FastMath dist4

Percentage Accurate: 88.2% → 100.0%

Time: 15.2s

Alternatives: 10

Speedup: 1.7×

• 31.9% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \left(\left(d1 \cdot d2 - d1 \cdot d3\right) + d4 \cdot d1\right) - d1 \cdot d1 \end{array} \]

FPCore

(FPCore (d1 d2 d3 d4)
 :precision binary64
 (- (+ (- (* d1 d2) (* d1 d3)) (* d4 d1)) (* d1 d1)))

C

double code(double d1, double d2, double d3, double d4) {
	return (((d1 * d2) - (d1 * d3)) + (d4 * d1)) - (d1 * d1);
}

Fortran

real(8) function code(d1, d2, d3, d4)
    real(8), intent (in) :: d1
    real(8), intent (in) :: d2
    real(8), intent (in) :: d3
    real(8), intent (in) :: d4
    code = (((d1 * d2) - (d1 * d3)) + (d4 * d1)) - (d1 * d1)
end function

Java

public static double code(double d1, double d2, double d3, double d4) {
	return (((d1 * d2) - (d1 * d3)) + (d4 * d1)) - (d1 * d1);
}

Python

def code(d1, d2, d3, d4):
	return (((d1 * d2) - (d1 * d3)) + (d4 * d1)) - (d1 * d1)

Julia

function code(d1, d2, d3, d4)
	return Float64(Float64(Float64(Float64(d1 * d2) - Float64(d1 * d3)) + Float64(d4 * d1)) - Float64(d1 * d1))
end

MATLAB

function tmp = code(d1, d2, d3, d4)
	tmp = (((d1 * d2) - (d1 * d3)) + (d4 * d1)) - (d1 * d1);
end

Wolfram

code[d1_, d2_, d3_, d4_] := N[(N[(N[(N[(d1 * d2), $MachinePrecision] - N[(d1 * d3), $MachinePrecision]), $MachinePrecision] + N[(d4 * d1), $MachinePrecision]), $MachinePrecision] - N[(d1 * d1), $MachinePrecision]), $MachinePrecision]

Initial Program: 88.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(\left(d1 \cdot d2 - d1 \cdot d3\right) + d4 \cdot d1\right) - d1 \cdot d1 \end{array} \]

FPCore

(FPCore (d1 d2 d3 d4)
 :precision binary64
 (- (+ (- (* d1 d2) (* d1 d3)) (* d4 d1)) (* d1 d1)))

C

double code(double d1, double d2, double d3, double d4) {
	return (((d1 * d2) - (d1 * d3)) + (d4 * d1)) - (d1 * d1);
}

Fortran

real(8) function code(d1, d2, d3, d4)
    real(8), intent (in) :: d1
    real(8), intent (in) :: d2
    real(8), intent (in) :: d3
    real(8), intent (in) :: d4
    code = (((d1 * d2) - (d1 * d3)) + (d4 * d1)) - (d1 * d1)
end function

Java

public static double code(double d1, double d2, double d3, double d4) {
	return (((d1 * d2) - (d1 * d3)) + (d4 * d1)) - (d1 * d1);
}

Python

def code(d1, d2, d3, d4):
	return (((d1 * d2) - (d1 * d3)) + (d4 * d1)) - (d1 * d1)

Julia

function code(d1, d2, d3, d4)
	return Float64(Float64(Float64(Float64(d1 * d2) - Float64(d1 * d3)) + Float64(d4 * d1)) - Float64(d1 * d1))
end

MATLAB

function tmp = code(d1, d2, d3, d4)
	tmp = (((d1 * d2) - (d1 * d3)) + (d4 * d1)) - (d1 * d1);
end

Wolfram

code[d1_, d2_, d3_, d4_] := N[(N[(N[(N[(d1 * d2), $MachinePrecision] - N[(d1 * d3), $MachinePrecision]), $MachinePrecision] + N[(d4 * d1), $MachinePrecision]), $MachinePrecision] - N[(d1 * d1), $MachinePrecision]), $MachinePrecision]

Alternative 1: 100.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ d1 \cdot \left(d2 + \left(d4 - \left(d1 + d3\right)\right)\right) \end{array} \]

FPCore

(FPCore (d1 d2 d3 d4) :precision binary64 (* d1 (+ d2 (- d4 (+ d1 d3)))))

C

double code(double d1, double d2, double d3, double d4) {
	return d1 * (d2 + (d4 - (d1 + d3)));
}

Fortran

real(8) function code(d1, d2, d3, d4)
    real(8), intent (in) :: d1
    real(8), intent (in) :: d2
    real(8), intent (in) :: d3
    real(8), intent (in) :: d4
    code = d1 * (d2 + (d4 - (d1 + d3)))
end function

Java

public static double code(double d1, double d2, double d3, double d4) {
	return d1 * (d2 + (d4 - (d1 + d3)));
}

Python

def code(d1, d2, d3, d4):
	return d1 * (d2 + (d4 - (d1 + d3)))

Julia

function code(d1, d2, d3, d4)
	return Float64(d1 * Float64(d2 + Float64(d4 - Float64(d1 + d3))))
end

MATLAB

function tmp = code(d1, d2, d3, d4)
	tmp = d1 * (d2 + (d4 - (d1 + d3)));
end

Wolfram

code[d1_, d2_, d3_, d4_] := N[(d1 * N[(d2 + N[(d4 - N[(d1 + d3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 86.3%

   \[\left(\left(d1 \cdot d2 - d1 \cdot d3\right) + d4 \cdot d1\right) - d1 \cdot d1 \]
2. Step-by-step derivation

   1. distribute-lft-out-- N/A

      \[\leadsto \left(d1 \cdot \left(d2 - d3\right) + d4 \cdot d1\right) - d1 \cdot d1 \]

   2. *-commutative N/A

      \[\leadsto \left(d1 \cdot \left(d2 - d3\right) + d1 \cdot d4\right) - d1 \cdot d1 \]

   3. distribute-lft-out N/A

      \[\leadsto d1 \cdot \left(\left(d2 - d3\right) + d4\right) - \color{blue}{d1} \cdot d1 \]

   4. distribute-lft-out-- N/A

      \[\leadsto d1 \cdot \color{blue}{\left(\left(\left(d2 - d3\right) + d4\right) - d1\right)} \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(d1, \color{blue}{\left(\left(\left(d2 - d3\right) + d4\right) - d1\right)}\right) \]

   6. associate-+r- N/A

      \[\leadsto \mathsf{*.f64}\left(d1, \left(\left(d2 - d3\right) + \color{blue}{\left(d4 - d1\right)}\right)\right) \]

   7. sub-neg N/A

      \[\leadsto \mathsf{*.f64}\left(d1, \left(\left(d2 + \left(\mathsf{neg}\left(d3\right)\right)\right) + \left(\color{blue}{d4} - d1\right)\right)\right) \]

   8. associate-+l+ N/A

      \[\leadsto \mathsf{*.f64}\left(d1, \left(d2 + \color{blue}{\left(\left(\mathsf{neg}\left(d3\right)\right) + \left(d4 - d1\right)\right)}\right)\right) \]

   9. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(d1, \mathsf{+.f64}\left(d2, \color{blue}{\left(\left(\mathsf{neg}\left(d3\right)\right) + \left(d4 - d1\right)\right)}\right)\right) \]

   10. associate-+r- N/A

       \[\leadsto \mathsf{*.f64}\left(d1, \mathsf{+.f64}\left(d2, \left(\left(\left(\mathsf{neg}\left(d3\right)\right) + d4\right) - \color{blue}{d1}\right)\right)\right) \]

   11. +-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(d1, \mathsf{+.f64}\left(d2, \left(\left(d4 + \left(\mathsf{neg}\left(d3\right)\right)\right) - d1\right)\right)\right) \]

   12. unsub-neg N/A

       \[\leadsto \mathsf{*.f64}\left(d1, \mathsf{+.f64}\left(d2, \left(\left(d4 - d3\right) - d1\right)\right)\right) \]

   13. associate--l- N/A

       \[\leadsto \mathsf{*.f64}\left(d1, \mathsf{+.f64}\left(d2, \left(d4 - \color{blue}{\left(d3 + d1\right)}\right)\right)\right) \]

   14. --lowering--.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(d1, \mathsf{+.f64}\left(d2, \mathsf{\_.f64}\left(d4, \color{blue}{\left(d3 + d1\right)}\right)\right)\right) \]

   15. +-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(d1, \mathsf{+.f64}\left(d2, \mathsf{\_.f64}\left(d4, \left(d1 + \color{blue}{d3}\right)\right)\right)\right) \]

   16. +-lowering-+.f64 100.0%

       \[\leadsto \mathsf{*.f64}\left(d1, \mathsf{+.f64}\left(d2, \mathsf{\_.f64}\left(d4, \mathsf{+.f64}\left(d1, \color{blue}{d3}\right)\right)\right)\right) \]

3. Simplified 100.0%

   \[\leadsto \color{blue}{d1 \cdot \left(d2 + \left(d4 - \left(d1 + d3\right)\right)\right)} \]
4. Add Preprocessing
5. Add Preprocessing

Alternative 2: 89.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d1 \leq -4.6 \cdot 10^{+117}:\\ \;\;\;\;d1 \cdot \left(d2 - d1\right)\\ \mathbf{elif}\;d1 \leq 9 \cdot 10^{+129}:\\ \;\;\;\;d1 \cdot \left(d2 + \left(d4 - d3\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d1 \cdot \left(d4 - d1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (d1 d2 d3 d4)
 :precision binary64
 (if (<= d1 -4.6e+117)
   (* d1 (- d2 d1))
   (if (<= d1 9e+129) (* d1 (+ d2 (- d4 d3))) (* d1 (- d4 d1)))))

C

double code(double d1, double d2, double d3, double d4) {
	double tmp;
	if (d1 <= -4.6e+117) {
		tmp = d1 * (d2 - d1);
	} else if (d1 <= 9e+129) {
		tmp = d1 * (d2 + (d4 - d3));
	} else {
		tmp = d1 * (d4 - d1);
	}
	return tmp;
}

Fortran

real(8) function code(d1, d2, d3, d4)
    real(8), intent (in) :: d1
    real(8), intent (in) :: d2
    real(8), intent (in) :: d3
    real(8), intent (in) :: d4
    real(8) :: tmp
    if (d1 <= (-4.6d+117)) then
        tmp = d1 * (d2 - d1)
    else if (d1 <= 9d+129) then
        tmp = d1 * (d2 + (d4 - d3))
    else
        tmp = d1 * (d4 - d1)
    end if
    code = tmp
end function

Java

public static double code(double d1, double d2, double d3, double d4) {
	double tmp;
	if (d1 <= -4.6e+117) {
		tmp = d1 * (d2 - d1);
	} else if (d1 <= 9e+129) {
		tmp = d1 * (d2 + (d4 - d3));
	} else {
		tmp = d1 * (d4 - d1);
	}
	return tmp;
}

Python

def code(d1, d2, d3, d4):
	tmp = 0
	if d1 <= -4.6e+117:
		tmp = d1 * (d2 - d1)
	elif d1 <= 9e+129:
		tmp = d1 * (d2 + (d4 - d3))
	else:
		tmp = d1 * (d4 - d1)
	return tmp

Julia

function code(d1, d2, d3, d4)
	tmp = 0.0
	if (d1 <= -4.6e+117)
		tmp = Float64(d1 * Float64(d2 - d1));
	elseif (d1 <= 9e+129)
		tmp = Float64(d1 * Float64(d2 + Float64(d4 - d3)));
	else
		tmp = Float64(d1 * Float64(d4 - d1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d1, d2, d3, d4)
	tmp = 0.0;
	if (d1 <= -4.6e+117)
		tmp = d1 * (d2 - d1);
	elseif (d1 <= 9e+129)
		tmp = d1 * (d2 + (d4 - d3));
	else
		tmp = d1 * (d4 - d1);
	end
	tmp_2 = tmp;
end

Wolfram

code[d1_, d2_, d3_, d4_] := If[LessEqual[d1, -4.6e+117], N[(d1 * N[(d2 - d1), $MachinePrecision]), $MachinePrecision], If[LessEqual[d1, 9e+129], N[(d1 * N[(d2 + N[(d4 - d3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d1 * N[(d4 - d1), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if d1 < -4.59999999999999976e117

   1. Initial program 51.4%

      \[\left(\left(d1 \cdot d2 - d1 \cdot d3\right) + d4 \cdot d1\right) - d1 \cdot d1 \]
   2. Step-by-step derivation

      1. distribute-lft-out-- N/A

         \[\leadsto \left(d1 \cdot \left(d2 - d3\right) + d4 \cdot d1\right) - d1 \cdot d1 \]

      2. *-commutative N/A

         \[\leadsto \left(d1 \cdot \left(d2 - d3\right) + d1 \cdot d4\right) - d1 \cdot d1 \]

      3. distribute-lft-out N/A

         \[\leadsto d1 \cdot \left(\left(d2 - d3\right) + d4\right) - \color{blue}{d1} \cdot d1 \]

      4. distribute-lft-out-- N/A

         \[\leadsto d1 \cdot \color{blue}{\left(\left(\left(d2 - d3\right) + d4\right) - d1\right)} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(d1, \color{blue}{\left(\left(\left(d2 - d3\right) + d4\right) - d1\right)}\right) \]

      6. associate-+r- N/A

         \[\leadsto \mathsf{*.f64}\left(d1, \left(\left(d2 - d3\right) + \color{blue}{\left(d4 - d1\right)}\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(d1, \left(\left(d2 + \left(\mathsf{neg}\left(d3\right)\right)\right) + \left(\color{blue}{d4} - d1\right)\right)\right) \]

      8. associate-+l+ N/A

         \[\leadsto \mathsf{*.f64}\left(d1, \left(d2 + \color{blue}{\left(\left(\mathsf{neg}\left(d3\right)\right) + \left(d4 - d1\right)\right)}\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(d1, \mathsf{+.f64}\left(d2, \color{blue}{\left(\left(\mathsf{neg}\left(d3\right)\right) + \left(d4 - d1\right)\right)}\right)\right) \]

      10. associate-+r- N/A

          \[\leadsto \mathsf{*.f64}\left(d1, \mathsf{+.f64}\left(d2, \left(\left(\left(\mathsf{neg}\left(d3\right)\right) + d4\right) - \color{blue}{d1}\right)\right)\right) \]

      11. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(d1, \mathsf{+.f64}\left(d2, \left(\left(d4 + \left(\mathsf{neg}\left(d3\right)\right)\right) - d1\right)\right)\right) \]

      12. unsub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(d1, \mathsf{+.f64}\left(d2, \left(\left(d4 - d3\right) - d1\right)\right)\right) \]

      13. associate--l- N/A

          \[\leadsto \mathsf{*.f64}\left(d1, \mathsf{+.f64}\left(d2, \left(d4 - \color{blue}{\left(d3 + d1\right)}\right)\right)\right) \]

      14. --lowering--.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(d1, \mathsf{+.f64}\left(d2, \mathsf{\_.f64}\left(d4, \color{blue}{\left(d3 + d1\right)}\right)\right)\right) \]

      15. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(d1, \mathsf{+.f64}\left(d2, \mathsf{\_.f64}\left(d4, \left(d1 + \color{blue}{d3}\right)\right)\right)\right) \]

      16. +-lowering-+.f64 100.0%

          \[\leadsto \mathsf{*.f64}\left(d1, \mathsf{+.f64}\left(d2, \mathsf{\_.f64}\left(d4, \mathsf{+.f64}\left(d1, \color{blue}{d3}\right)\right)\right)\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{d1 \cdot \left(d2 + \left(d4 - \left(d1 + d3\right)\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto d1 \cdot \left(\left(d4 - \left(d1 + d3\right)\right) + \color{blue}{d2}\right) \]

      2. distribute-rgt-in N/A

         \[\leadsto \left(d4 - \left(d1 + d3\right)\right) \cdot d1 + \color{blue}{d2 \cdot d1} \]

      3. *-commutative N/A

         \[\leadsto \left(d4 - \left(d1 + d3\right)\right) \cdot d1 + d1 \cdot \color{blue}{d2} \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(d4 - \left(d1 + d3\right)\right) \cdot d1\right), \color{blue}{\left(d1 \cdot d2\right)}\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\left(d1 \cdot \left(d4 - \left(d1 + d3\right)\right)\right), \left(\color{blue}{d1} \cdot d2\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(d1, \left(d4 - \left(d1 + d3\right)\right)\right), \left(\color{blue}{d1} \cdot d2\right)\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(d1, \mathsf{\_.f64}\left(d4, \left(d1 + d3\right)\right)\right), \left(d1 \cdot d2\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(d1, \mathsf{\_.f64}\left(d4, \mathsf{+.f64}\left(d1, d3\right)\right)\right), \left(d1 \cdot d2\right)\right) \]

