Graphics.Rasterific.CubicBezier:cachedBezierAt from Rasterific-0.6.1

Percentage Accurate: 91.9% → 96.7%

Time: 8.6s

Alternatives: 17

Speedup: 0.9×

• 31.9% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 91.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 96.7% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -20:\\ \;\;\;\;\mathsf{fma}\left(z, \mathsf{fma}\left(a, b, y\right), \mathsf{fma}\left(t, a, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, z, \mathsf{fma}\left(a, \mathsf{fma}\left(z, b, t\right), x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= z -20.0)
   (fma z (fma a b y) (fma t a x))
   (fma y z (fma a (fma z b t) x))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (z <= -20.0) {
		tmp = fma(z, fma(a, b, y), fma(t, a, x));
	} else {
		tmp = fma(y, z, fma(a, fma(z, b, t), x));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (z <= -20.0)
		tmp = fma(z, fma(a, b, y), fma(t, a, x));
	else
		tmp = fma(y, z, fma(a, fma(z, b, t), x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[z, -20.0], N[(z * N[(a * b + y), $MachinePrecision] + N[(t * a + x), $MachinePrecision]), $MachinePrecision], N[(y * z + N[(a * N[(z * b + t), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -20

   1. Initial program 74.8%

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

      1. +-commutative 74.8%

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

      2. +-commutative 74.8%

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

      3. associate-+l+ 74.8%

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

      4. associate-+r+ 74.8%

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

      5. *-commutative 74.8%

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

      6. associate-*l* 85.5%

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

      7. *-commutative 85.5%

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

      8. distribute-lft-out 93.5%

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

      9. fma-def 98.4%

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

      10. fma-def 98.4%

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

      11. +-commutative 98.4%

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

      12. fma-def 98.4%

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

   3. Simplified 98.4%

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

   if -20 < z

   1. Initial program 95.3%

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

      1. associate-+l+ 95.3%

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

      2. +-commutative 95.3%

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

      3. associate-+l+ 95.3%

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

      4. fma-def 95.8%

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

      5. +-commutative 95.8%

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

      6. *-commutative 95.8%

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

      7. associate-*l* 95.3%

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

      8. distribute-lft-out 97.9%

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

      9. fma-def 97.9%

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

      10. +-commutative 97.9%

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

      11. fma-def 97.9%

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

   3. Simplified 97.9%

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

4. Final simplification 98.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -20:\\ \;\;\;\;\mathsf{fma}\left(z, \mathsf{fma}\left(a, b, y\right), \mathsf{fma}\left(t, a, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, z, \mathsf{fma}\left(a, \mathsf{fma}\left(z, b, t\right), x\right)\right)\\ \end{array} \]

Alternative 2: 95.5% accurate, 0.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 90.3%

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

   1. associate-+l+ 90.3%

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

   2. +-commutative 90.3%

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

   3. associate-+l+ 90.3%

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

   4. fma-def 91.9%

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

   5. +-commutative 91.9%

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

   6. *-commutative 91.9%

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

   7. associate-*l* 92.6%

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

   8. distribute-lft-out 96.1%

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

   9. fma-def 96.1%

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

   10. +-commutative 96.1%

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

   11. fma-def 96.1%

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

3. Simplified 96.1%

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

4. Final simplification 96.1%

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

Alternative 3: 94.2% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 90.3%

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

   1. associate-+l+ 90.3%

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

   2. +-commutative 90.3%

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

   3. *-commutative 90.3%

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

   4. associate-*l* 91.0%

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

   5. distribute-lft-out 93.8%

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

   6. fma-def 95.3%

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

   7. +-commutative 95.3%

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

   8. fma-def 95.3%

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

3. Simplified 95.3%

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

4. Final simplification 95.3%

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

Alternative 4: 96.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(a \cdot t + \left(x + z \cdot y\right)\right) + b \cdot \left(z \cdot a\right)\\ \mathbf{if}\;t_1 \leq \infty:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;x + a \cdot \left(t + z \cdot b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ (+ (* a t) (+ x (* z y))) (* b (* z a)))))
   (if (<= t_1 INFINITY) t_1 (+ x (* a (+ t (* z b)))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = ((a * t) + (x + (z * y))) + (b * (z * a));
	double tmp;
	if (t_1 <= ((double) INFINITY)) {
		tmp = t_1;
	} else {
		tmp = x + (a * (t + (z * b)));
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = ((a * t) + (x + (z * y))) + (b * (z * a));
	double tmp;
	if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = t_1;
	} else {
		tmp = x + (a * (t + (z * b)));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = ((a * t) + (x + (z * y))) + (b * (z * a))
	tmp = 0
	if t_1 <= math.inf:
		tmp = t_1
	else:
		tmp = x + (a * (t + (z * b)))
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(Float64(a * t) + Float64(x + Float64(z * y))) + Float64(b * Float64(z * a)))
	tmp = 0.0
	if (t_1 <= Inf)
		tmp = t_1;
	else
		tmp = Float64(x + Float64(a * Float64(t + Float64(z * b))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = ((a * t) + (x + (z * y))) + (b * (z * a));
	tmp = 0.0;
	if (t_1 <= Inf)
		tmp = t_1;
	else
		tmp = x + (a * (t + (z * b)));
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (+.f64 (+.f64 (+.f64 x (*.f64 y z)) (*.f64 t a)) (*.f64 (*.f64 a z) b)) < +inf.0

   1. Initial program 97.6%

      \[\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b \]

   if +inf.0 < (+.f64 (+.f64 (+.f64 x (*.f64 y z)) (*.f64 t a)) (*.f64 (*.f64 a z) b))

   1. Initial program 0.0%

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

      1. associate-+l+ 0.0%

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

      2. +-commutative 0.0%

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

      3. associate-+l+ 0.0%

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

      4. fma-def 21.1%

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

      5. +-commutative 21.1%

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

      6. *-commutative 21.1%

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

      7. associate-*l* 31.6%

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

      8. distribute-lft-out 78.9%

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

      9. fma-def 78.9%

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

      10. +-commutative 78.9%

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

      11. fma-def 78.9%

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

   3. Simplified 78.9%

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

   4. Taylor expanded in y around 0 73.9%

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

4. Final simplification 95.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(a \cdot t + \left(x + z \cdot y\right)\right) + b \cdot \left(z \cdot a\right) \leq \infty:\\ \;\;\;\;\left(a \cdot t + \left(x + z \cdot y\right)\right) + b \cdot \left(z \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;x + a \cdot \left(t + z \cdot b\right)\\ \end{array} \]

Alternative 5: 92.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 3.6 \cdot 10^{+134}:\\ \;\;\;\;\left(a \cdot t + a \cdot \left(z \cdot b\right)\right) + \left(x + z \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;x + a \cdot \left(t + z \cdot b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= a 3.6e+134)
   (+ (+ (* a t) (* a (* z b))) (+ x (* z y)))
   (+ x (* a (+ t (* z b))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (a <= 3.6e+134) {
		tmp = ((a * t) + (a * (z * b))) + (x + (z * y));
	} else {
		tmp = x + (a * (t + (z * 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 <= 3.6d+134) then
        tmp = ((a * t) + (a * (z * b))) + (x + (z * y))
    else
        tmp = x + (a * (t + (z * 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 <= 3.6e+134) {
		tmp = ((a * t) + (a * (z * b))) + (x + (z * y));
	} else {
		tmp = x + (a * (t + (z * b)));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if a <= 3.6e+134:
		tmp = ((a * t) + (a * (z * b))) + (x + (z * y))
	else:
		tmp = x + (a * (t + (z * b)))
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (a <= 3.6e+134)
		tmp = Float64(Float64(Float64(a * t) + Float64(a * Float64(z * b))) + Float64(x + Float64(z * y)));
	else
		tmp = Float64(x + Float64(a * Float64(t + Float64(z * b))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (a <= 3.6e+134)
		tmp = ((a * t) + (a * (z * b))) + (x + (z * y));
	else
		tmp = x + (a * (t + (z * b)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[a, 3.6e+134], N[(N[(N[(a * t), $MachinePrecision] + N[(a * N[(z * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x + N[(z * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(a * N[(t + N[(z * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 3.59999999999999988e134

   1. Initial program 93.9%

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

      1. associate-+l+ 93.9%

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

      2. associate-*l* 93.9%

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

   3. Simplified 93.9%

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

   if 3.59999999999999988e134 < a

   1. Initial program 73.0%

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

      1. associate-+l+ 73.0%

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

      2. +-commutative 73.0%

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

      3. associate-+l+ 73.0%

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

      4. fma-def 82.1%

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

      5. +-commutative 82.1%

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

      6. *-commutative 82.1%

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

      7. associate-*l* 86.4%

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

      8. distribute-lft-out 93.2%

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

      9. fma-def 93.2%

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

      10. +-commutative 93.2%

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

      11. fma-def 93.2%

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

   3. Simplified 93.2%

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

   4. Taylor expanded in y around 0 91.0%

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

4. Final simplification 93.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 3.6 \cdot 10^{+134}:\\ \;\;\;\;\left(a \cdot t + a \cdot \left(z \cdot b\right)\right) + \left(x + z \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;x + a \cdot \left(t + z \cdot b\right)\\ \end{array} \]

