FastMath dist3

Percentage Accurate: 97.9% → 100.0%

Time: 4.3s

Alternatives: 6

Speedup: 1.9×

• 20.7% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 97.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.0%

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

   1. cancel-sign-sub 98.0%

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

   2. distribute-lft-neg-out 98.0%

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

   3. distribute-rgt-neg-in 98.0%

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

   4. *-commutative 98.0%

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

   5. distribute-lft-out 100.0%

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

   6. distribute-lft-out-- 100.0%

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

   7. associate-+r+ 100.0%

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

   8. associate--l+ 100.0%

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

   9. metadata-eval 100.0%

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

   10. metadata-eval 100.0%

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

3. Simplified 100.0%

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

5. Final simplification 100.0%

   \[\leadsto d1 \cdot \left(\left(d2 + d3\right) + 37\right) \]
6. Add Preprocessing

Alternative 2: 51.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d3 \leq 5.1 \cdot 10^{-189}:\\ \;\;\;\;d1 \cdot d2\\ \mathbf{elif}\;d3 \leq 4500:\\ \;\;\;\;d1 \cdot 37\\ \mathbf{else}:\\ \;\;\;\;d1 \cdot d3\\ \end{array} \end{array} \]

FPCore

(FPCore (d1 d2 d3)
 :precision binary64
 (if (<= d3 5.1e-189) (* d1 d2) (if (<= d3 4500.0) (* d1 37.0) (* d1 d3))))

C

double code(double d1, double d2, double d3) {
	double tmp;
	if (d3 <= 5.1e-189) {
		tmp = d1 * d2;
	} else if (d3 <= 4500.0) {
		tmp = d1 * 37.0;
	} else {
		tmp = d1 * d3;
	}
	return tmp;
}

Fortran

real(8) function code(d1, d2, d3)
    real(8), intent (in) :: d1
    real(8), intent (in) :: d2
    real(8), intent (in) :: d3
    real(8) :: tmp
    if (d3 <= 5.1d-189) then
        tmp = d1 * d2
    else if (d3 <= 4500.0d0) then
        tmp = d1 * 37.0d0
    else
        tmp = d1 * d3
    end if
    code = tmp
end function

Java

public static double code(double d1, double d2, double d3) {
	double tmp;
	if (d3 <= 5.1e-189) {
		tmp = d1 * d2;
	} else if (d3 <= 4500.0) {
		tmp = d1 * 37.0;
	} else {
		tmp = d1 * d3;
	}
	return tmp;
}

Python

def code(d1, d2, d3):
	tmp = 0
	if d3 <= 5.1e-189:
		tmp = d1 * d2
	elif d3 <= 4500.0:
		tmp = d1 * 37.0
	else:
		tmp = d1 * d3
	return tmp

Julia

function code(d1, d2, d3)
	tmp = 0.0
	if (d3 <= 5.1e-189)
		tmp = Float64(d1 * d2);
	elseif (d3 <= 4500.0)
		tmp = Float64(d1 * 37.0);
	else
		tmp = Float64(d1 * d3);
	end
	return tmp
end

MATLAB

function tmp_2 = code(d1, d2, d3)
	tmp = 0.0;
	if (d3 <= 5.1e-189)
		tmp = d1 * d2;
	elseif (d3 <= 4500.0)
		tmp = d1 * 37.0;
	else
		tmp = d1 * d3;
	end
	tmp_2 = tmp;
end

Wolfram

code[d1_, d2_, d3_] := If[LessEqual[d3, 5.1e-189], N[(d1 * d2), $MachinePrecision], If[LessEqual[d3, 4500.0], N[(d1 * 37.0), $MachinePrecision], N[(d1 * d3), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if d3 < 5.0999999999999998e-189