      9. *-lowering-*.f64 74.3%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(d1, \mathsf{\_.f64}\left(d4, \mathsf{+.f64}\left(d1, d3\right)\right)\right), \mathsf{*.f64}\left(d1, \color{blue}{d2}\right)\right) \]

   6. Applied egg-rr 74.3%

      \[\leadsto \color{blue}{d1 \cdot \left(d4 - \left(d1 + d3\right)\right) + d1 \cdot d2} \]

   7. Taylor expanded in d4 around 0

      \[\leadsto \color{blue}{-1 \cdot \left(d1 \cdot \left(d1 + d3\right)\right) + d1 \cdot d2} \]
   8. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto d1 \cdot d2 + \color{blue}{-1 \cdot \left(d1 \cdot \left(d1 + d3\right)\right)} \]

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. distribute-lft-out-- N/A

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

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(d1, \color{blue}{\left(d2 - \left(d1 + d3\right)\right)}\right) \]

      6. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(d1, \mathsf{\_.f64}\left(d2, \color{blue}{\left(d1 + d3\right)}\right)\right) \]

      7. +-lowering-+.f64 97.1%

         \[\leadsto \mathsf{*.f64}\left(d1, \mathsf{\_.f64}\left(d2, \mathsf{+.f64}\left(d1, \color{blue}{d3}\right)\right)\right) \]

   9. Simplified 97.1%

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

   10. Taylor expanded in d3 around 0

       \[\leadsto \color{blue}{d1 \cdot \left(d2 - d1\right)} \]
   11. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(d1, \color{blue}{\left(d2 - d1\right)}\right) \]

       2. --lowering--.f64 88.6%

          \[\leadsto \mathsf{*.f64}\left(d1, \mathsf{\_.f64}\left(d2, \color{blue}{d1}\right)\right) \]

   12. Simplified 88.6%

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

   if -4.59999999999999976e117 < d1 < 9.0000000000000003e129

   1. Initial program 99.4%

      \[\left(\left(d1 \cdot d2 - d1 \cdot d3\right) + d4 \cdot d1\right) - d1 \cdot d1 \]
   2. Step-by-step derivation

      1. distribute-lft-out-- N/A

         \[\leadsto \left(d1 \cdot \left(d2 - d3\right) + d4 \cdot d1\right) - d1 \cdot d1 \]

      2. *-commutative N/A

         \[\leadsto \left(d1 \cdot \left(d2 - d3\right) + d1 \cdot d4\right) - d1 \cdot d1 \]

      3. distribute-lft-out N/A

         \[\leadsto d1 \cdot \left(\left(d2 - d3\right) + d4\right) - \color{blue}{d1} \cdot d1 \]

      4. distribute-lft-out-- N/A

         \[\leadsto d1 \cdot \color{blue}{\left(\left(\left(d2 - d3\right) + d4\right) - d1\right)} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(d1, \color{blue}{\left(\left(\left(d2 - d3\right) + d4\right) - d1\right)}\right) \]

      6. associate-+r- N/A

         \[\leadsto \mathsf{*.f64}\left(d1, \left(\left(d2 - d3\right) + \color{blue}{\left(d4 - d1\right)}\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(d1, \left(\left(d2 + \left(\mathsf{neg}\left(d3\right)\right)\right) + \left(\color{blue}{d4} - d1\right)\right)\right) \]

      8. associate-+l+ N/A

         \[\leadsto \mathsf{*.f64}\left(d1, \left(d2 + \color{blue}{\left(\left(\mathsf{neg}\left(d3\right)\right) + \left(d4 - d1\right)\right)}\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(d1, \mathsf{+.f64}\left(d2, \color{blue}{\left(\left(\mathsf{neg}\left(d3\right)\right) + \left(d4 - d1\right)\right)}\right)\right) \]

      10. associate-+r- N/A

          \[\leadsto \mathsf{*.f64}\left(d1, \mathsf{+.f64}\left(d2, \left(\left(\left(\mathsf{neg}\left(d3\right)\right) + d4\right) - \color{blue}{d1}\right)\right)\right) \]

      11. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(d1, \mathsf{+.f64}\left(d2, \left(\left(d4 + \left(\mathsf{neg}\left(d3\right)\right)\right) - d1\right)\right)\right) \]

      12. unsub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(d1, \mathsf{+.f64}\left(d2, \left(\left(d4 - d3\right) - d1\right)\right)\right) \]

      13. associate--l- N/A

          \[\leadsto \mathsf{*.f64}\left(d1, \mathsf{+.f64}\left(d2, \left(d4 - \color{blue}{\left(d3 + d1\right)}\right)\right)\right) \]

      14. --lowering--.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(d1, \mathsf{+.f64}\left(d2, \mathsf{\_.f64}\left(d4, \color{blue}{\left(d3 + d1\right)}\right)\right)\right) \]

      15. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(d1, \mathsf{+.f64}\left(d2, \mathsf{\_.f64}\left(d4, \left(d1 + \color{blue}{d3}\right)\right)\right)\right) \]

      16. +-lowering-+.f64 100.0%

          \[\leadsto \mathsf{*.f64}\left(d1, \mathsf{+.f64}\left(d2, \mathsf{\_.f64}\left(d4, \mathsf{+.f64}\left(d1, \color{blue}{d3}\right)\right)\right)\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{d1 \cdot \left(d2 + \left(d4 - \left(d1 + d3\right)\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d1 around 0

      \[\leadsto \color{blue}{d1 \cdot \left(\left(d2 + d4\right) - d3\right)} \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(d1, \color{blue}{\left(\left(d2 + d4\right) - d3\right)}\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(d1, \left(\left(d2 + d4\right) + \color{blue}{\left(\mathsf{neg}\left(d3\right)\right)}\right)\right) \]

      3. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(d1, \left(\left(d2 + d4\right) + -1 \cdot \color{blue}{d3}\right)\right) \]

      4. associate-+r+ N/A

         \[\leadsto \mathsf{*.f64}\left(d1, \left(d2 + \color{blue}{\left(d4 + -1 \cdot d3\right)}\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(d1, \mathsf{+.f64}\left(d2, \color{blue}{\left(d4 + -1 \cdot d3\right)}\right)\right) \]

      6. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(d1, \mathsf{+.f64}\left(d2, \left(d4 + \left(\mathsf{neg}\left(d3\right)\right)\right)\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(d1, \mathsf{+.f64}\left(d2, \left(d4 - \color{blue}{d3}\right)\right)\right) \]

      8. --lowering--.f64 91.7%

         \[\leadsto \mathsf{*.f64}\left(d1, \mathsf{+.f64}\left(d2, \mathsf{\_.f64}\left(d4, \color{blue}{d3}\right)\right)\right) \]

   7. Simplified 91.7%

      \[\leadsto \color{blue}{d1 \cdot \left(d2 + \left(d4 - d3\right)\right)} \]

   if 9.0000000000000003e129 < d1

   1. Initial program 55.3%

      \[\left(\left(d1 \cdot d2 - d1 \cdot d3\right) + d4 \cdot d1\right) - d1 \cdot d1 \]
   2. Step-by-step derivation

      1. distribute-lft-out-- N/A

         \[\leadsto \left(d1 \cdot \left(d2 - d3\right) + d4 \cdot d1\right) - d1 \cdot d1 \]

      2. *-commutative N/A

         \[\leadsto \left(d1 \cdot \left(d2 - d3\right) + d1 \cdot d4\right) - d1 \cdot d1 \]

      3. distribute-lft-out N/A

         \[\leadsto d1 \cdot \left(\left(d2 - d3\right) + d4\right) - \color{blue}{d1} \cdot d1 \]

      4. distribute-lft-out-- N/A

         \[\leadsto d1 \cdot \color{blue}{\left(\left(\left(d2 - d3\right) + d4\right) - d1\right)} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(d1, \color{blue}{\left(\left(\left(d2 - d3\right) + d4\right) - d1\right)}\right) \]

      6. associate-+r- N/A

         \[\leadsto \mathsf{*.f64}\left(d1, \left(\left(d2 - d3\right) + \color{blue}{\left(d4 - d1\right)}\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(d1, \left(\left(d2 + \left(\mathsf{neg}\left(d3\right)\right)\right) + \left(\color{blue}{d4} - d1\right)\right)\right) \]

      8. associate-+l+ N/A

         \[\leadsto \mathsf{*.f64}\left(d1, \left(d2 + \color{blue}{\left(\left(\mathsf{neg}\left(d3\right)\right) + \left(d4 - d1\right)\right)}\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(d1, \mathsf{+.f64}\left(d2, \color{blue}{\left(\left(\mathsf{neg}\left(d3\right)\right) + \left(d4 - d1\right)\right)}\right)\right) \]

      10. associate-+r- N/A

          \[\leadsto \mathsf{*.f64}\left(d1, \mathsf{+.f64}\left(d2, \left(\left(\left(\mathsf{neg}\left(d3\right)\right) + d4\right) - \color{blue}{d1}\right)\right)\right) \]

      11. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(d1, \mathsf{+.f64}\left(d2, \left(\left(d4 + \left(\mathsf{neg}\left(d3\right)\right)\right) - d1\right)\right)\right) \]

      12. unsub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(d1, \mathsf{+.f64}\left(d2, \left(\left(d4 - d3\right) - d1\right)\right)\right) \]

      13. associate--l- N/A

          \[\leadsto \mathsf{*.f64}\left(d1, \mathsf{+.f64}\left(d2, \left(d4 - \color{blue}{\left(d3 + d1\right)}\right)\right)\right) \]

      14. --lowering--.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(d1, \mathsf{+.f64}\left(d2, \mathsf{\_.f64}\left(d4, \color{blue}{\left(d3 + d1\right)}\right)\right)\right) \]

      15. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(d1, \mathsf{+.f64}\left(d2, \mathsf{\_.f64}\left(d4, \left(d1 + \color{blue}{d3}\right)\right)\right)\right) \]

      16. +-lowering-+.f64 100.0%

          \[\leadsto \mathsf{*.f64}\left(d1, \mathsf{+.f64}\left(d2, \mathsf{\_.f64}\left(d4, \mathsf{+.f64}\left(d1, \color{blue}{d3}\right)\right)\right)\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{d1 \cdot \left(d2 + \left(d4 - \left(d1 + d3\right)\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d2 around 0

      \[\leadsto \color{blue}{d1 \cdot \left(d4 - \left(d1 + d3\right)\right)} \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(d1, \color{blue}{\left(d4 - \left(d1 + d3\right)\right)}\right) \]

      2. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(d1, \mathsf{\_.f64}\left(d4, \color{blue}{\left(d1 + d3\right)}\right)\right) \]

      3. +-lowering-+.f64 91.8%

         \[\leadsto \mathsf{*.f64}\left(d1, \mathsf{\_.f64}\left(d4, \mathsf{+.f64}\left(d1, \color{blue}{d3}\right)\right)\right) \]

   7. Simplified 91.8%

      \[\leadsto \color{blue}{d1 \cdot \left(d4 - \left(d1 + d3\right)\right)} \]

   8. Taylor expanded in d3 around 0

      \[\leadsto \color{blue}{d1 \cdot \left(d4 - d1\right)} \]
   9. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(d1, \color{blue}{\left(d4 - d1\right)}\right) \]

      2. --lowering--.f64 89.2%

         \[\leadsto \mathsf{*.f64}\left(d1, \mathsf{\_.f64}\left(d4, \color{blue}{d1}\right)\right) \]

   10. Simplified 89.2%

       \[\leadsto \color{blue}{d1 \cdot \left(d4 - d1\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 70.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := d1 \cdot \left(d2 - d3\right)\\ \mathbf{if}\;d3 \leq -1.2 \cdot 10^{+85}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d3 \leq 2.1 \cdot 10^{+65}:\\ \;\;\;\;d1 \cdot \left(d4 - d1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (d1 d2 d3 d4)
 :precision binary64
 (let* ((t_0 (* d1 (- d2 d3))))
   (if (<= d3 -1.2e+85) t_0 (if (<= d3 2.1e+65) (* d1 (- d4 d1)) t_0))))

C

double code(double d1, double d2, double d3, double d4) {
	double t_0 = d1 * (d2 - d3);
	double tmp;
	if (d3 <= -1.2e+85) {
		tmp = t_0;
	} else if (d3 <= 2.1e+65) {
		tmp = d1 * (d4 - d1);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(d1, d2, d3, d4)
    real(8), intent (in) :: d1
    real(8), intent (in) :: d2
    real(8), intent (in) :: d3
    real(8), intent (in) :: d4
    real(8) :: t_0
    real(8) :: tmp
    t_0 = d1 * (d2 - d3)
    if (d3 <= (-1.2d+85)) then
        tmp = t_0
    else if (d3 <= 2.1d+65) then
        tmp = d1 * (d4 - d1)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double d1, double d2, double d3, double d4) {
	double t_0 = d1 * (d2 - d3);
	double tmp;
	if (d3 <= -1.2e+85) {
		tmp = t_0;
	} else if (d3 <= 2.1e+65) {
		tmp = d1 * (d4 - d1);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(d1, d2, d3, d4):
	t_0 = d1 * (d2 - d3)
	tmp = 0
	if d3 <= -1.2e+85:
		tmp = t_0
	elif d3 <= 2.1e+65:
		tmp = d1 * (d4 - d1)
	else:
		tmp = t_0
	return tmp

Julia

function code(d1, d2, d3, d4)
	t_0 = Float64(d1 * Float64(d2 - d3))
	tmp = 0.0
	if (d3 <= -1.2e+85)
		tmp = t_0;
	elseif (d3 <= 2.1e+65)
		tmp = Float64(d1 * Float64(d4 - d1));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(d1, d2, d3, d4)
	t_0 = d1 * (d2 - d3);
	tmp = 0.0;
	if (d3 <= -1.2e+85)
		tmp = t_0;
	elseif (d3 <= 2.1e+65)
		tmp = d1 * (d4 - d1);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[d1_, d2_, d3_, d4_] := Block[{t$95$0 = N[(d1 * N[(d2 - d3), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d3, -1.2e+85], t$95$0, If[LessEqual[d3, 2.1e+65], N[(d1 * N[(d4 - d1), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if d3 < -1.19999999999999998e85 or 2.09999999999999991e65 < d3

   1. Initial program 84.0%

      \[\left(\left(d1 \cdot d2 - d1 \cdot d3\right) + d4 \cdot d1\right) - d1 \cdot d1 \]
   2. Step-by-step derivation

      1. distribute-lft-out-- N/A

         \[\leadsto \left(d1 \cdot \left(d2 - d3\right) + d4 \cdot d1\right) - d1 \cdot d1 \]

      2. *-commutative N/A

         \[\leadsto \left(d1 \cdot \left(d2 - d3\right) + d1 \cdot d4\right) - d1 \cdot d1 \]

      3. distribute-lft-out N/A

         \[\leadsto d1 \cdot \left(\left(d2 - d3\right) + d4\right) - \color{blue}{d1} \cdot d1 \]

      4. distribute-lft-out-- N/A

         \[\leadsto d1 \cdot \color{blue}{\left(\left(\left(d2 - d3\right) + d4\right) - d1\right)} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(d1, \color{blue}{\left(\left(\left(d2 - d3\right) + d4\right) - d1\right)}\right) \]

      6. associate-+r- N/A

         \[\leadsto \mathsf{*.f64}\left(d1, \left(\left(d2 - d3\right) + \color{blue}{\left(d4 - d1\right)}\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(d1, \left(\left(d2 + \left(\mathsf{neg}\left(d3\right)\right)\right) + \left(\color{blue}{d4} - d1\right)\right)\right) \]

      8. associate-+l+ N/A

         \[\leadsto \mathsf{*.f64}\left(d1, \left(d2 + \color{blue}{\left(\left(\mathsf{neg}\left(d3\right)\right) + \left(d4 - d1\right)\right)}\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(d1, \mathsf{+.f64}\left(d2, \color{blue}{\left(\left(\mathsf{neg}\left(d3\right)\right) + \left(d4 - d1\right)\right)}\right)\right) \]

