FastMath test3

Percentage Accurate: 97.7% → 99.9%

Time: 2.2min

Alternatives: 7

Speedup: 1.6×

• 21.0% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \left(d1 \cdot 3 + d1 \cdot d2\right) + d1 \cdot d3 \end{array} \]

FPCore

(FPCore (d1 d2 d3) :precision binary64 (+ (+ (* d1 3.0) (* d1 d2)) (* d1 d3)))

C

double code(double d1, double d2, double d3) {
	return ((d1 * 3.0) + (d1 * d2)) + (d1 * d3);
}

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 * 3.0d0) + (d1 * d2)) + (d1 * d3)
end function

Java

public static double code(double d1, double d2, double d3) {
	return ((d1 * 3.0) + (d1 * d2)) + (d1 * d3);
}

Python

def code(d1, d2, d3):
	return ((d1 * 3.0) + (d1 * d2)) + (d1 * d3)

Julia

function code(d1, d2, d3)
	return Float64(Float64(Float64(d1 * 3.0) + Float64(d1 * d2)) + Float64(d1 * d3))
end

MATLAB

function tmp = code(d1, d2, d3)
	tmp = ((d1 * 3.0) + (d1 * d2)) + (d1 * d3);
end

Wolfram

code[d1_, d2_, d3_] := N[(N[(N[(d1 * 3.0), $MachinePrecision] + N[(d1 * d2), $MachinePrecision]), $MachinePrecision] + N[(d1 * d3), $MachinePrecision]), $MachinePrecision]

Initial Program: 97.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(d1 \cdot 3 + d1 \cdot d2\right) + d1 \cdot d3 \end{array} \]

FPCore

(FPCore (d1 d2 d3) :precision binary64 (+ (+ (* d1 3.0) (* d1 d2)) (* d1 d3)))

C

double code(double d1, double d2, double d3) {
	return ((d1 * 3.0) + (d1 * d2)) + (d1 * d3);
}

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 * 3.0d0) + (d1 * d2)) + (d1 * d3)
end function

Java

public static double code(double d1, double d2, double d3) {
	return ((d1 * 3.0) + (d1 * d2)) + (d1 * d3);
}

Python

def code(d1, d2, d3):
	return ((d1 * 3.0) + (d1 * d2)) + (d1 * d3)

Julia

function code(d1, d2, d3)
	return Float64(Float64(Float64(d1 * 3.0) + Float64(d1 * d2)) + Float64(d1 * d3))
end

MATLAB

function tmp = code(d1, d2, d3)
	tmp = ((d1 * 3.0) + (d1 * d2)) + (d1 * d3);
end

Wolfram

code[d1_, d2_, d3_] := N[(N[(N[(d1 * 3.0), $MachinePrecision] + N[(d1 * d2), $MachinePrecision]), $MachinePrecision] + N[(d1 * d3), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ d1 \cdot \left(d2 + \left(3 + d3\right)\right) \end{array} \]

FPCore

(FPCore (d1 d2 d3) :precision binary64 (* d1 (+ d2 (+ 3.0 d3))))

C

double code(double d1, double d2, double d3) {
	return d1 * (d2 + (3.0 + d3));
}

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 + (3.0d0 + d3))
end function

Java

public static double code(double d1, double d2, double d3) {
	return d1 * (d2 + (3.0 + d3));
}

Python

def code(d1, d2, d3):
	return d1 * (d2 + (3.0 + d3))

Julia

function code(d1, d2, d3)
	return Float64(d1 * Float64(d2 + Float64(3.0 + d3)))
end

MATLAB

function tmp = code(d1, d2, d3)
	tmp = d1 * (d2 + (3.0 + d3));
end

Wolfram

code[d1_, d2_, d3_] := N[(d1 * N[(d2 + N[(3.0 + d3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 98.7%

   \[\left(d1 \cdot 3 + d1 \cdot d2\right) + d1 \cdot d3 \]
2. Step-by-step derivation

   1. distribute-lft-out N/A

      \[\leadsto d1 \cdot \left(3 + d2\right) + \color{blue}{d1} \cdot d3 \]

   2. distribute-lft-out N/A

      \[\leadsto d1 \cdot \color{blue}{\left(\left(3 + d2\right) + d3\right)} \]

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

      \[\leadsto \mathsf{*.f64}\left(d1, \color{blue}{\left(\left(3 + d2\right) + d3\right)}\right) \]

   4. +-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(d1, \left(\left(d2 + 3\right) + d3\right)\right) \]

   5. associate-+l+ N/A

      \[\leadsto \mathsf{*.f64}\left(d1, \left(d2 + \color{blue}{\left(3 + d3\right)}\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(d1, \mathsf{+.f64}\left(d2, \color{blue}{\left(3 + d3\right)}\right)\right) \]

