3

Percentage Accurate: 97.9% → 99.0%

Time: 3.4s

Alternatives: 7

Speedup: 1.9×

• 21.3% 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: 97.9% 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: 99.0% accurate, 0.1× speedup

Math

\[\begin{array}{l} [d2, d3] = \mathsf{sort}([d2, d3])\\ \\ \mathsf{fma}\left(d2 + 37, d1, d1 \cdot d3\right) \end{array} \]

FPCore

NOTE: d2 and d3 should be sorted in increasing order before calling this function.
(FPCore (d1 d2 d3) :precision binary64 (fma (+ d2 37.0) d1 (* d1 d3)))

C

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

Julia

d2, d3 = sort([d2, d3])
function code(d1, d2, d3)
	return fma(Float64(d2 + 37.0), d1, Float64(d1 * d3))
end

Wolfram

NOTE: d2 and d3 should be sorted in increasing order before calling this function.
code[d1_, d2_, d3_] := N[(N[(d2 + 37.0), $MachinePrecision] * d1 + N[(d1 * d3), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.2%

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

   1. +-commutative 99.2%

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

   2. +-commutative 99.2%

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

   3. *-commutative 99.2%

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

   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. Step-by-step derivation

   1. distribute-rgt-in 99.2%

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

   2. fma-def 99.6%

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

   3. *-commutative 99.6%

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

5. Applied egg-rr 99.6%

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

6. Final simplification 99.6%

   \[\leadsto \mathsf{fma}\left(d2 + 37, d1, d1 \cdot d3\right) \]

Alternative 2: 72.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} [d2, d3] = \mathsf{sort}([d2, d3])\\ \\ \begin{array}{l} \mathbf{if}\;d3 \leq -2.1 \cdot 10^{-282}:\\ \;\;\;\;d2 \cdot d1\\ \mathbf{elif}\;d3 \leq 6 \cdot 10^{-274}:\\ \;\;\;\;37 \cdot d1\\ \mathbf{elif}\;d3 \leq 1.35 \cdot 10^{-240}:\\ \;\;\;\;d2 \cdot d1\\ \mathbf{elif}\;d3 \leq 1.08 \cdot 10^{-186}:\\ \;\;\;\;37 \cdot d1\\ \mathbf{elif}\;d3 \leq 1.06 \cdot 10^{-135}:\\ \;\;\;\;d2 \cdot d1\\ \mathbf{elif}\;d3 \leq 5.5 \cdot 10^{-5}:\\ \;\;\;\;37 \cdot d1\\ \mathbf{else}:\\ \;\;\;\;d1 \cdot d3\\ \end{array} \end{array} \]

FPCore

NOTE: d2 and d3 should be sorted in increasing order before calling this function.
(FPCore (d1 d2 d3)
 :precision binary64
 (if (<= d3 -2.1e-282)
   (* d2 d1)
   (if (<= d3 6e-274)
     (* 37.0 d1)
     (if (<= d3 1.35e-240)
       (* d2 d1)
       (if (<= d3 1.08e-186)
         (* 37.0 d1)
         (if (<= d3 1.06e-135)
           (* d2 d1)
           (if (<= d3 5.5e-5) (* 37.0 d1) (* d1 d3))))))))

C

assert(d2 < d3);
double code(double d1, double d2, double d3) {
	double tmp;
	if (d3 <= -2.1e-282) {
		tmp = d2 * d1;
	} else if (d3 <= 6e-274) {
		tmp = 37.0 * d1;
	} else if (d3 <= 1.35e-240) {
		tmp = d2 * d1;
	} else if (d3 <= 1.08e-186) {
		tmp = 37.0 * d1;
	} else if (d3 <= 1.06e-135) {
		tmp = d2 * d1;
	} else if (d3 <= 5.5e-5) {
		tmp = 37.0 * d1;
	} else {
		tmp = d1 * d3;
	}
	return tmp;
}

Fortran

NOTE: d2 and d3 should be sorted in increasing order before calling this function.
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 <= (-2.1d-282)) then
        tmp = d2 * d1
    else if (d3 <= 6d-274) then
        tmp = 37.0d0 * d1
    else if (d3 <= 1.35d-240) then
        tmp = d2 * d1
    else if (d3 <= 1.08d-186) then
        tmp = 37.0d0 * d1
    else if (d3 <= 1.06d-135) then
        tmp = d2 * d1
    else if (d3 <= 5.5d-5) then
        tmp = 37.0d0 * d1
    else
        tmp = d1 * d3
    end if
    code = tmp
end function