Alternative 6: 61.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x + a \cdot t\\ \mathbf{if}\;a \leq -3.7 \cdot 10^{+240}:\\ \;\;\;\;b \cdot \left(z \cdot a\right)\\ \mathbf{elif}\;a \leq -9 \cdot 10^{+157}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq -1.65 \cdot 10^{+75}:\\ \;\;\;\;a \cdot \left(z \cdot b\right)\\ \mathbf{elif}\;a \leq -8.2 \cdot 10^{-54} \lor \neg \left(a \leq 4.4 \cdot 10^{-63}\right):\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;x + z \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ x (* a t))))
   (if (<= a -3.7e+240)
     (* b (* z a))
     (if (<= a -9e+157)
       t_1
       (if (<= a -1.65e+75)
         (* a (* z b))
         (if (or (<= a -8.2e-54) (not (<= a 4.4e-63))) t_1 (+ x (* z y))))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x + (a * t);
	double tmp;
	if (a <= -3.7e+240) {
		tmp = b * (z * a);
	} else if (a <= -9e+157) {
		tmp = t_1;
	} else if (a <= -1.65e+75) {
		tmp = a * (z * b);
	} else if ((a <= -8.2e-54) || !(a <= 4.4e-63)) {
		tmp = t_1;
	} else {
		tmp = x + (z * y);
	}
	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) :: t_1
    real(8) :: tmp
    t_1 = x + (a * t)
    if (a <= (-3.7d+240)) then
        tmp = b * (z * a)
    else if (a <= (-9d+157)) then
        tmp = t_1
    else if (a <= (-1.65d+75)) then
        tmp = a * (z * b)
    else if ((a <= (-8.2d-54)) .or. (.not. (a <= 4.4d-63))) then
        tmp = t_1
    else
        tmp = x + (z * y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x + (a * t);
	double tmp;
	if (a <= -3.7e+240) {
		tmp = b * (z * a);
	} else if (a <= -9e+157) {
		tmp = t_1;
	} else if (a <= -1.65e+75) {
		tmp = a * (z * b);
	} else if ((a <= -8.2e-54) || !(a <= 4.4e-63)) {
		tmp = t_1;
	} else {
		tmp = x + (z * y);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = x + (a * t)
	tmp = 0
	if a <= -3.7e+240:
		tmp = b * (z * a)
	elif a <= -9e+157:
		tmp = t_1
	elif a <= -1.65e+75:
		tmp = a * (z * b)
	elif (a <= -8.2e-54) or not (a <= 4.4e-63):
		tmp = t_1
	else:
		tmp = x + (z * y)
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(x + Float64(a * t))
	tmp = 0.0
	if (a <= -3.7e+240)
		tmp = Float64(b * Float64(z * a));
	elseif (a <= -9e+157)
		tmp = t_1;
	elseif (a <= -1.65e+75)
		tmp = Float64(a * Float64(z * b));
	elseif ((a <= -8.2e-54) || !(a <= 4.4e-63))
		tmp = t_1;
	else
		tmp = Float64(x + Float64(z * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = x + (a * t);
	tmp = 0.0;
	if (a <= -3.7e+240)
		tmp = b * (z * a);
	elseif (a <= -9e+157)
		tmp = t_1;
	elseif (a <= -1.65e+75)
		tmp = a * (z * b);
	elseif ((a <= -8.2e-54) || ~((a <= 4.4e-63)))
		tmp = t_1;
	else
		tmp = x + (z * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(x + N[(a * t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -3.7e+240], N[(b * N[(z * a), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, -9e+157], t$95$1, If[LessEqual[a, -1.65e+75], N[(a * N[(z * b), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[a, -8.2e-54], N[Not[LessEqual[a, 4.4e-63]], $MachinePrecision]], t$95$1, N[(x + N[(z * y), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if a < -3.7000000000000001e240

   1. Initial program 82.4%

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

      1. associate-+l+ 82.4%

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

      2. +-commutative 82.4%

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

      3. associate-+l+ 82.4%

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

      4. fma-def 82.4%

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

      5. +-commutative 82.4%

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

      6. *-commutative 82.4%

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

      7. associate-*l* 82.2%

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

      8. distribute-lft-out 93.9%

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

      9. fma-def 93.9%

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

      10. +-commutative 93.9%

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

      11. fma-def 93.9%

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

   3. Simplified 93.9%

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

   4. Taylor expanded in a around inf 93.9%

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

   5. Taylor expanded in z around inf 70.8%

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

      1. associate-*r* 76.5%

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

   7. Simplified 76.5%

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

   if -3.7000000000000001e240 < a < -8.9999999999999997e157 or -1.64999999999999999e75 < a < -8.2000000000000001e-54 or 4.3999999999999999e-63 < a

   1. Initial program 86.7%

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

      1. associate-+l+ 86.7%

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

      2. +-commutative 86.7%

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

      3. associate-+l+ 86.7%

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

      4. fma-def 89.9%

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

      5. +-commutative 89.9%

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

      6. *-commutative 89.9%

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

      7. associate-*l* 93.0%

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

      8. distribute-lft-out 96.1%

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

      9. fma-def 96.1%

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

      10. +-commutative 96.1%

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

      11. fma-def 96.1%

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

   3. Simplified 96.1%

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

   4. Taylor expanded in y around 0 86.5%

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

   5. Taylor expanded in z around 0 61.1%

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

   if -8.9999999999999997e157 < a < -1.64999999999999999e75

   1. Initial program 67.6%

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

      1. associate-+l+ 67.6%

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

      2. +-commutative 67.6%

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

      3. associate-+l+ 67.6%

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

      4. fma-def 67.6%

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

      5. +-commutative 67.6%

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

      6. *-commutative 67.6%

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

      7. associate-*l* 79.7%

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

      8. distribute-lft-out 99.7%

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

      9. fma-def 99.7%

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

      10. +-commutative 99.7%

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

      11. fma-def 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in a around inf 80.0%

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

   5. Taylor expanded in z around inf 56.1%

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

   if -8.2000000000000001e-54 < a < 4.3999999999999999e-63

   1. Initial program 100.0%

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

      1. associate-+l+ 100.0%

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

      2. +-commutative 100.0%

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

      3. associate-+l+ 100.0%

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

      4. fma-def 100.0%

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

      5. +-commutative 100.0%

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

      6. *-commutative 100.0%

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

      7. associate-*l* 96.0%

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

      8. distribute-lft-out 96.0%

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

      9. fma-def 96.0%

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

      10. +-commutative 96.0%

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

      11. fma-def 96.0%

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

   3. Simplified 96.0%

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

   4. Taylor expanded in a around 0 78.9%

      \[\leadsto \color{blue}{y \cdot z + x} \]
3. Recombined 4 regimes into one program.

4. Final simplification 68.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -3.7 \cdot 10^{+240}:\\ \;\;\;\;b \cdot \left(z \cdot a\right)\\ \mathbf{elif}\;a \leq -9 \cdot 10^{+157}:\\ \;\;\;\;x + a \cdot t\\ \mathbf{elif}\;a \leq -1.65 \cdot 10^{+75}:\\ \;\;\;\;a \cdot \left(z \cdot b\right)\\ \mathbf{elif}\;a \leq -8.2 \cdot 10^{-54} \lor \neg \left(a \leq 4.4 \cdot 10^{-63}\right):\\ \;\;\;\;x + a \cdot t\\ \mathbf{else}:\\ \;\;\;\;x + z \cdot y\\ \end{array} \]

