Linear.V3:$cdot from linear-1.19.1.3, B

Percentage Accurate: 97.7% → 98.9%

Time: 7.1s

Alternatives: 10

Speedup: 1.2×

• 33.2% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \left(x \cdot y + z \cdot t\right) + a \cdot b \end{array} \]

FPCore

(FPCore (x y z t a b) :precision binary64 (+ (+ (* x y) (* z t)) (* a b)))

C

double code(double x, double y, double z, double t, double a, double b) {
	return ((x * y) + (z * t)) + (a * b);
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = ((x * y) + (z * t)) + (a * b)
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	return ((x * y) + (z * t)) + (a * b);
}

Python

def code(x, y, z, t, a, b):
	return ((x * y) + (z * t)) + (a * b)

Julia

function code(x, y, z, t, a, b)
	return Float64(Float64(Float64(x * y) + Float64(z * t)) + Float64(a * b))
end

MATLAB

function tmp = code(x, y, z, t, a, b)
	tmp = ((x * y) + (z * t)) + (a * b);
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(N[(N[(x * y), $MachinePrecision] + N[(z * t), $MachinePrecision]), $MachinePrecision] + N[(a * b), $MachinePrecision]), $MachinePrecision]

Initial Program: 97.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(x \cdot y + z \cdot t\right) + a \cdot b \end{array} \]

FPCore

(FPCore (x y z t a b) :precision binary64 (+ (+ (* x y) (* z t)) (* a b)))

C

double code(double x, double y, double z, double t, double a, double b) {
	return ((x * y) + (z * t)) + (a * b);
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = ((x * y) + (z * t)) + (a * b)
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	return ((x * y) + (z * t)) + (a * b);
}

Python

def code(x, y, z, t, a, b):
	return ((x * y) + (z * t)) + (a * b)

Julia

function code(x, y, z, t, a, b)
	return Float64(Float64(Float64(x * y) + Float64(z * t)) + Float64(a * b))
end

MATLAB

function tmp = code(x, y, z, t, a, b)
	tmp = ((x * y) + (z * t)) + (a * b);
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(N[(N[(x * y), $MachinePrecision] + N[(z * t), $MachinePrecision]), $MachinePrecision] + N[(a * b), $MachinePrecision]), $MachinePrecision]

Alternative 1: 98.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(y, x, \mathsf{fma}\left(z, t, a \cdot b\right)\right) \end{array} \]

FPCore

(FPCore (x y z t a b) :precision binary64 (fma y x (fma z t (* a b))))

C

double code(double x, double y, double z, double t, double a, double b) {
	return fma(y, x, fma(z, t, (a * b)));
}

Julia

function code(x, y, z, t, a, b)
	return fma(y, x, fma(z, t, Float64(a * b)))
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(y * x + N[(z * t + N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 96.4%

   \[\left(x \cdot y + z \cdot t\right) + a \cdot b \]
2. Add Preprocessing
3. Step-by-step derivation

   1. associate-+l+ N/A

      \[\leadsto \color{blue}{x \cdot y + \left(z \cdot t + a \cdot b\right)} \]

   2. *-commutative N/A

      \[\leadsto \color{blue}{y \cdot x} + \left(z \cdot t + a \cdot b\right) \]

   3. accelerator-lowering-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, x, z \cdot t + a \cdot b\right)} \]

   4. accelerator-lowering-fma.f64 N/A

      \[\leadsto \mathsf{fma}\left(y, x, \color{blue}{\mathsf{fma}\left(z, t, a \cdot b\right)}\right) \]

   5. *-lowering-*.f64 98.8

      \[\leadsto \mathsf{fma}\left(y, x, \mathsf{fma}\left(z, t, \color{blue}{a \cdot b}\right)\right) \]

4. Applied egg-rr 98.8%

   \[\leadsto \color{blue}{\mathsf{fma}\left(y, x, \mathsf{fma}\left(z, t, a \cdot b\right)\right)} \]
5. Add Preprocessing