Java

assert d2 < d3;
public static double code(double d1, double d2, double d3) {
	double tmp;
	if (d3 <= -2.1e-282) {
		tmp = d2 * d1;
	} else if (d3 <= 6e-274) {
		tmp = 37.0 * d1;
	} else if (d3 <= 1.35e-240) {
		tmp = d2 * d1;
	} else if (d3 <= 1.08e-186) {
		tmp = 37.0 * d1;
	} else if (d3 <= 1.06e-135) {
		tmp = d2 * d1;
	} else if (d3 <= 5.5e-5) {
		tmp = 37.0 * d1;
	} else {
		tmp = d1 * d3;
	}
	return tmp;
}

Python

[d2, d3] = sort([d2, d3])
def code(d1, d2, d3):
	tmp = 0
	if d3 <= -2.1e-282:
		tmp = d2 * d1
	elif d3 <= 6e-274:
		tmp = 37.0 * d1
	elif d3 <= 1.35e-240:
		tmp = d2 * d1
	elif d3 <= 1.08e-186:
		tmp = 37.0 * d1
	elif d3 <= 1.06e-135:
		tmp = d2 * d1
	elif d3 <= 5.5e-5:
		tmp = 37.0 * d1
	else:
		tmp = d1 * d3
	return tmp

Julia

d2, d3 = sort([d2, d3])
function code(d1, d2, d3)
	tmp = 0.0
	if (d3 <= -2.1e-282)
		tmp = Float64(d2 * d1);
	elseif (d3 <= 6e-274)
		tmp = Float64(37.0 * d1);
	elseif (d3 <= 1.35e-240)
		tmp = Float64(d2 * d1);
	elseif (d3 <= 1.08e-186)
		tmp = Float64(37.0 * d1);
	elseif (d3 <= 1.06e-135)
		tmp = Float64(d2 * d1);
	elseif (d3 <= 5.5e-5)
		tmp = Float64(37.0 * d1);
	else
		tmp = Float64(d1 * d3);
	end
	return tmp
end

MATLAB

d2, d3 = num2cell(sort([d2, d3])){:}
function tmp_2 = code(d1, d2, d3)
	tmp = 0.0;
	if (d3 <= -2.1e-282)
		tmp = d2 * d1;
	elseif (d3 <= 6e-274)
		tmp = 37.0 * d1;
	elseif (d3 <= 1.35e-240)
		tmp = d2 * d1;
	elseif (d3 <= 1.08e-186)
		tmp = 37.0 * d1;
	elseif (d3 <= 1.06e-135)
		tmp = d2 * d1;
	elseif (d3 <= 5.5e-5)
		tmp = 37.0 * d1;
	else
		tmp = d1 * d3;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: d2 and d3 should be sorted in increasing order before calling this function.
code[d1_, d2_, d3_] := If[LessEqual[d3, -2.1e-282], N[(d2 * d1), $MachinePrecision], If[LessEqual[d3, 6e-274], N[(37.0 * d1), $MachinePrecision], If[LessEqual[d3, 1.35e-240], N[(d2 * d1), $MachinePrecision], If[LessEqual[d3, 1.08e-186], N[(37.0 * d1), $MachinePrecision], If[LessEqual[d3, 1.06e-135], N[(d2 * d1), $MachinePrecision], If[LessEqual[d3, 5.5e-5], N[(37.0 * d1), $MachinePrecision], N[(d1 * d3), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if d3 < -2.10000000000000012e-282 or 5.99999999999999954e-274 < d3 < 1.35000000000000009e-240 or 1.08e-186 < d3 < 1.06000000000000004e-135

   1. Initial program 99.3%

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

      1. +-commutative 99.3%

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

      2. +-commutative 99.3%

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

      3. *-commutative 99.3%

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

      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 42.3%

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

   if -2.10000000000000012e-282 < d3 < 5.99999999999999954e-274 or 1.35000000000000009e-240 < d3 < 1.08e-186 or 1.06000000000000004e-135 < d3 < 5.5000000000000002e-5

   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%

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

      2. +-commutative 99.9%

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

      3. *-commutative 99.9%

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

      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 0 100.0%

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

   5. Taylor expanded in d2 around 0 66.7%

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

   if 5.5000000000000002e-5 < d3

   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%

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

      2. +-commutative 98.0%

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

      3. *-commutative 98.0%

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

      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 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 84.9%