   7. +-lowering-+.f64 100.0%

      \[\leadsto \mathsf{*.f64}\left(d1, \mathsf{+.f64}\left(d2, \mathsf{+.f64}\left(3, \color{blue}{d3}\right)\right)\right) \]

3. Simplified 100.0%

   \[\leadsto \color{blue}{d1 \cdot \left(d2 + \left(3 + d3\right)\right)} \]
4. Add Preprocessing
5. Add Preprocessing

Alternative 2: 52.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d2 \leq -110:\\ \;\;\;\;d1 \cdot d2\\ \mathbf{elif}\;d2 \leq 2 \cdot 10^{-169}:\\ \;\;\;\;d1 \cdot 3\\ \mathbf{else}:\\ \;\;\;\;d1 \cdot d3\\ \end{array} \end{array} \]

FPCore

(FPCore (d1 d2 d3)
 :precision binary64
 (if (<= d2 -110.0) (* d1 d2) (if (<= d2 2e-169) (* d1 3.0) (* d1 d3))))

C

double code(double d1, double d2, double d3) {
	double tmp;
	if (d2 <= -110.0) {
		tmp = d1 * d2;
	} else if (d2 <= 2e-169) {
		tmp = d1 * 3.0;
	} 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 <= (-110.0d0)) then
        tmp = d1 * d2
    else if (d2 <= 2d-169) then
        tmp = d1 * 3.0d0
    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 <= -110.0) {
		tmp = d1 * d2;
	} else if (d2 <= 2e-169) {
		tmp = d1 * 3.0;
	} else {
		tmp = d1 * d3;
	}
	return tmp;
}

Python

def code(d1, d2, d3):
	tmp = 0
	if d2 <= -110.0:
		tmp = d1 * d2
	elif d2 <= 2e-169:
		tmp = d1 * 3.0
	else:
		tmp = d1 * d3
	return tmp

Julia

function code(d1, d2, d3)
	tmp = 0.0
	if (d2 <= -110.0)
		tmp = Float64(d1 * d2);
	elseif (d2 <= 2e-169)
		tmp = Float64(d1 * 3.0);
	else
		tmp = Float64(d1 * d3);
	end
	return tmp
end

MATLAB

function tmp_2 = code(d1, d2, d3)
	tmp = 0.0;
	if (d2 <= -110.0)
		tmp = d1 * d2;
	elseif (d2 <= 2e-169)
		tmp = d1 * 3.0;
	else
		tmp = d1 * d3;
	end
	tmp_2 = tmp;
end

Wolfram

code[d1_, d2_, d3_] := If[LessEqual[d2, -110.0], N[(d1 * d2), $MachinePrecision], If[LessEqual[d2, 2e-169], N[(d1 * 3.0), $MachinePrecision], N[(d1 * d3), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if d2 < -110

   1. Initial program 98.2%

      \[\left(d1 \cdot 3 + d1 \cdot d2\right) + d1 \cdot d3 \]
   2. Step-by-step derivation

      1. distribute-lft-out N/A

         \[\leadsto d1 \cdot \left(3 + d2\right) + \color{blue}{d1} \cdot d3 \]

      2. distribute-lft-out N/A

         \[\leadsto d1 \cdot \color{blue}{\left(\left(3 + d2\right) + d3\right)} \]

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

         \[\leadsto \mathsf{*.f64}\left(d1, \color{blue}{\left(\left(3 + d2\right) + d3\right)}\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(d1, \left(\left(d2 + 3\right) + d3\right)\right) \]

      5. associate-+l+ N/A

         \[\leadsto \mathsf{*.f64}\left(d1, \left(d2 + \color{blue}{\left(3 + d3\right)}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(d1, \mathsf{+.f64}\left(d2, \color{blue}{\left(3 + d3\right)}\right)\right) \]

      7. +-lowering-+.f64 100.0%

         \[\leadsto \mathsf{*.f64}\left(d1, \mathsf{+.f64}\left(d2, \mathsf{+.f64}\left(3, \color{blue}{d3}\right)\right)\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{d1 \cdot \left(d2 + \left(3 + d3\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d2 around inf

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

      1. *-lowering-*.f64 82.4%

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

   7. Simplified 82.4%

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

   if -110 < d2 < 2.00000000000000004e-169

   1. Initial program 99.8%

      \[\left(d1 \cdot 3 + d1 \cdot d2\right) + d1 \cdot d3 \]
   2. Step-by-step derivation

      1. distribute-lft-out N/A

         \[\leadsto d1 \cdot \left(3 + d2\right) + \color{blue}{d1} \cdot d3 \]

      2. distribute-lft-out N/A

         \[\leadsto d1 \cdot \color{blue}{\left(\left(3 + d2\right) + d3\right)} \]

