Graphics.Rasterific.CubicBezier:cachedBezierAt from Rasterific-0.6.1

Percentage Accurate: 92.5% → 98.3%

Time: 8.2s

Alternatives: 11

Speedup: 0.7×

• 31.3% 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: 92.5% 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: 98.3% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 3 regimes

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

   1. Initial program 80.5%

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

   3. Taylor expanded in x around 0

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

      1. associate-+r+ N/A

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

      2. distribute-lft-in N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 95.2

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

   5. Applied rewrites 95.2%

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

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

   1. Initial program 98.5%

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

   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. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{y \cdot \left(z + \left(\frac{x}{y} + \left(\frac{a \cdot t}{y} + \frac{a \cdot \left(b \cdot z\right)}{y}\right)\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

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

      2. associate-+r+ N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. associate-*l/ N/A

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

      6. associate-/l* N/A

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

      7. *-inverses N/A

         \[\leadsto \left(\left(\frac{a \cdot t}{y} + \frac{a \cdot \left(b \cdot z\right)}{y}\right) + z\right) \cdot y + x \cdot \color{blue}{1} \]

      8. *-rgt-identity N/A

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

      9. lower-fma.f64 N/A

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

   5. Applied rewrites 100.0%

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

4. Final simplification 98.1%

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

Alternative 2: 93.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\mathsf{fma}\left(\frac{a}{y}, \mathsf{fma}\left(b, z, t\right), z\right), y, x\right)\\ \mathbf{if}\;y \leq -3.5 \cdot 10^{-121}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 5.4 \cdot 10^{-55}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, z, t\right), a, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (fma (fma (/ a y) (fma b z t) z) y x)))
   (if (<= y -3.5e-121) t_1 (if (<= y 5.4e-55) (fma (fma b z t) a x) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = fma(fma((a / y), fma(b, z, t), z), y, x);
	double tmp;
	if (y <= -3.5e-121) {
		tmp = t_1;
	} else if (y <= 5.4e-55) {
		tmp = fma(fma(b, z, t), a, x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = fma(fma(Float64(a / y), fma(b, z, t), z), y, x)
	tmp = 0.0
	if (y <= -3.5e-121)
		tmp = t_1;
	elseif (y <= 5.4e-55)
		tmp = fma(fma(b, z, t), a, x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(N[(a / y), $MachinePrecision] * N[(b * z + t), $MachinePrecision] + z), $MachinePrecision] * y + x), $MachinePrecision]}, If[LessEqual[y, -3.5e-121], t$95$1, If[LessEqual[y, 5.4e-55], N[(N[(b * z + t), $MachinePrecision] * a + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -3.49999999999999993e-121 or 5.40000000000000008e-55 < y

   1. Initial program 90.7%

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

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{y \cdot \left(z + \left(\frac{x}{y} + \left(\frac{a \cdot t}{y} + \frac{a \cdot \left(b \cdot z\right)}{y}\right)\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

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

      2. associate-+r+ N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. associate-*l/ N/A

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

      6. associate-/l* N/A

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

      7. *-inverses N/A

         \[\leadsto \left(\left(\frac{a \cdot t}{y} + \frac{a \cdot \left(b \cdot z\right)}{y}\right) + z\right) \cdot y + x \cdot \color{blue}{1} \]

      8. *-rgt-identity N/A

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

      9. lower-fma.f64 N/A

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

   5. Applied rewrites 95.4%

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

   if -3.49999999999999993e-121 < y < 5.40000000000000008e-55

   1. Initial program 91.2%

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

   3. Taylor expanded in y around 0

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

      1. distribute-lft-in N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 92.7

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

   5. Applied rewrites 92.7%

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

Alternative 3: 63.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(a \cdot z, b, x\right)\\ \mathbf{if}\;b \leq -1.45 \cdot 10^{+68}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;b \leq -4.1 \cdot 10^{-112}:\\ \;\;\;\;\mathsf{fma}\left(a, t, x\right)\\ \mathbf{elif}\;b \leq 7.2 \cdot 10^{+24}:\\ \;\;\;\;\mathsf{fma}\left(z, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (fma (* a z) b x)))
   (if (<= b -1.45e+68)
     t_1
     (if (<= b -4.1e-112) (fma a t x) (if (<= b 7.2e+24) (fma z y x) t_1)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = fma((a * z), b, x);
	double tmp;
	if (b <= -1.45e+68) {
		tmp = t_1;
	} else if (b <= -4.1e-112) {
		tmp = fma(a, t, x);
	} else if (b <= 7.2e+24) {
		tmp = fma(z, y, x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = fma(Float64(a * z), b, x)
	tmp = 0.0
	if (b <= -1.45e+68)
		tmp = t_1;
	elseif (b <= -4.1e-112)
		tmp = fma(a, t, x);
	elseif (b <= 7.2e+24)
		tmp = fma(z, y, x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(a * z), $MachinePrecision] * b + x), $MachinePrecision]}, If[LessEqual[b, -1.45e+68], t$95$1, If[LessEqual[b, -4.1e-112], N[(a * t + x), $MachinePrecision], If[LessEqual[b, 7.2e+24], N[(z * y + x), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.45000000000000006e68 or 7.19999999999999966e24 < b