Alternative 2: 85.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot b \leq -0.1:\\ \;\;\;\;\mathsf{fma}\left(y, x, a \cdot b\right)\\ \mathbf{elif}\;a \cdot b \leq 2 \cdot 10^{+100}:\\ \;\;\;\;\mathsf{fma}\left(t, z, y \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, t, a \cdot b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= (* a b) -0.1)
   (fma y x (* a b))
   (if (<= (* a b) 2e+100) (fma t z (* y x)) (fma z t (* a b)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((a * b) <= -0.1) {
		tmp = fma(y, x, (a * b));
	} else if ((a * b) <= 2e+100) {
		tmp = fma(t, z, (y * x));
	} else {
		tmp = fma(z, t, (a * b));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (Float64(a * b) <= -0.1)
		tmp = fma(y, x, Float64(a * b));
	elseif (Float64(a * b) <= 2e+100)
		tmp = fma(t, z, Float64(y * x));
	else
		tmp = fma(z, t, Float64(a * b));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[N[(a * b), $MachinePrecision], -0.1], N[(y * x + N[(a * b), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(a * b), $MachinePrecision], 2e+100], N[(t * z + N[(y * x), $MachinePrecision]), $MachinePrecision], N[(z * t + N[(a * b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 a b) < -0.10000000000000001

   1. Initial program 90.2%

      \[\left(x \cdot y + z \cdot t\right) + a \cdot b \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. associate-+l+ N/A

         \[\leadsto \color{blue}{x \cdot y + \left(z \cdot t + a \cdot b\right)} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{y \cdot x} + \left(z \cdot t + a \cdot b\right) \]

      3. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(y, x, z \cdot t + a \cdot b\right)} \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, x, \color{blue}{\mathsf{fma}\left(z, t, a \cdot b\right)}\right) \]

      5. *-lowering-*.f64 95.1

         \[\leadsto \mathsf{fma}\left(y, x, \mathsf{fma}\left(z, t, \color{blue}{a \cdot b}\right)\right) \]

   4. Applied egg-rr 95.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, x, \mathsf{fma}\left(z, t, a \cdot b\right)\right)} \]

   5. Taylor expanded in z around 0

      \[\leadsto \mathsf{fma}\left(y, x, \color{blue}{a \cdot b}\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 82.7

         \[\leadsto \mathsf{fma}\left(y, x, \color{blue}{a \cdot b}\right) \]

   7. Simplified 82.7%

      \[\leadsto \mathsf{fma}\left(y, x, \color{blue}{a \cdot b}\right) \]

   if -0.10000000000000001 < (*.f64 a b) < 2.00000000000000003e100

   1. Initial program 100.0%

      \[\left(x \cdot y + z \cdot t\right) + a \cdot b \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{t \cdot z + x \cdot y} \]
   4. Step-by-step derivation

      1. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(t, z, x \cdot y\right)} \]

      2. *-lowering-*.f64 91.8

         \[\leadsto \mathsf{fma}\left(t, z, \color{blue}{x \cdot y}\right) \]

   5. Simplified 91.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(t, z, x \cdot y\right)} \]

   if 2.00000000000000003e100 < (*.f64 a b)

   1. Initial program 93.5%

      \[\left(x \cdot y + z \cdot t\right) + a \cdot b \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{a \cdot b + t \cdot z} \]
   4. Step-by-step derivation

      1. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(a, b, t \cdot z\right)} \]

      2. *-lowering-*.f64 85.1

         \[\leadsto \mathsf{fma}\left(a, b, \color{blue}{t \cdot z}\right) \]

   5. Simplified 85.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a, b, t \cdot z\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{t \cdot z + a \cdot b} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{z \cdot t} + a \cdot b \]

      3. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(z, t, a \cdot b\right)} \]

      4. *-lowering-*.f64 91.6

         \[\leadsto \mathsf{fma}\left(z, t, \color{blue}{a \cdot b}\right) \]