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

         \[\leadsto \mathsf{*.f64}\left(d1, \color{blue}{\left(\left(3 + d2\right) + d3\right)}\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(d1, \left(\left(d2 + 3\right) + d3\right)\right) \]

      5. associate-+l+ N/A

         \[\leadsto \mathsf{*.f64}\left(d1, \left(d2 + \color{blue}{\left(3 + d3\right)}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(d1, \mathsf{+.f64}\left(d2, \color{blue}{\left(3 + d3\right)}\right)\right) \]

      7. +-lowering-+.f64 99.9%

         \[\leadsto \mathsf{*.f64}\left(d1, \mathsf{+.f64}\left(d2, \mathsf{+.f64}\left(3, \color{blue}{d3}\right)\right)\right) \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{d1 \cdot \left(d2 + \left(3 + d3\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d2 around 0

      \[\leadsto \color{blue}{d1 \cdot \left(3 + d3\right)} \]
   6. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(d1, \color{blue}{\left(3 + d3\right)}\right) \]

      2. +-lowering-+.f64 98.6%

         \[\leadsto \mathsf{*.f64}\left(d1, \mathsf{+.f64}\left(3, \color{blue}{d3}\right)\right) \]

   7. Simplified 98.6%

      \[\leadsto \color{blue}{d1 \cdot \left(3 + d3\right)} \]

   8. Taylor expanded in d3 around 0

      \[\leadsto \color{blue}{3 \cdot d1} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto d1 \cdot \color{blue}{3} \]

      2. *-lowering-*.f64 61.5%

         \[\leadsto \mathsf{*.f64}\left(d1, \color{blue}{3}\right) \]

   10. Simplified 61.5%

       \[\leadsto \color{blue}{d1 \cdot 3} \]

   if 2.00000000000000004e-169 < d2

   1. Initial program 97.9%

      \[\left(d1 \cdot 3 + d1 \cdot d2\right) + d1 \cdot d3 \]
   2. Step-by-step derivation

      1. distribute-lft-out N/A

         \[\leadsto d1 \cdot \left(3 + d2\right) + \color{blue}{d1} \cdot d3 \]

      2. distribute-lft-out N/A

         \[\leadsto d1 \cdot \color{blue}{\left(\left(3 + d2\right) + d3\right)} \]

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

         \[\leadsto \mathsf{*.f64}\left(d1, \color{blue}{\left(\left(3 + d2\right) + d3\right)}\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(d1, \left(\left(d2 + 3\right) + d3\right)\right) \]

      5. associate-+l+ N/A

         \[\leadsto \mathsf{*.f64}\left(d1, \left(d2 + \color{blue}{\left(3 + d3\right)}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(d1, \mathsf{+.f64}\left(d2, \color{blue}{\left(3 + d3\right)}\right)\right) \]

      7. +-lowering-+.f64 100.0%

         \[\leadsto \mathsf{*.f64}\left(d1, \mathsf{+.f64}\left(d2, \mathsf{+.f64}\left(3, \color{blue}{d3}\right)\right)\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{d1 \cdot \left(d2 + \left(3 + d3\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d3 around inf

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

      1. *-lowering-*.f64 42.0%

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

   7. Simplified 42.0%

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

Alternative 3: 81.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d3 \leq 3:\\ \;\;\;\;d1 \cdot \left(d2 + 3\right)\\ \mathbf{else}:\\ \;\;\;\;d1 \cdot \left(d2 + d3\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (d1 d2 d3)
 :precision binary64
 (if (<= d3 3.0) (* d1 (+ d2 3.0)) (* d1 (+ d2 d3))))

C

double code(double d1, double d2, double d3) {
	double tmp;
	if (d3 <= 3.0) {
		tmp = d1 * (d2 + 3.0);
	} 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 <= 3.0d0) then
        tmp = d1 * (d2 + 3.0d0)
    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 <= 3.0) {
		tmp = d1 * (d2 + 3.0);
	} else {
		tmp = d1 * (d2 + d3);
	}
	return tmp;
}

Python

def code(d1, d2, d3):
	tmp = 0
	if d3 <= 3.0:
		tmp = d1 * (d2 + 3.0)
	else:
		tmp = d1 * (d2 + d3)
	return tmp

Julia

function code(d1, d2, d3)
	tmp = 0.0
	if (d3 <= 3.0)
		tmp = Float64(d1 * Float64(d2 + 3.0));
	else
		tmp = Float64(d1 * Float64(d2 + d3));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d1, d2, d3)
	tmp = 0.0;
	if (d3 <= 3.0)
		tmp = d1 * (d2 + 3.0);
	else
		tmp = d1 * (d2 + d3);
	end
	tmp_2 = tmp;
end

