FastMath dist3

Percentage Accurate: 97.7% → 100.0%

Time: 3.5s

Alternatives: 7

Speedup: 1.9×

• 20.8% 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 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. Final simplification 100.0%

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

Alternative 2: 50.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d2 \leq -5.3 \cdot 10^{+19}:\\ \;\;\;\;d1 \cdot d2\\ \mathbf{elif}\;d2 \leq -4.2 \cdot 10^{-199} \lor \neg \left(d2 \leq -2.4 \cdot 10^{-253}\right) \land d2 \leq 4.1 \cdot 10^{-307}:\\ \;\;\;\;d1 \cdot 37\\ \mathbf{else}:\\ \;\;\;\;d1 \cdot d3\\ \end{array} \end{array} \]

FPCore

(FPCore (d1 d2 d3)
 :precision binary64
 (if (<= d2 -5.3e+19)
   (* d1 d2)
   (if (or (<= d2 -4.2e-199) (and (not (<= d2 -2.4e-253)) (<= d2 4.1e-307)))
     (* d1 37.0)
     (* d1 d3))))

C

double code(double d1, double d2, double d3) {
	double tmp;
	if (d2 <= -5.3e+19) {
		tmp = d1 * d2;
	} else if ((d2 <= -4.2e-199) || (!(d2 <= -2.4e-253) && (d2 <= 4.1e-307))) {
		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 <= (-5.3d+19)) then
        tmp = d1 * d2
    else if ((d2 <= (-4.2d-199)) .or. (.not. (d2 <= (-2.4d-253))) .and. (d2 <= 4.1d-307)) 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 <= -5.3e+19) {
		tmp = d1 * d2;
	} else if ((d2 <= -4.2e-199) || (!(d2 <= -2.4e-253) && (d2 <= 4.1e-307))) {
		tmp = d1 * 37.0;
	} else {
		tmp = d1 * d3;
	}
	return tmp;
}

Python

def code(d1, d2, d3):
	tmp = 0
	if d2 <= -5.3e+19:
		tmp = d1 * d2
	elif (d2 <= -4.2e-199) or (not (d2 <= -2.4e-253) and (d2 <= 4.1e-307)):
		tmp = d1 * 37.0
	else:
		tmp = d1 * d3
	return tmp

Julia

function code(d1, d2, d3)
	tmp = 0.0
	if (d2 <= -5.3e+19)
		tmp = Float64(d1 * d2);
	elseif ((d2 <= -4.2e-199) || (!(d2 <= -2.4e-253) && (d2 <= 4.1e-307)))
		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 <= -5.3e+19)
		tmp = d1 * d2;
	elseif ((d2 <= -4.2e-199) || (~((d2 <= -2.4e-253)) && (d2 <= 4.1e-307)))
		tmp = d1 * 37.0;
	else
		tmp = d1 * d3;
	end
	tmp_2 = tmp;
end

Wolfram

code[d1_, d2_, d3_] := If[LessEqual[d2, -5.3e+19], N[(d1 * d2), $MachinePrecision], If[Or[LessEqual[d2, -4.2e-199], And[N[Not[LessEqual[d2, -2.4e-253]], $MachinePrecision], LessEqual[d2, 4.1e-307]]], N[(d1 * 37.0), $MachinePrecision], N[(d1 * d3), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if d2 < -5.3e19

   1. Initial program 96.7%

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

      1. *-commutative 96.7%

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

      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 d2 around inf 85.9%

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

   if -5.3e19 < d2 < -4.20000000000000004e-199 or -2.40000000000000009e-253 < d2 < 4.10000000000000032e-307

   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+ 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 0 60.1%

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

   5. Taylor expanded in d2 around 0 60.3%

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

   if -4.20000000000000004e-199 < d2 < -2.40000000000000009e-253 or 4.10000000000000032e-307 < d2

   1. Initial program 97.9%

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

      1. *-commutative 97.9%

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

      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 44.8%

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

4. Final simplification 57.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d2 \leq -5.3 \cdot 10^{+19}:\\ \;\;\;\;d1 \cdot d2\\ \mathbf{elif}\;d2 \leq -4.2 \cdot 10^{-199} \lor \neg \left(d2 \leq -2.4 \cdot 10^{-253}\right) \land d2 \leq 4.1 \cdot 10^{-307}:\\ \;\;\;\;d1 \cdot 37\\ \mathbf{else}:\\ \;\;\;\;d1 \cdot d3\\ \end{array} \]