   7. Applied egg-rr 91.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z, t, a \cdot b\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 89.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot b \leq -0.1:\\ \;\;\;\;\mathsf{fma}\left(y, x, a \cdot b\right)\\ \mathbf{elif}\;a \cdot b \leq 2 \cdot 10^{+100}:\\ \;\;\;\;\mathsf{fma}\left(t, z, y \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, t, a \cdot b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 86.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot t \leq -5 \cdot 10^{+50}:\\ \;\;\;\;\mathsf{fma}\left(y, x, z \cdot t\right)\\ \mathbf{elif}\;z \cdot t \leq 0.1:\\ \;\;\;\;\mathsf{fma}\left(a, b, y \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t, z, y \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= (* z t) -5e+50)
   (fma y x (* z t))
   (if (<= (* z t) 0.1) (fma a b (* y x)) (fma t z (* y x)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((z * t) <= -5e+50) {
		tmp = fma(y, x, (z * t));
	} else if ((z * t) <= 0.1) {
		tmp = fma(a, b, (y * x));
	} else {
		tmp = fma(t, z, (y * x));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (Float64(z * t) <= -5e+50)
		tmp = fma(y, x, Float64(z * t));
	elseif (Float64(z * t) <= 0.1)
		tmp = fma(a, b, Float64(y * x));
	else
		tmp = fma(t, z, Float64(y * x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[N[(z * t), $MachinePrecision], -5e+50], N[(y * x + N[(z * t), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(z * t), $MachinePrecision], 0.1], N[(a * b + N[(y * x), $MachinePrecision]), $MachinePrecision], N[(t * z + N[(y * x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 z t) < -5e50

   1. Initial program 92.3%

      \[\left(x \cdot y + z \cdot t\right) + a \cdot b \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. associate-+l+ N/A

         \[\leadsto \color{blue}{x \cdot y + \left(z \cdot t + a \cdot b\right)} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{y \cdot x} + \left(z \cdot t + a \cdot b\right) \]

      3. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(y, x, z \cdot t + a \cdot b\right)} \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, x, \color{blue}{\mathsf{fma}\left(z, t, a \cdot b\right)}\right) \]

      5. *-lowering-*.f64 100.0

         \[\leadsto \mathsf{fma}\left(y, x, \mathsf{fma}\left(z, t, \color{blue}{a \cdot b}\right)\right) \]

   4. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, x, \mathsf{fma}\left(z, t, a \cdot b\right)\right)} \]

   5. Taylor expanded in z around inf

      \[\leadsto \mathsf{fma}\left(y, x, \color{blue}{t \cdot z}\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 84.9

         \[\leadsto \mathsf{fma}\left(y, x, \color{blue}{t \cdot z}\right) \]

   7. Simplified 84.9%

      \[\leadsto \mathsf{fma}\left(y, x, \color{blue}{t \cdot z}\right) \]

   if -5e50 < (*.f64 z t) < 0.10000000000000001

   1. Initial program 99.2%

      \[\left(x \cdot y + z \cdot t\right) + a \cdot b \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{a \cdot b + x \cdot y} \]
   4. Step-by-step derivation

      1. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(a, b, x \cdot y\right)} \]

      2. *-lowering-*.f64 91.3

         \[\leadsto \mathsf{fma}\left(a, b, \color{blue}{x \cdot y}\right) \]

   5. Simplified 91.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a, b, x \cdot y\right)} \]

   if 0.10000000000000001 < (*.f64 z t)

   1. Initial program 94.0%

      \[\left(x \cdot y + z \cdot t\right) + a \cdot b \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{t \cdot z + x \cdot y} \]
   4. Step-by-step derivation

      1. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(t, z, x \cdot y\right)} \]

      2. *-lowering-*.f64 85.6

         \[\leadsto \mathsf{fma}\left(t, z, \color{blue}{x \cdot y}\right) \]

   5. Simplified 85.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(t, z, x \cdot y\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 88.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -5 \cdot 10^{+50}:\\ \;\;\;\;\mathsf{fma}\left(y, x, z \cdot t\right)\\ \mathbf{elif}\;z \cdot t \leq 0.1:\\ \;\;\;\;\mathsf{fma}\left(a, b, y \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t, z, y \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 85.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot b \leq -0.1:\\ \;\;\;\;\mathsf{fma}\left(y, x, a \cdot b\right)\\ \mathbf{elif}\;a \cdot b \leq 2 \cdot 10^{+100}:\\ \;\;\;\;\mathsf{fma}\left(t, z, y \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, b, z \cdot t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= (* a b) -0.1)
   (fma y x (* a b))
   (if (<= (* a b) 2e+100) (fma t z (* y x)) (fma a b (* z t)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((a * b) <= -0.1) {
		tmp = fma(y, x, (a * b));
	} else if ((a * b) <= 2e+100) {
		tmp = fma(t, z, (y * x));
	} else {
		tmp = fma(a, b, (z * t));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (Float64(a * b) <= -0.1)
		tmp = fma(y, x, Float64(a * b));
	elseif (Float64(a * b) <= 2e+100)
		tmp = fma(t, z, Float64(y * x));
	else
		tmp = fma(a, b, Float64(z * t));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[N[(a * b), $MachinePrecision], -0.1], N[(y * x + N[(a * b), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(a * b), $MachinePrecision], 2e+100], N[(t * z + N[(y * x), $MachinePrecision]), $MachinePrecision], N[(a * b + N[(z * t), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 a b) < -0.10000000000000001