Alternative 7: 55.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x + a \cdot t\\ \mathbf{if}\;z \leq -4.5 \cdot 10^{+262}:\\ \;\;\;\;z \cdot \left(a \cdot b\right)\\ \mathbf{elif}\;z \leq -1.02 \cdot 10^{+210}:\\ \;\;\;\;z \cdot y\\ \mathbf{elif}\;z \leq -1.38 \cdot 10^{+169}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -4.2 \cdot 10^{-63}:\\ \;\;\;\;z \cdot y\\ \mathbf{elif}\;z \leq 4.1 \cdot 10^{+79}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(z \cdot a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ x (* a t))))
   (if (<= z -4.5e+262)
     (* z (* a b))
     (if (<= z -1.02e+210)
       (* z y)
       (if (<= z -1.38e+169)
         t_1
         (if (<= z -4.2e-63)
           (* z y)
           (if (<= z 4.1e+79) t_1 (* b (* z a)))))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x + (a * t);
	double tmp;
	if (z <= -4.5e+262) {
		tmp = z * (a * b);
	} else if (z <= -1.02e+210) {
		tmp = z * y;
	} else if (z <= -1.38e+169) {
		tmp = t_1;
	} else if (z <= -4.2e-63) {
		tmp = z * y;
	} else if (z <= 4.1e+79) {
		tmp = t_1;
	} else {
		tmp = b * (z * a);
	}
	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) :: t_1
    real(8) :: tmp
    t_1 = x + (a * t)
    if (z <= (-4.5d+262)) then
        tmp = z * (a * b)
    else if (z <= (-1.02d+210)) then
        tmp = z * y
    else if (z <= (-1.38d+169)) then
        tmp = t_1
    else if (z <= (-4.2d-63)) then
        tmp = z * y
    else if (z <= 4.1d+79) then
        tmp = t_1
    else
        tmp = b * (z * a)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x + (a * t);
	double tmp;
	if (z <= -4.5e+262) {
		tmp = z * (a * b);
	} else if (z <= -1.02e+210) {
		tmp = z * y;
	} else if (z <= -1.38e+169) {
		tmp = t_1;
	} else if (z <= -4.2e-63) {
		tmp = z * y;
	} else if (z <= 4.1e+79) {
		tmp = t_1;
	} else {
		tmp = b * (z * a);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = x + (a * t)
	tmp = 0
	if z <= -4.5e+262:
		tmp = z * (a * b)
	elif z <= -1.02e+210:
		tmp = z * y
	elif z <= -1.38e+169:
		tmp = t_1
	elif z <= -4.2e-63:
		tmp = z * y
	elif z <= 4.1e+79:
		tmp = t_1
	else:
		tmp = b * (z * a)
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(x + Float64(a * t))
	tmp = 0.0
	if (z <= -4.5e+262)
		tmp = Float64(z * Float64(a * b));
	elseif (z <= -1.02e+210)
		tmp = Float64(z * y);
	elseif (z <= -1.38e+169)
		tmp = t_1;
	elseif (z <= -4.2e-63)
		tmp = Float64(z * y);
	elseif (z <= 4.1e+79)
		tmp = t_1;
	else
		tmp = Float64(b * Float64(z * a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = x + (a * t);
	tmp = 0.0;
	if (z <= -4.5e+262)
		tmp = z * (a * b);
	elseif (z <= -1.02e+210)
		tmp = z * y;
	elseif (z <= -1.38e+169)
		tmp = t_1;
	elseif (z <= -4.2e-63)
		tmp = z * y;
	elseif (z <= 4.1e+79)
		tmp = t_1;
	else
		tmp = b * (z * a);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(x + N[(a * t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -4.5e+262], N[(z * N[(a * b), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, -1.02e+210], N[(z * y), $MachinePrecision], If[LessEqual[z, -1.38e+169], t$95$1, If[LessEqual[z, -4.2e-63], N[(z * y), $MachinePrecision], If[LessEqual[z, 4.1e+79], t$95$1, N[(b * N[(z * a), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if z < -4.49999999999999972e262

   1. Initial program 35.2%

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

      1. associate-+l+ 35.2%

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

      2. +-commutative 35.2%

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

      3. associate-+l+ 35.2%

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

      4. fma-def 57.4%

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

      5. +-commutative 57.4%

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

      6. *-commutative 57.4%

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

      7. associate-*l* 78.4%

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

      8. distribute-lft-out 78.4%

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

      9. fma-def 78.4%

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

      10. +-commutative 78.4%

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

      11. fma-def 78.4%

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

   3. Simplified 78.4%

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

   4. Taylor expanded in z around inf 88.9%

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

   5. Taylor expanded in a around inf 67.3%

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

      1. associate-*r* 77.8%

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

      2. *-commutative 77.8%

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

   7. Simplified 77.8%

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

   if -4.49999999999999972e262 < z < -1.02000000000000005e210 or -1.38e169 < z < -4.2e-63

   1. Initial program 93.7%

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

      1. associate-+l+ 93.7%

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

      2. +-commutative 93.7%

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

      3. associate-+l+ 93.7%

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

      4. fma-def 93.7%

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

      5. +-commutative 93.7%

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

      6. *-commutative 93.7%

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

      7. associate-*l* 95.8%

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

      8. distribute-lft-out 95.8%

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

      9. fma-def 95.8%

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

      10. +-commutative 95.8%

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

      11. fma-def 95.8%

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

   3. Simplified 95.8%

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

   4. Taylor expanded in y around inf 56.3%

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

      1. *-commutative 56.3%

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

   6. Simplified 56.3%

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

   if -1.02000000000000005e210 < z < -1.38e169 or -4.2e-63 < z < 4.1e79

   1. Initial program 94.6%

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

      1. associate-+l+ 94.6%

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

      2. +-commutative 94.6%

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

      3. associate-+l+ 94.6%

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

      4. fma-def 95.2%

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

      5. +-commutative 95.2%

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

      6. *-commutative 95.2%

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

      7. associate-*l* 94.6%

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

      8. distribute-lft-out 98.2%

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

      9. fma-def 98.2%

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

      10. +-commutative 98.2%

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

      11. fma-def 98.2%

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

   3. Simplified 98.2%

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

   4. Taylor expanded in y around 0 87.2%

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

   5. Taylor expanded in z around 0 69.8%

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

   if 4.1e79 < z

   1. Initial program 78.8%

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

      1. associate-+l+ 78.8%

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

      2. +-commutative 78.8%

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

      3. associate-+l+ 78.8%

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

      4. fma-def 81.8%

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

      5. +-commutative 81.8%

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

      6. *-commutative 81.8%

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

      7. associate-*l* 81.8%

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

      8. distribute-lft-out 90.9%

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

      9. fma-def 90.9%

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

      10. +-commutative 90.9%

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

      11. fma-def 90.9%

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

   3. Simplified 90.9%

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

   4. Taylor expanded in a around inf 59.9%

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

   5. Taylor expanded in z around inf 49.4%

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

      1. associate-*r* 49.6%

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

   7. Simplified 49.6%

      \[\leadsto \color{blue}{\left(a \cdot z\right) \cdot b} \]
3. Recombined 4 regimes into one program.

4. Final simplification 65.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4.5 \cdot 10^{+262}:\\ \;\;\;\;z \cdot \left(a \cdot b\right)\\ \mathbf{elif}\;z \leq -1.02 \cdot 10^{+210}:\\ \;\;\;\;z \cdot y\\ \mathbf{elif}\;z \leq -1.38 \cdot 10^{+169}:\\ \;\;\;\;x + a \cdot t\\ \mathbf{elif}\;z \leq -4.2 \cdot 10^{-63}:\\ \;\;\;\;z \cdot y\\ \mathbf{elif}\;z \leq 4.1 \cdot 10^{+79}:\\ \;\;\;\;x + a \cdot t\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(z \cdot a\right)\\ \end{array} \]

Alternative 8: 74.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := a \cdot \left(t + z \cdot b\right)\\ \mathbf{if}\;a \leq -4.2 \cdot 10^{+219}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq -3.6 \cdot 10^{+202}:\\ \;\;\;\;x + a \cdot t\\ \mathbf{elif}\;a \leq -2.15 \cdot 10^{-54} \lor \neg \left(a \leq 2800000\right):\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;x + z \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* a (+ t (* z b)))))
   (if (<= a -4.2e+219)
     t_1
     (if (<= a -3.6e+202)
       (+ x (* a t))
       (if (or (<= a -2.15e-54) (not (<= a 2800000.0))) t_1 (+ x (* z y)))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = a * (t + (z * b));
	double tmp;
	if (a <= -4.2e+219) {
		tmp = t_1;
	} else if (a <= -3.6e+202) {
		tmp = x + (a * t);
	} else if ((a <= -2.15e-54) || !(a <= 2800000.0)) {
		tmp = t_1;
	} else {
		tmp = x + (z * y);
	}
	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) :: t_1
    real(8) :: tmp
    t_1 = a * (t + (z * b))
    if (a <= (-4.2d+219)) then
        tmp = t_1
    else if (a <= (-3.6d+202)) then
        tmp = x + (a * t)
    else if ((a <= (-2.15d-54)) .or. (.not. (a <= 2800000.0d0))) then
        tmp = t_1
    else
        tmp = x + (z * y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = a * (t + (z * b));
	double tmp;
	if (a <= -4.2e+219) {
		tmp = t_1;
	} else if (a <= -3.6e+202) {
		tmp = x + (a * t);
	} else if ((a <= -2.15e-54) || !(a <= 2800000.0)) {
		tmp = t_1;
	} else {
		tmp = x + (z * y);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = a * (t + (z * b))
	tmp = 0
	if a <= -4.2e+219:
		tmp = t_1
	elif a <= -3.6e+202:
		tmp = x + (a * t)
	elif (a <= -2.15e-54) or not (a <= 2800000.0):
		tmp = t_1
	else:
		tmp = x + (z * y)
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(a * Float64(t + Float64(z * b)))
	tmp = 0.0
	if (a <= -4.2e+219)
		tmp = t_1;
	elseif (a <= -3.6e+202)
		tmp = Float64(x + Float64(a * t));
	elseif ((a <= -2.15e-54) || !(a <= 2800000.0))
		tmp = t_1;
	else
		tmp = Float64(x + Float64(z * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = a * (t + (z * b));
	tmp = 0.0;
	if (a <= -4.2e+219)
		tmp = t_1;
	elseif (a <= -3.6e+202)
		tmp = x + (a * t);
	elseif ((a <= -2.15e-54) || ~((a <= 2800000.0)))
		tmp = t_1;
	else
		tmp = x + (z * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(a * N[(t + N[(z * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -4.2e+219], t$95$1, If[LessEqual[a, -3.6e+202], N[(x + N[(a * t), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[a, -2.15e-54], N[Not[LessEqual[a, 2800000.0]], $MachinePrecision]], t$95$1, N[(x + N[(z * y), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if a < -4.19999999999999976e219 or -3.60000000000000008e202 < a < -2.15e-54 or 2.8e6 < a