      10. associate-+r- N/A

          \[\leadsto \mathsf{*.f64}\left(d1, \mathsf{+.f64}\left(d2, \left(\left(\left(\mathsf{neg}\left(d3\right)\right) + d4\right) - \color{blue}{d1}\right)\right)\right) \]

      11. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(d1, \mathsf{+.f64}\left(d2, \left(\left(d4 + \left(\mathsf{neg}\left(d3\right)\right)\right) - d1\right)\right)\right) \]

      12. unsub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(d1, \mathsf{+.f64}\left(d2, \left(\left(d4 - d3\right) - d1\right)\right)\right) \]

      13. associate--l- N/A

          \[\leadsto \mathsf{*.f64}\left(d1, \mathsf{+.f64}\left(d2, \left(d4 - \color{blue}{\left(d3 + d1\right)}\right)\right)\right) \]

      14. --lowering--.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(d1, \mathsf{+.f64}\left(d2, \mathsf{\_.f64}\left(d4, \color{blue}{\left(d3 + d1\right)}\right)\right)\right) \]

      15. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(d1, \mathsf{+.f64}\left(d2, \mathsf{\_.f64}\left(d4, \left(d1 + \color{blue}{d3}\right)\right)\right)\right) \]

      16. +-lowering-+.f64 100.0%

          \[\leadsto \mathsf{*.f64}\left(d1, \mathsf{+.f64}\left(d2, \mathsf{\_.f64}\left(d4, \mathsf{+.f64}\left(d1, \color{blue}{d3}\right)\right)\right)\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{d1 \cdot \left(d2 + \left(d4 - \left(d1 + d3\right)\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto d1 \cdot \left(\left(d4 - \left(d1 + d3\right)\right) + \color{blue}{d2}\right) \]

      2. distribute-rgt-in N/A

         \[\leadsto \left(d4 - \left(d1 + d3\right)\right) \cdot d1 + \color{blue}{d2 \cdot d1} \]

      3. *-commutative N/A

         \[\leadsto \left(d4 - \left(d1 + d3\right)\right) \cdot d1 + d1 \cdot \color{blue}{d2} \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(d4 - \left(d1 + d3\right)\right) \cdot d1\right), \color{blue}{\left(d1 \cdot d2\right)}\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\left(d1 \cdot \left(d4 - \left(d1 + d3\right)\right)\right), \left(\color{blue}{d1} \cdot d2\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(d1, \left(d4 - \left(d1 + d3\right)\right)\right), \left(\color{blue}{d1} \cdot d2\right)\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(d1, \mathsf{\_.f64}\left(d4, \left(d1 + d3\right)\right)\right), \left(d1 \cdot d2\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(d1, \mathsf{\_.f64}\left(d4, \mathsf{+.f64}\left(d1, d3\right)\right)\right), \left(d1 \cdot d2\right)\right) \]

      9. *-lowering-*.f64 92.5%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(d1, \mathsf{\_.f64}\left(d4, \mathsf{+.f64}\left(d1, d3\right)\right)\right), \mathsf{*.f64}\left(d1, \color{blue}{d2}\right)\right) \]

   6. Applied egg-rr 92.5%

      \[\leadsto \color{blue}{d1 \cdot \left(d4 - \left(d1 + d3\right)\right) + d1 \cdot d2} \]

   7. Taylor expanded in d4 around 0

      \[\leadsto \color{blue}{-1 \cdot \left(d1 \cdot \left(d1 + d3\right)\right) + d1 \cdot d2} \]
   8. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto d1 \cdot d2 + \color{blue}{-1 \cdot \left(d1 \cdot \left(d1 + d3\right)\right)} \]

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. distribute-lft-out-- N/A

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

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(d1, \color{blue}{\left(d2 - \left(d1 + d3\right)\right)}\right) \]

      6. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(d1, \mathsf{\_.f64}\left(d2, \color{blue}{\left(d1 + d3\right)}\right)\right) \]

      7. +-lowering-+.f64 86.7%

         \[\leadsto \mathsf{*.f64}\left(d1, \mathsf{\_.f64}\left(d2, \mathsf{+.f64}\left(d1, \color{blue}{d3}\right)\right)\right) \]

   9. Simplified 86.7%

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

   10. Taylor expanded in d1 around 0

       \[\leadsto \color{blue}{d1 \cdot \left(d2 - d3\right)} \]
   11. Step-by-step derivation

       1. remove-double-neg N/A

          \[\leadsto \mathsf{neg}\left(\left(\mathsf{neg}\left(d1 \cdot \left(d2 - d3\right)\right)\right)\right) \]

       2. distribute-rgt-neg-out N/A

          \[\leadsto \mathsf{neg}\left(d1 \cdot \left(\mathsf{neg}\left(\left(d2 - d3\right)\right)\right)\right) \]

       3. mul-1-neg N/A

          \[\leadsto \mathsf{neg}\left(d1 \cdot \left(-1 \cdot \left(d2 - d3\right)\right)\right) \]

       4. distribute-rgt-neg-in N/A

          \[\leadsto d1 \cdot \color{blue}{\left(\mathsf{neg}\left(-1 \cdot \left(d2 - d3\right)\right)\right)} \]

       5. *-rgt-identity N/A

          \[\leadsto d1 \cdot \left(\mathsf{neg}\left(\left(-1 \cdot \left(d2 - d3\right)\right) \cdot 1\right)\right) \]

       6. *-inverses N/A

          \[\leadsto d1 \cdot \left(\mathsf{neg}\left(\left(-1 \cdot \left(d2 - d3\right)\right) \cdot \frac{d1}{d1}\right)\right) \]

       7. associate-/l* N/A

          \[\leadsto d1 \cdot \left(\mathsf{neg}\left(\frac{\left(-1 \cdot \left(d2 - d3\right)\right) \cdot d1}{d1}\right)\right) \]

       8. associate-*l/ N/A

          \[\leadsto d1 \cdot \left(\mathsf{neg}\left(\frac{-1 \cdot \left(d2 - d3\right)}{d1} \cdot d1\right)\right) \]

       9. associate-*r/ N/A

          \[\leadsto d1 \cdot \left(\mathsf{neg}\left(\left(-1 \cdot \frac{d2 - d3}{d1}\right) \cdot d1\right)\right) \]

       10. *-commutative N/A

           \[\leadsto d1 \cdot \left(\mathsf{neg}\left(d1 \cdot \left(-1 \cdot \frac{d2 - d3}{d1}\right)\right)\right) \]

       11. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(d1, \color{blue}{\left(\mathsf{neg}\left(d1 \cdot \left(-1 \cdot \frac{d2 - d3}{d1}\right)\right)\right)}\right) \]

       12. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(d1, \left(\mathsf{neg}\left(\left(-1 \cdot \frac{d2 - d3}{d1}\right) \cdot d1\right)\right)\right) \]

       13. associate-*r/ N/A

           \[\leadsto \mathsf{*.f64}\left(d1, \left(\mathsf{neg}\left(\frac{-1 \cdot \left(d2 - d3\right)}{d1} \cdot d1\right)\right)\right) \]

       14. associate-*l/ N/A

           \[\leadsto \mathsf{*.f64}\left(d1, \left(\mathsf{neg}\left(\frac{\left(-1 \cdot \left(d2 - d3\right)\right) \cdot d1}{d1}\right)\right)\right) \]

       15. associate-/l* N/A

           \[\leadsto \mathsf{*.f64}\left(d1, \left(\mathsf{neg}\left(\left(-1 \cdot \left(d2 - d3\right)\right) \cdot \frac{d1}{d1}\right)\right)\right) \]

       16. *-inverses N/A

           \[\leadsto \mathsf{*.f64}\left(d1, \left(\mathsf{neg}\left(\left(-1 \cdot \left(d2 - d3\right)\right) \cdot 1\right)\right)\right) \]

       17. *-rgt-identity N/A

           \[\leadsto \mathsf{*.f64}\left(d1, \left(\mathsf{neg}\left(-1 \cdot \left(d2 - d3\right)\right)\right)\right) \]

       18. sub-neg N/A

           \[\leadsto \mathsf{*.f64}\left(d1, \left(\mathsf{neg}\left(-1 \cdot \left(d2 + \left(\mathsf{neg}\left(d3\right)\right)\right)\right)\right)\right) \]

       19. distribute-lft-in N/A

           \[\leadsto \mathsf{*.f64}\left(d1, \left(\mathsf{neg}\left(\left(-1 \cdot d2 + -1 \cdot \left(\mathsf{neg}\left(d3\right)\right)\right)\right)\right)\right) \]

       20. mul-1-neg N/A

           \[\leadsto \mathsf{*.f64}\left(d1, \left(\mathsf{neg}\left(\left(-1 \cdot d2 + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(d3\right)\right)\right)\right)\right)\right)\right)\right) \]

       21. remove-double-neg N/A

           \[\leadsto \mathsf{*.f64}\left(d1, \left(\mathsf{neg}\left(\left(-1 \cdot d2 + d3\right)\right)\right)\right) \]

   12. Simplified 81.7%

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

   if -1.19999999999999998e85 < d3 < 2.09999999999999991e65

   1. Initial program 87.6%

      \[\left(\left(d1 \cdot d2 - d1 \cdot d3\right) + d4 \cdot d1\right) - d1 \cdot d1 \]
   2. Step-by-step derivation

      1. distribute-lft-out-- N/A

         \[\leadsto \left(d1 \cdot \left(d2 - d3\right) + d4 \cdot d1\right) - d1 \cdot d1 \]

      2. *-commutative N/A

         \[\leadsto \left(d1 \cdot \left(d2 - d3\right) + d1 \cdot d4\right) - d1 \cdot d1 \]

      3. distribute-lft-out N/A

         \[\leadsto d1 \cdot \left(\left(d2 - d3\right) + d4\right) - \color{blue}{d1} \cdot d1 \]

      4. distribute-lft-out-- N/A

         \[\leadsto d1 \cdot \color{blue}{\left(\left(\left(d2 - d3\right) + d4\right) - d1\right)} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(d1, \color{blue}{\left(\left(\left(d2 - d3\right) + d4\right) - d1\right)}\right) \]

      6. associate-+r- N/A

         \[\leadsto \mathsf{*.f64}\left(d1, \left(\left(d2 - d3\right) + \color{blue}{\left(d4 - d1\right)}\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(d1, \left(\left(d2 + \left(\mathsf{neg}\left(d3\right)\right)\right) + \left(\color{blue}{d4} - d1\right)\right)\right) \]

      8. associate-+l+ N/A

         \[\leadsto \mathsf{*.f64}\left(d1, \left(d2 + \color{blue}{\left(\left(\mathsf{neg}\left(d3\right)\right) + \left(d4 - d1\right)\right)}\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(d1, \mathsf{+.f64}\left(d2, \color{blue}{\left(\left(\mathsf{neg}\left(d3\right)\right) + \left(d4 - d1\right)\right)}\right)\right) \]

      10. associate-+r- N/A

          \[\leadsto \mathsf{*.f64}\left(d1, \mathsf{+.f64}\left(d2, \left(\left(\left(\mathsf{neg}\left(d3\right)\right) + d4\right) - \color{blue}{d1}\right)\right)\right) \]

      11. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(d1, \mathsf{+.f64}\left(d2, \left(\left(d4 + \left(\mathsf{neg}\left(d3\right)\right)\right) - d1\right)\right)\right) \]

      12. unsub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(d1, \mathsf{+.f64}\left(d2, \left(\left(d4 - d3\right) - d1\right)\right)\right) \]

      13. associate--l- N/A

          \[\leadsto \mathsf{*.f64}\left(d1, \mathsf{+.f64}\left(d2, \left(d4 - \color{blue}{\left(d3 + d1\right)}\right)\right)\right) \]

      14. --lowering--.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(d1, \mathsf{+.f64}\left(d2, \mathsf{\_.f64}\left(d4, \color{blue}{\left(d3 + d1\right)}\right)\right)\right) \]

      15. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(d1, \mathsf{+.f64}\left(d2, \mathsf{\_.f64}\left(d4, \left(d1 + \color{blue}{d3}\right)\right)\right)\right) \]

      16. +-lowering-+.f64 100.0%

          \[\leadsto \mathsf{*.f64}\left(d1, \mathsf{+.f64}\left(d2, \mathsf{\_.f64}\left(d4, \mathsf{+.f64}\left(d1, \color{blue}{d3}\right)\right)\right)\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{d1 \cdot \left(d2 + \left(d4 - \left(d1 + d3\right)\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d2 around 0

      \[\leadsto \color{blue}{d1 \cdot \left(d4 - \left(d1 + d3\right)\right)} \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(d1, \color{blue}{\left(d4 - \left(d1 + d3\right)\right)}\right) \]

      2. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(d1, \mathsf{\_.f64}\left(d4, \color{blue}{\left(d1 + d3\right)}\right)\right) \]

      3. +-lowering-+.f64 70.5%

         \[\leadsto \mathsf{*.f64}\left(d1, \mathsf{\_.f64}\left(d4, \mathsf{+.f64}\left(d1, \color{blue}{d3}\right)\right)\right) \]

   7. Simplified 70.5%

      \[\leadsto \color{blue}{d1 \cdot \left(d4 - \left(d1 + d3\right)\right)} \]

   8. Taylor expanded in d3 around 0

      \[\leadsto \color{blue}{d1 \cdot \left(d4 - d1\right)} \]
   9. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(d1, \color{blue}{\left(d4 - d1\right)}\right) \]

      2. --lowering--.f64 65.9%

         \[\leadsto \mathsf{*.f64}\left(d1, \mathsf{\_.f64}\left(d4, \color{blue}{d1}\right)\right) \]

   10. Simplified 65.9%

       \[\leadsto \color{blue}{d1 \cdot \left(d4 - d1\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 71.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := d1 \cdot \left(d2 - d3\right)\\ \mathbf{if}\;d3 \leq -2.9 \cdot 10^{+146}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d3 \leq 190000000000:\\ \;\;\;\;d1 \cdot \left(d2 + d4\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (d1 d2 d3 d4)
 :precision binary64
 (let* ((t_0 (* d1 (- d2 d3))))
   (if (<= d3 -2.9e+146)
     t_0
     (if (<= d3 190000000000.0) (* d1 (+ d2 d4)) t_0))))

C

double code(double d1, double d2, double d3, double d4) {
	double t_0 = d1 * (d2 - d3);
	double tmp;
	if (d3 <= -2.9e+146) {
		tmp = t_0;
	} else if (d3 <= 190000000000.0) {
		tmp = d1 * (d2 + d4);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(d1, d2, d3, d4)
    real(8), intent (in) :: d1
    real(8), intent (in) :: d2
    real(8), intent (in) :: d3
    real(8), intent (in) :: d4
    real(8) :: t_0
    real(8) :: tmp
    t_0 = d1 * (d2 - d3)
    if (d3 <= (-2.9d+146)) then
        tmp = t_0
    else if (d3 <= 190000000000.0d0) then
        tmp = d1 * (d2 + d4)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double d1, double d2, double d3, double d4) {
	double t_0 = d1 * (d2 - d3);
	double tmp;
	if (d3 <= -2.9e+146) {
		tmp = t_0;
	} else if (d3 <= 190000000000.0) {
		tmp = d1 * (d2 + d4);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(d1, d2, d3, d4):
	t_0 = d1 * (d2 - d3)
	tmp = 0
	if d3 <= -2.9e+146:
		tmp = t_0
	elif d3 <= 190000000000.0:
		tmp = d1 * (d2 + d4)
	else:
		tmp = t_0
	return tmp

Julia

function code(d1, d2, d3, d4)
	t_0 = Float64(d1 * Float64(d2 - d3))
	tmp = 0.0
	if (d3 <= -2.9e+146)
		tmp = t_0;
	elseif (d3 <= 190000000000.0)
		tmp = Float64(d1 * Float64(d2 + d4));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(d1, d2, d3, d4)
	t_0 = d1 * (d2 - d3);
	tmp = 0.0;
	if (d3 <= -2.9e+146)
		tmp = t_0;
	elseif (d3 <= 190000000000.0)
		tmp = d1 * (d2 + d4);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[d1_, d2_, d3_, d4_] := Block[{t$95$0 = N[(d1 * N[(d2 - d3), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d3, -2.9e+146], t$95$0, If[LessEqual[d3, 190000000000.0], N[(d1 * N[(d2 + d4), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if d3 < -2.8999999999999998e146 or 1.9e11 < d3

   1. Initial program 87.9%

      \[\left(\left(d1 \cdot d2 - d1 \cdot d3\right) + d4 \cdot d1\right) - d1 \cdot d1 \]
   2. Step-by-step derivation