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

4. Final simplification 55.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d3 \leq -2.1 \cdot 10^{-282}:\\ \;\;\;\;d2 \cdot d1\\ \mathbf{elif}\;d3 \leq 6 \cdot 10^{-274}:\\ \;\;\;\;37 \cdot d1\\ \mathbf{elif}\;d3 \leq 1.35 \cdot 10^{-240}:\\ \;\;\;\;d2 \cdot d1\\ \mathbf{elif}\;d3 \leq 1.08 \cdot 10^{-186}:\\ \;\;\;\;37 \cdot d1\\ \mathbf{elif}\;d3 \leq 1.06 \cdot 10^{-135}:\\ \;\;\;\;d2 \cdot d1\\ \mathbf{elif}\;d3 \leq 5.5 \cdot 10^{-5}:\\ \;\;\;\;37 \cdot d1\\ \mathbf{else}:\\ \;\;\;\;d1 \cdot d3\\ \end{array} \]

Alternative 3: 92.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} [d2, d3] = \mathsf{sort}([d2, d3])\\ \\ \begin{array}{l} \mathbf{if}\;d3 \leq 1.3 \cdot 10^{+23}:\\ \;\;\;\;\left(d2 + 37\right) \cdot d1\\ \mathbf{else}:\\ \;\;\;\;d1 \cdot d3\\ \end{array} \end{array} \]

FPCore

NOTE: d2 and d3 should be sorted in increasing order before calling this function.
(FPCore (d1 d2 d3)
 :precision binary64
 (if (<= d3 1.3e+23) (* (+ d2 37.0) d1) (* d1 d3)))

C

assert(d2 < d3);
double code(double d1, double d2, double d3) {
	double tmp;
	if (d3 <= 1.3e+23) {
		tmp = (d2 + 37.0) * d1;
	} else {
		tmp = d1 * d3;
	}
	return tmp;
}

Fortran

NOTE: d2 and d3 should be sorted in increasing order before calling this function.
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 <= 1.3d+23) then
        tmp = (d2 + 37.0d0) * d1
    else
        tmp = d1 * d3
    end if
    code = tmp
end function

Java

assert d2 < d3;
public static double code(double d1, double d2, double d3) {
	double tmp;
	if (d3 <= 1.3e+23) {
		tmp = (d2 + 37.0) * d1;
	} else {
		tmp = d1 * d3;
	}
	return tmp;
}

Python

[d2, d3] = sort([d2, d3])
def code(d1, d2, d3):
	tmp = 0
	if d3 <= 1.3e+23:
		tmp = (d2 + 37.0) * d1
	else:
		tmp = d1 * d3
	return tmp

Julia

d2, d3 = sort([d2, d3])
function code(d1, d2, d3)
	tmp = 0.0
	if (d3 <= 1.3e+23)
		tmp = Float64(Float64(d2 + 37.0) * d1);
	else
		tmp = Float64(d1 * d3);
	end
	return tmp
end

MATLAB

d2, d3 = num2cell(sort([d2, d3])){:}
function tmp_2 = code(d1, d2, d3)
	tmp = 0.0;
	if (d3 <= 1.3e+23)
		tmp = (d2 + 37.0) * d1;
	else
		tmp = d1 * d3;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: d2 and d3 should be sorted in increasing order before calling this function.
code[d1_, d2_, d3_] := If[LessEqual[d3, 1.3e+23], N[(N[(d2 + 37.0), $MachinePrecision] * d1), $MachinePrecision], N[(d1 * d3), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d3 < 1.29999999999999996e23

   1. Initial program 99.5%

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

      1. +-commutative 99.5%

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

      2. +-commutative 99.5%

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

      3. *-commutative 99.5%

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

      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 d3 around 0 73.6%

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

   if 1.29999999999999996e23 < d3

   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 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 88.4%

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

4. Final simplification 76.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d3 \leq 1.3 \cdot 10^{+23}:\\ \;\;\;\;\left(d2 + 37\right) \cdot d1\\ \mathbf{else}:\\ \;\;\;\;d1 \cdot d3\\ \end{array} \]

Alternative 4: 91.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} [d2, d3] = \mathsf{sort}([d2, d3])\\ \\ \begin{array}{l} \mathbf{if}\;d2 \leq -2.8 \cdot 10^{+27}:\\ \;\;\;\;d2 \cdot d1\\ \mathbf{else}:\\ \;\;\;\;d1 \cdot \left(37 + d3\right)\\ \end{array} \end{array} \]