   1. Initial program 82.1%

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

      1. associate-+l+ 82.1%

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

      2. +-commutative 82.1%

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

      3. associate-+l+ 82.1%

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

      4. fma-def 85.0%

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

      5. +-commutative 85.0%

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

      6. *-commutative 85.0%

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

      7. associate-*l* 90.5%

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

      8. distribute-lft-out 97.0%

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

      9. fma-def 97.0%

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

      10. +-commutative 97.0%

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

      11. fma-def 97.0%

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

   3. Simplified 97.0%

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

   4. Taylor expanded in a around inf 77.1%

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

   if -4.19999999999999976e219 < a < -3.60000000000000008e202

   1. Initial program 100.0%

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

      1. associate-+l+ 100.0%

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

      2. +-commutative 100.0%

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

      3. associate-+l+ 100.0%

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

      4. fma-def 100.0%

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

      5. +-commutative 100.0%

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

      6. *-commutative 100.0%

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

      7. associate-*l* 100.0%

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

      8. distribute-lft-out 100.0%

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

      9. fma-def 100.0%

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

      10. +-commutative 100.0%

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

      11. fma-def 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around 0 86.7%

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

   5. Taylor expanded in z around 0 86.7%

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

   if -2.15e-54 < a < 2.8e6

   1. Initial program 100.0%

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

      1. associate-+l+ 100.0%

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

      2. +-commutative 100.0%

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

      3. associate-+l+ 100.0%

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

      4. fma-def 100.0%

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

      5. +-commutative 100.0%

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

      6. *-commutative 100.0%

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

      7. associate-*l* 94.7%

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

      8. distribute-lft-out 94.8%

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

      9. fma-def 94.8%

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

      10. +-commutative 94.8%

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

      11. fma-def 94.8%

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

   3. Simplified 94.8%

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

   4. Taylor expanded in a around 0 77.3%

      \[\leadsto \color{blue}{y \cdot z + x} \]
3. Recombined 3 regimes into one program.

4. Final simplification 77.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -4.2 \cdot 10^{+219}:\\ \;\;\;\;a \cdot \left(t + z \cdot b\right)\\ \mathbf{elif}\;a \leq -3.6 \cdot 10^{+202}:\\ \;\;\;\;x + a \cdot t\\ \mathbf{elif}\;a \leq -2.15 \cdot 10^{-54} \lor \neg \left(a \leq 2800000\right):\\ \;\;\;\;a \cdot \left(t + z \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;x + z \cdot y\\ \end{array} \]

Alternative 9: 82.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -9.2 \cdot 10^{-144} \lor \neg \left(a \leq 1.9 \cdot 10^{-125}\right):\\ \;\;\;\;x + a \cdot \left(t + z \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;x + z \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= a -9.2e-144) (not (<= a 1.9e-125)))
   (+ x (* a (+ t (* z b))))
   (+ x (* z y))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((a <= -9.2e-144) || !(a <= 1.9e-125)) {
		tmp = x + (a * (t + (z * b)));
	} else {
		tmp = x + (z * y);
	}
	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 <= (-9.2d-144)) .or. (.not. (a <= 1.9d-125))) then
        tmp = x + (a * (t + (z * b)))
    else
        tmp = x + (z * y)
    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 <= -9.2e-144) || !(a <= 1.9e-125)) {
		tmp = x + (a * (t + (z * b)));
	} else {
		tmp = x + (z * y);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (a <= -9.2e-144) or not (a <= 1.9e-125):
		tmp = x + (a * (t + (z * b)))
	else:
		tmp = x + (z * y)
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((a <= -9.2e-144) || !(a <= 1.9e-125))
		tmp = Float64(x + Float64(a * Float64(t + Float64(z * b))));
	else
		tmp = Float64(x + Float64(z * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((a <= -9.2e-144) || ~((a <= 1.9e-125)))
		tmp = x + (a * (t + (z * b)));
	else
		tmp = x + (z * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[a, -9.2e-144], N[Not[LessEqual[a, 1.9e-125]], $MachinePrecision]], N[(x + N[(a * N[(t + N[(z * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(z * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -9.2e-144 or 1.9000000000000001e-125 < a

   1. Initial program 86.6%

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

      1. associate-+l+ 86.6%

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

      2. +-commutative 86.6%

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

      3. associate-+l+ 86.6%

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

      4. fma-def 88.8%

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

      5. +-commutative 88.8%

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

      6. *-commutative 88.8%

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

      7. associate-*l* 91.4%

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

      8. distribute-lft-out 96.2%

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

      9. fma-def 96.2%

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

      10. +-commutative 96.2%

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

      11. fma-def 96.2%

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

   3. Simplified 96.2%

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

   4. Taylor expanded in y around 0 84.9%

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

   if -9.2e-144 < a < 1.9000000000000001e-125

   1. Initial program 100.0%

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

      1. associate-+l+ 100.0%

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

      2. +-commutative 100.0%

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

      3. associate-+l+ 100.0%

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

      4. fma-def 100.0%

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

      5. +-commutative 100.0%

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

      6. *-commutative 100.0%

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

      7. associate-*l* 95.8%

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

      8. distribute-lft-out 95.9%

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

      9. fma-def 95.9%

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

      10. +-commutative 95.9%

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

      11. fma-def 95.9%

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

   3. Simplified 95.9%

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

   4. Taylor expanded in a around 0 88.6%

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

4. Final simplification 86.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -9.2 \cdot 10^{-144} \lor \neg \left(a \leq 1.9 \cdot 10^{-125}\right):\\ \;\;\;\;x + a \cdot \left(t + z \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;x + z \cdot y\\ \end{array} \]

Alternative 10: 87.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.12 \cdot 10^{-54} \lor \neg \left(a \leq 3350000\right):\\ \;\;\;\;x + a \cdot \left(t + z \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x + a \cdot t\right) + z \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= a -1.12e-54) (not (<= a 3350000.0)))
   (+ x (* a (+ t (* z b))))
   (+ (+ x (* a t)) (* z y))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((a <= -1.12e-54) || !(a <= 3350000.0)) {
		tmp = x + (a * (t + (z * b)));
	} else {
		tmp = (x + (a * t)) + (z * y);
	}
	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 <= (-1.12d-54)) .or. (.not. (a <= 3350000.0d0))) then
        tmp = x + (a * (t + (z * b)))
    else
        tmp = (x + (a * t)) + (z * y)
    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 <= -1.12e-54) || !(a <= 3350000.0)) {
		tmp = x + (a * (t + (z * b)));
	} else {
		tmp = (x + (a * t)) + (z * y);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (a <= -1.12e-54) or not (a <= 3350000.0):
		tmp = x + (a * (t + (z * b)))
	else:
		tmp = (x + (a * t)) + (z * y)
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((a <= -1.12e-54) || !(a <= 3350000.0))
		tmp = Float64(x + Float64(a * Float64(t + Float64(z * b))));
	else
		tmp = Float64(Float64(x + Float64(a * t)) + Float64(z * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((a <= -1.12e-54) || ~((a <= 3350000.0)))
		tmp = x + (a * (t + (z * b)));
	else
		tmp = (x + (a * t)) + (z * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[a, -1.12e-54], N[Not[LessEqual[a, 3350000.0]], $MachinePrecision]], N[(x + N[(a * N[(t + N[(z * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x + N[(a * t), $MachinePrecision]), $MachinePrecision] + N[(z * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -1.11999999999999994e-54 or 3.35e6 < a