      1. distribute-lft-out-- N/A

         \[\leadsto \left(d1 \cdot \left(d2 - d3\right) + d4 \cdot d1\right) - d1 \cdot d1 \]

      2. *-commutative N/A

         \[\leadsto \left(d1 \cdot \left(d2 - d3\right) + d1 \cdot d4\right) - d1 \cdot d1 \]

      3. distribute-lft-out N/A

         \[\leadsto d1 \cdot \left(\left(d2 - d3\right) + d4\right) - \color{blue}{d1} \cdot d1 \]

      4. distribute-lft-out-- N/A

         \[\leadsto d1 \cdot \color{blue}{\left(\left(\left(d2 - d3\right) + d4\right) - d1\right)} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(d1, \color{blue}{\left(\left(\left(d2 - d3\right) + d4\right) - d1\right)}\right) \]

      6. associate-+r- N/A

         \[\leadsto \mathsf{*.f64}\left(d1, \left(\left(d2 - d3\right) + \color{blue}{\left(d4 - d1\right)}\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(d1, \left(\left(d2 + \left(\mathsf{neg}\left(d3\right)\right)\right) + \left(\color{blue}{d4} - d1\right)\right)\right) \]

      8. associate-+l+ N/A

         \[\leadsto \mathsf{*.f64}\left(d1, \left(d2 + \color{blue}{\left(\left(\mathsf{neg}\left(d3\right)\right) + \left(d4 - d1\right)\right)}\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(d1, \mathsf{+.f64}\left(d2, \color{blue}{\left(\left(\mathsf{neg}\left(d3\right)\right) + \left(d4 - d1\right)\right)}\right)\right) \]

      10. associate-+r- N/A

          \[\leadsto \mathsf{*.f64}\left(d1, \mathsf{+.f64}\left(d2, \left(\left(\left(\mathsf{neg}\left(d3\right)\right) + d4\right) - \color{blue}{d1}\right)\right)\right) \]

      11. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(d1, \mathsf{+.f64}\left(d2, \left(\left(d4 + \left(\mathsf{neg}\left(d3\right)\right)\right) - d1\right)\right)\right) \]

      12. unsub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(d1, \mathsf{+.f64}\left(d2, \left(\left(d4 - d3\right) - d1\right)\right)\right) \]

      13. associate--l- N/A

          \[\leadsto \mathsf{*.f64}\left(d1, \mathsf{+.f64}\left(d2, \left(d4 - \color{blue}{\left(d3 + d1\right)}\right)\right)\right) \]

      14. --lowering--.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(d1, \mathsf{+.f64}\left(d2, \mathsf{\_.f64}\left(d4, \color{blue}{\left(d3 + d1\right)}\right)\right)\right) \]

      15. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(d1, \mathsf{+.f64}\left(d2, \mathsf{\_.f64}\left(d4, \left(d1 + \color{blue}{d3}\right)\right)\right)\right) \]

      16. +-lowering-+.f64 100.0%

          \[\leadsto \mathsf{*.f64}\left(d1, \mathsf{+.f64}\left(d2, \mathsf{\_.f64}\left(d4, \mathsf{+.f64}\left(d1, \color{blue}{d3}\right)\right)\right)\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{d1 \cdot \left(d2 + \left(d4 - \left(d1 + d3\right)\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto d1 \cdot \left(\left(d4 - \left(d1 + d3\right)\right) + \color{blue}{d2}\right) \]

      2. distribute-rgt-in N/A

         \[\leadsto \left(d4 - \left(d1 + d3\right)\right) \cdot d1 + \color{blue}{d2 \cdot d1} \]

      3. *-commutative N/A

         \[\leadsto \left(d4 - \left(d1 + d3\right)\right) \cdot d1 + d1 \cdot \color{blue}{d2} \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(d4 - \left(d1 + d3\right)\right) \cdot d1\right), \color{blue}{\left(d1 \cdot d2\right)}\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\left(d1 \cdot \left(d4 - \left(d1 + d3\right)\right)\right), \left(\color{blue}{d1} \cdot d2\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(d1, \left(d4 - \left(d1 + d3\right)\right)\right), \left(\color{blue}{d1} \cdot d2\right)\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(d1, \mathsf{\_.f64}\left(d4, \left(d1 + d3\right)\right)\right), \left(d1 \cdot d2\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(d1, \mathsf{\_.f64}\left(d4, \mathsf{+.f64}\left(d1, d3\right)\right)\right), \left(d1 \cdot d2\right)\right) \]

      9. *-lowering-*.f64 92.9%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(d1, \mathsf{\_.f64}\left(d4, \mathsf{+.f64}\left(d1, d3\right)\right)\right), \mathsf{*.f64}\left(d1, \color{blue}{d2}\right)\right) \]

   6. Applied egg-rr 92.9%

      \[\leadsto \color{blue}{d1 \cdot \left(d4 - \left(d1 + d3\right)\right) + d1 \cdot d2} \]

   7. Taylor expanded in d4 around 0

      \[\leadsto \color{blue}{-1 \cdot \left(d1 \cdot \left(d1 + d3\right)\right) + d1 \cdot d2} \]
   8. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto d1 \cdot d2 + \color{blue}{-1 \cdot \left(d1 \cdot \left(d1 + d3\right)\right)} \]

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. distribute-lft-out-- N/A

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

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(d1, \color{blue}{\left(d2 - \left(d1 + d3\right)\right)}\right) \]

      6. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(d1, \mathsf{\_.f64}\left(d2, \color{blue}{\left(d1 + d3\right)}\right)\right) \]

      7. +-lowering-+.f64 86.1%

         \[\leadsto \mathsf{*.f64}\left(d1, \mathsf{\_.f64}\left(d2, \mathsf{+.f64}\left(d1, \color{blue}{d3}\right)\right)\right) \]

   9. Simplified 86.1%

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

   10. Taylor expanded in d1 around 0

       \[\leadsto \color{blue}{d1 \cdot \left(d2 - d3\right)} \]
   11. Step-by-step derivation

       1. remove-double-neg N/A

          \[\leadsto \mathsf{neg}\left(\left(\mathsf{neg}\left(d1 \cdot \left(d2 - d3\right)\right)\right)\right) \]

       2. distribute-rgt-neg-out N/A

          \[\leadsto \mathsf{neg}\left(d1 \cdot \left(\mathsf{neg}\left(\left(d2 - d3\right)\right)\right)\right) \]

       3. mul-1-neg N/A

          \[\leadsto \mathsf{neg}\left(d1 \cdot \left(-1 \cdot \left(d2 - d3\right)\right)\right) \]

       4. distribute-rgt-neg-in N/A

          \[\leadsto d1 \cdot \color{blue}{\left(\mathsf{neg}\left(-1 \cdot \left(d2 - d3\right)\right)\right)} \]

       5. *-rgt-identity N/A

          \[\leadsto d1 \cdot \left(\mathsf{neg}\left(\left(-1 \cdot \left(d2 - d3\right)\right) \cdot 1\right)\right) \]

       6. *-inverses N/A

          \[\leadsto d1 \cdot \left(\mathsf{neg}\left(\left(-1 \cdot \left(d2 - d3\right)\right) \cdot \frac{d1}{d1}\right)\right) \]

       7. associate-/l* N/A

          \[\leadsto d1 \cdot \left(\mathsf{neg}\left(\frac{\left(-1 \cdot \left(d2 - d3\right)\right) \cdot d1}{d1}\right)\right) \]

       8. associate-*l/ N/A

          \[\leadsto d1 \cdot \left(\mathsf{neg}\left(\frac{-1 \cdot \left(d2 - d3\right)}{d1} \cdot d1\right)\right) \]

       9. associate-*r/ N/A

          \[\leadsto d1 \cdot \left(\mathsf{neg}\left(\left(-1 \cdot \frac{d2 - d3}{d1}\right) \cdot d1\right)\right) \]

       10. *-commutative N/A

           \[\leadsto d1 \cdot \left(\mathsf{neg}\left(d1 \cdot \left(-1 \cdot \frac{d2 - d3}{d1}\right)\right)\right) \]

       11. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(d1, \color{blue}{\left(\mathsf{neg}\left(d1 \cdot \left(-1 \cdot \frac{d2 - d3}{d1}\right)\right)\right)}\right) \]

       12. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(d1, \left(\mathsf{neg}\left(\left(-1 \cdot \frac{d2 - d3}{d1}\right) \cdot d1\right)\right)\right) \]

       13. associate-*r/ N/A

           \[\leadsto \mathsf{*.f64}\left(d1, \left(\mathsf{neg}\left(\frac{-1 \cdot \left(d2 - d3\right)}{d1} \cdot d1\right)\right)\right) \]

       14. associate-*l/ N/A

           \[\leadsto \mathsf{*.f64}\left(d1, \left(\mathsf{neg}\left(\frac{\left(-1 \cdot \left(d2 - d3\right)\right) \cdot d1}{d1}\right)\right)\right) \]

       15. associate-/l* N/A

           \[\leadsto \mathsf{*.f64}\left(d1, \left(\mathsf{neg}\left(\left(-1 \cdot \left(d2 - d3\right)\right) \cdot \frac{d1}{d1}\right)\right)\right) \]

       16. *-inverses N/A

           \[\leadsto \mathsf{*.f64}\left(d1, \left(\mathsf{neg}\left(\left(-1 \cdot \left(d2 - d3\right)\right) \cdot 1\right)\right)\right) \]

       17. *-rgt-identity N/A

           \[\leadsto \mathsf{*.f64}\left(d1, \left(\mathsf{neg}\left(-1 \cdot \left(d2 - d3\right)\right)\right)\right) \]

       18. sub-neg N/A

           \[\leadsto \mathsf{*.f64}\left(d1, \left(\mathsf{neg}\left(-1 \cdot \left(d2 + \left(\mathsf{neg}\left(d3\right)\right)\right)\right)\right)\right) \]

       19. distribute-lft-in N/A

           \[\leadsto \mathsf{*.f64}\left(d1, \left(\mathsf{neg}\left(\left(-1 \cdot d2 + -1 \cdot \left(\mathsf{neg}\left(d3\right)\right)\right)\right)\right)\right) \]

       20. mul-1-neg N/A

           \[\leadsto \mathsf{*.f64}\left(d1, \left(\mathsf{neg}\left(\left(-1 \cdot d2 + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(d3\right)\right)\right)\right)\right)\right)\right)\right) \]

       21. remove-double-neg N/A

           \[\leadsto \mathsf{*.f64}\left(d1, \left(\mathsf{neg}\left(\left(-1 \cdot d2 + d3\right)\right)\right)\right) \]

   12. Simplified 79.4%

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

   if -2.8999999999999998e146 < d3 < 1.9e11

   1. Initial program 85.3%

      \[\left(\left(d1 \cdot d2 - d1 \cdot d3\right) + d4 \cdot d1\right) - d1 \cdot d1 \]
   2. Step-by-step derivation

      1. distribute-lft-out-- N/A

         \[\leadsto \left(d1 \cdot \left(d2 - d3\right) + d4 \cdot d1\right) - d1 \cdot d1 \]

      2. *-commutative N/A

         \[\leadsto \left(d1 \cdot \left(d2 - d3\right) + d1 \cdot d4\right) - d1 \cdot d1 \]

      3. distribute-lft-out N/A

         \[\leadsto d1 \cdot \left(\left(d2 - d3\right) + d4\right) - \color{blue}{d1} \cdot d1 \]

      4. distribute-lft-out-- N/A

         \[\leadsto d1 \cdot \color{blue}{\left(\left(\left(d2 - d3\right) + d4\right) - d1\right)} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(d1, \color{blue}{\left(\left(\left(d2 - d3\right) + d4\right) - d1\right)}\right) \]

      6. associate-+r- N/A

         \[\leadsto \mathsf{*.f64}\left(d1, \left(\left(d2 - d3\right) + \color{blue}{\left(d4 - d1\right)}\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(d1, \left(\left(d2 + \left(\mathsf{neg}\left(d3\right)\right)\right) + \left(\color{blue}{d4} - d1\right)\right)\right) \]

      8. associate-+l+ N/A

         \[\leadsto \mathsf{*.f64}\left(d1, \left(d2 + \color{blue}{\left(\left(\mathsf{neg}\left(d3\right)\right) + \left(d4 - d1\right)\right)}\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(d1, \mathsf{+.f64}\left(d2, \color{blue}{\left(\left(\mathsf{neg}\left(d3\right)\right) + \left(d4 - d1\right)\right)}\right)\right) \]

      10. associate-+r- N/A

          \[\leadsto \mathsf{*.f64}\left(d1, \mathsf{+.f64}\left(d2, \left(\left(\left(\mathsf{neg}\left(d3\right)\right) + d4\right) - \color{blue}{d1}\right)\right)\right) \]

      11. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(d1, \mathsf{+.f64}\left(d2, \left(\left(d4 + \left(\mathsf{neg}\left(d3\right)\right)\right) - d1\right)\right)\right) \]

      12. unsub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(d1, \mathsf{+.f64}\left(d2, \left(\left(d4 - d3\right) - d1\right)\right)\right) \]

      13. associate--l- N/A

          \[\leadsto \mathsf{*.f64}\left(d1, \mathsf{+.f64}\left(d2, \left(d4 - \color{blue}{\left(d3 + d1\right)}\right)\right)\right) \]

      14. --lowering--.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(d1, \mathsf{+.f64}\left(d2, \mathsf{\_.f64}\left(d4, \color{blue}{\left(d3 + d1\right)}\right)\right)\right) \]

      15. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(d1, \mathsf{+.f64}\left(d2, \mathsf{\_.f64}\left(d4, \left(d1 + \color{blue}{d3}\right)\right)\right)\right) \]

      16. +-lowering-+.f64 100.0%

          \[\leadsto \mathsf{*.f64}\left(d1, \mathsf{+.f64}\left(d2, \mathsf{\_.f64}\left(d4, \mathsf{+.f64}\left(d1, \color{blue}{d3}\right)\right)\right)\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{d1 \cdot \left(d2 + \left(d4 - \left(d1 + d3\right)\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d4 around inf

      \[\leadsto \mathsf{*.f64}\left(d1, \mathsf{+.f64}\left(d2, \color{blue}{d4}\right)\right) \]
   6. Step-by-step derivation

   7. Simplified 68.3%

      \[\leadsto d1 \cdot \left(d2 + \color{blue}{d4}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 67.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0 - d1 \cdot d3\\ \mathbf{if}\;d3 \leq -1.25 \cdot 10^{+156}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d3 \leq 4.6 \cdot 10^{+66}:\\ \;\;\;\;d1 \cdot \left(d2 + d4\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (d1 d2 d3 d4)
 :precision binary64
 (let* ((t_0 (- 0.0 (* d1 d3))))
   (if (<= d3 -1.25e+156) t_0 (if (<= d3 4.6e+66) (* d1 (+ d2 d4)) t_0))))

C

double code(double d1, double d2, double d3, double d4) {
	double t_0 = 0.0 - (d1 * d3);
	double tmp;
	if (d3 <= -1.25e+156) {
		tmp = t_0;
	} else if (d3 <= 4.6e+66) {
		tmp = d1 * (d2 + d4);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(d1, d2, d3, d4)
    real(8), intent (in) :: d1
    real(8), intent (in) :: d2
    real(8), intent (in) :: d3
    real(8), intent (in) :: d4
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 0.0d0 - (d1 * d3)
    if (d3 <= (-1.25d+156)) then
        tmp = t_0
    else if (d3 <= 4.6d+66) then
        tmp = d1 * (d2 + d4)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double d1, double d2, double d3, double d4) {
	double t_0 = 0.0 - (d1 * d3);
	double tmp;
	if (d3 <= -1.25e+156) {
		tmp = t_0;
	} else if (d3 <= 4.6e+66) {
		tmp = d1 * (d2 + d4);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(d1, d2, d3, d4):
	t_0 = 0.0 - (d1 * d3)
	tmp = 0
	if d3 <= -1.25e+156:
		tmp = t_0
	elif d3 <= 4.6e+66:
		tmp = d1 * (d2 + d4)
	else:
		tmp = t_0
	return tmp