Wolfram

code[d1_, d2_, d3_] := If[LessEqual[d3, 3.0], N[(d1 * N[(d2 + 3.0), $MachinePrecision]), $MachinePrecision], N[(d1 * N[(d2 + d3), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d3 < 3

   1. Initial program 98.9%

      \[\left(d1 \cdot 3 + d1 \cdot d2\right) + d1 \cdot d3 \]
   2. Step-by-step derivation

      1. distribute-lft-out N/A

         \[\leadsto d1 \cdot \left(3 + d2\right) + \color{blue}{d1} \cdot d3 \]

      2. distribute-lft-out N/A

         \[\leadsto d1 \cdot \color{blue}{\left(\left(3 + d2\right) + d3\right)} \]

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

         \[\leadsto \mathsf{*.f64}\left(d1, \color{blue}{\left(\left(3 + d2\right) + d3\right)}\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(d1, \left(\left(d2 + 3\right) + d3\right)\right) \]

      5. associate-+l+ N/A

         \[\leadsto \mathsf{*.f64}\left(d1, \left(d2 + \color{blue}{\left(3 + d3\right)}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(d1, \mathsf{+.f64}\left(d2, \color{blue}{\left(3 + d3\right)}\right)\right) \]

      7. +-lowering-+.f64 99.9%

         \[\leadsto \mathsf{*.f64}\left(d1, \mathsf{+.f64}\left(d2, \mathsf{+.f64}\left(3, \color{blue}{d3}\right)\right)\right) \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{d1 \cdot \left(d2 + \left(3 + d3\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d3 around 0

      \[\leadsto \mathsf{*.f64}\left(d1, \mathsf{+.f64}\left(d2, \color{blue}{3}\right)\right) \]
   6. Step-by-step derivation

   7. Simplified 77.2%

      \[\leadsto d1 \cdot \left(d2 + \color{blue}{3}\right) \]

   if 3 < d3

   1. Initial program 98.1%

      \[\left(d1 \cdot 3 + d1 \cdot d2\right) + d1 \cdot d3 \]
   2. Step-by-step derivation

      1. distribute-lft-out N/A

         \[\leadsto d1 \cdot \left(3 + d2\right) + \color{blue}{d1} \cdot d3 \]

      2. distribute-lft-out N/A

         \[\leadsto d1 \cdot \color{blue}{\left(\left(3 + d2\right) + d3\right)} \]

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

         \[\leadsto \mathsf{*.f64}\left(d1, \color{blue}{\left(\left(3 + d2\right) + d3\right)}\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(d1, \left(\left(d2 + 3\right) + d3\right)\right) \]

      5. associate-+l+ N/A

         \[\leadsto \mathsf{*.f64}\left(d1, \left(d2 + \color{blue}{\left(3 + d3\right)}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(d1, \mathsf{+.f64}\left(d2, \color{blue}{\left(3 + d3\right)}\right)\right) \]

      7. +-lowering-+.f64 100.0%

         \[\leadsto \mathsf{*.f64}\left(d1, \mathsf{+.f64}\left(d2, \mathsf{+.f64}\left(3, \color{blue}{d3}\right)\right)\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{d1 \cdot \left(d2 + \left(3 + d3\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d3 around inf

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

   7. Simplified 98.7%

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

Alternative 4: 76.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d2 \leq -2.4 \cdot 10^{-14}:\\ \;\;\;\;d1 \cdot \left(d2 + 3\right)\\ \mathbf{else}:\\ \;\;\;\;d1 \cdot \left(3 + d3\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (d1 d2 d3)
 :precision binary64
 (if (<= d2 -2.4e-14) (* d1 (+ d2 3.0)) (* d1 (+ 3.0 d3))))

C

double code(double d1, double d2, double d3) {
	double tmp;
	if (d2 <= -2.4e-14) {
		tmp = d1 * (d2 + 3.0);
	} else {
		tmp = d1 * (3.0 + 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 <= (-2.4d-14)) then
        tmp = d1 * (d2 + 3.0d0)
    else
        tmp = d1 * (3.0d0 + d3)
    end if
    code = tmp
end function

Java

public static double code(double d1, double d2, double d3) {
	double tmp;
	if (d2 <= -2.4e-14) {
		tmp = d1 * (d2 + 3.0);
	} else {
		tmp = d1 * (3.0 + d3);
	}
	return tmp;
}

Python

def code(d1, d2, d3):
	tmp = 0
	if d2 <= -2.4e-14:
		tmp = d1 * (d2 + 3.0)
	else:
		tmp = d1 * (3.0 + d3)
	return tmp