   1. Initial program 82.9%

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

      1. associate-+l+ 82.9%

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

      2. +-commutative 82.9%

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

      3. associate-+l+ 82.9%

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

      4. fma-def 85.7%

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

      5. +-commutative 85.7%

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

      6. *-commutative 85.7%

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

      7. associate-*l* 91.0%

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

      8. distribute-lft-out 97.2%

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

      9. fma-def 97.2%

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

      10. +-commutative 97.2%

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

      11. fma-def 97.2%

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

   3. Simplified 97.2%

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

   4. Taylor expanded in y around 0 88.5%

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

   if -1.11999999999999994e-54 < a < 3.35e6

   1. Initial program 100.0%

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

      1. associate-+l+ 100.0%

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

      2. +-commutative 100.0%

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

      3. associate-+l+ 100.0%

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

      4. fma-def 100.0%

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

      5. +-commutative 100.0%

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

      6. *-commutative 100.0%

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

      7. associate-*l* 94.7%

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

      8. distribute-lft-out 94.8%

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

      9. fma-def 94.8%

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

      10. +-commutative 94.8%

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

      11. fma-def 94.8%

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

   3. Simplified 94.8%

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

   4. Taylor expanded in b around 0 90.2%

      \[\leadsto \color{blue}{y \cdot z + \left(a \cdot t + x\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 89.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.12 \cdot 10^{-54} \lor \neg \left(a \leq 3350000\right):\\ \;\;\;\;x + a \cdot \left(t + z \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x + a \cdot t\right) + z \cdot y\\ \end{array} \]

Alternative 11: 39.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -9.5 \cdot 10^{-34}:\\ \;\;\;\;a \cdot t\\ \mathbf{elif}\;t \leq -1.3 \cdot 10^{-278}:\\ \;\;\;\;a \cdot \left(z \cdot b\right)\\ \mathbf{elif}\;t \leq 2.95 \cdot 10^{-161}:\\ \;\;\;\;x\\ \mathbf{elif}\;t \leq 1350:\\ \;\;\;\;z \cdot y\\ \mathbf{else}:\\ \;\;\;\;a \cdot t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= t -9.5e-34)
   (* a t)
   (if (<= t -1.3e-278)
     (* a (* z b))
     (if (<= t 2.95e-161) x (if (<= t 1350.0) (* z y) (* a t))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (t <= -9.5e-34) {
		tmp = a * t;
	} else if (t <= -1.3e-278) {
		tmp = a * (z * b);
	} else if (t <= 2.95e-161) {
		tmp = x;
	} else if (t <= 1350.0) {
		tmp = z * y;
	} else {
		tmp = a * t;
	}
	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 (t <= (-9.5d-34)) then
        tmp = a * t
    else if (t <= (-1.3d-278)) then
        tmp = a * (z * b)
    else if (t <= 2.95d-161) then
        tmp = x
    else if (t <= 1350.0d0) then
        tmp = z * y
    else
        tmp = a * t
    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 (t <= -9.5e-34) {
		tmp = a * t;
	} else if (t <= -1.3e-278) {
		tmp = a * (z * b);
	} else if (t <= 2.95e-161) {
		tmp = x;
	} else if (t <= 1350.0) {
		tmp = z * y;
	} else {
		tmp = a * t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if t <= -9.5e-34:
		tmp = a * t
	elif t <= -1.3e-278:
		tmp = a * (z * b)
	elif t <= 2.95e-161:
		tmp = x
	elif t <= 1350.0:
		tmp = z * y
	else:
		tmp = a * t
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (t <= -9.5e-34)
		tmp = Float64(a * t);
	elseif (t <= -1.3e-278)
		tmp = Float64(a * Float64(z * b));
	elseif (t <= 2.95e-161)
		tmp = x;
	elseif (t <= 1350.0)
		tmp = Float64(z * y);
	else
		tmp = Float64(a * t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (t <= -9.5e-34)
		tmp = a * t;
	elseif (t <= -1.3e-278)
		tmp = a * (z * b);
	elseif (t <= 2.95e-161)
		tmp = x;
	elseif (t <= 1350.0)
		tmp = z * y;
	else
		tmp = a * t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[t, -9.5e-34], N[(a * t), $MachinePrecision], If[LessEqual[t, -1.3e-278], N[(a * N[(z * b), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 2.95e-161], x, If[LessEqual[t, 1350.0], N[(z * y), $MachinePrecision], N[(a * t), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if t < -9.49999999999999985e-34 or 1350 < t

   1. Initial program 86.7%

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

      1. associate-+l+ 86.7%

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

      2. +-commutative 86.7%

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

      3. associate-+l+ 86.7%

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

      4. fma-def 87.5%

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

      5. +-commutative 87.5%

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

      6. *-commutative 87.5%

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

      7. associate-*l* 88.3%

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

      8. distribute-lft-out 95.8%

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

      9. fma-def 95.8%

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

      10. +-commutative 95.8%

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

      11. fma-def 95.8%

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

   3. Simplified 95.8%

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

   4. Taylor expanded in a around inf 63.4%

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

   5. Taylor expanded in z around 0 51.4%

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

      1. *-commutative 51.4%

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

   7. Simplified 51.4%

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

   if -9.49999999999999985e-34 < t < -1.2999999999999999e-278

   1. Initial program 93.5%

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

      1. associate-+l+ 93.5%

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

      2. +-commutative 93.5%

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

      3. associate-+l+ 93.5%

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

      4. fma-def 95.1%

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

      5. +-commutative 95.1%

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

      6. *-commutative 95.1%

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

      7. associate-*l* 95.1%

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

      8. distribute-lft-out 95.1%

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

      9. fma-def 95.1%

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

      10. +-commutative 95.1%

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

      11. fma-def 95.1%

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

   3. Simplified 95.1%

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

   4. Taylor expanded in a around inf 55.3%

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

   5. Taylor expanded in z around inf 52.2%

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

   if -1.2999999999999999e-278 < t < 2.9500000000000001e-161

   1. Initial program 94.4%

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

      1. associate-+l+ 94.4%

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

      2. +-commutative 94.4%

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

      3. associate-+l+ 94.4%

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

      4. fma-def 97.3%

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

      5. +-commutative 97.3%

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

      6. *-commutative 97.3%

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

      7. associate-*l* 94.5%

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

      8. distribute-lft-out 94.5%

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

      9. fma-def 94.5%

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

      10. +-commutative 94.5%

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

      11. fma-def 94.5%

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

   3. Simplified 94.5%

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

   4. Taylor expanded in y around 0 75.4%

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

   5. Taylor expanded in x around inf 52.9%

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

   if 2.9500000000000001e-161 < t < 1350

   1. Initial program 92.7%

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

      1. associate-+l+ 92.7%

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

      2. +-commutative 92.7%

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

      3. associate-+l+ 92.7%

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

      4. fma-def 95.3%

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

      5. +-commutative 95.3%

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

      6. *-commutative 95.3%

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

      7. associate-*l* 100.0%

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

      8. distribute-lft-out 100.0%

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

      9. fma-def 100.0%

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

      10. +-commutative 100.0%

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

      11. fma-def 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around inf 46.9%

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

      1. *-commutative 46.9%

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

   6. Simplified 46.9%

      \[\leadsto \color{blue}{z \cdot y} \]
3. Recombined 4 regimes into one program.

4. Final simplification 51.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -9.5 \cdot 10^{-34}:\\ \;\;\;\;a \cdot t\\ \mathbf{elif}\;t \leq -1.3 \cdot 10^{-278}:\\ \;\;\;\;a \cdot \left(z \cdot b\right)\\ \mathbf{elif}\;t \leq 2.95 \cdot 10^{-161}:\\ \;\;\;\;x\\ \mathbf{elif}\;t \leq 1350:\\ \;\;\;\;z \cdot y\\ \mathbf{else}:\\ \;\;\;\;a \cdot t\\ \end{array} \]

