FastMath dist3

Percentage Accurate: 97.7% → 100.0%

Time: 3.4s

Alternatives: 6

Speedup: 1.9×

• 21.4% 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: 97.7% 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

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

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(d2 + Float64(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[(d2 + N[(d3 + 37.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.2%

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

   1. *-commutative 99.2%

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

   2. distribute-lft-out 99.9%

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

   3. distribute-lft-out 100.0%

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

   4. associate-+r+ 100.0%

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

   5. associate-+l+ 100.0%

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

   6. metadata-eval 100.0%

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

3. Simplified 100.0%

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

4. Final simplification 100.0%

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

Alternative 2: 52.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d3 \leq -1.02 \cdot 10^{-166}:\\ \;\;\;\;d1 \cdot d2\\ \mathbf{elif}\;d3 \leq -1.1 \cdot 10^{-200}:\\ \;\;\;\;d1 \cdot 37\\ \mathbf{elif}\;d3 \leq 2.35 \cdot 10^{-120}:\\ \;\;\;\;d1 \cdot d2\\ \mathbf{elif}\;d3 \leq 31:\\ \;\;\;\;d1 \cdot 37\\ \mathbf{else}:\\ \;\;\;\;d1 \cdot d3\\ \end{array} \end{array} \]

FPCore

(FPCore (d1 d2 d3)
 :precision binary64
 (if (<= d3 -1.02e-166)
   (* d1 d2)
   (if (<= d3 -1.1e-200)
     (* d1 37.0)
     (if (<= d3 2.35e-120)
       (* d1 d2)
       (if (<= d3 31.0) (* d1 37.0) (* d1 d3))))))

C

double code(double d1, double d2, double d3) {
	double tmp;
	if (d3 <= -1.02e-166) {
		tmp = d1 * d2;
	} else if (d3 <= -1.1e-200) {
		tmp = d1 * 37.0;
	} else if (d3 <= 2.35e-120) {
		tmp = d1 * d2;
	} else if (d3 <= 31.0) {
		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 <= (-1.02d-166)) then
        tmp = d1 * d2
    else if (d3 <= (-1.1d-200)) then
        tmp = d1 * 37.0d0
    else if (d3 <= 2.35d-120) then
        tmp = d1 * d2
    else if (d3 <= 31.0d0) 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 <= -1.02e-166) {
		tmp = d1 * d2;
	} else if (d3 <= -1.1e-200) {
		tmp = d1 * 37.0;
	} else if (d3 <= 2.35e-120) {
		tmp = d1 * d2;
	} else if (d3 <= 31.0) {
		tmp = d1 * 37.0;
	} else {
		tmp = d1 * d3;
	}
	return tmp;
}

Python

def code(d1, d2, d3):
	tmp = 0
	if d3 <= -1.02e-166:
		tmp = d1 * d2
	elif d3 <= -1.1e-200:
		tmp = d1 * 37.0
	elif d3 <= 2.35e-120:
		tmp = d1 * d2
	elif d3 <= 31.0:
		tmp = d1 * 37.0
	else:
		tmp = d1 * d3
	return tmp

Julia

function code(d1, d2, d3)
	tmp = 0.0
	if (d3 <= -1.02e-166)
		tmp = Float64(d1 * d2);
	elseif (d3 <= -1.1e-200)
		tmp = Float64(d1 * 37.0);
	elseif (d3 <= 2.35e-120)
		tmp = Float64(d1 * d2);
	elseif (d3 <= 31.0)
		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 <= -1.02e-166)
		tmp = d1 * d2;
	elseif (d3 <= -1.1e-200)
		tmp = d1 * 37.0;
	elseif (d3 <= 2.35e-120)
		tmp = d1 * d2;
	elseif (d3 <= 31.0)
		tmp = d1 * 37.0;
	else
		tmp = d1 * d3;
	end
	tmp_2 = tmp;
end

Wolfram

code[d1_, d2_, d3_] := If[LessEqual[d3, -1.02e-166], N[(d1 * d2), $MachinePrecision], If[LessEqual[d3, -1.1e-200], N[(d1 * 37.0), $MachinePrecision], If[LessEqual[d3, 2.35e-120], N[(d1 * d2), $MachinePrecision], If[LessEqual[d3, 31.0], N[(d1 * 37.0), $MachinePrecision], N[(d1 * d3), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if d3 < -1.02000000000000007e-166 or -1.10000000000000007e-200 < d3 < 2.35000000000000008e-120

   1. Initial program 99.3%

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

      1. *-commutative 99.3%

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

      2. distribute-lft-out 100.0%

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

      3. distribute-lft-out 100.0%

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

      4. associate-+r+ 100.0%

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

      5. associate-+l+ 100.0%

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

      6. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in d2 around inf 44.2%

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

   if -1.02000000000000007e-166 < d3 < -1.10000000000000007e-200 or 2.35000000000000008e-120 < d3 < 31