Julia

function code(d1, d2, d3)
	tmp = 0.0
	if (d2 <= -2.4e-14)
		tmp = Float64(d1 * Float64(d2 + 3.0));
	else
		tmp = Float64(d1 * Float64(3.0 + d3));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d1, d2, d3)
	tmp = 0.0;
	if (d2 <= -2.4e-14)
		tmp = d1 * (d2 + 3.0);
	else
		tmp = d1 * (3.0 + d3);
	end
	tmp_2 = tmp;
end

Wolfram

code[d1_, d2_, d3_] := If[LessEqual[d2, -2.4e-14], N[(d1 * N[(d2 + 3.0), $MachinePrecision]), $MachinePrecision], N[(d1 * N[(3.0 + d3), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d2 < -2.4e-14

   1. Initial program 98.3%

      \[\left(d1 \cdot 3 + d1 \cdot d2\right) + d1 \cdot d3 \]
   2. Step-by-step derivation

      1. distribute-lft-out N/A

         \[\leadsto d1 \cdot \left(3 + d2\right) + \color{blue}{d1} \cdot d3 \]

      2. distribute-lft-out N/A

         \[\leadsto d1 \cdot \color{blue}{\left(\left(3 + d2\right) + d3\right)} \]

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

         \[\leadsto \mathsf{*.f64}\left(d1, \color{blue}{\left(\left(3 + d2\right) + d3\right)}\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(d1, \left(\left(d2 + 3\right) + d3\right)\right) \]

      5. associate-+l+ N/A

         \[\leadsto \mathsf{*.f64}\left(d1, \left(d2 + \color{blue}{\left(3 + d3\right)}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(d1, \mathsf{+.f64}\left(d2, \color{blue}{\left(3 + d3\right)}\right)\right) \]

      7. +-lowering-+.f64 100.0%

         \[\leadsto \mathsf{*.f64}\left(d1, \mathsf{+.f64}\left(d2, \mathsf{+.f64}\left(3, \color{blue}{d3}\right)\right)\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{d1 \cdot \left(d2 + \left(3 + d3\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d3 around 0

      \[\leadsto \mathsf{*.f64}\left(d1, \mathsf{+.f64}\left(d2, \color{blue}{3}\right)\right) \]
   6. Step-by-step derivation

   7. Simplified 84.4%

      \[\leadsto d1 \cdot \left(d2 + \color{blue}{3}\right) \]

   if -2.4e-14 < d2

   1. Initial program 98.9%

      \[\left(d1 \cdot 3 + d1 \cdot d2\right) + d1 \cdot d3 \]
   2. Step-by-step derivation

      1. distribute-lft-out N/A

         \[\leadsto d1 \cdot \left(3 + d2\right) + \color{blue}{d1} \cdot d3 \]

      2. distribute-lft-out N/A

         \[\leadsto d1 \cdot \color{blue}{\left(\left(3 + d2\right) + d3\right)} \]

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

         \[\leadsto \mathsf{*.f64}\left(d1, \color{blue}{\left(\left(3 + d2\right) + d3\right)}\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(d1, \left(\left(d2 + 3\right) + d3\right)\right) \]

      5. associate-+l+ N/A

         \[\leadsto \mathsf{*.f64}\left(d1, \left(d2 + \color{blue}{\left(3 + d3\right)}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(d1, \mathsf{+.f64}\left(d2, \color{blue}{\left(3 + d3\right)}\right)\right) \]

      7. +-lowering-+.f64 99.9%

         \[\leadsto \mathsf{*.f64}\left(d1, \mathsf{+.f64}\left(d2, \mathsf{+.f64}\left(3, \color{blue}{d3}\right)\right)\right) \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{d1 \cdot \left(d2 + \left(3 + d3\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d2 around 0

      \[\leadsto \color{blue}{d1 \cdot \left(3 + d3\right)} \]
   6. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(d1, \color{blue}{\left(3 + d3\right)}\right) \]

      2. +-lowering-+.f64 79.7%

         \[\leadsto \mathsf{*.f64}\left(d1, \mathsf{+.f64}\left(3, \color{blue}{d3}\right)\right) \]

   7. Simplified 79.7%

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

Alternative 5: 76.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d2 \leq -7800000000000:\\ \;\;\;\;d1 \cdot d2\\ \mathbf{else}:\\ \;\;\;\;d1 \cdot \left(3 + d3\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (d1 d2 d3)
 :precision binary64
 (if (<= d2 -7800000000000.0) (* d1 d2) (* d1 (+ 3.0 d3))))

C

double code(double d1, double d2, double d3) {
	double tmp;
	if (d2 <= -7800000000000.0) {
		tmp = d1 * d2;
	} else {
		tmp = d1 * (3.0 + 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 <= (-7800000000000.0d0)) then
        tmp = d1 * d2
    else
        tmp = d1 * (3.0d0 + d3)
    end if
    code = tmp
end function

