FastMath test3

Percentage Accurate: 98.0% → 99.9%

Time: 14.1s

Alternatives: 7

Speedup: 1.6×

• 20.9% 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: 98.0% 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

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

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(3.0 + Float64(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[(3.0 + N[(d2 + d3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 96.8%

   \[\left(d1 \cdot 3 + d1 \cdot d2\right) + d1 \cdot d3 \]
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. associate-+l+ N/A

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

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

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

   6. +-lowering-+.f64 100.0%

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

3. Simplified 100.0%

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

Alternative 2: 52.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d2 \leq -3:\\ \;\;\;\;d1 \cdot d2\\ \mathbf{elif}\;d2 \leq -4.2 \cdot 10^{-192}:\\ \;\;\;\;d1 \cdot 3\\ \mathbf{else}:\\ \;\;\;\;d1 \cdot d3\\ \end{array} \end{array} \]

FPCore

(FPCore (d1 d2 d3)
 :precision binary64
 (if (<= d2 -3.0) (* d1 d2) (if (<= d2 -4.2e-192) (* d1 3.0) (* d1 d3))))

C

double code(double d1, double d2, double d3) {
	double tmp;
	if (d2 <= -3.0) {
		tmp = d1 * d2;
	} else if (d2 <= -4.2e-192) {
		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 <= (-3.0d0)) then
        tmp = d1 * d2
    else if (d2 <= (-4.2d-192)) 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 <= -3.0) {
		tmp = d1 * d2;
	} else if (d2 <= -4.2e-192) {
		tmp = d1 * 3.0;
	} else {
		tmp = d1 * d3;
	}
	return tmp;
}

Python

def code(d1, d2, d3):
	tmp = 0
	if d2 <= -3.0:
		tmp = d1 * d2
	elif d2 <= -4.2e-192:
		tmp = d1 * 3.0
	else:
		tmp = d1 * d3
	return tmp

Julia

function code(d1, d2, d3)
	tmp = 0.0
	if (d2 <= -3.0)
		tmp = Float64(d1 * d2);
	elseif (d2 <= -4.2e-192)
		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 <= -3.0)
		tmp = d1 * d2;
	elseif (d2 <= -4.2e-192)
		tmp = d1 * 3.0;
	else
		tmp = d1 * d3;
	end
	tmp_2 = tmp;
end

Wolfram

code[d1_, d2_, d3_] := If[LessEqual[d2, -3.0], N[(d1 * d2), $MachinePrecision], If[LessEqual[d2, -4.2e-192], N[(d1 * 3.0), $MachinePrecision], N[(d1 * d3), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if d2 < -3

   1. Initial program 93.7%

      \[\left(d1 \cdot 3 + d1 \cdot d2\right) + d1 \cdot d3 \]
   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. associate-+l+ N/A

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

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

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

      6. +-lowering-+.f64 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in d2 around inf

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

      1. *-lowering-*.f64 86.9%

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

   7. Simplified 86.9%

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

   if -3 < d2 < -4.19999999999999986e-192

   1. Initial program 99.9%

      \[\left(d1 \cdot 3 + d1 \cdot d2\right) + d1 \cdot d3 \]
   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. associate-+l+ N/A

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

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

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

      6. +-lowering-+.f64 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in d3 around 0

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

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

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

      2. +-lowering-+.f64 53.1%

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

   7. Simplified 53.1%

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

   8. Taylor expanded in d2 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 50.7%

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

   10. Simplified 50.7%

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

   if -4.19999999999999986e-192 < d2

   1. Initial program 97.2%

      \[\left(d1 \cdot 3 + d1 \cdot d2\right) + d1 \cdot d3 \]
   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. associate-+l+ N/A

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

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

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

      6. +-lowering-+.f64 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in d3 around inf