Alternative 3: 74.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d3 \leq 4.4 \cdot 10^{+17} \lor \neg \left(d3 \leq 2.4 \cdot 10^{+100}\right) \land d3 \leq 9 \cdot 10^{+121}:\\ \;\;\;\;d1 \cdot \left(d2 + 37\right)\\ \mathbf{else}:\\ \;\;\;\;d1 \cdot d3\\ \end{array} \end{array} \]

FPCore

(FPCore (d1 d2 d3)
 :precision binary64
 (if (or (<= d3 4.4e+17) (and (not (<= d3 2.4e+100)) (<= d3 9e+121)))
   (* d1 (+ d2 37.0))
   (* d1 d3)))

C

double code(double d1, double d2, double d3) {
	double tmp;
	if ((d3 <= 4.4e+17) || (!(d3 <= 2.4e+100) && (d3 <= 9e+121))) {
		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 <= 4.4d+17) .or. (.not. (d3 <= 2.4d+100)) .and. (d3 <= 9d+121)) 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 <= 4.4e+17) || (!(d3 <= 2.4e+100) && (d3 <= 9e+121))) {
		tmp = d1 * (d2 + 37.0);
	} else {
		tmp = d1 * d3;
	}
	return tmp;
}

Python

def code(d1, d2, d3):
	tmp = 0
	if (d3 <= 4.4e+17) or (not (d3 <= 2.4e+100) and (d3 <= 9e+121)):
		tmp = d1 * (d2 + 37.0)
	else:
		tmp = d1 * d3
	return tmp

Julia

function code(d1, d2, d3)
	tmp = 0.0
	if ((d3 <= 4.4e+17) || (!(d3 <= 2.4e+100) && (d3 <= 9e+121)))
		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 <= 4.4e+17) || (~((d3 <= 2.4e+100)) && (d3 <= 9e+121)))
		tmp = d1 * (d2 + 37.0);
	else
		tmp = d1 * d3;
	end
	tmp_2 = tmp;
end

Wolfram

code[d1_, d2_, d3_] := If[Or[LessEqual[d3, 4.4e+17], And[N[Not[LessEqual[d3, 2.4e+100]], $MachinePrecision], LessEqual[d3, 9e+121]]], N[(d1 * N[(d2 + 37.0), $MachinePrecision]), $MachinePrecision], N[(d1 * d3), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d3 < 4.4e17 or 2.40000000000000012e100 < d3 < 9.0000000000000007e121

   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. *-commutative 98.5%

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

      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 76.8%

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

   if 4.4e17 < d3 < 2.40000000000000012e100 or 9.0000000000000007e121 < d3

   1. Initial program 95.8%

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

      1. *-commutative 95.8%

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

      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 inf 86.3%

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

4. Final simplification 78.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d3 \leq 4.4 \cdot 10^{+17} \lor \neg \left(d3 \leq 2.4 \cdot 10^{+100}\right) \land d3 \leq 9 \cdot 10^{+121}:\\ \;\;\;\;d1 \cdot \left(d2 + 37\right)\\ \mathbf{else}:\\ \;\;\;\;d1 \cdot d3\\ \end{array} \]

Alternative 4: 60.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d3 \leq -8.4 \cdot 10^{-66}:\\ \;\;\;\;d1 \cdot d3\\ \mathbf{elif}\;d3 \leq 37:\\ \;\;\;\;d1 \cdot 37\\ \mathbf{else}:\\ \;\;\;\;d1 \cdot d3\\ \end{array} \end{array} \]

FPCore

(FPCore (d1 d2 d3)
 :precision binary64
 (if (<= d3 -8.4e-66) (* d1 d3) (if (<= d3 37.0) (* d1 37.0) (* d1 d3))))

C

double code(double d1, double d2, double d3) {
	double tmp;
	if (d3 <= -8.4e-66) {
		tmp = d1 * d3;
	} else if (d3 <= 37.0) {
		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 <= (-8.4d-66)) then
        tmp = d1 * d3
    else if (d3 <= 37.0d0) 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 <= -8.4e-66) {
		tmp = d1 * d3;
	} else if (d3 <= 37.0) {
		tmp = d1 * 37.0;
	} else {
		tmp = d1 * d3;
	}
	return tmp;
}

Python

def code(d1, d2, d3):
	tmp = 0
	if d3 <= -8.4e-66:
		tmp = d1 * d3
	elif d3 <= 37.0:
		tmp = d1 * 37.0
	else:
		tmp = d1 * d3
	return tmp