Alternative 12: 38.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -2.9 \cdot 10^{-33}:\\ \;\;\;\;a \cdot t\\ \mathbf{elif}\;t \leq -3.9 \cdot 10^{-281}:\\ \;\;\;\;z \cdot \left(a \cdot b\right)\\ \mathbf{elif}\;t \leq 1.15 \cdot 10^{-160}:\\ \;\;\;\;x\\ \mathbf{elif}\;t \leq 1.25:\\ \;\;\;\;z \cdot y\\ \mathbf{else}:\\ \;\;\;\;a \cdot t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= t -2.9e-33)
   (* a t)
   (if (<= t -3.9e-281)
     (* z (* a b))
     (if (<= t 1.15e-160) x (if (<= t 1.25) (* z y) (* a t))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (t <= -2.9e-33) {
		tmp = a * t;
	} else if (t <= -3.9e-281) {
		tmp = z * (a * b);
	} else if (t <= 1.15e-160) {
		tmp = x;
	} else if (t <= 1.25) {
		tmp = z * y;
	} else {
		tmp = a * t;
	}
	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 (t <= (-2.9d-33)) then
        tmp = a * t
    else if (t <= (-3.9d-281)) then
        tmp = z * (a * b)
    else if (t <= 1.15d-160) then
        tmp = x
    else if (t <= 1.25d0) then
        tmp = z * y
    else
        tmp = a * t
    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 (t <= -2.9e-33) {
		tmp = a * t;
	} else if (t <= -3.9e-281) {
		tmp = z * (a * b);
	} else if (t <= 1.15e-160) {
		tmp = x;
	} else if (t <= 1.25) {
		tmp = z * y;
	} else {
		tmp = a * t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if t <= -2.9e-33:
		tmp = a * t
	elif t <= -3.9e-281:
		tmp = z * (a * b)
	elif t <= 1.15e-160:
		tmp = x
	elif t <= 1.25:
		tmp = z * y
	else:
		tmp = a * t
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (t <= -2.9e-33)
		tmp = Float64(a * t);
	elseif (t <= -3.9e-281)
		tmp = Float64(z * Float64(a * b));
	elseif (t <= 1.15e-160)
		tmp = x;
	elseif (t <= 1.25)
		tmp = Float64(z * y);
	else
		tmp = Float64(a * t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (t <= -2.9e-33)
		tmp = a * t;
	elseif (t <= -3.9e-281)
		tmp = z * (a * b);
	elseif (t <= 1.15e-160)
		tmp = x;
	elseif (t <= 1.25)
		tmp = z * y;
	else
		tmp = a * t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[t, -2.9e-33], N[(a * t), $MachinePrecision], If[LessEqual[t, -3.9e-281], N[(z * N[(a * b), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.15e-160], x, If[LessEqual[t, 1.25], N[(z * y), $MachinePrecision], N[(a * t), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if t < -2.90000000000000003e-33 or 1.25 < t

   1. Initial program 86.7%

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

      1. associate-+l+ 86.7%

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

      2. +-commutative 86.7%

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

      3. associate-+l+ 86.7%

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

      4. fma-def 87.5%

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

      5. +-commutative 87.5%

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

      6. *-commutative 87.5%

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

      7. associate-*l* 88.3%

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

      8. distribute-lft-out 95.8%

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

      9. fma-def 95.8%

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

      10. +-commutative 95.8%

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

      11. fma-def 95.8%

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

   3. Simplified 95.8%

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

   4. Taylor expanded in a around inf 63.4%

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

   5. Taylor expanded in z around 0 51.4%

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

      1. *-commutative 51.4%

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

   7. Simplified 51.4%

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

   if -2.90000000000000003e-33 < t < -3.9000000000000002e-281

   1. Initial program 93.5%

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

      1. associate-+l+ 93.5%

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

      2. +-commutative 93.5%

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

      3. associate-+l+ 93.5%

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

      4. fma-def 95.1%

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

      5. +-commutative 95.1%

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

      6. *-commutative 95.1%

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

      7. associate-*l* 95.1%

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

      8. distribute-lft-out 95.1%

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

      9. fma-def 95.1%

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

      10. +-commutative 95.1%

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

      11. fma-def 95.1%

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

   3. Simplified 95.1%

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

   4. Taylor expanded in z around inf 69.8%

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

   5. Taylor expanded in a around inf 52.2%

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

      1. associate-*r* 52.3%

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

      2. *-commutative 52.3%

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

   7. Simplified 52.3%

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

   if -3.9000000000000002e-281 < t < 1.14999999999999992e-160

   1. Initial program 94.4%

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

      1. associate-+l+ 94.4%

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

      2. +-commutative 94.4%

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

      3. associate-+l+ 94.4%

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

      4. fma-def 97.3%

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

      5. +-commutative 97.3%

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

      6. *-commutative 97.3%

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

      7. associate-*l* 94.5%

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

      8. distribute-lft-out 94.5%

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

      9. fma-def 94.5%

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

      10. +-commutative 94.5%

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

      11. fma-def 94.5%

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

   3. Simplified 94.5%

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

   4. Taylor expanded in y around 0 75.4%

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

   5. Taylor expanded in x around inf 52.9%

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

   if 1.14999999999999992e-160 < t < 1.25

   1. Initial program 92.7%

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

      1. associate-+l+ 92.7%

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

      2. +-commutative 92.7%

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

      3. associate-+l+ 92.7%

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

      4. fma-def 95.3%

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

      5. +-commutative 95.3%

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

      6. *-commutative 95.3%

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

      7. associate-*l* 100.0%

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

      8. distribute-lft-out 100.0%

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

      9. fma-def 100.0%

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

      10. +-commutative 100.0%

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

      11. fma-def 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around inf 46.9%

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

      1. *-commutative 46.9%

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

   6. Simplified 46.9%

      \[\leadsto \color{blue}{z \cdot y} \]
3. Recombined 4 regimes into one program.

4. Final simplification 51.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2.9 \cdot 10^{-33}:\\ \;\;\;\;a \cdot t\\ \mathbf{elif}\;t \leq -3.9 \cdot 10^{-281}:\\ \;\;\;\;z \cdot \left(a \cdot b\right)\\ \mathbf{elif}\;t \leq 1.15 \cdot 10^{-160}:\\ \;\;\;\;x\\ \mathbf{elif}\;t \leq 1.25:\\ \;\;\;\;z \cdot y\\ \mathbf{else}:\\ \;\;\;\;a \cdot t\\ \end{array} \]

Alternative 13: 39.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -3 \cdot 10^{-33}:\\ \;\;\;\;a \cdot t\\ \mathbf{elif}\;t \leq -8.4 \cdot 10^{-283}:\\ \;\;\;\;b \cdot \left(z \cdot a\right)\\ \mathbf{elif}\;t \leq 5.3 \cdot 10^{-160}:\\ \;\;\;\;x\\ \mathbf{elif}\;t \leq 22000:\\ \;\;\;\;z \cdot y\\ \mathbf{else}:\\ \;\;\;\;a \cdot t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= t -3e-33)
   (* a t)
   (if (<= t -8.4e-283)
     (* b (* z a))
     (if (<= t 5.3e-160) x (if (<= t 22000.0) (* z y) (* a t))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (t <= -3e-33) {
		tmp = a * t;
	} else if (t <= -8.4e-283) {
		tmp = b * (z * a);
	} else if (t <= 5.3e-160) {
		tmp = x;
	} else if (t <= 22000.0) {
		tmp = z * y;
	} else {
		tmp = a * t;
	}
	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 (t <= (-3d-33)) then
        tmp = a * t
    else if (t <= (-8.4d-283)) then
        tmp = b * (z * a)
    else if (t <= 5.3d-160) then
        tmp = x
    else if (t <= 22000.0d0) then
        tmp = z * y
    else
        tmp = a * t
    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 (t <= -3e-33) {
		tmp = a * t;
	} else if (t <= -8.4e-283) {
		tmp = b * (z * a);
	} else if (t <= 5.3e-160) {
		tmp = x;
	} else if (t <= 22000.0) {
		tmp = z * y;
	} else {
		tmp = a * t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if t <= -3e-33:
		tmp = a * t
	elif t <= -8.4e-283:
		tmp = b * (z * a)
	elif t <= 5.3e-160:
		tmp = x
	elif t <= 22000.0:
		tmp = z * y
	else:
		tmp = a * t
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (t <= -3e-33)
		tmp = Float64(a * t);
	elseif (t <= -8.4e-283)
		tmp = Float64(b * Float64(z * a));
	elseif (t <= 5.3e-160)
		tmp = x;
	elseif (t <= 22000.0)
		tmp = Float64(z * y);
	else
		tmp = Float64(a * t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (t <= -3e-33)
		tmp = a * t;
	elseif (t <= -8.4e-283)
		tmp = b * (z * a);
	elseif (t <= 5.3e-160)
		tmp = x;
	elseif (t <= 22000.0)
		tmp = z * y;
	else
		tmp = a * t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[t, -3e-33], N[(a * t), $MachinePrecision], If[LessEqual[t, -8.4e-283], N[(b * N[(z * a), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 5.3e-160], x, If[LessEqual[t, 22000.0], N[(z * y), $MachinePrecision], N[(a * t), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if t < -3.0000000000000002e-33 or 22000 < t