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

      1. *-lowering-*.f64 45.1%

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

   7. Simplified 45.1%

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

Alternative 3: 80.0% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

function code(d1, d2, d3)
	tmp = 0.0
	if (d3 <= 3.0)
		tmp = Float64(d1 * Float64(3.0 + d2));
	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 * (3.0 + d2);
	else
		tmp = d1 * (d2 + d3);
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if d3 < 3

   1. Initial program 97.7%

      \[\left(d1 \cdot 3 + d1 \cdot d2\right) + d1 \cdot d3 \]
   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. associate-+l+ N/A

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

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

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

      6. +-lowering-+.f64 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in d3 around 0

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

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

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

      2. +-lowering-+.f64 75.6%

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

   7. Simplified 75.6%

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

   if 3 < d3

   1. Initial program 94.8%

      \[\left(d1 \cdot 3 + d1 \cdot d2\right) + d1 \cdot d3 \]
   2. Add Preprocessing

   3. Taylor expanded in d2 around inf

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

      1. *-lowering-*.f64 94.7%

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

   5. Simplified 94.7%

      \[\leadsto \color{blue}{d1 \cdot d2} + d1 \cdot d3 \]
   6. Step-by-step derivation

      1. distribute-lft-out N/A

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

      2. *-commutative N/A

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

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

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

      4. +-lowering-+.f64 99.9%

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

   7. Applied egg-rr 99.9%

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

4. Final simplification 82.9%

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

Alternative 4: 76.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d2 \leq -1.05 \cdot 10^{-15}:\\ \;\;\;\;d1 \cdot \left(3 + d2\right)\\ \mathbf{else}:\\ \;\;\;\;d1 \cdot \left(3 + d3\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (d1 d2 d3)
 :precision binary64
 (if (<= d2 -1.05e-15) (* d1 (+ 3.0 d2)) (* d1 (+ 3.0 d3))))

C

double code(double d1, double d2, double d3) {
	double tmp;
	if (d2 <= -1.05e-15) {
		tmp = d1 * (3.0 + 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 <= (-1.05d-15)) then
        tmp = d1 * (3.0d0 + 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 <= -1.05e-15) {
		tmp = d1 * (3.0 + d2);
	} else {
		tmp = d1 * (3.0 + d3);
	}
	return tmp;
}

Python

def code(d1, d2, d3):
	tmp = 0
	if d2 <= -1.05e-15:
		tmp = d1 * (3.0 + d2)
	else:
		tmp = d1 * (3.0 + d3)
	return tmp

Julia

function code(d1, d2, d3)
	tmp = 0.0
	if (d2 <= -1.05e-15)
		tmp = Float64(d1 * Float64(3.0 + 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 <= -1.05e-15)
		tmp = d1 * (3.0 + d2);
	else
		tmp = d1 * (3.0 + d3);
	end
	tmp_2 = tmp;
end

Wolfram

code[d1_, d2_, d3_] := If[LessEqual[d2, -1.05e-15], N[(d1 * N[(3.0 + d2), $MachinePrecision]), $MachinePrecision], N[(d1 * N[(3.0 + d3), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d2 < -1.0499999999999999e-15

   1. Initial program 93.9%

      \[\left(d1 \cdot 3 + d1 \cdot d2\right) + d1 \cdot d3 \]
   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. associate-+l+ N/A

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

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

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

      6. +-lowering-+.f64 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in d3 around 0

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

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

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

      2. +-lowering-+.f64 87.5%

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

   7. Simplified 87.5%

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

   if -1.0499999999999999e-15 < d2

   1. Initial program 97.8%

      \[\left(d1 \cdot 3 + d1 \cdot d2\right) + d1 \cdot d3 \]
   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. associate-+l+ N/A

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

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

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

      6. +-lowering-+.f64 99.9%

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

   3. Simplified 99.9%

      \[\leadsto \color{blue}{d1 \cdot \left(3 + \left(d2 + d3\right)\right)} \]
   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 73.0%

