FastMath dist3

Percentage Accurate: 97.8% → 100.0%

Time: 2.8s

Alternatives: 6

Speedup: 1.9×

• 20.1% 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: 97.8% 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 98.8%

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

   1. *-commutative 98.8%

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

   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. Final simplification 100.0%

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

Alternative 2: 74.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d3 \leq 190000 \lor \neg \left(d3 \leq 2.2 \cdot 10^{+42}\right) \land d3 \leq 4.5 \cdot 10^{+58}:\\ \;\;\;\;d1 \cdot \left(d2 + 37\right)\\ \mathbf{else}:\\ \;\;\;\;d1 \cdot d3\\ \end{array} \end{array} \]

FPCore

(FPCore (d1 d2 d3)
 :precision binary64
 (if (or (<= d3 190000.0) (and (not (<= d3 2.2e+42)) (<= d3 4.5e+58)))
   (* d1 (+ d2 37.0))
   (* d1 d3)))

C

double code(double d1, double d2, double d3) {
	double tmp;
	if ((d3 <= 190000.0) || (!(d3 <= 2.2e+42) && (d3 <= 4.5e+58))) {
		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 <= 190000.0d0) .or. (.not. (d3 <= 2.2d+42)) .and. (d3 <= 4.5d+58)) 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 <= 190000.0) || (!(d3 <= 2.2e+42) && (d3 <= 4.5e+58))) {
		tmp = d1 * (d2 + 37.0);
	} else {
		tmp = d1 * d3;
	}
	return tmp;
}

Python

def code(d1, d2, d3):
	tmp = 0
	if (d3 <= 190000.0) or (not (d3 <= 2.2e+42) and (d3 <= 4.5e+58)):
		tmp = d1 * (d2 + 37.0)
	else:
		tmp = d1 * d3
	return tmp

Julia

function code(d1, d2, d3)
	tmp = 0.0
	if ((d3 <= 190000.0) || (!(d3 <= 2.2e+42) && (d3 <= 4.5e+58)))
		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 <= 190000.0) || (~((d3 <= 2.2e+42)) && (d3 <= 4.5e+58)))
		tmp = d1 * (d2 + 37.0);
	else
		tmp = d1 * d3;
	end
	tmp_2 = tmp;
end

Wolfram

code[d1_, d2_, d3_] := If[Or[LessEqual[d3, 190000.0], And[N[Not[LessEqual[d3, 2.2e+42]], $MachinePrecision], LessEqual[d3, 4.5e+58]]], N[(d1 * N[(d2 + 37.0), $MachinePrecision]), $MachinePrecision], N[(d1 * d3), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d3 < 1.9e5 or 2.2000000000000001e42 < d3 < 4.4999999999999998e58

   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 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 73.8%

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

   if 1.9e5 < d3 < 2.2000000000000001e42 or 4.4999999999999998e58 < d3

   1. Initial program 95.6%

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

      1. *-commutative 95.6%

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

      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 78.4%

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

4. Final simplification 75.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d3 \leq 190000 \lor \neg \left(d3 \leq 2.2 \cdot 10^{+42}\right) \land d3 \leq 4.5 \cdot 10^{+58}:\\ \;\;\;\;d1 \cdot \left(d2 + 37\right)\\ \mathbf{else}:\\ \;\;\;\;d1 \cdot d3\\ \end{array} \]

Alternative 3: 60.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d3 \leq -6.9 \cdot 10^{-65}:\\ \;\;\;\;d1 \cdot d3\\ \mathbf{elif}\;d3 \leq 2100:\\ \;\;\;\;d1 \cdot 37\\ \mathbf{else}:\\ \;\;\;\;d1 \cdot d3\\ \end{array} \end{array} \]

FPCore

(FPCore (d1 d2 d3)
 :precision binary64
 (if (<= d3 -6.9e-65) (* d1 d3) (if (<= d3 2100.0) (* d1 37.0) (* d1 d3))))