   1. Initial program 90.2%

      \[\left(x \cdot y + z \cdot t\right) + a \cdot b \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. associate-+l+ N/A

         \[\leadsto \color{blue}{x \cdot y + \left(z \cdot t + a \cdot b\right)} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{y \cdot x} + \left(z \cdot t + a \cdot b\right) \]

      3. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(y, x, z \cdot t + a \cdot b\right)} \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, x, \color{blue}{\mathsf{fma}\left(z, t, a \cdot b\right)}\right) \]

      5. *-lowering-*.f64 95.1

         \[\leadsto \mathsf{fma}\left(y, x, \mathsf{fma}\left(z, t, \color{blue}{a \cdot b}\right)\right) \]

   4. Applied egg-rr 95.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, x, \mathsf{fma}\left(z, t, a \cdot b\right)\right)} \]

   5. Taylor expanded in z around 0

      \[\leadsto \mathsf{fma}\left(y, x, \color{blue}{a \cdot b}\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 82.7

         \[\leadsto \mathsf{fma}\left(y, x, \color{blue}{a \cdot b}\right) \]

   7. Simplified 82.7%

      \[\leadsto \mathsf{fma}\left(y, x, \color{blue}{a \cdot b}\right) \]

   if -0.10000000000000001 < (*.f64 a b) < 2.00000000000000003e100

   1. Initial program 100.0%

      \[\left(x \cdot y + z \cdot t\right) + a \cdot b \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{t \cdot z + x \cdot y} \]
   4. Step-by-step derivation

      1. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(t, z, x \cdot y\right)} \]

      2. *-lowering-*.f64 91.8

         \[\leadsto \mathsf{fma}\left(t, z, \color{blue}{x \cdot y}\right) \]

   5. Simplified 91.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(t, z, x \cdot y\right)} \]

   if 2.00000000000000003e100 < (*.f64 a b)

   1. Initial program 93.5%

      \[\left(x \cdot y + z \cdot t\right) + a \cdot b \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{a \cdot b + t \cdot z} \]
   4. Step-by-step derivation

      1. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(a, b, t \cdot z\right)} \]

      2. *-lowering-*.f64 85.1

         \[\leadsto \mathsf{fma}\left(a, b, \color{blue}{t \cdot z}\right) \]

   5. Simplified 85.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a, b, t \cdot z\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 88.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot b \leq -0.1:\\ \;\;\;\;\mathsf{fma}\left(y, x, a \cdot b\right)\\ \mathbf{elif}\;a \cdot b \leq 2 \cdot 10^{+100}:\\ \;\;\;\;\mathsf{fma}\left(t, z, y \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, b, z \cdot t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 85.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot t \leq -1 \cdot 10^{+101}:\\ \;\;\;\;\mathsf{fma}\left(a, b, z \cdot t\right)\\ \mathbf{elif}\;z \cdot t \leq 0.1:\\ \;\;\;\;\mathsf{fma}\left(a, b, y \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t, z, y \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= (* z t) -1e+101)
   (fma a b (* z t))
   (if (<= (* z t) 0.1) (fma a b (* y x)) (fma t z (* y x)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((z * t) <= -1e+101) {
		tmp = fma(a, b, (z * t));
	} else if ((z * t) <= 0.1) {
		tmp = fma(a, b, (y * x));
	} else {
		tmp = fma(t, z, (y * x));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (Float64(z * t) <= -1e+101)
		tmp = fma(a, b, Float64(z * t));
	elseif (Float64(z * t) <= 0.1)
		tmp = fma(a, b, Float64(y * x));
	else
		tmp = fma(t, z, Float64(y * x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[N[(z * t), $MachinePrecision], -1e+101], N[(a * b + N[(z * t), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(z * t), $MachinePrecision], 0.1], N[(a * b + N[(y * x), $MachinePrecision]), $MachinePrecision], N[(t * z + N[(y * x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 z t) < -9.9999999999999998e100