   1. Initial program 90.2%

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

   3. Taylor expanded in y around 0

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

      1. distribute-lft-in N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 84.6

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

   5. Applied rewrites 84.6%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 68.2%

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

   10. Applied rewrites 73.5%

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

   if -1.45000000000000006e68 < b < -4.09999999999999996e-112

   1. Initial program 97.0%

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

   3. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. +-commutative N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt-in N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 64.6

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

   5. Applied rewrites 64.6%

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

   6. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 86.1

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

   8. Applied rewrites 86.1%

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

   if -4.09999999999999996e-112 < b < 7.19999999999999966e24

   1. Initial program 89.9%

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

   3. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. +-commutative N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt-in N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 71.7

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

   5. Applied rewrites 71.7%

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

   6. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 64.0

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

   8. Applied rewrites 64.0%

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

Alternative 4: 63.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -4.1 \cdot 10^{+79}:\\ \;\;\;\;\mathsf{fma}\left(a, t, z \cdot y\right)\\ \mathbf{elif}\;t \leq -2.1 \cdot 10^{-67}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot a, z, x\right)\\ \mathbf{elif}\;t \leq 2.8 \cdot 10^{+53}:\\ \;\;\;\;\mathsf{fma}\left(z, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, t, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= t -4.1e+79)
   (fma a t (* z y))
   (if (<= t -2.1e-67)
     (fma (* b a) z x)
     (if (<= t 2.8e+53) (fma z y x) (fma a t x)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (t <= -4.1e+79) {
		tmp = fma(a, t, (z * y));
	} else if (t <= -2.1e-67) {
		tmp = fma((b * a), z, x);
	} else if (t <= 2.8e+53) {
		tmp = fma(z, y, x);
	} else {
		tmp = fma(a, t, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (t <= -4.1e+79)
		tmp = fma(a, t, Float64(z * y));
	elseif (t <= -2.1e-67)
		tmp = fma(Float64(b * a), z, x);
	elseif (t <= 2.8e+53)
		tmp = fma(z, y, x);
	else
		tmp = fma(a, t, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[t, -4.1e+79], N[(a * t + N[(z * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, -2.1e-67], N[(N[(b * a), $MachinePrecision] * z + x), $MachinePrecision], If[LessEqual[t, 2.8e+53], N[(z * y + x), $MachinePrecision], N[(a * t + x), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if t < -4.1e79

   1. Initial program 92.1%

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

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{y \cdot \left(z + \left(\frac{x}{y} + \left(\frac{a \cdot t}{y} + \frac{a \cdot \left(b \cdot z\right)}{y}\right)\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

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

      2. associate-+r+ N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. associate-*l/ N/A

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

      6. associate-/l* N/A

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

      7. *-inverses N/A

         \[\leadsto \left(\left(\frac{a \cdot t}{y} + \frac{a \cdot \left(b \cdot z\right)}{y}\right) + z\right) \cdot y + x \cdot \color{blue}{1} \]

      8. *-rgt-identity N/A

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

      9. lower-fma.f64 N/A

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

   5. Applied rewrites 88.3%

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

   6. Taylor expanded in x around 0

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

      1. associate-+r+ N/A

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

      2. distribute-lft-in N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 87.3

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

   8. Applied rewrites 87.3%

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

   9. Taylor expanded in b around 0

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

   11. Applied rewrites 81.6%

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

   if -4.1e79 < t < -2.1000000000000002e-67

   1. Initial program 81.6%

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

   3. Taylor expanded in y around 0

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

      1. distribute-lft-in N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 78.1

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

   5. Applied rewrites 78.1%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 67.3%

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

   if -2.1000000000000002e-67 < t < 2.8e53

   1. Initial program 93.6%

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

   3. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. +-commutative N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt-in N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 90.5