C

double code(double d1, double d2, double d3) {
	double tmp;
	if (d3 <= -6.9e-65) {
		tmp = d1 * d3;
	} else if (d3 <= 2100.0) {
		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 <= (-6.9d-65)) then
        tmp = d1 * d3
    else if (d3 <= 2100.0d0) 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 <= -6.9e-65) {
		tmp = d1 * d3;
	} else if (d3 <= 2100.0) {
		tmp = d1 * 37.0;
	} else {
		tmp = d1 * d3;
	}
	return tmp;
}

Python

def code(d1, d2, d3):
	tmp = 0
	if d3 <= -6.9e-65:
		tmp = d1 * d3
	elif d3 <= 2100.0:
		tmp = d1 * 37.0
	else:
		tmp = d1 * d3
	return tmp

Julia

function code(d1, d2, d3)
	tmp = 0.0
	if (d3 <= -6.9e-65)
		tmp = Float64(d1 * d3);
	elseif (d3 <= 2100.0)
		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 <= -6.9e-65)
		tmp = d1 * d3;
	elseif (d3 <= 2100.0)
		tmp = d1 * 37.0;
	else
		tmp = d1 * d3;
	end
	tmp_2 = tmp;
end

Wolfram

code[d1_, d2_, d3_] := If[LessEqual[d3, -6.9e-65], N[(d1 * d3), $MachinePrecision], If[LessEqual[d3, 2100.0], N[(d1 * 37.0), $MachinePrecision], N[(d1 * d3), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if d3 < -6.89999999999999991e-65 or 2100 < d3

   1. Initial program 98.0%

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

      1. *-commutative 98.0%

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

      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 69.4%

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

   if -6.89999999999999991e-65 < d3 < 2100

   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 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 99.9%

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

   5. Taylor expanded in d2 around 0 57.3%

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

4. Final simplification 64.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d3 \leq -6.9 \cdot 10^{-65}:\\ \;\;\;\;d1 \cdot d3\\ \mathbf{elif}\;d3 \leq 2100:\\ \;\;\;\;d1 \cdot 37\\ \mathbf{else}:\\ \;\;\;\;d1 \cdot d3\\ \end{array} \]

Alternative 4: 52.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d2 \leq -38:\\ \;\;\;\;d1 \cdot d2\\ \mathbf{elif}\;d2 \leq -2 \cdot 10^{-286}:\\ \;\;\;\;d1 \cdot 37\\ \mathbf{else}:\\ \;\;\;\;d1 \cdot d3\\ \end{array} \end{array} \]

FPCore

(FPCore (d1 d2 d3)
 :precision binary64
 (if (<= d2 -38.0) (* d1 d2) (if (<= d2 -2e-286) (* d1 37.0) (* d1 d3))))

C

double code(double d1, double d2, double d3) {
	double tmp;
	if (d2 <= -38.0) {
		tmp = d1 * d2;
	} else if (d2 <= -2e-286) {
		tmp = d1 * 37.0;
	} else {
		tmp = d1 * d3;
	}
	return tmp;
}

Fortran

real(8) function code(d1, d2, d3)
    real(8), intent (in) :: d1
    real(8), intent (in) :: d2
    real(8), intent (in) :: d3
    real(8) :: tmp
    if (d2 <= (-38.0d0)) then
        tmp = d1 * d2
    else if (d2 <= (-2d-286)) then
        tmp = d1 * 37.0d0
    else
        tmp = d1 * d3
    end if
    code = tmp
end function

Java

public static double code(double d1, double d2, double d3) {
	double tmp;
	if (d2 <= -38.0) {
		tmp = d1 * d2;
	} else if (d2 <= -2e-286) {
		tmp = d1 * 37.0;
	} else {
		tmp = d1 * d3;
	}
	return tmp;
}

Python

def code(d1, d2, d3):
	tmp = 0
	if d2 <= -38.0:
		tmp = d1 * d2
	elif d2 <= -2e-286:
		tmp = d1 * 37.0
	else:
		tmp = d1 * d3
	return tmp