   1. Initial program 90.5%

      \[\left(x \cdot y + z \cdot t\right) + a \cdot b \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{a \cdot b + t \cdot z} \]
   4. Step-by-step derivation

      1. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(a, b, t \cdot z\right)} \]

      2. *-lowering-*.f64 83.7

         \[\leadsto \mathsf{fma}\left(a, b, \color{blue}{t \cdot z}\right) \]

   5. Simplified 83.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a, b, t \cdot z\right)} \]

   if -9.9999999999999998e100 < (*.f64 z t) < 0.10000000000000001

   1. Initial program 99.3%

      \[\left(x \cdot y + z \cdot t\right) + a \cdot b \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{a \cdot b + x \cdot y} \]
   4. Step-by-step derivation

      1. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(a, b, x \cdot y\right)} \]

      2. *-lowering-*.f64 90.6

         \[\leadsto \mathsf{fma}\left(a, b, \color{blue}{x \cdot y}\right) \]

   5. Simplified 90.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a, b, x \cdot y\right)} \]

   if 0.10000000000000001 < (*.f64 z t)

   1. Initial program 94.0%

      \[\left(x \cdot y + z \cdot t\right) + a \cdot b \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{t \cdot z + x \cdot y} \]
   4. Step-by-step derivation

      1. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(t, z, x \cdot y\right)} \]

      2. *-lowering-*.f64 85.6

         \[\leadsto \mathsf{fma}\left(t, z, \color{blue}{x \cdot y}\right) \]

   5. Simplified 85.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(t, z, x \cdot y\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 88.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -1 \cdot 10^{+101}:\\ \;\;\;\;\mathsf{fma}\left(a, b, z \cdot t\right)\\ \mathbf{elif}\;z \cdot t \leq 0.1:\\ \;\;\;\;\mathsf{fma}\left(a, b, y \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t, z, y \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 85.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(a, b, y \cdot x\right)\\ \mathbf{if}\;y \cdot x \leq -0.0061:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \cdot x \leq 2.5:\\ \;\;\;\;\mathsf{fma}\left(a, b, z \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (fma a b (* y x))))
   (if (<= (* y x) -0.0061) t_1 (if (<= (* y x) 2.5) (fma a b (* z t)) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = fma(a, b, (y * x));
	double tmp;
	if ((y * x) <= -0.0061) {
		tmp = t_1;
	} else if ((y * x) <= 2.5) {
		tmp = fma(a, b, (z * t));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = fma(a, b, Float64(y * x))
	tmp = 0.0
	if (Float64(y * x) <= -0.0061)
		tmp = t_1;
	elseif (Float64(y * x) <= 2.5)
		tmp = fma(a, b, Float64(z * t));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(a * b + N[(y * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(y * x), $MachinePrecision], -0.0061], t$95$1, If[LessEqual[N[(y * x), $MachinePrecision], 2.5], N[(a * b + N[(z * t), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < -0.00610000000000000039 or 2.5 < (*.f64 x y)

   1. Initial program 93.8%

      \[\left(x \cdot y + z \cdot t\right) + a \cdot b \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{a \cdot b + x \cdot y} \]
   4. Step-by-step derivation

      1. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(a, b, x \cdot y\right)} \]

      2. *-lowering-*.f64 81.3

         \[\leadsto \mathsf{fma}\left(a, b, \color{blue}{x \cdot y}\right) \]

   5. Simplified 81.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a, b, x \cdot y\right)} \]

   if -0.00610000000000000039 < (*.f64 x y) < 2.5

   1. Initial program 99.2%

      \[\left(x \cdot y + z \cdot t\right) + a \cdot b \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{a \cdot b + t \cdot z} \]
   4. Step-by-step derivation

      1. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(a, b, t \cdot z\right)} \]

      2. *-lowering-*.f64 93.2

         \[\leadsto \mathsf{fma}\left(a, b, \color{blue}{t \cdot z}\right) \]