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. distribute-lft-out 100.0%

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

      3. distribute-lft-out 99.9%

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

      4. associate-+r+ 100.0%

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

      5. associate-+l+ 100.0%

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

      6. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in d3 around 0 97.9%

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

   5. Taylor expanded in d2 around 0 54.4%

      \[\leadsto \color{blue}{37 \cdot d1} \]
   6. Step-by-step derivation

      1. *-commutative 54.4%

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

   7. Simplified 54.4%

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

   if 31 < d3

   1. Initial program 98.4%

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

      1. *-commutative 98.4%

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

      2. distribute-lft-out 99.9%

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

      3. distribute-lft-out 100.0%

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

      4. associate-+r+ 100.0%

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

      5. associate-+l+ 100.0%

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

      6. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in d3 around inf 78.3%

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

4. Final simplification 54.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d3 \leq -1.02 \cdot 10^{-166}:\\ \;\;\;\;d1 \cdot d2\\ \mathbf{elif}\;d3 \leq -1.1 \cdot 10^{-200}:\\ \;\;\;\;d1 \cdot 37\\ \mathbf{elif}\;d3 \leq 2.35 \cdot 10^{-120}:\\ \;\;\;\;d1 \cdot d2\\ \mathbf{elif}\;d3 \leq 31:\\ \;\;\;\;d1 \cdot 37\\ \mathbf{else}:\\ \;\;\;\;d1 \cdot d3\\ \end{array} \]

Alternative 3: 58.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d3 \leq -3.6 \cdot 10^{-106}:\\ \;\;\;\;d1 \cdot d3\\ \mathbf{elif}\;d3 \leq 31:\\ \;\;\;\;d1 \cdot 37\\ \mathbf{else}:\\ \;\;\;\;d1 \cdot d3\\ \end{array} \end{array} \]

FPCore

(FPCore (d1 d2 d3)
 :precision binary64
 (if (<= d3 -3.6e-106) (* d1 d3) (if (<= d3 31.0) (* d1 37.0) (* d1 d3))))

C

double code(double d1, double d2, double d3) {
	double tmp;
	if (d3 <= -3.6e-106) {
		tmp = d1 * d3;
	} else if (d3 <= 31.0) {
		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 <= (-3.6d-106)) then
        tmp = d1 * d3
    else if (d3 <= 31.0d0) 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 <= -3.6e-106) {
		tmp = d1 * d3;
	} else if (d3 <= 31.0) {
		tmp = d1 * 37.0;
	} else {
		tmp = d1 * d3;
	}
	return tmp;
}

Python

def code(d1, d2, d3):
	tmp = 0
	if d3 <= -3.6e-106:
		tmp = d1 * d3
	elif d3 <= 31.0:
		tmp = d1 * 37.0
	else:
		tmp = d1 * d3
	return tmp

Julia

function code(d1, d2, d3)
	tmp = 0.0
	if (d3 <= -3.6e-106)
		tmp = Float64(d1 * d3);
	elseif (d3 <= 31.0)
		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 <= -3.6e-106)
		tmp = d1 * d3;
	elseif (d3 <= 31.0)
		tmp = d1 * 37.0;
	else
		tmp = d1 * d3;
	end
	tmp_2 = tmp;
end

Wolfram

code[d1_, d2_, d3_] := If[LessEqual[d3, -3.6e-106], N[(d1 * d3), $MachinePrecision], If[LessEqual[d3, 31.0], N[(d1 * 37.0), $MachinePrecision], N[(d1 * d3), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if d3 < -3.60000000000000013e-106 or 31 < d3

   1. Initial program 98.7%

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

      1. *-commutative 98.7%

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

      2. distribute-lft-out 100.0%

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

      3. distribute-lft-out 100.0%

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

      4. associate-+r+ 100.0%

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

      5. associate-+l+ 100.0%

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

      6. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in d3 around inf 68.2%

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

   if -3.60000000000000013e-106 < d3 < 31

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

      2. distribute-lft-out 99.9%

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

      3. distribute-lft-out 99.9%

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

      4. associate-+r+ 100.0%

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

      5. associate-+l+ 100.0%

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

      6. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in d3 around 0 99.2%

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

   5. Taylor expanded in d2 around 0 46.3%

      \[\leadsto \color{blue}{37 \cdot d1} \]
   6. Step-by-step derivation

      1. *-commutative 46.3%

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

   7. Simplified 46.3%

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

4. Final simplification 59.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d3 \leq -3.6 \cdot 10^{-106}:\\ \;\;\;\;d1 \cdot d3\\ \mathbf{elif}\;d3 \leq 31:\\ \;\;\;\;d1 \cdot 37\\ \mathbf{else}:\\ \;\;\;\;d1 \cdot d3\\ \end{array} \]

Alternative 4: 76.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d3 \leq 6.5 \cdot 10^{+25}:\\ \;\;\;\;d1 \cdot \left(d2 + 37\right)\\ \mathbf{else}:\\ \;\;\;\;d1 \cdot d3\\ \end{array} \end{array} \]

FPCore

(FPCore (d1 d2 d3)
 :precision binary64
 (if (<= d3 6.5e+25) (* d1 (+ d2 37.0)) (* d1 d3)))