Julia

function code(d1, d2, d3)
	tmp = 0.0
	if (d2 <= -38.0)
		tmp = Float64(d1 * d2);
	elseif (d2 <= -2e-286)
		tmp = Float64(d1 * 37.0);
	else
		tmp = Float64(d1 * d3);
	end
	return tmp
end

MATLAB

function tmp_2 = code(d1, d2, d3)
	tmp = 0.0;
	if (d2 <= -38.0)
		tmp = d1 * d2;
	elseif (d2 <= -2e-286)
		tmp = d1 * 37.0;
	else
		tmp = d1 * d3;
	end
	tmp_2 = tmp;
end

Wolfram

code[d1_, d2_, d3_] := If[LessEqual[d2, -38.0], N[(d1 * d2), $MachinePrecision], If[LessEqual[d2, -2e-286], N[(d1 * 37.0), $MachinePrecision], N[(d1 * d3), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if d2 < -38

   1. Initial program 95.3%

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

      1. *-commutative 95.3%

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

      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 66.2%

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

   if -38 < d2 < -2.0000000000000001e-286

   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 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 57.5%

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

   5. Taylor expanded in d2 around 0 56.2%

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

   if -2.0000000000000001e-286 < d2

   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 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 44.3%

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

4. Final simplification 53.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d2 \leq -38:\\ \;\;\;\;d1 \cdot d2\\ \mathbf{elif}\;d2 \leq -2 \cdot 10^{-286}:\\ \;\;\;\;d1 \cdot 37\\ \mathbf{else}:\\ \;\;\;\;d1 \cdot d3\\ \end{array} \]

Alternative 5: 76.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d2 \leq -1.55 \cdot 10^{+39}:\\ \;\;\;\;d1 \cdot d2\\ \mathbf{else}:\\ \;\;\;\;d1 \cdot \left(d3 + 37\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (d1 d2 d3)
 :precision binary64
 (if (<= d2 -1.55e+39) (* d1 d2) (* d1 (+ d3 37.0))))

C

double code(double d1, double d2, double d3) {
	double tmp;
	if (d2 <= -1.55e+39) {
		tmp = d1 * d2;
	} 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 (d2 <= (-1.55d+39)) then
        tmp = d1 * d2
    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 (d2 <= -1.55e+39) {
		tmp = d1 * d2;
	} else {
		tmp = d1 * (d3 + 37.0);
	}
	return tmp;
}

Python

def code(d1, d2, d3):
	tmp = 0
	if d2 <= -1.55e+39:
		tmp = d1 * d2
	else:
		tmp = d1 * (d3 + 37.0)
	return tmp

Julia

function code(d1, d2, d3)
	tmp = 0.0
	if (d2 <= -1.55e+39)
		tmp = Float64(d1 * d2);
	else
		tmp = Float64(d1 * Float64(d3 + 37.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d1, d2, d3)
	tmp = 0.0;
	if (d2 <= -1.55e+39)
		tmp = d1 * d2;
	else
		tmp = d1 * (d3 + 37.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[d1_, d2_, d3_] := If[LessEqual[d2, -1.55e+39], N[(d1 * d2), $MachinePrecision], N[(d1 * N[(d3 + 37.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d2 < -1.5500000000000001e39

   1. Initial program 94.5%

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

      1. *-commutative 94.5%

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

      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 75.8%

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

   if -1.5500000000000001e39 < d2

   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 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 77.1%

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

4. Final simplification 76.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d2 \leq -1.55 \cdot 10^{+39}:\\ \;\;\;\;d1 \cdot d2\\ \mathbf{else}:\\ \;\;\;\;d1 \cdot \left(d3 + 37\right)\\ \end{array} \]

Alternative 6: 27.0% 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 98.8%

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

   1. *-commutative 98.8%

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

   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 61.4%

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

5. Taylor expanded in d2 around 0 26.6%

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

6. Final simplification 26.6%

   \[\leadsto d1 \cdot 37 \]

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 2023201 
(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)))