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

   7. Simplified 73.0%

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

Alternative 5: 75.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d3 \leq 70000000000:\\ \;\;\;\;d1 \cdot \left(3 + d2\right)\\ \mathbf{else}:\\ \;\;\;\;d1 \cdot d3\\ \end{array} \end{array} \]

FPCore

(FPCore (d1 d2 d3)
 :precision binary64
 (if (<= d3 70000000000.0) (* d1 (+ 3.0 d2)) (* d1 d3)))

C

double code(double d1, double d2, double d3) {
	double tmp;
	if (d3 <= 70000000000.0) {
		tmp = d1 * (3.0 + d2);
	} 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 <= 70000000000.0d0) then
        tmp = d1 * (3.0d0 + d2)
    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 <= 70000000000.0) {
		tmp = d1 * (3.0 + d2);
	} else {
		tmp = d1 * d3;
	}
	return tmp;
}

Python

def code(d1, d2, d3):
	tmp = 0
	if d3 <= 70000000000.0:
		tmp = d1 * (3.0 + d2)
	else:
		tmp = d1 * d3
	return tmp

Julia

function code(d1, d2, d3)
	tmp = 0.0
	if (d3 <= 70000000000.0)
		tmp = Float64(d1 * Float64(3.0 + d2));
	else
		tmp = Float64(d1 * d3);
	end
	return tmp
end

MATLAB

function tmp_2 = code(d1, d2, d3)
	tmp = 0.0;
	if (d3 <= 70000000000.0)
		tmp = d1 * (3.0 + d2);
	else
		tmp = d1 * d3;
	end
	tmp_2 = tmp;
end

Wolfram

code[d1_, d2_, d3_] := If[LessEqual[d3, 70000000000.0], N[(d1 * N[(3.0 + d2), $MachinePrecision]), $MachinePrecision], N[(d1 * d3), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d3 < 7e10

   1. Initial program 97.7%

      \[\left(d1 \cdot 3 + d1 \cdot d2\right) + d1 \cdot d3 \]
   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. associate-+l+ N/A

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

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

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

      6. +-lowering-+.f64 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in d3 around 0

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

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

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

      2. +-lowering-+.f64 76.0%

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

   7. Simplified 76.0%

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

   if 7e10 < d3

   1. Initial program 94.6%

      \[\left(d1 \cdot 3 + d1 \cdot d2\right) + d1 \cdot d3 \]
   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. associate-+l+ N/A

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

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

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

      6. +-lowering-+.f64 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in d3 around inf

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

      1. *-lowering-*.f64 71.2%

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

   7. Simplified 71.2%

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

Alternative 6: 44.7% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if d2 < -3

   1. Initial program 93.7%

      \[\left(d1 \cdot 3 + d1 \cdot d2\right) + d1 \cdot d3 \]
   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. associate-+l+ N/A

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

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

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

      6. +-lowering-+.f64 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in d2 around inf

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

      1. *-lowering-*.f64 86.9%

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

   7. Simplified 86.9%

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

   if -3 < 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. associate-+l+ N/A

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

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

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

      6. +-lowering-+.f64 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in d3 around 0

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

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

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

      2. +-lowering-+.f64 57.2%

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

   7. Simplified 57.2%

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

   8. Taylor expanded in d2 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 29.3%

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

   10. Simplified 29.3%

       \[\leadsto \color{blue}{d1 \cdot 3} \]
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 96.8%

   \[\left(d1 \cdot 3 + d1 \cdot d2\right) + d1 \cdot d3 \]
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. associate-+l+ N/A

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

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

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

   6. +-lowering-+.f64 100.0%

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

3. Simplified 100.0%

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

5. Taylor expanded in d3 around 0

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

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

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

   2. +-lowering-+.f64 64.5%

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

7. Simplified 64.5%

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

8. Taylor expanded in d2 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 22.5%

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

10. Simplified 22.5%

    \[\leadsto \color{blue}{d1 \cdot 3} \]
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 2024191 
(FPCore (d1 d2 d3)
  :name "FastMath test3"
  :precision binary64

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

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