Julia

function code(d1, d2, d3, d4)
	t_0 = Float64(0.0 - Float64(d1 * d3))
	tmp = 0.0
	if (d3 <= -1.25e+156)
		tmp = t_0;
	elseif (d3 <= 4.6e+66)
		tmp = Float64(d1 * Float64(d2 + d4));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(d1, d2, d3, d4)
	t_0 = 0.0 - (d1 * d3);
	tmp = 0.0;
	if (d3 <= -1.25e+156)
		tmp = t_0;
	elseif (d3 <= 4.6e+66)
		tmp = d1 * (d2 + d4);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[d1_, d2_, d3_, d4_] := Block[{t$95$0 = N[(0.0 - N[(d1 * d3), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d3, -1.25e+156], t$95$0, If[LessEqual[d3, 4.6e+66], N[(d1 * N[(d2 + d4), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if d3 < -1.24999999999999998e156 or 4.6e66 < d3

   1. Initial program 87.0%

      \[\left(\left(d1 \cdot d2 - d1 \cdot d3\right) + d4 \cdot d1\right) - d1 \cdot d1 \]
   2. Step-by-step derivation

      1. distribute-lft-out-- N/A

         \[\leadsto \left(d1 \cdot \left(d2 - d3\right) + d4 \cdot d1\right) - d1 \cdot d1 \]

      2. *-commutative N/A

         \[\leadsto \left(d1 \cdot \left(d2 - d3\right) + d1 \cdot d4\right) - d1 \cdot d1 \]

      3. distribute-lft-out N/A

         \[\leadsto d1 \cdot \left(\left(d2 - d3\right) + d4\right) - \color{blue}{d1} \cdot d1 \]

      4. distribute-lft-out-- N/A

         \[\leadsto d1 \cdot \color{blue}{\left(\left(\left(d2 - d3\right) + d4\right) - d1\right)} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(d1, \color{blue}{\left(\left(\left(d2 - d3\right) + d4\right) - d1\right)}\right) \]

      6. associate-+r- N/A

         \[\leadsto \mathsf{*.f64}\left(d1, \left(\left(d2 - d3\right) + \color{blue}{\left(d4 - d1\right)}\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(d1, \left(\left(d2 + \left(\mathsf{neg}\left(d3\right)\right)\right) + \left(\color{blue}{d4} - d1\right)\right)\right) \]

      8. associate-+l+ N/A

         \[\leadsto \mathsf{*.f64}\left(d1, \left(d2 + \color{blue}{\left(\left(\mathsf{neg}\left(d3\right)\right) + \left(d4 - d1\right)\right)}\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(d1, \mathsf{+.f64}\left(d2, \color{blue}{\left(\left(\mathsf{neg}\left(d3\right)\right) + \left(d4 - d1\right)\right)}\right)\right) \]

      10. associate-+r- N/A

          \[\leadsto \mathsf{*.f64}\left(d1, \mathsf{+.f64}\left(d2, \left(\left(\left(\mathsf{neg}\left(d3\right)\right) + d4\right) - \color{blue}{d1}\right)\right)\right) \]

      11. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(d1, \mathsf{+.f64}\left(d2, \left(\left(d4 + \left(\mathsf{neg}\left(d3\right)\right)\right) - d1\right)\right)\right) \]

      12. unsub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(d1, \mathsf{+.f64}\left(d2, \left(\left(d4 - d3\right) - d1\right)\right)\right) \]

      13. associate--l- N/A

          \[\leadsto \mathsf{*.f64}\left(d1, \mathsf{+.f64}\left(d2, \left(d4 - \color{blue}{\left(d3 + d1\right)}\right)\right)\right) \]

      14. --lowering--.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(d1, \mathsf{+.f64}\left(d2, \mathsf{\_.f64}\left(d4, \color{blue}{\left(d3 + d1\right)}\right)\right)\right) \]

      15. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(d1, \mathsf{+.f64}\left(d2, \mathsf{\_.f64}\left(d4, \left(d1 + \color{blue}{d3}\right)\right)\right)\right) \]

      16. +-lowering-+.f64 100.0%

          \[\leadsto \mathsf{*.f64}\left(d1, \mathsf{+.f64}\left(d2, \mathsf{\_.f64}\left(d4, \mathsf{+.f64}\left(d1, \color{blue}{d3}\right)\right)\right)\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{d1 \cdot \left(d2 + \left(d4 - \left(d1 + d3\right)\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d3 around inf

      \[\leadsto \color{blue}{-1 \cdot \left(d1 \cdot d3\right)} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(d1 \cdot d3\right) \]

      2. neg-sub0 N/A

         \[\leadsto 0 - \color{blue}{d1 \cdot d3} \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{\left(d1 \cdot d3\right)}\right) \]

      4. *-lowering-*.f64 84.4%

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(d1, \color{blue}{d3}\right)\right) \]

   7. Simplified 84.4%

      \[\leadsto \color{blue}{0 - d1 \cdot d3} \]
   8. Step-by-step derivation

      1. sub0-neg N/A

         \[\leadsto \mathsf{neg}\left(d1 \cdot d3\right) \]

      2. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\left(d1 \cdot d3\right)\right) \]

      3. *-lowering-*.f64 84.4%

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(d1, d3\right)\right) \]

   9. Applied egg-rr 84.4%

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

   if -1.24999999999999998e156 < d3 < 4.6e66

   1. Initial program 86.0%

      \[\left(\left(d1 \cdot d2 - d1 \cdot d3\right) + d4 \cdot d1\right) - d1 \cdot d1 \]
   2. Step-by-step derivation

      1. distribute-lft-out-- N/A

         \[\leadsto \left(d1 \cdot \left(d2 - d3\right) + d4 \cdot d1\right) - d1 \cdot d1 \]

      2. *-commutative N/A

         \[\leadsto \left(d1 \cdot \left(d2 - d3\right) + d1 \cdot d4\right) - d1 \cdot d1 \]

      3. distribute-lft-out N/A

         \[\leadsto d1 \cdot \left(\left(d2 - d3\right) + d4\right) - \color{blue}{d1} \cdot d1 \]

      4. distribute-lft-out-- N/A

         \[\leadsto d1 \cdot \color{blue}{\left(\left(\left(d2 - d3\right) + d4\right) - d1\right)} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(d1, \color{blue}{\left(\left(\left(d2 - d3\right) + d4\right) - d1\right)}\right) \]

      6. associate-+r- N/A

         \[\leadsto \mathsf{*.f64}\left(d1, \left(\left(d2 - d3\right) + \color{blue}{\left(d4 - d1\right)}\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(d1, \left(\left(d2 + \left(\mathsf{neg}\left(d3\right)\right)\right) + \left(\color{blue}{d4} - d1\right)\right)\right) \]

      8. associate-+l+ N/A

         \[\leadsto \mathsf{*.f64}\left(d1, \left(d2 + \color{blue}{\left(\left(\mathsf{neg}\left(d3\right)\right) + \left(d4 - d1\right)\right)}\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(d1, \mathsf{+.f64}\left(d2, \color{blue}{\left(\left(\mathsf{neg}\left(d3\right)\right) + \left(d4 - d1\right)\right)}\right)\right) \]

      10. associate-+r- N/A

          \[\leadsto \mathsf{*.f64}\left(d1, \mathsf{+.f64}\left(d2, \left(\left(\left(\mathsf{neg}\left(d3\right)\right) + d4\right) - \color{blue}{d1}\right)\right)\right) \]

      11. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(d1, \mathsf{+.f64}\left(d2, \left(\left(d4 + \left(\mathsf{neg}\left(d3\right)\right)\right) - d1\right)\right)\right) \]

      12. unsub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(d1, \mathsf{+.f64}\left(d2, \left(\left(d4 - d3\right) - d1\right)\right)\right) \]

      13. associate--l- N/A

          \[\leadsto \mathsf{*.f64}\left(d1, \mathsf{+.f64}\left(d2, \left(d4 - \color{blue}{\left(d3 + d1\right)}\right)\right)\right) \]

      14. --lowering--.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(d1, \mathsf{+.f64}\left(d2, \mathsf{\_.f64}\left(d4, \color{blue}{\left(d3 + d1\right)}\right)\right)\right) \]

      15. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(d1, \mathsf{+.f64}\left(d2, \mathsf{\_.f64}\left(d4, \left(d1 + \color{blue}{d3}\right)\right)\right)\right) \]

      16. +-lowering-+.f64 100.0%

          \[\leadsto \mathsf{*.f64}\left(d1, \mathsf{+.f64}\left(d2, \mathsf{\_.f64}\left(d4, \mathsf{+.f64}\left(d1, \color{blue}{d3}\right)\right)\right)\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{d1 \cdot \left(d2 + \left(d4 - \left(d1 + d3\right)\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d4 around inf

      \[\leadsto \mathsf{*.f64}\left(d1, \mathsf{+.f64}\left(d2, \color{blue}{d4}\right)\right) \]
   6. Step-by-step derivation

   7. Simplified 66.3%

      \[\leadsto d1 \cdot \left(d2 + \color{blue}{d4}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 71.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d3 \leq -1.25 \cdot 10^{+156}:\\ \;\;\;\;0 - d1 \cdot d3\\ \mathbf{elif}\;d3 \leq 4.6 \cdot 10^{+66}:\\ \;\;\;\;d1 \cdot \left(d2 + d4\right)\\ \mathbf{else}:\\ \;\;\;\;0 - d1 \cdot d3\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 39.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d4 \leq -2.85 \cdot 10^{-148}:\\ \;\;\;\;d1 \cdot d2\\ \mathbf{elif}\;d4 \leq 1.25 \cdot 10^{+107}:\\ \;\;\;\;0 - d1 \cdot d3\\ \mathbf{else}:\\ \;\;\;\;d1 \cdot d4\\ \end{array} \end{array} \]

FPCore

(FPCore (d1 d2 d3 d4)
 :precision binary64
 (if (<= d4 -2.85e-148)
   (* d1 d2)
   (if (<= d4 1.25e+107) (- 0.0 (* d1 d3)) (* d1 d4))))

C

double code(double d1, double d2, double d3, double d4) {
	double tmp;
	if (d4 <= -2.85e-148) {
		tmp = d1 * d2;
	} else if (d4 <= 1.25e+107) {
		tmp = 0.0 - (d1 * d3);
	} else {
		tmp = d1 * d4;
	}
	return tmp;
}

Fortran

real(8) function code(d1, d2, d3, d4)
    real(8), intent (in) :: d1
    real(8), intent (in) :: d2
    real(8), intent (in) :: d3
    real(8), intent (in) :: d4
    real(8) :: tmp
    if (d4 <= (-2.85d-148)) then
        tmp = d1 * d2
    else if (d4 <= 1.25d+107) then
        tmp = 0.0d0 - (d1 * d3)
    else
        tmp = d1 * d4
    end if
    code = tmp
end function

Java

public static double code(double d1, double d2, double d3, double d4) {
	double tmp;
	if (d4 <= -2.85e-148) {
		tmp = d1 * d2;
	} else if (d4 <= 1.25e+107) {
		tmp = 0.0 - (d1 * d3);
	} else {
		tmp = d1 * d4;
	}
	return tmp;
}

Python

def code(d1, d2, d3, d4):
	tmp = 0
	if d4 <= -2.85e-148:
		tmp = d1 * d2
	elif d4 <= 1.25e+107:
		tmp = 0.0 - (d1 * d3)
	else:
		tmp = d1 * d4
	return tmp

Julia

function code(d1, d2, d3, d4)
	tmp = 0.0
	if (d4 <= -2.85e-148)
		tmp = Float64(d1 * d2);
	elseif (d4 <= 1.25e+107)
		tmp = Float64(0.0 - Float64(d1 * d3));
	else
		tmp = Float64(d1 * d4);
	end
	return tmp
end

MATLAB

function tmp_2 = code(d1, d2, d3, d4)
	tmp = 0.0;
	if (d4 <= -2.85e-148)
		tmp = d1 * d2;
	elseif (d4 <= 1.25e+107)
		tmp = 0.0 - (d1 * d3);
	else
		tmp = d1 * d4;
	end
	tmp_2 = tmp;
end

Wolfram

code[d1_, d2_, d3_, d4_] := If[LessEqual[d4, -2.85e-148], N[(d1 * d2), $MachinePrecision], If[LessEqual[d4, 1.25e+107], N[(0.0 - N[(d1 * d3), $MachinePrecision]), $MachinePrecision], N[(d1 * d4), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if d4 < -2.8499999999999999e-148

   1. Initial program 84.2%

      \[\left(\left(d1 \cdot d2 - d1 \cdot d3\right) + d4 \cdot d1\right) - d1 \cdot d1 \]
   2. Step-by-step derivation

      1. distribute-lft-out-- N/A

         \[\leadsto \left(d1 \cdot \left(d2 - d3\right) + d4 \cdot d1\right) - d1 \cdot d1 \]

      2. *-commutative N/A

         \[\leadsto \left(d1 \cdot \left(d2 - d3\right) + d1 \cdot d4\right) - d1 \cdot d1 \]

      3. distribute-lft-out N/A

         \[\leadsto d1 \cdot \left(\left(d2 - d3\right) + d4\right) - \color{blue}{d1} \cdot d1 \]

      4. distribute-lft-out-- N/A

         \[\leadsto d1 \cdot \color{blue}{\left(\left(\left(d2 - d3\right) + d4\right) - d1\right)} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(d1, \color{blue}{\left(\left(\left(d2 - d3\right) + d4\right) - d1\right)}\right) \]

      6. associate-+r- N/A

         \[\leadsto \mathsf{*.f64}\left(d1, \left(\left(d2 - d3\right) + \color{blue}{\left(d4 - d1\right)}\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(d1, \left(\left(d2 + \left(\mathsf{neg}\left(d3\right)\right)\right) + \left(\color{blue}{d4} - d1\right)\right)\right) \]

      8. associate-+l+ N/A

         \[\leadsto \mathsf{*.f64}\left(d1, \left(d2 + \color{blue}{\left(\left(\mathsf{neg}\left(d3\right)\right) + \left(d4 - d1\right)\right)}\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(d1, \mathsf{+.f64}\left(d2, \color{blue}{\left(\left(\mathsf{neg}\left(d3\right)\right) + \left(d4 - d1\right)\right)}\right)\right) \]

      10. associate-+r- N/A

          \[\leadsto \mathsf{*.f64}\left(d1, \mathsf{+.f64}\left(d2, \left(\left(\left(\mathsf{neg}\left(d3\right)\right) + d4\right) - \color{blue}{d1}\right)\right)\right) \]

      11. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(d1, \mathsf{+.f64}\left(d2, \left(\left(d4 + \left(\mathsf{neg}\left(d3\right)\right)\right) - d1\right)\right)\right) \]

      12. unsub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(d1, \mathsf{+.f64}\left(d2, \left(\left(d4 - d3\right) - d1\right)\right)\right) \]

      13. associate--l- N/A

          \[\leadsto \mathsf{*.f64}\left(d1, \mathsf{+.f64}\left(d2, \left(d4 - \color{blue}{\left(d3 + d1\right)}\right)\right)\right) \]

      14. --lowering--.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(d1, \mathsf{+.f64}\left(d2, \mathsf{\_.f64}\left(d4, \color{blue}{\left(d3 + d1\right)}\right)\right)\right) \]

      15. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(d1, \mathsf{+.f64}\left(d2, \mathsf{\_.f64}\left(d4, \left(d1 + \color{blue}{d3}\right)\right)\right)\right) \]

      16. +-lowering-+.f64 100.0%

          \[\leadsto \mathsf{*.f64}\left(d1, \mathsf{+.f64}\left(d2, \mathsf{\_.f64}\left(d4, \mathsf{+.f64}\left(d1, \color{blue}{d3}\right)\right)\right)\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{d1 \cdot \left(d2 + \left(d4 - \left(d1 + d3\right)\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d2 around inf

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

      1. *-lowering-*.f64 26.8%

         \[\leadsto \mathsf{*.f64}\left(d1, \color{blue}{d2}\right) \]

   7. Simplified 26.8%

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

   if -2.8499999999999999e-148 < d4 < 1.25e107

   1. Initial program 88.9%

      \[\left(\left(d1 \cdot d2 - d1 \cdot d3\right) + d4 \cdot d1\right) - d1 \cdot d1 \]
   2. Step-by-step derivation

      1. distribute-lft-out-- N/A

         \[\leadsto \left(d1 \cdot \left(d2 - d3\right) + d4 \cdot d1\right) - d1 \cdot d1 \]

      2. *-commutative N/A

         \[\leadsto \left(d1 \cdot \left(d2 - d3\right) + d1 \cdot d4\right) - d1 \cdot d1 \]

      3. distribute-lft-out N/A

         \[\leadsto d1 \cdot \left(\left(d2 - d3\right) + d4\right) - \color{blue}{d1} \cdot d1 \]