   1. Initial program 97.5%

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

      1. cancel-sign-sub 97.5%

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

      2. distribute-lft-neg-out 97.5%

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

      3. distribute-rgt-neg-in 97.5%

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

      4. *-commutative 97.5%

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

      5. distribute-lft-out 100.0%

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

      6. distribute-lft-out-- 100.0%

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

      7. associate-+r+ 100.0%

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

      8. associate--l+ 100.0%

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

      9. metadata-eval 100.0%

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

      10. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in d2 around inf 46.5%

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

   if 5.0999999999999998e-189 < d3 < 4500

   1. Initial program 100.0%

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

      1. cancel-sign-sub 100.0%

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

      2. distribute-lft-neg-out 100.0%

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

      3. distribute-rgt-neg-in 100.0%

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

      4. *-commutative 100.0%

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

      5. distribute-lft-out 100.0%

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

      6. distribute-lft-out-- 100.0%

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

      7. associate-+r+ 100.0%

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

      8. associate--l+ 100.0%

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

      9. metadata-eval 100.0%

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

      10. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in d3 around 0 94.8%

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

   6. Taylor expanded in d2 around 0 49.1%

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

   if 4500 < d3

   1. Initial program 98.3%

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

      1. cancel-sign-sub 98.3%

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

      2. distribute-lft-neg-out 98.3%

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

      3. distribute-rgt-neg-in 98.3%

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

      4. *-commutative 98.3%

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

      5. distribute-lft-out 100.0%

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

      6. distribute-lft-out-- 100.0%

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

      7. associate-+r+ 100.0%

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

      8. associate--l+ 100.0%

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

      9. metadata-eval 100.0%

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

      10. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in d3 around inf 77.3%

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

4. Final simplification 53.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d3 \leq 5.1 \cdot 10^{-189}:\\ \;\;\;\;d1 \cdot d2\\ \mathbf{elif}\;d3 \leq 4500:\\ \;\;\;\;d1 \cdot 37\\ \mathbf{else}:\\ \;\;\;\;d1 \cdot d3\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 76.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d3 \leq 3.3 \cdot 10^{+33}:\\ \;\;\;\;d1 \cdot \left(d2 + 37\right)\\ \mathbf{else}:\\ \;\;\;\;d1 \cdot d3\\ \end{array} \end{array} \]

FPCore

(FPCore (d1 d2 d3)
 :precision binary64
 (if (<= d3 3.3e+33) (* d1 (+ d2 37.0)) (* d1 d3)))

C

double code(double d1, double d2, double d3) {
	double tmp;
	if (d3 <= 3.3e+33) {
		tmp = d1 * (d2 + 37.0);
	} else {
		tmp = d1 * d3;
	}
	return tmp;
}

Fortran

real(8) function code(d1, d2, d3)
    real(8), intent (in) :: d1
    real(8), intent (in) :: d2
    real(8), intent (in) :: d3
    real(8) :: tmp
    if (d3 <= 3.3d+33) then
        tmp = d1 * (d2 + 37.0d0)
    else
        tmp = d1 * d3
    end if
    code = tmp
end function

Java

public static double code(double d1, double d2, double d3) {
	double tmp;
	if (d3 <= 3.3e+33) {
		tmp = d1 * (d2 + 37.0);
	} else {
		tmp = d1 * d3;
	}
	return tmp;
}

Python

def code(d1, d2, d3):
	tmp = 0
	if d3 <= 3.3e+33:
		tmp = d1 * (d2 + 37.0)
	else:
		tmp = d1 * d3
	return tmp

Julia

function code(d1, d2, d3)
	tmp = 0.0
	if (d3 <= 3.3e+33)
		tmp = Float64(d1 * Float64(d2 + 37.0));
	else
		tmp = Float64(d1 * d3);
	end
	return tmp
end

MATLAB

function tmp_2 = code(d1, d2, d3)
	tmp = 0.0;
	if (d3 <= 3.3e+33)
		tmp = d1 * (d2 + 37.0);
	else
		tmp = d1 * d3;
	end
	tmp_2 = tmp;
end

Wolfram

code[d1_, d2_, d3_] := If[LessEqual[d3, 3.3e+33], N[(d1 * N[(d2 + 37.0), $MachinePrecision]), $MachinePrecision], N[(d1 * d3), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d3 < 3.29999999999999976e33

   1. Initial program 98.0%

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

      1. cancel-sign-sub 98.0%

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

      2. distribute-lft-neg-out 98.0%

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

      3. distribute-rgt-neg-in 98.0%

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

      4. *-commutative 98.0%

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

      5. distribute-lft-out 100.0%

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

      6. distribute-lft-out-- 100.0%

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

      7. associate-+r+ 100.0%

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

      8. associate--l+ 100.0%

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

      9. metadata-eval 100.0%

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

      10. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in d3 around 0 75.0%

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

   if 3.29999999999999976e33 < d3

   1. Initial program 98.1%

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

      1. cancel-sign-sub 98.1%

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

      2. distribute-lft-neg-out 98.1%

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

      3. distribute-rgt-neg-in 98.1%

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

      4. *-commutative 98.1%

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

      5. distribute-lft-out 100.0%

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

      6. distribute-lft-out-- 100.0%

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

      7. associate-+r+ 100.0%

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

      8. associate--l+ 100.0%

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

      9. metadata-eval 100.0%

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

      10. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in d3 around inf 81.5%

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

4. Final simplification 76.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d3 \leq 3.3 \cdot 10^{+33}:\\ \;\;\;\;d1 \cdot \left(d2 + 37\right)\\ \mathbf{else}:\\ \;\;\;\;d1 \cdot d3\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 75.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d3 \leq 6.7 \cdot 10^{-6}:\\ \;\;\;\;d1 \cdot \left(d2 + 37\right)\\ \mathbf{else}:\\ \;\;\;\;d1 \cdot \left(d3 + 37\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (d1 d2 d3)
 :precision binary64
 (if (<= d3 6.7e-6) (* d1 (+ d2 37.0)) (* d1 (+ d3 37.0))))

C

double code(double d1, double d2, double d3) {
	double tmp;
	if (d3 <= 6.7e-6) {
		tmp = d1 * (d2 + 37.0);
	} else {
		tmp = d1 * (d3 + 37.0);
	}
	return tmp;
}