Julia

function code(d1, d2, d3)
	tmp = 0.0
	if (d3 <= -8.4e-66)
		tmp = Float64(d1 * d3);
	elseif (d3 <= 37.0)
		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 <= -8.4e-66)
		tmp = d1 * d3;
	elseif (d3 <= 37.0)
		tmp = d1 * 37.0;
	else
		tmp = d1 * d3;
	end
	tmp_2 = tmp;
end

Wolfram

code[d1_, d2_, d3_] := If[LessEqual[d3, -8.4e-66], N[(d1 * d3), $MachinePrecision], If[LessEqual[d3, 37.0], N[(d1 * 37.0), $MachinePrecision], N[(d1 * d3), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if d3 < -8.4000000000000001e-66 or 37 < d3

   1. Initial program 96.5%

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

      1. *-commutative 96.5%

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

      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 67.0%

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

   if -8.4000000000000001e-66 < d3 < 37

   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 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 0 100.0%

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

   5. Taylor expanded in d2 around 0 51.2%

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

4. Final simplification 60.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d3 \leq -8.4 \cdot 10^{-66}:\\ \;\;\;\;d1 \cdot d3\\ \mathbf{elif}\;d3 \leq 37:\\ \;\;\;\;d1 \cdot 37\\ \mathbf{else}:\\ \;\;\;\;d1 \cdot d3\\ \end{array} \]

Alternative 5: 81.6% accurate, 1.8× speedup

Math

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

FPCore

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

C

double code(double d1, double d2, double d3) {
	double tmp;
	if (d3 <= 37.0) {
		tmp = d1 * (d2 + 37.0);
	} else {
		tmp = d1 * (d2 + 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 <= 37.0d0) then
        tmp = d1 * (d2 + 37.0d0)
    else
        tmp = d1 * (d2 + d3)
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if d3 < 37

   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 0 76.2%

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

   if 37 < d3

   1. Initial program 96.5%

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

      1. associate-+l+ 96.5%

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

      2. fma-def 100.0%

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

      3. *-commutative 100.0%

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

      4. distribute-lft-out 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in d3 around inf 100.0%

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

   5. Taylor expanded in d1 around 0 100.0%

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

4. Final simplification 81.5%

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

Alternative 6: 80.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d2 \leq -5.3 \cdot 10^{+19}:\\ \;\;\;\;d1 \cdot \left(d2 + d3\right)\\ \mathbf{else}:\\ \;\;\;\;d1 \cdot \left(d3 + 37\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (d1 d2 d3)
 :precision binary64
 (if (<= d2 -5.3e+19) (* d1 (+ d2 d3)) (* d1 (+ d3 37.0))))

C

double code(double d1, double d2, double d3) {
	double tmp;
	if (d2 <= -5.3e+19) {
		tmp = d1 * (d2 + d3);
	} 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 <= (-5.3d+19)) then
        tmp = d1 * (d2 + d3)
    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 <= -5.3e+19) {
		tmp = d1 * (d2 + d3);
	} else {
		tmp = d1 * (d3 + 37.0);
	}
	return tmp;
}

Python

def code(d1, d2, d3):
	tmp = 0
	if d2 <= -5.3e+19:
		tmp = d1 * (d2 + d3)
	else:
		tmp = d1 * (d3 + 37.0)
	return tmp

Julia

function code(d1, d2, d3)
	tmp = 0.0
	if (d2 <= -5.3e+19)
		tmp = Float64(d1 * Float64(d2 + d3));
	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 <= -5.3e+19)
		tmp = d1 * (d2 + d3);
	else
		tmp = d1 * (d3 + 37.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[d1_, d2_, d3_] := If[LessEqual[d2, -5.3e+19], N[(d1 * N[(d2 + d3), $MachinePrecision]), $MachinePrecision], N[(d1 * N[(d3 + 37.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d2 < -5.3e19

   1. Initial program 96.7%

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

      1. associate-+l+ 96.7%

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

      2. fma-def 99.9%

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

      3. *-commutative 99.9%

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

      4. distribute-lft-out 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in d3 around inf 99.9%

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

   5. Taylor expanded in d1 around 0 100.0%

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

   if -5.3e19 < d2

   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 76.7%

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

4. Final simplification 82.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d2 \leq -5.3 \cdot 10^{+19}:\\ \;\;\;\;d1 \cdot \left(d2 + d3\right)\\ \mathbf{else}:\\ \;\;\;\;d1 \cdot \left(d3 + 37\right)\\ \end{array} \]

Alternative 7: 26.7% 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. *-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 0 66.6%

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

5. Taylor expanded in d2 around 0 27.6%

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

6. Final simplification 27.6%

   \[\leadsto d1 \cdot 37 \]

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 2023188 
(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)))