Java

public static double code(double d1, double d2, double d3) {
	double tmp;
	if (d2 <= -7800000000000.0) {
		tmp = d1 * d2;
	} else {
		tmp = d1 * (3.0 + d3);
	}
	return tmp;
}

Python

def code(d1, d2, d3):
	tmp = 0
	if d2 <= -7800000000000.0:
		tmp = d1 * d2
	else:
		tmp = d1 * (3.0 + d3)
	return tmp

Julia

function code(d1, d2, d3)
	tmp = 0.0
	if (d2 <= -7800000000000.0)
		tmp = Float64(d1 * d2);
	else
		tmp = Float64(d1 * Float64(3.0 + d3));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d1, d2, d3)
	tmp = 0.0;
	if (d2 <= -7800000000000.0)
		tmp = d1 * d2;
	else
		tmp = d1 * (3.0 + d3);
	end
	tmp_2 = tmp;
end

Wolfram

code[d1_, d2_, d3_] := If[LessEqual[d2, -7800000000000.0], N[(d1 * d2), $MachinePrecision], N[(d1 * N[(3.0 + d3), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d2 < -7.8e12

   1. Initial program 98.2%

      \[\left(d1 \cdot 3 + d1 \cdot d2\right) + d1 \cdot d3 \]
   2. Step-by-step derivation

      1. distribute-lft-out N/A

         \[\leadsto d1 \cdot \left(3 + d2\right) + \color{blue}{d1} \cdot d3 \]

      2. distribute-lft-out N/A

         \[\leadsto d1 \cdot \color{blue}{\left(\left(3 + d2\right) + d3\right)} \]

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

         \[\leadsto \mathsf{*.f64}\left(d1, \color{blue}{\left(\left(3 + d2\right) + d3\right)}\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(d1, \left(\left(d2 + 3\right) + d3\right)\right) \]

      5. associate-+l+ N/A

         \[\leadsto \mathsf{*.f64}\left(d1, \left(d2 + \color{blue}{\left(3 + d3\right)}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(d1, \mathsf{+.f64}\left(d2, \color{blue}{\left(3 + d3\right)}\right)\right) \]

      7. +-lowering-+.f64 100.0%

         \[\leadsto \mathsf{*.f64}\left(d1, \mathsf{+.f64}\left(d2, \mathsf{+.f64}\left(3, \color{blue}{d3}\right)\right)\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{d1 \cdot \left(d2 + \left(3 + d3\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d2 around inf

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

      1. *-lowering-*.f64 84.5%

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

   7. Simplified 84.5%

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

   if -7.8e12 < d2

   1. Initial program 98.9%

      \[\left(d1 \cdot 3 + d1 \cdot d2\right) + d1 \cdot d3 \]
   2. Step-by-step derivation

      1. distribute-lft-out N/A

         \[\leadsto d1 \cdot \left(3 + d2\right) + \color{blue}{d1} \cdot d3 \]

      2. distribute-lft-out N/A

         \[\leadsto d1 \cdot \color{blue}{\left(\left(3 + d2\right) + d3\right)} \]

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

         \[\leadsto \mathsf{*.f64}\left(d1, \color{blue}{\left(\left(3 + d2\right) + d3\right)}\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(d1, \left(\left(d2 + 3\right) + d3\right)\right) \]

      5. associate-+l+ N/A

         \[\leadsto \mathsf{*.f64}\left(d1, \left(d2 + \color{blue}{\left(3 + d3\right)}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(d1, \mathsf{+.f64}\left(d2, \color{blue}{\left(3 + d3\right)}\right)\right) \]

      7. +-lowering-+.f64 99.9%

         \[\leadsto \mathsf{*.f64}\left(d1, \mathsf{+.f64}\left(d2, \mathsf{+.f64}\left(3, \color{blue}{d3}\right)\right)\right) \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{d1 \cdot \left(d2 + \left(3 + d3\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d2 around 0

      \[\leadsto \color{blue}{d1 \cdot \left(3 + d3\right)} \]
   6. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(d1, \color{blue}{\left(3 + d3\right)}\right) \]

      2. +-lowering-+.f64 79.2%

         \[\leadsto \mathsf{*.f64}\left(d1, \mathsf{+.f64}\left(3, \color{blue}{d3}\right)\right) \]

   7. Simplified 79.2%

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

Alternative 6: 45.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d2 \leq -110:\\ \;\;\;\;d1 \cdot d2\\ \mathbf{else}:\\ \;\;\;\;d1 \cdot 3\\ \end{array} \end{array} \]

FPCore

(FPCore (d1 d2 d3)
 :precision binary64
 (if (<= d2 -110.0) (* d1 d2) (* d1 3.0)))