      4. distribute-lft-out-- N/A

         \[\leadsto d1 \cdot \color{blue}{\left(\left(\left(d2 - d3\right) + d4\right) - d1\right)} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(d1, \color{blue}{\left(\left(\left(d2 - d3\right) + d4\right) - d1\right)}\right) \]

      6. associate-+r- N/A

         \[\leadsto \mathsf{*.f64}\left(d1, \left(\left(d2 - d3\right) + \color{blue}{\left(d4 - d1\right)}\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(d1, \left(\left(d2 + \left(\mathsf{neg}\left(d3\right)\right)\right) + \left(\color{blue}{d4} - d1\right)\right)\right) \]

      8. associate-+l+ N/A

         \[\leadsto \mathsf{*.f64}\left(d1, \left(d2 + \color{blue}{\left(\left(\mathsf{neg}\left(d3\right)\right) + \left(d4 - d1\right)\right)}\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(d1, \mathsf{+.f64}\left(d2, \color{blue}{\left(\left(\mathsf{neg}\left(d3\right)\right) + \left(d4 - d1\right)\right)}\right)\right) \]

      10. associate-+r- N/A

          \[\leadsto \mathsf{*.f64}\left(d1, \mathsf{+.f64}\left(d2, \left(\left(\left(\mathsf{neg}\left(d3\right)\right) + d4\right) - \color{blue}{d1}\right)\right)\right) \]

      11. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(d1, \mathsf{+.f64}\left(d2, \left(\left(d4 + \left(\mathsf{neg}\left(d3\right)\right)\right) - d1\right)\right)\right) \]

      12. unsub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(d1, \mathsf{+.f64}\left(d2, \left(\left(d4 - d3\right) - d1\right)\right)\right) \]

      13. associate--l- N/A

          \[\leadsto \mathsf{*.f64}\left(d1, \mathsf{+.f64}\left(d2, \left(d4 - \color{blue}{\left(d3 + d1\right)}\right)\right)\right) \]

      14. --lowering--.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(d1, \mathsf{+.f64}\left(d2, \mathsf{\_.f64}\left(d4, \color{blue}{\left(d3 + d1\right)}\right)\right)\right) \]

      15. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(d1, \mathsf{+.f64}\left(d2, \mathsf{\_.f64}\left(d4, \left(d1 + \color{blue}{d3}\right)\right)\right)\right) \]

      16. +-lowering-+.f64 100.0%

          \[\leadsto \mathsf{*.f64}\left(d1, \mathsf{+.f64}\left(d2, \mathsf{\_.f64}\left(d4, \mathsf{+.f64}\left(d1, \color{blue}{d3}\right)\right)\right)\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{d1 \cdot \left(d2 + \left(d4 - \left(d1 + d3\right)\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d3 around inf

      \[\leadsto \color{blue}{-1 \cdot \left(d1 \cdot d3\right)} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(d1 \cdot d3\right) \]

      2. neg-sub0 N/A

         \[\leadsto 0 - \color{blue}{d1 \cdot d3} \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{\left(d1 \cdot d3\right)}\right) \]

      4. *-lowering-*.f64 43.5%

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(d1, \color{blue}{d3}\right)\right) \]

   7. Simplified 43.5%

      \[\leadsto \color{blue}{0 - d1 \cdot d3} \]
   8. Step-by-step derivation

      1. sub0-neg N/A

         \[\leadsto \mathsf{neg}\left(d1 \cdot d3\right) \]

      2. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\left(d1 \cdot d3\right)\right) \]

      3. *-lowering-*.f64 43.5%

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(d1, d3\right)\right) \]

   9. Applied egg-rr 43.5%

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

   if 1.25e107 < d4

   1. Initial program 84.2%

      \[\left(\left(d1 \cdot d2 - d1 \cdot d3\right) + d4 \cdot d1\right) - d1 \cdot d1 \]
   2. Step-by-step derivation

      1. distribute-lft-out-- N/A

         \[\leadsto \left(d1 \cdot \left(d2 - d3\right) + d4 \cdot d1\right) - d1 \cdot d1 \]

      2. *-commutative N/A

         \[\leadsto \left(d1 \cdot \left(d2 - d3\right) + d1 \cdot d4\right) - d1 \cdot d1 \]

      3. distribute-lft-out N/A

         \[\leadsto d1 \cdot \left(\left(d2 - d3\right) + d4\right) - \color{blue}{d1} \cdot d1 \]

      4. distribute-lft-out-- N/A

         \[\leadsto d1 \cdot \color{blue}{\left(\left(\left(d2 - d3\right) + d4\right) - d1\right)} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(d1, \color{blue}{\left(\left(\left(d2 - d3\right) + d4\right) - d1\right)}\right) \]

      6. associate-+r- N/A

         \[\leadsto \mathsf{*.f64}\left(d1, \left(\left(d2 - d3\right) + \color{blue}{\left(d4 - d1\right)}\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(d1, \left(\left(d2 + \left(\mathsf{neg}\left(d3\right)\right)\right) + \left(\color{blue}{d4} - d1\right)\right)\right) \]

      8. associate-+l+ N/A

         \[\leadsto \mathsf{*.f64}\left(d1, \left(d2 + \color{blue}{\left(\left(\mathsf{neg}\left(d3\right)\right) + \left(d4 - d1\right)\right)}\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(d1, \mathsf{+.f64}\left(d2, \color{blue}{\left(\left(\mathsf{neg}\left(d3\right)\right) + \left(d4 - d1\right)\right)}\right)\right) \]

      10. associate-+r- N/A

          \[\leadsto \mathsf{*.f64}\left(d1, \mathsf{+.f64}\left(d2, \left(\left(\left(\mathsf{neg}\left(d3\right)\right) + d4\right) - \color{blue}{d1}\right)\right)\right) \]

      11. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(d1, \mathsf{+.f64}\left(d2, \left(\left(d4 + \left(\mathsf{neg}\left(d3\right)\right)\right) - d1\right)\right)\right) \]

      12. unsub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(d1, \mathsf{+.f64}\left(d2, \left(\left(d4 - d3\right) - d1\right)\right)\right) \]

      13. associate--l- N/A

          \[\leadsto \mathsf{*.f64}\left(d1, \mathsf{+.f64}\left(d2, \left(d4 - \color{blue}{\left(d3 + d1\right)}\right)\right)\right) \]

      14. --lowering--.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(d1, \mathsf{+.f64}\left(d2, \mathsf{\_.f64}\left(d4, \color{blue}{\left(d3 + d1\right)}\right)\right)\right) \]

      15. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(d1, \mathsf{+.f64}\left(d2, \mathsf{\_.f64}\left(d4, \left(d1 + \color{blue}{d3}\right)\right)\right)\right) \]

      16. +-lowering-+.f64 100.0%

          \[\leadsto \mathsf{*.f64}\left(d1, \mathsf{+.f64}\left(d2, \mathsf{\_.f64}\left(d4, \mathsf{+.f64}\left(d1, \color{blue}{d3}\right)\right)\right)\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{d1 \cdot \left(d2 + \left(d4 - \left(d1 + d3\right)\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d4 around inf

      \[\leadsto \color{blue}{d1 \cdot d4} \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 76.8%

         \[\leadsto \mathsf{*.f64}\left(d1, \color{blue}{d4}\right) \]

   7. Simplified 76.8%

      \[\leadsto \color{blue}{d1 \cdot d4} \]
3. Recombined 3 regimes into one program.

4. Final simplification 41.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d4 \leq -2.85 \cdot 10^{-148}:\\ \;\;\;\;d1 \cdot d2\\ \mathbf{elif}\;d4 \leq 1.25 \cdot 10^{+107}:\\ \;\;\;\;0 - d1 \cdot d3\\ \mathbf{else}:\\ \;\;\;\;d1 \cdot d4\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 83.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d4 \leq 4.5 \cdot 10^{-11}:\\ \;\;\;\;d1 \cdot \left(d2 - \left(d1 + d3\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d1 \cdot \left(d4 - \left(d1 + d3\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (d1 d2 d3 d4)
 :precision binary64
 (if (<= d4 4.5e-11) (* d1 (- d2 (+ d1 d3))) (* d1 (- d4 (+ d1 d3)))))

C

double code(double d1, double d2, double d3, double d4) {
	double tmp;
	if (d4 <= 4.5e-11) {
		tmp = d1 * (d2 - (d1 + d3));
	} else {
		tmp = d1 * (d4 - (d1 + d3));
	}
	return tmp;
}

Fortran

real(8) function code(d1, d2, d3, d4)
    real(8), intent (in) :: d1
    real(8), intent (in) :: d2
    real(8), intent (in) :: d3
    real(8), intent (in) :: d4
    real(8) :: tmp
    if (d4 <= 4.5d-11) then
        tmp = d1 * (d2 - (d1 + d3))
    else
        tmp = d1 * (d4 - (d1 + d3))
    end if
    code = tmp
end function

Java

public static double code(double d1, double d2, double d3, double d4) {
	double tmp;
	if (d4 <= 4.5e-11) {
		tmp = d1 * (d2 - (d1 + d3));
	} else {
		tmp = d1 * (d4 - (d1 + d3));
	}
	return tmp;
}

Python

def code(d1, d2, d3, d4):
	tmp = 0
	if d4 <= 4.5e-11:
		tmp = d1 * (d2 - (d1 + d3))
	else:
		tmp = d1 * (d4 - (d1 + d3))
	return tmp

Julia

function code(d1, d2, d3, d4)
	tmp = 0.0
	if (d4 <= 4.5e-11)
		tmp = Float64(d1 * Float64(d2 - Float64(d1 + d3)));
	else
		tmp = Float64(d1 * Float64(d4 - Float64(d1 + d3)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d1, d2, d3, d4)
	tmp = 0.0;
	if (d4 <= 4.5e-11)
		tmp = d1 * (d2 - (d1 + d3));
	else
		tmp = d1 * (d4 - (d1 + d3));
	end
	tmp_2 = tmp;
end

Wolfram

code[d1_, d2_, d3_, d4_] := If[LessEqual[d4, 4.5e-11], N[(d1 * N[(d2 - N[(d1 + d3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d1 * N[(d4 - N[(d1 + d3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d4 < 4.5e-11

   1. Initial program 87.0%

      \[\left(\left(d1 \cdot d2 - d1 \cdot d3\right) + d4 \cdot d1\right) - d1 \cdot d1 \]
   2. Step-by-step derivation

      1. distribute-lft-out-- N/A

         \[\leadsto \left(d1 \cdot \left(d2 - d3\right) + d4 \cdot d1\right) - d1 \cdot d1 \]

      2. *-commutative N/A

         \[\leadsto \left(d1 \cdot \left(d2 - d3\right) + d1 \cdot d4\right) - d1 \cdot d1 \]

      3. distribute-lft-out N/A

         \[\leadsto d1 \cdot \left(\left(d2 - d3\right) + d4\right) - \color{blue}{d1} \cdot d1 \]

      4. distribute-lft-out-- N/A

         \[\leadsto d1 \cdot \color{blue}{\left(\left(\left(d2 - d3\right) + d4\right) - d1\right)} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(d1, \color{blue}{\left(\left(\left(d2 - d3\right) + d4\right) - d1\right)}\right) \]

      6. associate-+r- N/A

         \[\leadsto \mathsf{*.f64}\left(d1, \left(\left(d2 - d3\right) + \color{blue}{\left(d4 - d1\right)}\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(d1, \left(\left(d2 + \left(\mathsf{neg}\left(d3\right)\right)\right) + \left(\color{blue}{d4} - d1\right)\right)\right) \]

      8. associate-+l+ N/A

         \[\leadsto \mathsf{*.f64}\left(d1, \left(d2 + \color{blue}{\left(\left(\mathsf{neg}\left(d3\right)\right) + \left(d4 - d1\right)\right)}\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(d1, \mathsf{+.f64}\left(d2, \color{blue}{\left(\left(\mathsf{neg}\left(d3\right)\right) + \left(d4 - d1\right)\right)}\right)\right) \]

      10. associate-+r- N/A

          \[\leadsto \mathsf{*.f64}\left(d1, \mathsf{+.f64}\left(d2, \left(\left(\left(\mathsf{neg}\left(d3\right)\right) + d4\right) - \color{blue}{d1}\right)\right)\right) \]

      11. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(d1, \mathsf{+.f64}\left(d2, \left(\left(d4 + \left(\mathsf{neg}\left(d3\right)\right)\right) - d1\right)\right)\right) \]

      12. unsub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(d1, \mathsf{+.f64}\left(d2, \left(\left(d4 - d3\right) - d1\right)\right)\right) \]

      13. associate--l- N/A

          \[\leadsto \mathsf{*.f64}\left(d1, \mathsf{+.f64}\left(d2, \left(d4 - \color{blue}{\left(d3 + d1\right)}\right)\right)\right) \]

      14. --lowering--.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(d1, \mathsf{+.f64}\left(d2, \mathsf{\_.f64}\left(d4, \color{blue}{\left(d3 + d1\right)}\right)\right)\right) \]

      15. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(d1, \mathsf{+.f64}\left(d2, \mathsf{\_.f64}\left(d4, \left(d1 + \color{blue}{d3}\right)\right)\right)\right) \]

      16. +-lowering-+.f64 100.0%

          \[\leadsto \mathsf{*.f64}\left(d1, \mathsf{+.f64}\left(d2, \mathsf{\_.f64}\left(d4, \mathsf{+.f64}\left(d1, \color{blue}{d3}\right)\right)\right)\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{d1 \cdot \left(d2 + \left(d4 - \left(d1 + d3\right)\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto d1 \cdot \left(\left(d4 - \left(d1 + d3\right)\right) + \color{blue}{d2}\right) \]

      2. distribute-rgt-in N/A

         \[\leadsto \left(d4 - \left(d1 + d3\right)\right) \cdot d1 + \color{blue}{d2 \cdot d1} \]

      3. *-commutative N/A

         \[\leadsto \left(d4 - \left(d1 + d3\right)\right) \cdot d1 + d1 \cdot \color{blue}{d2} \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(d4 - \left(d1 + d3\right)\right) \cdot d1\right), \color{blue}{\left(d1 \cdot d2\right)}\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\left(d1 \cdot \left(d4 - \left(d1 + d3\right)\right)\right), \left(\color{blue}{d1} \cdot d2\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(d1, \left(d4 - \left(d1 + d3\right)\right)\right), \left(\color{blue}{d1} \cdot d2\right)\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(d1, \mathsf{\_.f64}\left(d4, \left(d1 + d3\right)\right)\right), \left(d1 \cdot d2\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(d1, \mathsf{\_.f64}\left(d4, \mathsf{+.f64}\left(d1, d3\right)\right)\right), \left(d1 \cdot d2\right)\right) \]

      9. *-lowering-*.f64 93.5%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(d1, \mathsf{\_.f64}\left(d4, \mathsf{+.f64}\left(d1, d3\right)\right)\right), \mathsf{*.f64}\left(d1, \color{blue}{d2}\right)\right) \]

   6. Applied egg-rr 93.5%

      \[\leadsto \color{blue}{d1 \cdot \left(d4 - \left(d1 + d3\right)\right) + d1 \cdot d2} \]

   7. Taylor expanded in d4 around 0

      \[\leadsto \color{blue}{-1 \cdot \left(d1 \cdot \left(d1 + d3\right)\right) + d1 \cdot d2} \]
   8. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto d1 \cdot d2 + \color{blue}{-1 \cdot \left(d1 \cdot \left(d1 + d3\right)\right)} \]

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. distribute-lft-out-- N/A

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

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(d1, \color{blue}{\left(d2 - \left(d1 + d3\right)\right)}\right) \]

      6. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(d1, \mathsf{\_.f64}\left(d2, \color{blue}{\left(d1 + d3\right)}\right)\right) \]

      7. +-lowering-+.f64 87.9%

         \[\leadsto \mathsf{*.f64}\left(d1, \mathsf{\_.f64}\left(d2, \mathsf{+.f64}\left(d1, \color{blue}{d3}\right)\right)\right) \]

   9. Simplified 87.9%

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

   if 4.5e-11 < d4

   1. Initial program 83.9%

      \[\left(\left(d1 \cdot d2 - d1 \cdot d3\right) + d4 \cdot d1\right) - d1 \cdot d1 \]
   2. Step-by-step derivation

      1. distribute-lft-out-- N/A

         \[\leadsto \left(d1 \cdot \left(d2 - d3\right) + d4 \cdot d1\right) - d1 \cdot d1 \]

      2. *-commutative N/A

         \[\leadsto \left(d1 \cdot \left(d2 - d3\right) + d1 \cdot d4\right) - d1 \cdot d1 \]