C

double code(double d1, double d2, double d3) {
	double tmp;
	if (d3 <= 6.5e+25) {
		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 <= 6.5d+25) 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 <= 6.5e+25) {
		tmp = d1 * (d2 + 37.0);
	} else {
		tmp = d1 * d3;
	}
	return tmp;
}

Python

def code(d1, d2, d3):
	tmp = 0
	if d3 <= 6.5e+25:
		tmp = d1 * (d2 + 37.0)
	else:
		tmp = d1 * d3
	return tmp

Julia

function code(d1, d2, d3)
	tmp = 0.0
	if (d3 <= 6.5e+25)
		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 <= 6.5e+25)
		tmp = d1 * (d2 + 37.0);
	else
		tmp = d1 * d3;
	end
	tmp_2 = tmp;
end

Wolfram

code[d1_, d2_, d3_] := If[LessEqual[d3, 6.5e+25], N[(d1 * N[(d2 + 37.0), $MachinePrecision]), $MachinePrecision], N[(d1 * d3), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d3 < 6.50000000000000005e25

   1. Initial program 99.4%

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

      1. *-commutative 99.4%

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

      2. distribute-lft-out 99.9%

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

      3. distribute-lft-out 100.0%

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

      4. associate-+r+ 100.0%

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

      5. associate-+l+ 100.0%

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

      6. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in d3 around 0 72.2%

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

   if 6.50000000000000005e25 < d3

   1. Initial program 98.4%

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

      1. *-commutative 98.4%

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

      2. distribute-lft-out 99.9%

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

      3. distribute-lft-out 99.9%

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

      4. associate-+r+ 99.9%

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

      5. associate-+l+ 99.9%

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

      6. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in d3 around inf 79.5%

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

4. Final simplification 73.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d3 \leq 6.5 \cdot 10^{+25}:\\ \;\;\;\;d1 \cdot \left(d2 + 37\right)\\ \mathbf{else}:\\ \;\;\;\;d1 \cdot d3\\ \end{array} \]

Alternative 5: 76.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d3 \leq 6.5 \cdot 10^{-5}:\\ \;\;\;\;d1 \cdot \left(d2 + 37\right)\\ \mathbf{else}:\\ \;\;\;\;d1 \cdot \left(d3 + 37\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (d1 d2 d3)
 :precision binary64
 (if (<= d3 6.5e-5) (* d1 (+ d2 37.0)) (* d1 (+ d3 37.0))))

C

double code(double d1, double d2, double d3) {
	double tmp;
	if (d3 <= 6.5e-5) {
		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 <= 6.5d-5) 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 <= 6.5e-5) {
		tmp = d1 * (d2 + 37.0);
	} else {
		tmp = d1 * (d3 + 37.0);
	}
	return tmp;
}

Python

def code(d1, d2, d3):
	tmp = 0
	if d3 <= 6.5e-5:
		tmp = d1 * (d2 + 37.0)
	else:
		tmp = d1 * (d3 + 37.0)
	return tmp

Julia

function code(d1, d2, d3)
	tmp = 0.0
	if (d3 <= 6.5e-5)
		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 <= 6.5e-5)
		tmp = d1 * (d2 + 37.0);
	else
		tmp = d1 * (d3 + 37.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[d1_, d2_, d3_] := If[LessEqual[d3, 6.5e-5], 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 < 6.49999999999999943e-5

   1. Initial program 99.4%

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

      1. *-commutative 99.4%

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

      2. distribute-lft-out 100.0%

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

      3. distribute-lft-out 100.0%

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

      4. associate-+r+ 100.0%

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

      5. associate-+l+ 100.0%

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

      6. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in d3 around 0 72.9%

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

   if 6.49999999999999943e-5 < d3

   1. Initial program 98.4%

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

      1. *-commutative 98.4%

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

      2. distribute-lft-out 99.9%

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

      3. distribute-lft-out 100.0%

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

      4. associate-+r+ 100.0%

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

      5. associate-+l+ 100.0%

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

      6. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in d2 around 0 79.3%

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

4. Final simplification 74.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d3 \leq 6.5 \cdot 10^{-5}:\\ \;\;\;\;d1 \cdot \left(d2 + 37\right)\\ \mathbf{else}:\\ \;\;\;\;d1 \cdot \left(d3 + 37\right)\\ \end{array} \]

Alternative 6: 26.6% 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 99.2%

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

   1. *-commutative 99.2%

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

   2. distribute-lft-out 99.9%

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

   3. distribute-lft-out 100.0%

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

   4. associate-+r+ 100.0%

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

   5. associate-+l+ 100.0%

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

   6. metadata-eval 100.0%

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

3. Simplified 100.0%

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

4. Taylor expanded in d3 around 0 61.6%

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

5. Taylor expanded in d2 around 0 24.4%

   \[\leadsto \color{blue}{37 \cdot d1} \]
6. Step-by-step derivation

   1. *-commutative 24.4%

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

7. Simplified 24.4%

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

8. Final simplification 24.4%

   \[\leadsto d1 \cdot 37 \]

Developer target: 100.0% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2023182 
(FPCore (d1 d2 d3)
  :name "FastMath dist3"
  :precision binary64

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

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