C

double code(double d1, double d2, double d3) {
	double tmp;
	if (d2 <= -110.0) {
		tmp = d1 * d2;
	} else {
		tmp = d1 * 3.0;
	}
	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 <= (-110.0d0)) then
        tmp = d1 * d2
    else
        tmp = d1 * 3.0d0
    end if
    code = tmp
end function

Java

public static double code(double d1, double d2, double d3) {
	double tmp;
	if (d2 <= -110.0) {
		tmp = d1 * d2;
	} else {
		tmp = d1 * 3.0;
	}
	return tmp;
}

Python

def code(d1, d2, d3):
	tmp = 0
	if d2 <= -110.0:
		tmp = d1 * d2
	else:
		tmp = d1 * 3.0
	return tmp

Julia

function code(d1, d2, d3)
	tmp = 0.0
	if (d2 <= -110.0)
		tmp = Float64(d1 * d2);
	else
		tmp = Float64(d1 * 3.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(d1, d2, d3)
	tmp = 0.0;
	if (d2 <= -110.0)
		tmp = d1 * d2;
	else
		tmp = d1 * 3.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[d1_, d2_, d3_] := If[LessEqual[d2, -110.0], N[(d1 * d2), $MachinePrecision], N[(d1 * 3.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d2 < -110

   1. Initial program 98.2%

      \[\left(d1 \cdot 3 + d1 \cdot d2\right) + d1 \cdot d3 \]
   2. Step-by-step derivation

      1. distribute-lft-out N/A

         \[\leadsto d1 \cdot \left(3 + d2\right) + \color{blue}{d1} \cdot d3 \]

      2. distribute-lft-out N/A

         \[\leadsto d1 \cdot \color{blue}{\left(\left(3 + d2\right) + d3\right)} \]

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

         \[\leadsto \mathsf{*.f64}\left(d1, \color{blue}{\left(\left(3 + d2\right) + d3\right)}\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(d1, \left(\left(d2 + 3\right) + d3\right)\right) \]

      5. associate-+l+ N/A

         \[\leadsto \mathsf{*.f64}\left(d1, \left(d2 + \color{blue}{\left(3 + d3\right)}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(d1, \mathsf{+.f64}\left(d2, \color{blue}{\left(3 + d3\right)}\right)\right) \]

      7. +-lowering-+.f64 100.0%

         \[\leadsto \mathsf{*.f64}\left(d1, \mathsf{+.f64}\left(d2, \mathsf{+.f64}\left(3, \color{blue}{d3}\right)\right)\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{d1 \cdot \left(d2 + \left(3 + d3\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d2 around inf

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

      1. *-lowering-*.f64 82.4%

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

   7. Simplified 82.4%

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

   if -110 < d2

   1. Initial program 98.9%

      \[\left(d1 \cdot 3 + d1 \cdot d2\right) + d1 \cdot d3 \]
   2. Step-by-step derivation

      1. distribute-lft-out N/A

         \[\leadsto d1 \cdot \left(3 + d2\right) + \color{blue}{d1} \cdot d3 \]

      2. distribute-lft-out N/A

         \[\leadsto d1 \cdot \color{blue}{\left(\left(3 + d2\right) + d3\right)} \]

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

         \[\leadsto \mathsf{*.f64}\left(d1, \color{blue}{\left(\left(3 + d2\right) + d3\right)}\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(d1, \left(\left(d2 + 3\right) + d3\right)\right) \]

      5. associate-+l+ N/A

         \[\leadsto \mathsf{*.f64}\left(d1, \left(d2 + \color{blue}{\left(3 + d3\right)}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(d1, \mathsf{+.f64}\left(d2, \color{blue}{\left(3 + d3\right)}\right)\right) \]

      7. +-lowering-+.f64 99.9%

         \[\leadsto \mathsf{*.f64}\left(d1, \mathsf{+.f64}\left(d2, \mathsf{+.f64}\left(3, \color{blue}{d3}\right)\right)\right) \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{d1 \cdot \left(d2 + \left(3 + d3\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d2 around 0

      \[\leadsto \color{blue}{d1 \cdot \left(3 + d3\right)} \]
   6. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(d1, \color{blue}{\left(3 + d3\right)}\right) \]

      2. +-lowering-+.f64 79.4%

         \[\leadsto \mathsf{*.f64}\left(d1, \mathsf{+.f64}\left(3, \color{blue}{d3}\right)\right) \]

   7. Simplified 79.4%

      \[\leadsto \color{blue}{d1 \cdot \left(3 + d3\right)} \]

   8. Taylor expanded in d3 around 0

      \[\leadsto \color{blue}{3 \cdot d1} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto d1 \cdot \color{blue}{3} \]