      3. distribute-lft-out N/A

         \[\leadsto d1 \cdot \left(\left(d2 - d3\right) + d4\right) - \color{blue}{d1} \cdot d1 \]

      4. distribute-lft-out-- N/A

         \[\leadsto d1 \cdot \color{blue}{\left(\left(\left(d2 - d3\right) + d4\right) - d1\right)} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(d1, \color{blue}{\left(\left(\left(d2 - d3\right) + d4\right) - d1\right)}\right) \]

      6. associate-+r- N/A

         \[\leadsto \mathsf{*.f64}\left(d1, \left(\left(d2 - d3\right) + \color{blue}{\left(d4 - d1\right)}\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(d1, \left(\left(d2 + \left(\mathsf{neg}\left(d3\right)\right)\right) + \left(\color{blue}{d4} - d1\right)\right)\right) \]

      8. associate-+l+ N/A

         \[\leadsto \mathsf{*.f64}\left(d1, \left(d2 + \color{blue}{\left(\left(\mathsf{neg}\left(d3\right)\right) + \left(d4 - d1\right)\right)}\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(d1, \mathsf{+.f64}\left(d2, \color{blue}{\left(\left(\mathsf{neg}\left(d3\right)\right) + \left(d4 - d1\right)\right)}\right)\right) \]

      10. associate-+r- N/A

          \[\leadsto \mathsf{*.f64}\left(d1, \mathsf{+.f64}\left(d2, \left(\left(\left(\mathsf{neg}\left(d3\right)\right) + d4\right) - \color{blue}{d1}\right)\right)\right) \]

      11. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(d1, \mathsf{+.f64}\left(d2, \left(\left(d4 + \left(\mathsf{neg}\left(d3\right)\right)\right) - d1\right)\right)\right) \]

      12. unsub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(d1, \mathsf{+.f64}\left(d2, \left(\left(d4 - d3\right) - d1\right)\right)\right) \]

      13. associate--l- N/A

          \[\leadsto \mathsf{*.f64}\left(d1, \mathsf{+.f64}\left(d2, \left(d4 - \color{blue}{\left(d3 + d1\right)}\right)\right)\right) \]

      14. --lowering--.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(d1, \mathsf{+.f64}\left(d2, \mathsf{\_.f64}\left(d4, \color{blue}{\left(d3 + d1\right)}\right)\right)\right) \]

      15. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(d1, \mathsf{+.f64}\left(d2, \mathsf{\_.f64}\left(d4, \left(d1 + \color{blue}{d3}\right)\right)\right)\right) \]

      16. +-lowering-+.f64 100.0%

          \[\leadsto \mathsf{*.f64}\left(d1, \mathsf{+.f64}\left(d2, \mathsf{\_.f64}\left(d4, \mathsf{+.f64}\left(d1, \color{blue}{d3}\right)\right)\right)\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{d1 \cdot \left(d2 + \left(d4 - \left(d1 + d3\right)\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d2 around 0

      \[\leadsto \color{blue}{d1 \cdot \left(d4 - \left(d1 + d3\right)\right)} \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(d1, \color{blue}{\left(d4 - \left(d1 + d3\right)\right)}\right) \]

      2. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(d1, \mathsf{\_.f64}\left(d4, \color{blue}{\left(d1 + d3\right)}\right)\right) \]

      3. +-lowering-+.f64 83.6%

         \[\leadsto \mathsf{*.f64}\left(d1, \mathsf{\_.f64}\left(d4, \mathsf{+.f64}\left(d1, \color{blue}{d3}\right)\right)\right) \]

   7. Simplified 83.6%

      \[\leadsto \color{blue}{d1 \cdot \left(d4 - \left(d1 + d3\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 84.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d4 \leq 0.017:\\ \;\;\;\;d1 \cdot \left(d2 - \left(d1 + d3\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d1 \cdot \left(d2 + \left(d4 - d3\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (d1 d2 d3 d4)
 :precision binary64
 (if (<= d4 0.017) (* d1 (- d2 (+ d1 d3))) (* d1 (+ d2 (- d4 d3)))))

C

double code(double d1, double d2, double d3, double d4) {
	double tmp;
	if (d4 <= 0.017) {
		tmp = d1 * (d2 - (d1 + d3));
	} else {
		tmp = d1 * (d2 + (d4 - d3));
	}
	return tmp;
}

Fortran

real(8) function code(d1, d2, d3, d4)
    real(8), intent (in) :: d1
    real(8), intent (in) :: d2
    real(8), intent (in) :: d3
    real(8), intent (in) :: d4
    real(8) :: tmp
    if (d4 <= 0.017d0) then
        tmp = d1 * (d2 - (d1 + d3))
    else
        tmp = d1 * (d2 + (d4 - d3))
    end if
    code = tmp
end function

Java

public static double code(double d1, double d2, double d3, double d4) {
	double tmp;
	if (d4 <= 0.017) {
		tmp = d1 * (d2 - (d1 + d3));
	} else {
		tmp = d1 * (d2 + (d4 - d3));
	}
	return tmp;
}

Python

def code(d1, d2, d3, d4):
	tmp = 0
	if d4 <= 0.017:
		tmp = d1 * (d2 - (d1 + d3))
	else:
		tmp = d1 * (d2 + (d4 - d3))
	return tmp

Julia

function code(d1, d2, d3, d4)
	tmp = 0.0
	if (d4 <= 0.017)
		tmp = Float64(d1 * Float64(d2 - Float64(d1 + d3)));
	else
		tmp = Float64(d1 * Float64(d2 + Float64(d4 - d3)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d1, d2, d3, d4)
	tmp = 0.0;
	if (d4 <= 0.017)
		tmp = d1 * (d2 - (d1 + d3));
	else
		tmp = d1 * (d2 + (d4 - d3));
	end
	tmp_2 = tmp;
end

Wolfram

code[d1_, d2_, d3_, d4_] := If[LessEqual[d4, 0.017], N[(d1 * N[(d2 - N[(d1 + d3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d1 * N[(d2 + N[(d4 - d3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d4 < 0.017000000000000001

   1. Initial program 87.0%

      \[\left(\left(d1 \cdot d2 - d1 \cdot d3\right) + d4 \cdot d1\right) - d1 \cdot d1 \]
   2. Step-by-step derivation

      1. distribute-lft-out-- N/A

         \[\leadsto \left(d1 \cdot \left(d2 - d3\right) + d4 \cdot d1\right) - d1 \cdot d1 \]

      2. *-commutative N/A

         \[\leadsto \left(d1 \cdot \left(d2 - d3\right) + d1 \cdot d4\right) - d1 \cdot d1 \]

      3. distribute-lft-out N/A

         \[\leadsto d1 \cdot \left(\left(d2 - d3\right) + d4\right) - \color{blue}{d1} \cdot d1 \]

      4. distribute-lft-out-- N/A

         \[\leadsto d1 \cdot \color{blue}{\left(\left(\left(d2 - d3\right) + d4\right) - d1\right)} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(d1, \color{blue}{\left(\left(\left(d2 - d3\right) + d4\right) - d1\right)}\right) \]

      6. associate-+r- N/A

         \[\leadsto \mathsf{*.f64}\left(d1, \left(\left(d2 - d3\right) + \color{blue}{\left(d4 - d1\right)}\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(d1, \left(\left(d2 + \left(\mathsf{neg}\left(d3\right)\right)\right) + \left(\color{blue}{d4} - d1\right)\right)\right) \]

      8. associate-+l+ N/A

         \[\leadsto \mathsf{*.f64}\left(d1, \left(d2 + \color{blue}{\left(\left(\mathsf{neg}\left(d3\right)\right) + \left(d4 - d1\right)\right)}\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(d1, \mathsf{+.f64}\left(d2, \color{blue}{\left(\left(\mathsf{neg}\left(d3\right)\right) + \left(d4 - d1\right)\right)}\right)\right) \]

      10. associate-+r- N/A

          \[\leadsto \mathsf{*.f64}\left(d1, \mathsf{+.f64}\left(d2, \left(\left(\left(\mathsf{neg}\left(d3\right)\right) + d4\right) - \color{blue}{d1}\right)\right)\right) \]

      11. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(d1, \mathsf{+.f64}\left(d2, \left(\left(d4 + \left(\mathsf{neg}\left(d3\right)\right)\right) - d1\right)\right)\right) \]

      12. unsub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(d1, \mathsf{+.f64}\left(d2, \left(\left(d4 - d3\right) - d1\right)\right)\right) \]

      13. associate--l- N/A

          \[\leadsto \mathsf{*.f64}\left(d1, \mathsf{+.f64}\left(d2, \left(d4 - \color{blue}{\left(d3 + d1\right)}\right)\right)\right) \]

      14. --lowering--.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(d1, \mathsf{+.f64}\left(d2, \mathsf{\_.f64}\left(d4, \color{blue}{\left(d3 + d1\right)}\right)\right)\right) \]

      15. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(d1, \mathsf{+.f64}\left(d2, \mathsf{\_.f64}\left(d4, \left(d1 + \color{blue}{d3}\right)\right)\right)\right) \]

      16. +-lowering-+.f64 100.0%

          \[\leadsto \mathsf{*.f64}\left(d1, \mathsf{+.f64}\left(d2, \mathsf{\_.f64}\left(d4, \mathsf{+.f64}\left(d1, \color{blue}{d3}\right)\right)\right)\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{d1 \cdot \left(d2 + \left(d4 - \left(d1 + d3\right)\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto d1 \cdot \left(\left(d4 - \left(d1 + d3\right)\right) + \color{blue}{d2}\right) \]

      2. distribute-rgt-in N/A

         \[\leadsto \left(d4 - \left(d1 + d3\right)\right) \cdot d1 + \color{blue}{d2 \cdot d1} \]

      3. *-commutative N/A

         \[\leadsto \left(d4 - \left(d1 + d3\right)\right) \cdot d1 + d1 \cdot \color{blue}{d2} \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(d4 - \left(d1 + d3\right)\right) \cdot d1\right), \color{blue}{\left(d1 \cdot d2\right)}\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\left(d1 \cdot \left(d4 - \left(d1 + d3\right)\right)\right), \left(\color{blue}{d1} \cdot d2\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(d1, \left(d4 - \left(d1 + d3\right)\right)\right), \left(\color{blue}{d1} \cdot d2\right)\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(d1, \mathsf{\_.f64}\left(d4, \left(d1 + d3\right)\right)\right), \left(d1 \cdot d2\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(d1, \mathsf{\_.f64}\left(d4, \mathsf{+.f64}\left(d1, d3\right)\right)\right), \left(d1 \cdot d2\right)\right) \]

      9. *-lowering-*.f64 93.5%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(d1, \mathsf{\_.f64}\left(d4, \mathsf{+.f64}\left(d1, d3\right)\right)\right), \mathsf{*.f64}\left(d1, \color{blue}{d2}\right)\right) \]

   6. Applied egg-rr 93.5%

      \[\leadsto \color{blue}{d1 \cdot \left(d4 - \left(d1 + d3\right)\right) + d1 \cdot d2} \]

   7. Taylor expanded in d4 around 0

      \[\leadsto \color{blue}{-1 \cdot \left(d1 \cdot \left(d1 + d3\right)\right) + d1 \cdot d2} \]
   8. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto d1 \cdot d2 + \color{blue}{-1 \cdot \left(d1 \cdot \left(d1 + d3\right)\right)} \]

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. distribute-lft-out-- N/A

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

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(d1, \color{blue}{\left(d2 - \left(d1 + d3\right)\right)}\right) \]

      6. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(d1, \mathsf{\_.f64}\left(d2, \color{blue}{\left(d1 + d3\right)}\right)\right) \]

      7. +-lowering-+.f64 87.9%

         \[\leadsto \mathsf{*.f64}\left(d1, \mathsf{\_.f64}\left(d2, \mathsf{+.f64}\left(d1, \color{blue}{d3}\right)\right)\right) \]

   9. Simplified 87.9%

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

   if 0.017000000000000001 < d4

   1. Initial program 83.9%

      \[\left(\left(d1 \cdot d2 - d1 \cdot d3\right) + d4 \cdot d1\right) - d1 \cdot d1 \]
   2. Step-by-step derivation

      1. distribute-lft-out-- N/A

         \[\leadsto \left(d1 \cdot \left(d2 - d3\right) + d4 \cdot d1\right) - d1 \cdot d1 \]

      2. *-commutative N/A

         \[\leadsto \left(d1 \cdot \left(d2 - d3\right) + d1 \cdot d4\right) - d1 \cdot d1 \]

      3. distribute-lft-out N/A

         \[\leadsto d1 \cdot \left(\left(d2 - d3\right) + d4\right) - \color{blue}{d1} \cdot d1 \]

      4. distribute-lft-out-- N/A

         \[\leadsto d1 \cdot \color{blue}{\left(\left(\left(d2 - d3\right) + d4\right) - d1\right)} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(d1, \color{blue}{\left(\left(\left(d2 - d3\right) + d4\right) - d1\right)}\right) \]

      6. associate-+r- N/A

         \[\leadsto \mathsf{*.f64}\left(d1, \left(\left(d2 - d3\right) + \color{blue}{\left(d4 - d1\right)}\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(d1, \left(\left(d2 + \left(\mathsf{neg}\left(d3\right)\right)\right) + \left(\color{blue}{d4} - d1\right)\right)\right) \]

      8. associate-+l+ N/A

         \[\leadsto \mathsf{*.f64}\left(d1, \left(d2 + \color{blue}{\left(\left(\mathsf{neg}\left(d3\right)\right) + \left(d4 - d1\right)\right)}\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(d1, \mathsf{+.f64}\left(d2, \color{blue}{\left(\left(\mathsf{neg}\left(d3\right)\right) + \left(d4 - d1\right)\right)}\right)\right) \]

      10. associate-+r- N/A

          \[\leadsto \mathsf{*.f64}\left(d1, \mathsf{+.f64}\left(d2, \left(\left(\left(\mathsf{neg}\left(d3\right)\right) + d4\right) - \color{blue}{d1}\right)\right)\right) \]

      11. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(d1, \mathsf{+.f64}\left(d2, \left(\left(d4 + \left(\mathsf{neg}\left(d3\right)\right)\right) - d1\right)\right)\right) \]

      12. unsub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(d1, \mathsf{+.f64}\left(d2, \left(\left(d4 - d3\right) - d1\right)\right)\right) \]

      13. associate--l- N/A

          \[\leadsto \mathsf{*.f64}\left(d1, \mathsf{+.f64}\left(d2, \left(d4 - \color{blue}{\left(d3 + d1\right)}\right)\right)\right) \]

      14. --lowering--.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(d1, \mathsf{+.f64}\left(d2, \mathsf{\_.f64}\left(d4, \color{blue}{\left(d3 + d1\right)}\right)\right)\right) \]

      15. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(d1, \mathsf{+.f64}\left(d2, \mathsf{\_.f64}\left(d4, \left(d1 + \color{blue}{d3}\right)\right)\right)\right) \]

      16. +-lowering-+.f64 100.0%

          \[\leadsto \mathsf{*.f64}\left(d1, \mathsf{+.f64}\left(d2, \mathsf{\_.f64}\left(d4, \mathsf{+.f64}\left(d1, \color{blue}{d3}\right)\right)\right)\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{d1 \cdot \left(d2 + \left(d4 - \left(d1 + d3\right)\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d1 around 0

      \[\leadsto \color{blue}{d1 \cdot \left(\left(d2 + d4\right) - d3\right)} \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(d1, \color{blue}{\left(\left(d2 + d4\right) - d3\right)}\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(d1, \left(\left(d2 + d4\right) + \color{blue}{\left(\mathsf{neg}\left(d3\right)\right)}\right)\right) \]

      3. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(d1, \left(\left(d2 + d4\right) + -1 \cdot \color{blue}{d3}\right)\right) \]

      4. associate-+r+ N/A

         \[\leadsto \mathsf{*.f64}\left(d1, \left(d2 + \color{blue}{\left(d4 + -1 \cdot d3\right)}\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(d1, \mathsf{+.f64}\left(d2, \color{blue}{\left(d4 + -1 \cdot d3\right)}\right)\right) \]

      6. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(d1, \mathsf{+.f64}\left(d2, \left(d4 + \left(\mathsf{neg}\left(d3\right)\right)\right)\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(d1, \mathsf{+.f64}\left(d2, \left(d4 - \color{blue}{d3}\right)\right)\right) \]