   5. Simplified 93.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a, b, t \cdot z\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 87.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot x \leq -0.0061:\\ \;\;\;\;\mathsf{fma}\left(a, b, y \cdot x\right)\\ \mathbf{elif}\;y \cdot x \leq 2.5:\\ \;\;\;\;\mathsf{fma}\left(a, b, z \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, b, y \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 79.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot x \leq -2.9 \cdot 10^{+125}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \cdot x \leq 4.8 \cdot 10^{+49}:\\ \;\;\;\;\mathsf{fma}\left(a, b, z \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= (* y x) -2.9e+125)
   (* y x)
   (if (<= (* y x) 4.8e+49) (fma a b (* z t)) (* y x))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((y * x) <= -2.9e+125) {
		tmp = y * x;
	} else if ((y * x) <= 4.8e+49) {
		tmp = fma(a, b, (z * t));
	} else {
		tmp = y * x;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (Float64(y * x) <= -2.9e+125)
		tmp = Float64(y * x);
	elseif (Float64(y * x) <= 4.8e+49)
		tmp = fma(a, b, Float64(z * t));
	else
		tmp = Float64(y * x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[N[(y * x), $MachinePrecision], -2.9e+125], N[(y * x), $MachinePrecision], If[LessEqual[N[(y * x), $MachinePrecision], 4.8e+49], N[(a * b + N[(z * t), $MachinePrecision]), $MachinePrecision], N[(y * x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < -2.89999999999999993e125 or 4.8e49 < (*.f64 x y)

   1. Initial program 94.2%

      \[\left(x \cdot y + z \cdot t\right) + a \cdot b \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot y} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 71.4

         \[\leadsto \color{blue}{x \cdot y} \]

   5. Simplified 71.4%

      \[\leadsto \color{blue}{x \cdot y} \]

   if -2.89999999999999993e125 < (*.f64 x y) < 4.8e49

   1. Initial program 98.0%

      \[\left(x \cdot y + z \cdot t\right) + a \cdot b \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{a \cdot b + t \cdot z} \]
   4. Step-by-step derivation

      1. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(a, b, t \cdot z\right)} \]

      2. *-lowering-*.f64 86.7

         \[\leadsto \mathsf{fma}\left(a, b, \color{blue}{t \cdot z}\right) \]

   5. Simplified 86.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a, b, t \cdot z\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 80.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot x \leq -2.9 \cdot 10^{+125}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \cdot x \leq 4.8 \cdot 10^{+49}:\\ \;\;\;\;\mathsf{fma}\left(a, b, z \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 54.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot x \leq -0.072:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \cdot x \leq 2.05 \cdot 10^{+43}:\\ \;\;\;\;z \cdot t\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= (* y x) -0.072) (* y x) (if (<= (* y x) 2.05e+43) (* z t) (* y x))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((y * x) <= -0.072) {
		tmp = y * x;
	} else if ((y * x) <= 2.05e+43) {
		tmp = z * t;
	} else {
		tmp = y * x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((y * x) <= (-0.072d0)) then
        tmp = y * x
    else if ((y * x) <= 2.05d+43) then
        tmp = z * t
    else
        tmp = y * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((y * x) <= -0.072) {
		tmp = y * x;
	} else if ((y * x) <= 2.05e+43) {
		tmp = z * t;
	} else {
		tmp = y * x;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (y * x) <= -0.072:
		tmp = y * x
	elif (y * x) <= 2.05e+43:
		tmp = z * t
	else:
		tmp = y * x
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (Float64(y * x) <= -0.072)
		tmp = Float64(y * x);
	elseif (Float64(y * x) <= 2.05e+43)
		tmp = Float64(z * t);
	else
		tmp = Float64(y * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((y * x) <= -0.072)
		tmp = y * x;
	elseif ((y * x) <= 2.05e+43)
		tmp = z * t;
	else
		tmp = y * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[N[(y * x), $MachinePrecision], -0.072], N[(y * x), $MachinePrecision], If[LessEqual[N[(y * x), $MachinePrecision], 2.05e+43], N[(z * t), $MachinePrecision], N[(y * x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < -0.0719999999999999946 or 2.05e43 < (*.f64 x y)

   1. Initial program 93.3%

      \[\left(x \cdot y + z \cdot t\right) + a \cdot b \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot y} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 67.4

         \[\leadsto \color{blue}{x \cdot y} \]

   5. Simplified 67.4%

      \[\leadsto \color{blue}{x \cdot y} \]

   if -0.0719999999999999946 < (*.f64 x y) < 2.05e43

   1. Initial program 99.2%

      \[\left(x \cdot y + z \cdot t\right) + a \cdot b \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{t \cdot z} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 52.7

         \[\leadsto \color{blue}{t \cdot z} \]