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

   5. Applied rewrites 90.5%

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

   6. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 68.3

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

   8. Applied rewrites 68.3%

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

   if 2.8e53 < t

   1. Initial program 89.7%

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

   3. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. +-commutative N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt-in N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 52.3

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

   5. Applied rewrites 52.3%

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

   6. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 76.1

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

   8. Applied rewrites 76.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a, t, x\right)} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 5: 86.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\mathsf{fma}\left(b, z, t\right), a, x\right)\\ \mathbf{if}\;a \leq -0.0085:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 3.9 \cdot 10^{-82}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, a, y\right), z, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (fma (fma b z t) a x)))
   (if (<= a -0.0085) t_1 (if (<= a 3.9e-82) (fma (fma b a y) z x) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = fma(fma(b, z, t), a, x);
	double tmp;
	if (a <= -0.0085) {
		tmp = t_1;
	} else if (a <= 3.9e-82) {
		tmp = fma(fma(b, a, y), z, x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = fma(fma(b, z, t), a, x)
	tmp = 0.0
	if (a <= -0.0085)
		tmp = t_1;
	elseif (a <= 3.9e-82)
		tmp = fma(fma(b, a, y), z, x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(b * z + t), $MachinePrecision] * a + x), $MachinePrecision]}, If[LessEqual[a, -0.0085], t$95$1, If[LessEqual[a, 3.9e-82], N[(N[(b * a + y), $MachinePrecision] * z + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if a < -0.0085000000000000006 or 3.89999999999999973e-82 < a

   1. Initial program 83.4%

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

   3. Taylor expanded in y around 0

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

      1. distribute-lft-in N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 90.4

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

   5. Applied rewrites 90.4%

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

   if -0.0085000000000000006 < a < 3.89999999999999973e-82

   1. Initial program 99.9%

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

   3. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. +-commutative N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt-in N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 89.9

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

   5. Applied rewrites 89.9%

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

Alternative 6: 79.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -6 \cdot 10^{+124}:\\ \;\;\;\;\mathsf{fma}\left(a, t, z \cdot y\right)\\ \mathbf{elif}\;t \leq 2.55 \cdot 10^{+57}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, a, y\right), z, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, t, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= t -6e+124)
   (fma a t (* z y))
   (if (<= t 2.55e+57) (fma (fma b a y) z x) (fma a t x))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (t <= -6e+124) {
		tmp = fma(a, t, (z * y));
	} else if (t <= 2.55e+57) {
		tmp = fma(fma(b, a, y), z, x);
	} else {
		tmp = fma(a, t, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (t <= -6e+124)
		tmp = fma(a, t, Float64(z * y));
	elseif (t <= 2.55e+57)
		tmp = fma(fma(b, a, y), z, x);
	else
		tmp = fma(a, t, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[t, -6e+124], N[(a * t + N[(z * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 2.55e+57], N[(N[(b * a + y), $MachinePrecision] * z + x), $MachinePrecision], N[(a * t + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -5.9999999999999999e124

   1. Initial program 92.5%

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

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{y \cdot \left(z + \left(\frac{x}{y} + \left(\frac{a \cdot t}{y} + \frac{a \cdot \left(b \cdot z\right)}{y}\right)\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

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

      2. associate-+r+ N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. associate-*l/ N/A

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

      6. associate-/l* N/A

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

      7. *-inverses N/A

         \[\leadsto \left(\left(\frac{a \cdot t}{y} + \frac{a \cdot \left(b \cdot z\right)}{y}\right) + z\right) \cdot y + x \cdot \color{blue}{1} \]

      8. *-rgt-identity N/A

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

      9. lower-fma.f64 N/A

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

   5. Applied rewrites 90.2%

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

   6. Taylor expanded in x around 0

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

      1. associate-+r+ N/A

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

      2. distribute-lft-in N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 86.4

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

   8. Applied rewrites 86.4%

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

   9. Taylor expanded in b around 0

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

   11. Applied rewrites 86.4%

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

   if -5.9999999999999999e124 < t < 2.55000000000000011e57

   1. Initial program 90.8%

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

   3. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. +-commutative N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt-in N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 88.3

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

   5. Applied rewrites 88.3%

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

   if 2.55000000000000011e57 < t

   1. Initial program 89.7%

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

   3. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. +-commutative N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt-in N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 52.3