      2. *-lowering-*.f64 41.8%

         \[\leadsto \mathsf{*.f64}\left(d1, \color{blue}{3}\right) \]

   10. Simplified 41.8%

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

Alternative 7: 26.1% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ d1 \cdot 3 \end{array} \]

FPCore

(FPCore (d1 d2 d3) :precision binary64 (* d1 3.0))

C

double code(double d1, double d2, double d3) {
	return d1 * 3.0;
}

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 * 3.0d0
end function

Java

public static double code(double d1, double d2, double d3) {
	return d1 * 3.0;
}

Python

def code(d1, d2, d3):
	return d1 * 3.0

Julia

function code(d1, d2, d3)
	return Float64(d1 * 3.0)
end

MATLAB

function tmp = code(d1, d2, d3)
	tmp = d1 * 3.0;
end

Wolfram

code[d1_, d2_, d3_] := N[(d1 * 3.0), $MachinePrecision]

Derivation

1. Initial program 98.7%

   \[\left(d1 \cdot 3 + d1 \cdot d2\right) + d1 \cdot d3 \]
2. Step-by-step derivation

   1. distribute-lft-out N/A

      \[\leadsto d1 \cdot \left(3 + d2\right) + \color{blue}{d1} \cdot d3 \]

   2. distribute-lft-out N/A

      \[\leadsto d1 \cdot \color{blue}{\left(\left(3 + d2\right) + d3\right)} \]

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

      \[\leadsto \mathsf{*.f64}\left(d1, \color{blue}{\left(\left(3 + d2\right) + d3\right)}\right) \]

   4. +-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(d1, \left(\left(d2 + 3\right) + d3\right)\right) \]

   5. associate-+l+ N/A

      \[\leadsto \mathsf{*.f64}\left(d1, \left(d2 + \color{blue}{\left(3 + d3\right)}\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(d1, \mathsf{+.f64}\left(d2, \color{blue}{\left(3 + d3\right)}\right)\right) \]

   7. +-lowering-+.f64 100.0%

      \[\leadsto \mathsf{*.f64}\left(d1, \mathsf{+.f64}\left(d2, \mathsf{+.f64}\left(3, \color{blue}{d3}\right)\right)\right) \]

3. Simplified 100.0%

   \[\leadsto \color{blue}{d1 \cdot \left(d2 + \left(3 + d3\right)\right)} \]
4. Add Preprocessing

5. Taylor expanded in d2 around 0

   \[\leadsto \color{blue}{d1 \cdot \left(3 + d3\right)} \]
6. Step-by-step derivation

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

      \[\leadsto \mathsf{*.f64}\left(d1, \color{blue}{\left(3 + d3\right)}\right) \]

   2. +-lowering-+.f64 67.0%

      \[\leadsto \mathsf{*.f64}\left(d1, \mathsf{+.f64}\left(3, \color{blue}{d3}\right)\right) \]

7. Simplified 67.0%

   \[\leadsto \color{blue}{d1 \cdot \left(3 + d3\right)} \]

8. Taylor expanded in d3 around 0

   \[\leadsto \color{blue}{3 \cdot d1} \]
9. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto d1 \cdot \color{blue}{3} \]

   2. *-lowering-*.f64 32.9%

      \[\leadsto \mathsf{*.f64}\left(d1, \color{blue}{3}\right) \]

10. Simplified 32.9%

    \[\leadsto \color{blue}{d1 \cdot 3} \]
11. Add Preprocessing

Developer Target 1: 99.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ d1 \cdot \left(\left(3 + d2\right) + d3\right) \end{array} \]

FPCore

(FPCore (d1 d2 d3) :precision binary64 (* d1 (+ (+ 3.0 d2) d3)))

C

double code(double d1, double d2, double d3) {
	return d1 * ((3.0 + d2) + d3);
}

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 * ((3.0d0 + d2) + d3)
end function

Java

public static double code(double d1, double d2, double d3) {
	return d1 * ((3.0 + d2) + d3);
}

Python

def code(d1, d2, d3):
	return d1 * ((3.0 + d2) + d3)

Julia

function code(d1, d2, d3)
	return Float64(d1 * Float64(Float64(3.0 + d2) + d3))
end

MATLAB

function tmp = code(d1, d2, d3)
	tmp = d1 * ((3.0 + d2) + d3);
end

Wolfram

code[d1_, d2_, d3_] := N[(d1 * N[(N[(3.0 + d2), $MachinePrecision] + d3), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2024138 
(FPCore (d1 d2 d3)
  :name "FastMath test3"
  :precision binary64

  :alt
  (! :herbie-platform default (* d1 (+ 3 d2 d3)))

  (+ (+ (* d1 3.0) (* d1 d2)) (* d1 d3)))