FPCore

NOTE: d2 and d3 should be sorted in increasing order before calling this function.
(FPCore (d1 d2 d3)
 :precision binary64
 (if (<= d2 -2.8e+27) (* d2 d1) (* d1 (+ 37.0 d3))))

C

assert(d2 < d3);
double code(double d1, double d2, double d3) {
	double tmp;
	if (d2 <= -2.8e+27) {
		tmp = d2 * d1;
	} else {
		tmp = d1 * (37.0 + d3);
	}
	return tmp;
}

Fortran

NOTE: d2 and d3 should be sorted in increasing order before calling this function.
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 <= (-2.8d+27)) then
        tmp = d2 * d1
    else
        tmp = d1 * (37.0d0 + d3)
    end if
    code = tmp
end function

Java

assert d2 < d3;
public static double code(double d1, double d2, double d3) {
	double tmp;
	if (d2 <= -2.8e+27) {
		tmp = d2 * d1;
	} else {
		tmp = d1 * (37.0 + d3);
	}
	return tmp;
}

Python

[d2, d3] = sort([d2, d3])
def code(d1, d2, d3):
	tmp = 0
	if d2 <= -2.8e+27:
		tmp = d2 * d1
	else:
		tmp = d1 * (37.0 + d3)
	return tmp

Julia

d2, d3 = sort([d2, d3])
function code(d1, d2, d3)
	tmp = 0.0
	if (d2 <= -2.8e+27)
		tmp = Float64(d2 * d1);
	else
		tmp = Float64(d1 * Float64(37.0 + d3));
	end
	return tmp
end

MATLAB

d2, d3 = num2cell(sort([d2, d3])){:}
function tmp_2 = code(d1, d2, d3)
	tmp = 0.0;
	if (d2 <= -2.8e+27)
		tmp = d2 * d1;
	else
		tmp = d1 * (37.0 + d3);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: d2 and d3 should be sorted in increasing order before calling this function.
code[d1_, d2_, d3_] := If[LessEqual[d2, -2.8e+27], N[(d2 * d1), $MachinePrecision], N[(d1 * N[(37.0 + d3), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d2 < -2.7999999999999999e27

   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 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 83.9%

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

   if -2.7999999999999999e27 < d2

   1. Initial program 99.0%

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

      1. +-commutative 99.0%

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

      2. +-commutative 99.0%

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

      3. *-commutative 99.0%

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

      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 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 76.8%

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

4. Final simplification 78.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d2 \leq -2.8 \cdot 10^{+27}:\\ \;\;\;\;d2 \cdot d1\\ \mathbf{else}:\\ \;\;\;\;d1 \cdot \left(37 + d3\right)\\ \end{array} \]

Alternative 5: 100.0% accurate, 1.9× speedup

Math

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

FPCore

NOTE: d2 and d3 should be sorted in increasing order before calling this function.
(FPCore (d1 d2 d3) :precision binary64 (* d1 (+ (+ d2 37.0) d3)))

C

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

Fortran

NOTE: d2 and d3 should be sorted in increasing order before calling this function.
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

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

Python

[d2, d3] = sort([d2, d3])
def code(d1, d2, d3):
	return d1 * ((d2 + 37.0) + d3)

Julia

d2, d3 = sort([d2, d3])
function code(d1, d2, d3)
	return Float64(d1 * Float64(Float64(d2 + 37.0) + d3))
end

MATLAB

d2, d3 = num2cell(sort([d2, d3])){:}
function tmp = code(d1, d2, d3)
	tmp = d1 * ((d2 + 37.0) + d3);
end

Wolfram

NOTE: d2 and d3 should be sorted in increasing order before calling this function.
code[d1_, d2_, d3_] := N[(d1 * N[(N[(d2 + 37.0), $MachinePrecision] + d3), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.2%

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

   1. +-commutative 99.2%

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

   2. +-commutative 99.2%

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

   3. *-commutative 99.2%

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

   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 6: 62.8% accurate, 2.6× speedup

Math

\[\begin{array}{l} [d2, d3] = \mathsf{sort}([d2, d3])\\ \\ \begin{array}{l} \mathbf{if}\;d2 \leq -3300:\\ \;\;\;\;d2 \cdot d1\\ \mathbf{else}:\\ \;\;\;\;37 \cdot d1\\ \end{array} \end{array} \]