   1. Initial program 86.7%

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

      1. associate-+l+ 86.7%

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

      2. +-commutative 86.7%

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

      3. associate-+l+ 86.7%

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

      4. fma-def 87.5%

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

      5. +-commutative 87.5%

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

      6. *-commutative 87.5%

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

      7. associate-*l* 88.3%

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

      8. distribute-lft-out 95.8%

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

      9. fma-def 95.8%

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

      10. +-commutative 95.8%

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

      11. fma-def 95.8%

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

   3. Simplified 95.8%

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

   4. Taylor expanded in a around inf 63.4%

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

   5. Taylor expanded in z around 0 51.4%

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

      1. *-commutative 51.4%

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

   7. Simplified 51.4%

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

   if -3.0000000000000002e-33 < t < -8.39999999999999989e-283

   1. Initial program 93.5%

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

      1. associate-+l+ 93.5%

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

      2. +-commutative 93.5%

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

      3. associate-+l+ 93.5%

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

      4. fma-def 95.1%

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

      5. +-commutative 95.1%

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

      6. *-commutative 95.1%

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

      7. associate-*l* 95.1%

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

      8. distribute-lft-out 95.1%

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

      9. fma-def 95.1%

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

      10. +-commutative 95.1%

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

      11. fma-def 95.1%

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

   3. Simplified 95.1%

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

   4. Taylor expanded in a around inf 55.3%

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

   5. Taylor expanded in z around inf 52.2%

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

      1. associate-*r* 55.4%

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

   7. Simplified 55.4%

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

   if -8.39999999999999989e-283 < t < 5.3000000000000001e-160

   1. Initial program 94.4%

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

      1. associate-+l+ 94.4%

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

      2. +-commutative 94.4%

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

      3. associate-+l+ 94.4%

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

      4. fma-def 97.3%

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

      5. +-commutative 97.3%

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

      6. *-commutative 97.3%

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

      7. associate-*l* 94.5%

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

      8. distribute-lft-out 94.5%

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

      9. fma-def 94.5%

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

      10. +-commutative 94.5%

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

      11. fma-def 94.5%

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

   3. Simplified 94.5%

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

   4. Taylor expanded in y around 0 75.4%

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

   5. Taylor expanded in x around inf 52.9%

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

   if 5.3000000000000001e-160 < t < 22000

   1. Initial program 92.7%

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

      1. associate-+l+ 92.7%

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

      2. +-commutative 92.7%

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

      3. associate-+l+ 92.7%

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

      4. fma-def 95.3%

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

      5. +-commutative 95.3%

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

      6. *-commutative 95.3%

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

      7. associate-*l* 100.0%

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

      8. distribute-lft-out 100.0%

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

      9. fma-def 100.0%

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

      10. +-commutative 100.0%

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

      11. fma-def 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around inf 46.9%

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

      1. *-commutative 46.9%

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

   6. Simplified 46.9%

      \[\leadsto \color{blue}{z \cdot y} \]
3. Recombined 4 regimes into one program.

4. Final simplification 51.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -3 \cdot 10^{-33}:\\ \;\;\;\;a \cdot t\\ \mathbf{elif}\;t \leq -8.4 \cdot 10^{-283}:\\ \;\;\;\;b \cdot \left(z \cdot a\right)\\ \mathbf{elif}\;t \leq 5.3 \cdot 10^{-160}:\\ \;\;\;\;x\\ \mathbf{elif}\;t \leq 22000:\\ \;\;\;\;z \cdot y\\ \mathbf{else}:\\ \;\;\;\;a \cdot t\\ \end{array} \]

Alternative 14: 74.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -4.1 \cdot 10^{-63} \lor \neg \left(z \leq 3.2 \cdot 10^{-38}\right):\\ \;\;\;\;z \cdot \left(y + a \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;x + a \cdot t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= z -4.1e-63) (not (<= z 3.2e-38)))
   (* z (+ y (* a b)))
   (+ x (* a t))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((z <= -4.1e-63) || !(z <= 3.2e-38)) {
		tmp = z * (y + (a * b));
	} else {
		tmp = x + (a * t);
	}
	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 ((z <= (-4.1d-63)) .or. (.not. (z <= 3.2d-38))) then
        tmp = z * (y + (a * b))
    else
        tmp = x + (a * t)
    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 ((z <= -4.1e-63) || !(z <= 3.2e-38)) {
		tmp = z * (y + (a * b));
	} else {
		tmp = x + (a * t);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (z <= -4.1e-63) or not (z <= 3.2e-38):
		tmp = z * (y + (a * b))
	else:
		tmp = x + (a * t)
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((z <= -4.1e-63) || !(z <= 3.2e-38))
		tmp = Float64(z * Float64(y + Float64(a * b)));
	else
		tmp = Float64(x + Float64(a * t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((z <= -4.1e-63) || ~((z <= 3.2e-38)))
		tmp = z * (y + (a * b));
	else
		tmp = x + (a * t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[z, -4.1e-63], N[Not[LessEqual[z, 3.2e-38]], $MachinePrecision]], N[(z * N[(y + N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(a * t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -4.0999999999999998e-63 or 3.19999999999999977e-38 < z

   1. Initial program 83.5%

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

      1. associate-+l+ 83.5%

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

      2. +-commutative 83.5%

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

      3. associate-+l+ 83.5%

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

      4. fma-def 86.4%

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

      5. +-commutative 86.4%

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

      6. *-commutative 86.4%

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

      7. associate-*l* 87.8%

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

      8. distribute-lft-out 92.9%

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

      9. fma-def 92.9%

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

      10. +-commutative 92.9%

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

      11. fma-def 92.9%

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

   3. Simplified 92.9%

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

   4. Taylor expanded in z around inf 74.6%

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

   if -4.0999999999999998e-63 < z < 3.19999999999999977e-38

   1. Initial program 98.3%

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

      1. associate-+l+ 98.3%

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

      2. +-commutative 98.3%

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

      3. associate-+l+ 98.3%

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

      4. fma-def 98.3%

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

      5. +-commutative 98.3%

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

      6. *-commutative 98.3%

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

      7. associate-*l* 98.2%

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

      8. distribute-lft-out 99.9%

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

      9. fma-def 99.9%

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

      10. +-commutative 99.9%

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

      11. fma-def 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in y around 0 89.4%

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

   5. Taylor expanded in z around 0 78.6%

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

4. Final simplification 76.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4.1 \cdot 10^{-63} \lor \neg \left(z \leq 3.2 \cdot 10^{-38}\right):\\ \;\;\;\;z \cdot \left(y + a \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;x + a \cdot t\\ \end{array} \]

Alternative 15: 39.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.05 \cdot 10^{-58}:\\ \;\;\;\;a \cdot t\\ \mathbf{elif}\;t \leq 3 \cdot 10^{-162}:\\ \;\;\;\;x\\ \mathbf{elif}\;t \leq 2.15:\\ \;\;\;\;z \cdot y\\ \mathbf{else}:\\ \;\;\;\;a \cdot t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= t -1.05e-58)
   (* a t)
   (if (<= t 3e-162) x (if (<= t 2.15) (* z y) (* a t)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (t <= -1.05e-58) {
		tmp = a * t;
	} else if (t <= 3e-162) {
		tmp = x;
	} else if (t <= 2.15) {
		tmp = z * y;
	} else {
		tmp = a * t;
	}
	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 (t <= (-1.05d-58)) then
        tmp = a * t
    else if (t <= 3d-162) then
        tmp = x
    else if (t <= 2.15d0) then
        tmp = z * y
    else
        tmp = a * t
    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 (t <= -1.05e-58) {
		tmp = a * t;
	} else if (t <= 3e-162) {
		tmp = x;
	} else if (t <= 2.15) {
		tmp = z * y;
	} else {
		tmp = a * t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if t <= -1.05e-58:
		tmp = a * t
	elif t <= 3e-162:
		tmp = x
	elif t <= 2.15:
		tmp = z * y
	else:
		tmp = a * t
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (t <= -1.05e-58)
		tmp = Float64(a * t);
	elseif (t <= 3e-162)
		tmp = x;
	elseif (t <= 2.15)
		tmp = Float64(z * y);
	else
		tmp = Float64(a * t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (t <= -1.05e-58)
		tmp = a * t;
	elseif (t <= 3e-162)
		tmp = x;
	elseif (t <= 2.15)
		tmp = z * y;
	else
		tmp = a * t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[t, -1.05e-58], N[(a * t), $MachinePrecision], If[LessEqual[t, 3e-162], x, If[LessEqual[t, 2.15], N[(z * y), $MachinePrecision], N[(a * t), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if t < -1.04999999999999994e-58 or 2.14999999999999991 < t