      8. --lowering--.f64 89.7%

         \[\leadsto \mathsf{*.f64}\left(d1, \mathsf{+.f64}\left(d2, \mathsf{\_.f64}\left(d4, \color{blue}{d3}\right)\right)\right) \]

   7. Simplified 89.7%

      \[\leadsto \color{blue}{d1 \cdot \left(d2 + \left(d4 - d3\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 39.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d4 \leq 3.1 \cdot 10^{-7}:\\ \;\;\;\;d1 \cdot d2\\ \mathbf{else}:\\ \;\;\;\;d1 \cdot d4\\ \end{array} \end{array} \]

FPCore

(FPCore (d1 d2 d3 d4)
 :precision binary64
 (if (<= d4 3.1e-7) (* d1 d2) (* d1 d4)))

C

double code(double d1, double d2, double d3, double d4) {
	double tmp;
	if (d4 <= 3.1e-7) {
		tmp = d1 * d2;
	} else {
		tmp = d1 * d4;
	}
	return tmp;
}

Fortran

real(8) function code(d1, d2, d3, d4)
    real(8), intent (in) :: d1
    real(8), intent (in) :: d2
    real(8), intent (in) :: d3
    real(8), intent (in) :: d4
    real(8) :: tmp
    if (d4 <= 3.1d-7) then
        tmp = d1 * d2
    else
        tmp = d1 * d4
    end if
    code = tmp
end function

Java

public static double code(double d1, double d2, double d3, double d4) {
	double tmp;
	if (d4 <= 3.1e-7) {
		tmp = d1 * d2;
	} else {
		tmp = d1 * d4;
	}
	return tmp;
}

Python

def code(d1, d2, d3, d4):
	tmp = 0
	if d4 <= 3.1e-7:
		tmp = d1 * d2
	else:
		tmp = d1 * d4
	return tmp

Julia

function code(d1, d2, d3, d4)
	tmp = 0.0
	if (d4 <= 3.1e-7)
		tmp = Float64(d1 * d2);
	else
		tmp = Float64(d1 * d4);
	end
	return tmp
end

MATLAB

function tmp_2 = code(d1, d2, d3, d4)
	tmp = 0.0;
	if (d4 <= 3.1e-7)
		tmp = d1 * d2;
	else
		tmp = d1 * d4;
	end
	tmp_2 = tmp;
end

Wolfram

code[d1_, d2_, d3_, d4_] := If[LessEqual[d4, 3.1e-7], N[(d1 * d2), $MachinePrecision], N[(d1 * d4), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d4 < 3.1e-7

   1. Initial program 87.0%

      \[\left(\left(d1 \cdot d2 - d1 \cdot d3\right) + d4 \cdot d1\right) - d1 \cdot d1 \]
   2. Step-by-step derivation

      1. distribute-lft-out-- N/A

         \[\leadsto \left(d1 \cdot \left(d2 - d3\right) + d4 \cdot d1\right) - d1 \cdot d1 \]

      2. *-commutative N/A

         \[\leadsto \left(d1 \cdot \left(d2 - d3\right) + d1 \cdot d4\right) - d1 \cdot d1 \]

      3. distribute-lft-out N/A

         \[\leadsto d1 \cdot \left(\left(d2 - d3\right) + d4\right) - \color{blue}{d1} \cdot d1 \]

      4. distribute-lft-out-- N/A

         \[\leadsto d1 \cdot \color{blue}{\left(\left(\left(d2 - d3\right) + d4\right) - d1\right)} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(d1, \color{blue}{\left(\left(\left(d2 - d3\right) + d4\right) - d1\right)}\right) \]

      6. associate-+r- N/A

         \[\leadsto \mathsf{*.f64}\left(d1, \left(\left(d2 - d3\right) + \color{blue}{\left(d4 - d1\right)}\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(d1, \left(\left(d2 + \left(\mathsf{neg}\left(d3\right)\right)\right) + \left(\color{blue}{d4} - d1\right)\right)\right) \]

      8. associate-+l+ N/A

         \[\leadsto \mathsf{*.f64}\left(d1, \left(d2 + \color{blue}{\left(\left(\mathsf{neg}\left(d3\right)\right) + \left(d4 - d1\right)\right)}\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(d1, \mathsf{+.f64}\left(d2, \color{blue}{\left(\left(\mathsf{neg}\left(d3\right)\right) + \left(d4 - d1\right)\right)}\right)\right) \]

      10. associate-+r- N/A

          \[\leadsto \mathsf{*.f64}\left(d1, \mathsf{+.f64}\left(d2, \left(\left(\left(\mathsf{neg}\left(d3\right)\right) + d4\right) - \color{blue}{d1}\right)\right)\right) \]

      11. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(d1, \mathsf{+.f64}\left(d2, \left(\left(d4 + \left(\mathsf{neg}\left(d3\right)\right)\right) - d1\right)\right)\right) \]

      12. unsub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(d1, \mathsf{+.f64}\left(d2, \left(\left(d4 - d3\right) - d1\right)\right)\right) \]

      13. associate--l- N/A

          \[\leadsto \mathsf{*.f64}\left(d1, \mathsf{+.f64}\left(d2, \left(d4 - \color{blue}{\left(d3 + d1\right)}\right)\right)\right) \]

      14. --lowering--.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(d1, \mathsf{+.f64}\left(d2, \mathsf{\_.f64}\left(d4, \color{blue}{\left(d3 + d1\right)}\right)\right)\right) \]

      15. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(d1, \mathsf{+.f64}\left(d2, \mathsf{\_.f64}\left(d4, \left(d1 + \color{blue}{d3}\right)\right)\right)\right) \]

      16. +-lowering-+.f64 100.0%

          \[\leadsto \mathsf{*.f64}\left(d1, \mathsf{+.f64}\left(d2, \mathsf{\_.f64}\left(d4, \mathsf{+.f64}\left(d1, \color{blue}{d3}\right)\right)\right)\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{d1 \cdot \left(d2 + \left(d4 - \left(d1 + d3\right)\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d2 around inf

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

      1. *-lowering-*.f64 32.1%

         \[\leadsto \mathsf{*.f64}\left(d1, \color{blue}{d2}\right) \]

   7. Simplified 32.1%

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

   if 3.1e-7 < d4

   1. Initial program 83.9%

      \[\left(\left(d1 \cdot d2 - d1 \cdot d3\right) + d4 \cdot d1\right) - d1 \cdot d1 \]
   2. Step-by-step derivation

      1. distribute-lft-out-- N/A

         \[\leadsto \left(d1 \cdot \left(d2 - d3\right) + d4 \cdot d1\right) - d1 \cdot d1 \]

      2. *-commutative N/A

         \[\leadsto \left(d1 \cdot \left(d2 - d3\right) + d1 \cdot d4\right) - d1 \cdot d1 \]

      3. distribute-lft-out N/A

         \[\leadsto d1 \cdot \left(\left(d2 - d3\right) + d4\right) - \color{blue}{d1} \cdot d1 \]

      4. distribute-lft-out-- N/A

         \[\leadsto d1 \cdot \color{blue}{\left(\left(\left(d2 - d3\right) + d4\right) - d1\right)} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(d1, \color{blue}{\left(\left(\left(d2 - d3\right) + d4\right) - d1\right)}\right) \]

      6. associate-+r- N/A

         \[\leadsto \mathsf{*.f64}\left(d1, \left(\left(d2 - d3\right) + \color{blue}{\left(d4 - d1\right)}\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(d1, \left(\left(d2 + \left(\mathsf{neg}\left(d3\right)\right)\right) + \left(\color{blue}{d4} - d1\right)\right)\right) \]

      8. associate-+l+ N/A

         \[\leadsto \mathsf{*.f64}\left(d1, \left(d2 + \color{blue}{\left(\left(\mathsf{neg}\left(d3\right)\right) + \left(d4 - d1\right)\right)}\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(d1, \mathsf{+.f64}\left(d2, \color{blue}{\left(\left(\mathsf{neg}\left(d3\right)\right) + \left(d4 - d1\right)\right)}\right)\right) \]

      10. associate-+r- N/A

          \[\leadsto \mathsf{*.f64}\left(d1, \mathsf{+.f64}\left(d2, \left(\left(\left(\mathsf{neg}\left(d3\right)\right) + d4\right) - \color{blue}{d1}\right)\right)\right) \]

      11. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(d1, \mathsf{+.f64}\left(d2, \left(\left(d4 + \left(\mathsf{neg}\left(d3\right)\right)\right) - d1\right)\right)\right) \]

      12. unsub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(d1, \mathsf{+.f64}\left(d2, \left(\left(d4 - d3\right) - d1\right)\right)\right) \]

      13. associate--l- N/A

          \[\leadsto \mathsf{*.f64}\left(d1, \mathsf{+.f64}\left(d2, \left(d4 - \color{blue}{\left(d3 + d1\right)}\right)\right)\right) \]

      14. --lowering--.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(d1, \mathsf{+.f64}\left(d2, \mathsf{\_.f64}\left(d4, \color{blue}{\left(d3 + d1\right)}\right)\right)\right) \]

      15. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(d1, \mathsf{+.f64}\left(d2, \mathsf{\_.f64}\left(d4, \left(d1 + \color{blue}{d3}\right)\right)\right)\right) \]

      16. +-lowering-+.f64 100.0%

          \[\leadsto \mathsf{*.f64}\left(d1, \mathsf{+.f64}\left(d2, \mathsf{\_.f64}\left(d4, \mathsf{+.f64}\left(d1, \color{blue}{d3}\right)\right)\right)\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{d1 \cdot \left(d2 + \left(d4 - \left(d1 + d3\right)\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d4 around inf

      \[\leadsto \color{blue}{d1 \cdot d4} \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 58.7%

         \[\leadsto \mathsf{*.f64}\left(d1, \color{blue}{d4}\right) \]

   7. Simplified 58.7%

      \[\leadsto \color{blue}{d1 \cdot d4} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 31.1% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ d1 \cdot d2 \end{array} \]

FPCore

(FPCore (d1 d2 d3 d4) :precision binary64 (* d1 d2))

C

double code(double d1, double d2, double d3, double d4) {
	return d1 * d2;
}

Fortran

real(8) function code(d1, d2, d3, d4)
    real(8), intent (in) :: d1
    real(8), intent (in) :: d2
    real(8), intent (in) :: d3
    real(8), intent (in) :: d4
    code = d1 * d2
end function

Java

public static double code(double d1, double d2, double d3, double d4) {
	return d1 * d2;
}

Python

def code(d1, d2, d3, d4):
	return d1 * d2

Julia

function code(d1, d2, d3, d4)
	return Float64(d1 * d2)
end

MATLAB

function tmp = code(d1, d2, d3, d4)
	tmp = d1 * d2;
end

Wolfram

code[d1_, d2_, d3_, d4_] := N[(d1 * d2), $MachinePrecision]

Derivation

1. Initial program 86.3%

   \[\left(\left(d1 \cdot d2 - d1 \cdot d3\right) + d4 \cdot d1\right) - d1 \cdot d1 \]
2. Step-by-step derivation

   1. distribute-lft-out-- N/A

      \[\leadsto \left(d1 \cdot \left(d2 - d3\right) + d4 \cdot d1\right) - d1 \cdot d1 \]

   2. *-commutative N/A

      \[\leadsto \left(d1 \cdot \left(d2 - d3\right) + d1 \cdot d4\right) - d1 \cdot d1 \]

   3. distribute-lft-out N/A

      \[\leadsto d1 \cdot \left(\left(d2 - d3\right) + d4\right) - \color{blue}{d1} \cdot d1 \]

   4. distribute-lft-out-- N/A

      \[\leadsto d1 \cdot \color{blue}{\left(\left(\left(d2 - d3\right) + d4\right) - d1\right)} \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(d1, \color{blue}{\left(\left(\left(d2 - d3\right) + d4\right) - d1\right)}\right) \]

   6. associate-+r- N/A

      \[\leadsto \mathsf{*.f64}\left(d1, \left(\left(d2 - d3\right) + \color{blue}{\left(d4 - d1\right)}\right)\right) \]

   7. sub-neg N/A

      \[\leadsto \mathsf{*.f64}\left(d1, \left(\left(d2 + \left(\mathsf{neg}\left(d3\right)\right)\right) + \left(\color{blue}{d4} - d1\right)\right)\right) \]

   8. associate-+l+ N/A

      \[\leadsto \mathsf{*.f64}\left(d1, \left(d2 + \color{blue}{\left(\left(\mathsf{neg}\left(d3\right)\right) + \left(d4 - d1\right)\right)}\right)\right) \]

   9. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(d1, \mathsf{+.f64}\left(d2, \color{blue}{\left(\left(\mathsf{neg}\left(d3\right)\right) + \left(d4 - d1\right)\right)}\right)\right) \]

   10. associate-+r- N/A

       \[\leadsto \mathsf{*.f64}\left(d1, \mathsf{+.f64}\left(d2, \left(\left(\left(\mathsf{neg}\left(d3\right)\right) + d4\right) - \color{blue}{d1}\right)\right)\right) \]

   11. +-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(d1, \mathsf{+.f64}\left(d2, \left(\left(d4 + \left(\mathsf{neg}\left(d3\right)\right)\right) - d1\right)\right)\right) \]

   12. unsub-neg N/A

       \[\leadsto \mathsf{*.f64}\left(d1, \mathsf{+.f64}\left(d2, \left(\left(d4 - d3\right) - d1\right)\right)\right) \]

   13. associate--l- N/A

       \[\leadsto \mathsf{*.f64}\left(d1, \mathsf{+.f64}\left(d2, \left(d4 - \color{blue}{\left(d3 + d1\right)}\right)\right)\right) \]

   14. --lowering--.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(d1, \mathsf{+.f64}\left(d2, \mathsf{\_.f64}\left(d4, \color{blue}{\left(d3 + d1\right)}\right)\right)\right) \]

   15. +-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(d1, \mathsf{+.f64}\left(d2, \mathsf{\_.f64}\left(d4, \left(d1 + \color{blue}{d3}\right)\right)\right)\right) \]

   16. +-lowering-+.f64 100.0%

       \[\leadsto \mathsf{*.f64}\left(d1, \mathsf{+.f64}\left(d2, \mathsf{\_.f64}\left(d4, \mathsf{+.f64}\left(d1, \color{blue}{d3}\right)\right)\right)\right) \]

3. Simplified 100.0%

   \[\leadsto \color{blue}{d1 \cdot \left(d2 + \left(d4 - \left(d1 + d3\right)\right)\right)} \]
4. Add Preprocessing

5. Taylor expanded in d2 around inf

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

   1. *-lowering-*.f64 29.2%

      \[\leadsto \mathsf{*.f64}\left(d1, \color{blue}{d2}\right) \]

7. Simplified 29.2%

   \[\leadsto \color{blue}{d1 \cdot d2} \]
8. Add Preprocessing

Developer Target 1: 100.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ d1 \cdot \left(\left(\left(d2 - d3\right) + d4\right) - d1\right) \end{array} \]

FPCore

(FPCore (d1 d2 d3 d4) :precision binary64 (* d1 (- (+ (- d2 d3) d4) d1)))

C

double code(double d1, double d2, double d3, double d4) {
	return d1 * (((d2 - d3) + d4) - d1);
}

Fortran

real(8) function code(d1, d2, d3, d4)
    real(8), intent (in) :: d1
    real(8), intent (in) :: d2
    real(8), intent (in) :: d3
    real(8), intent (in) :: d4
    code = d1 * (((d2 - d3) + d4) - d1)
end function

Java

public static double code(double d1, double d2, double d3, double d4) {
	return d1 * (((d2 - d3) + d4) - d1);
}

Python

def code(d1, d2, d3, d4):
	return d1 * (((d2 - d3) + d4) - d1)

Julia

function code(d1, d2, d3, d4)
	return Float64(d1 * Float64(Float64(Float64(d2 - d3) + d4) - d1))
end

MATLAB

function tmp = code(d1, d2, d3, d4)
	tmp = d1 * (((d2 - d3) + d4) - d1);
end

Wolfram

code[d1_, d2_, d3_, d4_] := N[(d1 * N[(N[(N[(d2 - d3), $MachinePrecision] + d4), $MachinePrecision] - d1), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2024139 
(FPCore (d1 d2 d3 d4)
  :name "FastMath dist4"
  :precision binary64

  :alt
  (! :herbie-platform default (* d1 (- (+ (- d2 d3) d4) d1)))

  (- (+ (- (* d1 d2) (* d1 d3)) (* d4 d1)) (* d1 d1)))