   5. Simplified 52.7%

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

4. Final simplification 59.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot x \leq -0.072:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \cdot x \leq 2.05 \cdot 10^{+43}:\\ \;\;\;\;z \cdot t\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 54.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot b \leq -3700000:\\ \;\;\;\;a \cdot b\\ \mathbf{elif}\;a \cdot b \leq 1.6 \cdot 10^{+134}:\\ \;\;\;\;z \cdot t\\ \mathbf{else}:\\ \;\;\;\;a \cdot b\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= (* a b) -3700000.0)
   (* a b)
   (if (<= (* a b) 1.6e+134) (* z t) (* a b))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((a * b) <= -3700000.0) {
		tmp = a * b;
	} else if ((a * b) <= 1.6e+134) {
		tmp = z * t;
	} else {
		tmp = a * b;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((a * b) <= (-3700000.0d0)) then
        tmp = a * b
    else if ((a * b) <= 1.6d+134) then
        tmp = z * t
    else
        tmp = a * b
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((a * b) <= -3700000.0) {
		tmp = a * b;
	} else if ((a * b) <= 1.6e+134) {
		tmp = z * t;
	} else {
		tmp = a * b;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (a * b) <= -3700000.0:
		tmp = a * b
	elif (a * b) <= 1.6e+134:
		tmp = z * t
	else:
		tmp = a * b
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (Float64(a * b) <= -3700000.0)
		tmp = Float64(a * b);
	elseif (Float64(a * b) <= 1.6e+134)
		tmp = Float64(z * t);
	else
		tmp = Float64(a * b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((a * b) <= -3700000.0)
		tmp = a * b;
	elseif ((a * b) <= 1.6e+134)
		tmp = z * t;
	else
		tmp = a * b;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[N[(a * b), $MachinePrecision], -3700000.0], N[(a * b), $MachinePrecision], If[LessEqual[N[(a * b), $MachinePrecision], 1.6e+134], N[(z * t), $MachinePrecision], N[(a * b), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 a b) < -3.7e6 or 1.6e134 < (*.f64 a b)

   1. Initial program 90.8%

      \[\left(x \cdot y + z \cdot t\right) + a \cdot b \]
   2. Add Preprocessing

   3. Taylor expanded in a around inf

      \[\leadsto \color{blue}{a \cdot b} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 63.6

         \[\leadsto \color{blue}{a \cdot b} \]

   5. Simplified 63.6%

      \[\leadsto \color{blue}{a \cdot b} \]

   if -3.7e6 < (*.f64 a b) < 1.6e134

   1. Initial program 100.0%

      \[\left(x \cdot y + z \cdot t\right) + a \cdot b \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{t \cdot z} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 46.5

         \[\leadsto \color{blue}{t \cdot z} \]

   5. Simplified 46.5%

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

4. Final simplification 53.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot b \leq -3700000:\\ \;\;\;\;a \cdot b\\ \mathbf{elif}\;a \cdot b \leq 1.6 \cdot 10^{+134}:\\ \;\;\;\;z \cdot t\\ \mathbf{else}:\\ \;\;\;\;a \cdot b\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 35.2% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ a \cdot b \end{array} \]

FPCore

(FPCore (x y z t a b) :precision binary64 (* a b))

C

double code(double x, double y, double z, double t, double a, double b) {
	return a * b;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = a * b
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	return a * b;
}

Python

def code(x, y, z, t, a, b):
	return a * b

Julia

function code(x, y, z, t, a, b)
	return Float64(a * b)
end

MATLAB

function tmp = code(x, y, z, t, a, b)
	tmp = a * b;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(a * b), $MachinePrecision]

Derivation

1. Initial program 96.4%

   \[\left(x \cdot y + z \cdot t\right) + a \cdot b \]
2. Add Preprocessing

3. Taylor expanded in a around inf

   \[\leadsto \color{blue}{a \cdot b} \]
4. Step-by-step derivation

   1. *-lowering-*.f64 31.3

      \[\leadsto \color{blue}{a \cdot b} \]

5. Simplified 31.3%

   \[\leadsto \color{blue}{a \cdot b} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024198 
(FPCore (x y z t a b)
  :name "Linear.V3:$cdot from linear-1.19.1.3, B"
  :precision binary64
  (+ (+ (* x y) (* z t)) (* a b)))