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

   5. Applied rewrites 52.3%

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

   6. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 76.1

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

   8. Applied rewrites 76.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a, t, x\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 7: 74.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(b, a, y\right) \cdot z\\ \mathbf{if}\;z \leq -5.4 \cdot 10^{-29}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 195000:\\ \;\;\;\;\mathsf{fma}\left(a, t, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (fma b a y) z)))
   (if (<= z -5.4e-29) t_1 (if (<= z 195000.0) (fma a t x) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = fma(b, a, y) * z;
	double tmp;
	if (z <= -5.4e-29) {
		tmp = t_1;
	} else if (z <= 195000.0) {
		tmp = fma(a, t, x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(fma(b, a, y) * z)
	tmp = 0.0
	if (z <= -5.4e-29)
		tmp = t_1;
	elseif (z <= 195000.0)
		tmp = fma(a, t, x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(b * a + y), $MachinePrecision] * z), $MachinePrecision]}, If[LessEqual[z, -5.4e-29], t$95$1, If[LessEqual[z, 195000.0], N[(a * t + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -5.40000000000000045e-29 or 195000 < z

   1. Initial program 82.0%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 77.1

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

   5. Applied rewrites 77.1%

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

   if -5.40000000000000045e-29 < z < 195000

   1. Initial program 99.2%

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

   3. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. +-commutative N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt-in N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 60.5

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

   5. Applied rewrites 60.5%

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

   6. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 77.2

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

   8. Applied rewrites 77.2%

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

Alternative 8: 63.7% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -0.000135:\\ \;\;\;\;\mathsf{fma}\left(a, t, x\right)\\ \mathbf{elif}\;a \leq 3.9 \cdot 10^{-82}:\\ \;\;\;\;\mathsf{fma}\left(z, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, t, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= a -0.000135) (fma a t x) (if (<= a 3.9e-82) (fma z y x) (fma a t x))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (a <= -0.000135) {
		tmp = fma(a, t, x);
	} else if (a <= 3.9e-82) {
		tmp = fma(z, y, x);
	} else {
		tmp = fma(a, t, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (a <= -0.000135)
		tmp = fma(a, t, x);
	elseif (a <= 3.9e-82)
		tmp = fma(z, y, x);
	else
		tmp = fma(a, t, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[a, -0.000135], N[(a * t + x), $MachinePrecision], If[LessEqual[a, 3.9e-82], N[(z * y + x), $MachinePrecision], N[(a * t + x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if a < -1.35000000000000002e-4 or 3.89999999999999973e-82 < a

   1. Initial program 83.6%

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

   3. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. +-commutative N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt-in N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 61.4

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

   5. Applied rewrites 61.4%

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

   6. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 59.5

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

   8. Applied rewrites 59.5%

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

   if -1.35000000000000002e-4 < a < 3.89999999999999973e-82

   1. Initial program 99.9%

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

   3. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. +-commutative N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt-in N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 89.8

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

   5. Applied rewrites 89.8%

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

   6. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 79.0

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

   8. Applied rewrites 79.0%

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

Alternative 9: 58.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.4 \cdot 10^{+227}:\\ \;\;\;\;z \cdot y\\ \mathbf{elif}\;z \leq 1.55 \cdot 10^{+56}:\\ \;\;\;\;\mathsf{fma}\left(a, t, x\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= z -1.4e+227) (* z y) (if (<= z 1.55e+56) (fma a t x) (* z y))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (z <= -1.4e+227) {
		tmp = z * y;
	} else if (z <= 1.55e+56) {
		tmp = fma(a, t, x);
	} else {
		tmp = z * y;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (z <= -1.4e+227)
		tmp = Float64(z * y);
	elseif (z <= 1.55e+56)
		tmp = fma(a, t, x);
	else
		tmp = Float64(z * y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[z, -1.4e+227], N[(z * y), $MachinePrecision], If[LessEqual[z, 1.55e+56], N[(a * t + x), $MachinePrecision], N[(z * y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.39999999999999992e227 or 1.55000000000000002e56 < z

   1. Initial program 75.1%

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

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{y \cdot \left(z + \left(\frac{x}{y} + \left(\frac{a \cdot t}{y} + \frac{a \cdot \left(b \cdot z\right)}{y}\right)\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

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

      2. associate-+r+ N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. associate-*l/ N/A

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

      6. associate-/l* N/A

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

      7. *-inverses N/A

         \[\leadsto \left(\left(\frac{a \cdot t}{y} + \frac{a \cdot \left(b \cdot z\right)}{y}\right) + z\right) \cdot y + x \cdot \color{blue}{1} \]