Fortran

real(8) function code(d1, d2, d3)
    real(8), intent (in) :: d1
    real(8), intent (in) :: d2
    real(8), intent (in) :: d3
    real(8) :: tmp
    if (d3 <= 6.7d-6) then
        tmp = d1 * (d2 + 37.0d0)
    else
        tmp = d1 * (d3 + 37.0d0)
    end if
    code = tmp
end function

Java

public static double code(double d1, double d2, double d3) {
	double tmp;
	if (d3 <= 6.7e-6) {
		tmp = d1 * (d2 + 37.0);
	} else {
		tmp = d1 * (d3 + 37.0);
	}
	return tmp;
}

Python

def code(d1, d2, d3):
	tmp = 0
	if d3 <= 6.7e-6:
		tmp = d1 * (d2 + 37.0)
	else:
		tmp = d1 * (d3 + 37.0)
	return tmp

Julia

function code(d1, d2, d3)
	tmp = 0.0
	if (d3 <= 6.7e-6)
		tmp = Float64(d1 * Float64(d2 + 37.0));
	else
		tmp = Float64(d1 * Float64(d3 + 37.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d1, d2, d3)
	tmp = 0.0;
	if (d3 <= 6.7e-6)
		tmp = d1 * (d2 + 37.0);
	else
		tmp = d1 * (d3 + 37.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[d1_, d2_, d3_] := If[LessEqual[d3, 6.7e-6], N[(d1 * N[(d2 + 37.0), $MachinePrecision]), $MachinePrecision], N[(d1 * N[(d3 + 37.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d3 < 6.7e-6

   1. Initial program 97.9%

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

      1. cancel-sign-sub 97.9%

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

      2. distribute-lft-neg-out 97.9%

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

      3. distribute-rgt-neg-in 97.9%

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

      4. *-commutative 97.9%

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

      5. distribute-lft-out 100.0%

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

      6. distribute-lft-out-- 100.0%

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

      7. associate-+r+ 100.0%

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

      8. associate--l+ 100.0%

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

      9. metadata-eval 100.0%

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

      10. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in d3 around 0 76.0%

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

   if 6.7e-6 < d3

   1. Initial program 98.3%

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

      1. cancel-sign-sub 98.3%

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

      2. distribute-lft-neg-out 98.3%

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

      3. distribute-rgt-neg-in 98.3%

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

      4. *-commutative 98.3%

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

      5. distribute-lft-out 100.0%

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

      6. distribute-lft-out-- 100.0%

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

      7. associate-+r+ 100.0%

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

      8. associate--l+ 100.0%

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

      9. metadata-eval 100.0%

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

      10. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in d2 around 0 79.1%

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

4. Final simplification 76.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d3 \leq 6.7 \cdot 10^{-6}:\\ \;\;\;\;d1 \cdot \left(d2 + 37\right)\\ \mathbf{else}:\\ \;\;\;\;d1 \cdot \left(d3 + 37\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 45.3% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if d2 < -37

   1. Initial program 98.5%

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

      1. cancel-sign-sub 98.5%

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

      2. distribute-lft-neg-out 98.5%

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

      3. distribute-rgt-neg-in 98.5%

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

      4. *-commutative 98.5%

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

      5. distribute-lft-out 100.0%

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

      6. distribute-lft-out-- 100.0%

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

      7. associate-+r+ 100.0%

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

      8. associate--l+ 100.0%

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

      9. metadata-eval 100.0%

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

      10. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in d2 around inf 76.4%

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

   if -37 < d2

   1. Initial program 97.8%

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

      1. cancel-sign-sub 97.8%

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

      2. distribute-lft-neg-out 97.8%

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

      3. distribute-rgt-neg-in 97.8%

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

      4. *-commutative 97.8%

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

      5. distribute-lft-out 100.0%

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

      6. distribute-lft-out-- 100.0%

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

      7. associate-+r+ 100.0%

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

      8. associate--l+ 100.0%

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

      9. metadata-eval 100.0%

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

      10. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in d3 around 0 60.3%

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

   6. Taylor expanded in d2 around 0 35.0%

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

4. Final simplification 46.3%

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

Alternative 6: 26.5% accurate, 4.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.0%

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

   1. cancel-sign-sub 98.0%

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

   2. distribute-lft-neg-out 98.0%

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

   3. distribute-rgt-neg-in 98.0%

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

   4. *-commutative 98.0%

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

   5. distribute-lft-out 100.0%

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

   6. distribute-lft-out-- 100.0%

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

   7. associate-+r+ 100.0%

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

   8. associate--l+ 100.0%

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

   9. metadata-eval 100.0%

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

   10. metadata-eval 100.0%

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

3. Simplified 100.0%

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

5. Taylor expanded in d3 around 0 64.7%

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

6. Taylor expanded in d2 around 0 26.0%

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

7. Final simplification 26.0%

   \[\leadsto d1 \cdot 37 \]
8. Add Preprocessing

Developer target: 100.0% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024046 
(FPCore (d1 d2 d3)
  :name "FastMath dist3"
  :precision binary64

  :alt
  (* d1 (+ (+ 37.0 d3) d2))

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