FPCore

NOTE: d2 and d3 should be sorted in increasing order before calling this function.
(FPCore (d1 d2 d3)
 :precision binary64
 (if (<= d2 -3300.0) (* d2 d1) (* 37.0 d1)))

C

assert(d2 < d3);
double code(double d1, double d2, double d3) {
	double tmp;
	if (d2 <= -3300.0) {
		tmp = d2 * d1;
	} else {
		tmp = 37.0 * d1;
	}
	return tmp;
}

Fortran

NOTE: d2 and d3 should be sorted in increasing order before calling this function.
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 <= (-3300.0d0)) then
        tmp = d2 * d1
    else
        tmp = 37.0d0 * d1
    end if
    code = tmp
end function

Java

assert d2 < d3;
public static double code(double d1, double d2, double d3) {
	double tmp;
	if (d2 <= -3300.0) {
		tmp = d2 * d1;
	} else {
		tmp = 37.0 * d1;
	}
	return tmp;
}

Python

[d2, d3] = sort([d2, d3])
def code(d1, d2, d3):
	tmp = 0
	if d2 <= -3300.0:
		tmp = d2 * d1
	else:
		tmp = 37.0 * d1
	return tmp

Julia

d2, d3 = sort([d2, d3])
function code(d1, d2, d3)
	tmp = 0.0
	if (d2 <= -3300.0)
		tmp = Float64(d2 * d1);
	else
		tmp = Float64(37.0 * d1);
	end
	return tmp
end

MATLAB

d2, d3 = num2cell(sort([d2, d3])){:}
function tmp_2 = code(d1, d2, d3)
	tmp = 0.0;
	if (d2 <= -3300.0)
		tmp = d2 * d1;
	else
		tmp = 37.0 * d1;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: d2 and d3 should be sorted in increasing order before calling this function.
code[d1_, d2_, d3_] := If[LessEqual[d2, -3300.0], N[(d2 * d1), $MachinePrecision], N[(37.0 * d1), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d2 < -3300

   1. Initial program 98.1%

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

      1. +-commutative 98.1%

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

      2. +-commutative 98.1%

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

      3. *-commutative 98.1%

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

      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 d2 around inf 77.9%

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

   if -3300 < d2

   1. Initial program 99.5%

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

      1. +-commutative 99.5%

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

      2. +-commutative 99.5%

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

      3. *-commutative 99.5%

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

      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 d3 around 0 60.2%

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

   5. Taylor expanded in d2 around 0 36.6%

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

4. Final simplification 45.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d2 \leq -3300:\\ \;\;\;\;d2 \cdot d1\\ \mathbf{else}:\\ \;\;\;\;37 \cdot d1\\ \end{array} \]

Alternative 7: 26.6% accurate, 4.3× speedup

Math

\[\begin{array}{l} [d2, d3] = \mathsf{sort}([d2, d3])\\ \\ 37 \cdot d1 \end{array} \]

FPCore

NOTE: d2 and d3 should be sorted in increasing order before calling this function.
(FPCore (d1 d2 d3) :precision binary64 (* 37.0 d1))

C

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

Fortran

NOTE: d2 and d3 should be sorted in increasing order before calling this function.
real(8) function code(d1, d2, d3)
    real(8), intent (in) :: d1
    real(8), intent (in) :: d2
    real(8), intent (in) :: d3
    code = 37.0d0 * d1
end function

Java

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

Python

[d2, d3] = sort([d2, d3])
def code(d1, d2, d3):
	return 37.0 * d1

Julia

d2, d3 = sort([d2, d3])
function code(d1, d2, d3)
	return Float64(37.0 * d1)
end

MATLAB

d2, d3 = num2cell(sort([d2, d3])){:}
function tmp = code(d1, d2, d3)
	tmp = 37.0 * d1;
end

Wolfram

NOTE: d2 and d3 should be sorted in increasing order before calling this function.
code[d1_, d2_, d3_] := N[(37.0 * d1), $MachinePrecision]

Derivation

1. Initial program 99.2%

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

   1. +-commutative 99.2%

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

   2. +-commutative 99.2%

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

   3. *-commutative 99.2%

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

   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 63.9%

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

5. Taylor expanded in d2 around 0 29.3%

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

6. Final simplification 29.3%

   \[\leadsto 37 \cdot d1 \]

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 2023293 
(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)))