      8. *-rgt-identity N/A

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

      9. lower-fma.f64 N/A

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

   5. Applied rewrites 85.8%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 56.5%

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

   if -1.39999999999999992e227 < z < 1.55000000000000002e56

   1. Initial program 96.3%

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

   3. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. +-commutative N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt-in N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 67.9

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

   5. Applied rewrites 67.9%

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

   6. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 67.1

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

   8. Applied rewrites 67.1%

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

Alternative 10: 39.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -2 \cdot 10^{-11}:\\ \;\;\;\;a \cdot t\\ \mathbf{elif}\;a \leq 3.9 \cdot 10^{-82}:\\ \;\;\;\;z \cdot y\\ \mathbf{else}:\\ \;\;\;\;a \cdot t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= a -2e-11) (* a t) (if (<= a 3.9e-82) (* z y) (* a t))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (a <= -2e-11) {
		tmp = a * t;
	} else if (a <= 3.9e-82) {
		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 (a <= (-2d-11)) then
        tmp = a * t
    else if (a <= 3.9d-82) 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 (a <= -2e-11) {
		tmp = a * t;
	} else if (a <= 3.9e-82) {
		tmp = z * y;
	} else {
		tmp = a * t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if a <= -2e-11:
		tmp = a * t
	elif a <= 3.9e-82:
		tmp = z * y
	else:
		tmp = a * t
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (a <= -2e-11)
		tmp = Float64(a * t);
	elseif (a <= 3.9e-82)
		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 (a <= -2e-11)
		tmp = a * t;
	elseif (a <= 3.9e-82)
		tmp = z * y;
	else
		tmp = a * t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[a, -2e-11], N[(a * t), $MachinePrecision], If[LessEqual[a, 3.9e-82], N[(z * y), $MachinePrecision], N[(a * t), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if a < -1.99999999999999988e-11 or 3.89999999999999973e-82 < a

   1. Initial program 83.7%

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

   3. Taylor expanded in t around inf

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

      1. lower-*.f64 43.1

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

   5. Applied rewrites 43.1%

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

   if -1.99999999999999988e-11 < a < 3.89999999999999973e-82

   1. Initial program 99.9%

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

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{y \cdot \left(z + \left(\frac{x}{y} + \left(\frac{a \cdot t}{y} + \frac{a \cdot \left(b \cdot z\right)}{y}\right)\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

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

      2. associate-+r+ N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. associate-*l/ N/A

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

      6. associate-/l* N/A

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

      7. *-inverses N/A

         \[\leadsto \left(\left(\frac{a \cdot t}{y} + \frac{a \cdot \left(b \cdot z\right)}{y}\right) + z\right) \cdot y + x \cdot \color{blue}{1} \]

      8. *-rgt-identity N/A

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

      9. lower-fma.f64 N/A

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

   5. Applied rewrites 89.9%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 41.3%

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

Alternative 11: 28.0% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ z \cdot y \end{array} \]

FPCore

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

C

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

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 = z * y
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 90.8%

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

3. Taylor expanded in y around inf

   \[\leadsto \color{blue}{y \cdot \left(z + \left(\frac{x}{y} + \left(\frac{a \cdot t}{y} + \frac{a \cdot \left(b \cdot z\right)}{y}\right)\right)\right)} \]
4. Step-by-step derivation

   1. +-commutative N/A

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

   2. associate-+r+ N/A

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

   3. distribute-rgt-in N/A

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

   4. +-commutative N/A

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

   5. associate-*l/ N/A

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

   6. associate-/l* N/A

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

   7. *-inverses N/A

      \[\leadsto \left(\left(\frac{a \cdot t}{y} + \frac{a \cdot \left(b \cdot z\right)}{y}\right) + z\right) \cdot y + x \cdot \color{blue}{1} \]

   8. *-rgt-identity N/A

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

   9. lower-fma.f64 N/A

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

5. Applied rewrites 84.1%

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

6. Taylor expanded in y around inf

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

8. Applied rewrites 27.0%

   \[\leadsto z \cdot \color{blue}{y} \]
9. Add Preprocessing

Developer Target 1: 97.5% accurate, 0.8× 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 2024332 
(FPCore (x y z t a b)
  :name "Graphics.Rasterific.CubicBezier:cachedBezierAt from Rasterific-0.6.1"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< z -11820553527347888000) (+ (* z (+ (* b a) y)) (+ x (* t a))) (if (< z 47589743188364287/1000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (+ (* (+ (* b z) t) a) (+ (* z y) x)) (+ (* z (+ (* b a) y)) (+ x (* t a))))))

  (+ (+ (+ x (* y z)) (* t a)) (* (* a z) b)))