   1. Initial program 87.4%

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

      1. associate-+l+ 87.4%

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

      2. +-commutative 87.4%

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

      3. associate-+l+ 87.4%

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

      4. fma-def 88.2%

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

      5. +-commutative 88.2%

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

      6. *-commutative 88.2%

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

      7. associate-*l* 88.9%

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

      8. distribute-lft-out 96.0%

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

      9. fma-def 96.1%

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

      10. +-commutative 96.1%

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

      11. fma-def 96.1%

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

   3. Simplified 96.1%

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

   4. Taylor expanded in a around inf 64.7%

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

   5. Taylor expanded in z around 0 49.4%

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

      1. *-commutative 49.4%

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

   7. Simplified 49.4%

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

   if -1.04999999999999994e-58 < t < 2.99999999999999999e-162

   1. Initial program 93.4%

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

      1. associate-+l+ 93.4%

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

      2. +-commutative 93.4%

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

      3. associate-+l+ 93.4%

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

      4. fma-def 95.6%

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

      5. +-commutative 95.6%

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

      6. *-commutative 95.6%

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

      7. associate-*l* 94.5%

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

      8. distribute-lft-out 94.5%

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

      9. fma-def 94.5%

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

      10. +-commutative 94.5%

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

      11. fma-def 94.5%

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

   3. Simplified 94.5%

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

   4. Taylor expanded in y around 0 77.2%

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

   5. Taylor expanded in x around inf 38.4%

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

   if 2.99999999999999999e-162 < t < 2.14999999999999991

   1. Initial program 92.7%

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

      1. associate-+l+ 92.7%

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

      2. +-commutative 92.7%

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

      3. associate-+l+ 92.7%

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

      4. fma-def 95.3%

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

      5. +-commutative 95.3%

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

      6. *-commutative 95.3%

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

      7. associate-*l* 100.0%

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

      8. distribute-lft-out 100.0%

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

      9. fma-def 100.0%

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

      10. +-commutative 100.0%

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

      11. fma-def 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around inf 46.9%

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

      1. *-commutative 46.9%

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

   6. Simplified 46.9%

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

4. Final simplification 45.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.05 \cdot 10^{-58}:\\ \;\;\;\;a \cdot t\\ \mathbf{elif}\;t \leq 3 \cdot 10^{-162}:\\ \;\;\;\;x\\ \mathbf{elif}\;t \leq 2.15:\\ \;\;\;\;z \cdot y\\ \mathbf{else}:\\ \;\;\;\;a \cdot t\\ \end{array} \]

Alternative 16: 38.9% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -8.5 \cdot 10^{+92}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 2.35 \cdot 10^{-42}:\\ \;\;\;\;a \cdot t\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= x -8.5e+92) x (if (<= x 2.35e-42) (* a t) x)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (x <= -8.5e+92) {
		tmp = x;
	} else if (x <= 2.35e-42) {
		tmp = a * t;
	} else {
		tmp = 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 (x <= (-8.5d+92)) then
        tmp = x
    else if (x <= 2.35d-42) then
        tmp = a * t
    else
        tmp = 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 (x <= -8.5e+92) {
		tmp = x;
	} else if (x <= 2.35e-42) {
		tmp = a * t;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if x <= -8.5e+92:
		tmp = x
	elif x <= 2.35e-42:
		tmp = a * t
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (x <= -8.5e+92)
		tmp = x;
	elseif (x <= 2.35e-42)
		tmp = Float64(a * t);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (x <= -8.5e+92)
		tmp = x;
	elseif (x <= 2.35e-42)
		tmp = a * t;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[x, -8.5e+92], x, If[LessEqual[x, 2.35e-42], N[(a * t), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if x < -8.5000000000000001e92 or 2.35e-42 < x

   1. Initial program 89.9%

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

      1. associate-+l+ 89.9%

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

      2. +-commutative 89.9%

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

      3. associate-+l+ 89.9%

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

      4. fma-def 92.5%

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

      5. +-commutative 92.5%

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

      6. *-commutative 92.5%

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

      7. associate-*l* 92.5%

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

      8. distribute-lft-out 95.9%

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

      9. fma-def 95.9%

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

      10. +-commutative 95.9%

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

      11. fma-def 95.9%

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

   3. Simplified 95.9%

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

   4. Taylor expanded in y around 0 84.9%

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

   5. Taylor expanded in x around inf 47.6%

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

   if -8.5000000000000001e92 < x < 2.35e-42

   1. Initial program 90.7%

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

      1. associate-+l+ 90.7%

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

      2. +-commutative 90.7%

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

      3. associate-+l+ 90.7%

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

      4. fma-def 91.4%

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

      5. +-commutative 91.4%

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

      6. *-commutative 91.4%

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

      7. associate-*l* 92.7%

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

      8. distribute-lft-out 96.3%

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

      9. fma-def 96.3%

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

      10. +-commutative 96.3%

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

      11. fma-def 96.3%

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

   3. Simplified 96.3%

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

   4. Taylor expanded in a around inf 62.5%

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

   5. Taylor expanded in z around 0 37.1%

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

      1. *-commutative 37.1%

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

   7. Simplified 37.1%

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

4. Final simplification 42.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -8.5 \cdot 10^{+92}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 2.35 \cdot 10^{-42}:\\ \;\;\;\;a \cdot t\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 17: 26.7% accurate, 15.0× speedup

Math

\[\begin{array}{l} \\ x \end{array} \]

FPCore

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

C

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

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
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_, a_, b_] := x

Derivation

1. Initial program 90.3%

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

   1. associate-+l+ 90.3%

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

   2. +-commutative 90.3%

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

   3. associate-+l+ 90.3%

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

   4. fma-def 91.9%

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

   5. +-commutative 91.9%

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

   6. *-commutative 91.9%

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

   7. associate-*l* 92.6%

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

   8. distribute-lft-out 96.1%

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

   9. fma-def 96.1%

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

   10. +-commutative 96.1%

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

   11. fma-def 96.1%

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

3. Simplified 96.1%

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

4. Taylor expanded in y around 0 76.1%

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

5. Taylor expanded in x around inf 26.4%

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

6. Final simplification 26.4%

   \[\leadsto x \]

Developer target: 97.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := z \cdot \left(b \cdot a + y\right) + \left(x + t \cdot a\right)\\ \mathbf{if}\;z < -11820553527347888000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z < 4.7589743188364287 \cdot 10^{-122}:\\ \;\;\;\;\left(b \cdot z + t\right) \cdot a + \left(z \cdot y + x\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ (* z (+ (* b a) y)) (+ x (* t a)))))
   (if (< z -11820553527347888000.0)
     t_1
     (if (< z 4.7589743188364287e-122)
       (+ (* (+ (* b z) t) a) (+ (* z y) x))
       t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (z * ((b * a) + y)) + (x + (t * a));
	double tmp;
	if (z < -11820553527347888000.0) {
		tmp = t_1;
	} else if (z < 4.7589743188364287e-122) {
		tmp = (((b * z) + t) * a) + ((z * y) + x);
	} else {
		tmp = t_1;
	}
	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) :: t_1
    real(8) :: tmp
    t_1 = (z * ((b * a) + y)) + (x + (t * a))
    if (z < (-11820553527347888000.0d0)) then
        tmp = t_1
    else if (z < 4.7589743188364287d-122) then
        tmp = (((b * z) + t) * a) + ((z * y) + x)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (z * ((b * a) + y)) + (x + (t * a));
	double tmp;
	if (z < -11820553527347888000.0) {
		tmp = t_1;
	} else if (z < 4.7589743188364287e-122) {
		tmp = (((b * z) + t) * a) + ((z * y) + x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = (z * ((b * a) + y)) + (x + (t * a))
	tmp = 0
	if z < -11820553527347888000.0:
		tmp = t_1
	elif z < 4.7589743188364287e-122:
		tmp = (((b * z) + t) * a) + ((z * y) + x)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(z * Float64(Float64(b * a) + y)) + Float64(x + Float64(t * a)))
	tmp = 0.0
	if (z < -11820553527347888000.0)
		tmp = t_1;
	elseif (z < 4.7589743188364287e-122)
		tmp = Float64(Float64(Float64(Float64(b * z) + t) * a) + Float64(Float64(z * y) + x));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = (z * ((b * a) + y)) + (x + (t * a));
	tmp = 0.0;
	if (z < -11820553527347888000.0)
		tmp = t_1;
	elseif (z < 4.7589743188364287e-122)
		tmp = (((b * z) + t) * a) + ((z * y) + x);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(z * N[(N[(b * a), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision] + N[(x + N[(t * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Less[z, -11820553527347888000.0], t$95$1, If[Less[z, 4.7589743188364287e-122], N[(N[(N[(N[(b * z), $MachinePrecision] + t), $MachinePrecision] * a), $MachinePrecision] + N[(N[(z * y), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Reproduce

herbie shell --seed 2023200 
(FPCore (x y z t a b)
  :name "Graphics.Rasterific.CubicBezier:cachedBezierAt from Rasterific-0.6.1"
  :precision binary64

  :herbie-target
  (if (< z -11820553527347888000.0) (+ (* z (+ (* b a) y)) (+ x (* t a))) (if (< z 4.7589743188364287e-122) (+ (* (+ (* b z) t) a) (+ (* z y) x)) (+ (* z (+ (* b a) y)) (+ x (* t a)))))

  (+ (+ (+ x (* y z)) (* t a)) (* (* a z) b)))