Graphics.Rasterific.CubicBezier:cachedBezierAt from Rasterific-0.6.1

Percentage Accurate: 92.7% → 98.4%

Time: 13.1s

Alternatives: 15

Speedup: 0.7×

• 32.0% 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: 92.7% 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: 98.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(z \cdot a\right) \cdot b\\ \mathbf{if}\;t\_1 \leq 5 \cdot 10^{+295}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, y + a \cdot \left(b + \frac{t}{z}\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ (+ (+ x (* y z)) (* t a)) (* (* z a) b))))
   (if (<= t_1 5e+295) t_1 (fma z (+ y (* a (+ b (/ t z)))) x))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = ((x + (y * z)) + (t * a)) + ((z * a) * b);
	double tmp;
	if (t_1 <= 5e+295) {
		tmp = t_1;
	} else {
		tmp = fma(z, (y + (a * (b + (t / z)))), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(Float64(x + Float64(y * z)) + Float64(t * a)) + Float64(Float64(z * a) * b))
	tmp = 0.0
	if (t_1 <= 5e+295)
		tmp = t_1;
	else
		tmp = fma(z, Float64(y + Float64(a * Float64(b + Float64(t / z)))), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(N[(x + N[(y * z), $MachinePrecision]), $MachinePrecision] + N[(t * a), $MachinePrecision]), $MachinePrecision] + N[(N[(z * a), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 5e+295], t$95$1, N[(z * N[(y + N[(a * N[(b + N[(t / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x), $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)) < 4.99999999999999991e295

   1. Initial program 97.8%

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

   if 4.99999999999999991e295 < (+.f64 (+.f64 (+.f64 x (*.f64 y z)) (*.f64 t a)) (*.f64 (*.f64 a z) b))

   1. Initial program 65.1%

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

   3. Taylor expanded in z around inf

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. associate-+r+ N/A

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

      4. distribute-lft-in N/A

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

      5. *-commutative N/A

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

      6. associate-*l/ N/A

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

      7. associate-/l* N/A

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

      8. *-inverses N/A

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

      9. *-rgt-identity N/A

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

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

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

      11. associate-+l+ N/A

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

      12. +-commutative N/A

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

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

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

      14. associate-/l* N/A

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

      15. distribute-lft-out N/A

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

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

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

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

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

      18. /-lowering-/.f64 100.0

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

   5. Simplified 100.0%

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

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(z \cdot a\right) \cdot b \leq 5 \cdot 10^{+295}:\\ \;\;\;\;\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(z \cdot a\right) \cdot b\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, y + a \cdot \left(b + \frac{t}{z}\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 94.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(z, y + a \cdot \left(b + \frac{t}{z}\right), x\right)\\ \mathbf{if}\;z \leq -1.22 \cdot 10^{-79}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 6.7 \cdot 10^{-135}:\\ \;\;\;\;\mathsf{fma}\left(a, \mathsf{fma}\left(b, z, t\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (fma z (+ y (* a (+ b (/ t z)))) x)))
   (if (<= z -1.22e-79) t_1 (if (<= z 6.7e-135) (fma a (fma b z t) x) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = fma(z, (y + (a * (b + (t / z)))), x);
	double tmp;
	if (z <= -1.22e-79) {
		tmp = t_1;
	} else if (z <= 6.7e-135) {
		tmp = fma(a, fma(b, z, t), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = fma(z, Float64(y + Float64(a * Float64(b + Float64(t / z)))), x)
	tmp = 0.0
	if (z <= -1.22e-79)
		tmp = t_1;
	elseif (z <= 6.7e-135)
		tmp = fma(a, fma(b, z, t), x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(z * N[(y + N[(a * N[(b + N[(t / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[z, -1.22e-79], t$95$1, If[LessEqual[z, 6.7e-135], N[(a * N[(b * z + t), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.22e-79 or 6.7000000000000004e-135 < z

   1. Initial program 86.8%

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

   3. Taylor expanded in z around inf

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. associate-+r+ N/A

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

      4. distribute-lft-in N/A

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

      5. *-commutative N/A

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

      6. associate-*l/ N/A

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

      7. associate-/l* N/A

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

      8. *-inverses N/A

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

      9. *-rgt-identity N/A

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

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

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

      11. associate-+l+ N/A

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

      12. +-commutative N/A

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

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

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

      14. associate-/l* N/A

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

      15. distribute-lft-out N/A

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

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

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

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

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

      18. /-lowering-/.f64 98.8

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

   5. Simplified 98.8%

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

   if -1.22e-79 < z < 6.7000000000000004e-135

   1. Initial program 99.4%

      \[\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b \]
   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. accelerator-lowering-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. accelerator-lowering-fma.f64 93.1

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

   5. Simplified 93.1%

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

4. Final simplification 97.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.22 \cdot 10^{-79}:\\ \;\;\;\;\mathsf{fma}\left(z, y + a \cdot \left(b + \frac{t}{z}\right), x\right)\\ \mathbf{elif}\;z \leq 6.7 \cdot 10^{-135}:\\ \;\;\;\;\mathsf{fma}\left(a, \mathsf{fma}\left(b, z, t\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, y + a \cdot \left(b + \frac{t}{z}\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 83.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.55 \cdot 10^{+186}:\\ \;\;\;\;z \cdot \mathsf{fma}\left(a, b, y\right)\\ \mathbf{elif}\;z \leq -1.35 \cdot 10^{-78}:\\ \;\;\;\;\mathsf{fma}\left(z, y, a \cdot \mathsf{fma}\left(b, z, t\right)\right)\\ \mathbf{elif}\;z \leq 8.4 \cdot 10^{-135}:\\ \;\;\;\;\mathsf{fma}\left(a, \mathsf{fma}\left(b, z, t\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, y + a \cdot b, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= z -1.55e+186)
   (* z (fma a b y))
   (if (<= z -1.35e-78)
     (fma z y (* a (fma b z t)))
     (if (<= z 8.4e-135) (fma a (fma b z t) x) (fma z (+ y (* a b)) x)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (z <= -1.55e+186) {
		tmp = z * fma(a, b, y);
	} else if (z <= -1.35e-78) {
		tmp = fma(z, y, (a * fma(b, z, t)));
	} else if (z <= 8.4e-135) {
		tmp = fma(a, fma(b, z, t), x);
	} else {
		tmp = fma(z, (y + (a * b)), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (z <= -1.55e+186)
		tmp = Float64(z * fma(a, b, y));
	elseif (z <= -1.35e-78)
		tmp = fma(z, y, Float64(a * fma(b, z, t)));
	elseif (z <= 8.4e-135)
		tmp = fma(a, fma(b, z, t), x);
	else
		tmp = fma(z, Float64(y + Float64(a * b)), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[z, -1.55e+186], N[(z * N[(a * b + y), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, -1.35e-78], N[(z * y + N[(a * N[(b * z + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 8.4e-135], N[(a * N[(b * z + t), $MachinePrecision] + x), $MachinePrecision], N[(z * N[(y + N[(a * b), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if z < -1.5500000000000001e186

   1. Initial program 85.3%

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

   3. Taylor expanded in z around inf

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

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

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

      2. +-commutative N/A

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

      3. accelerator-lowering-fma.f64 96.5

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

   5. Simplified 96.5%

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

   if -1.5500000000000001e186 < z < -1.34999999999999997e-78

   1. Initial program 88.2%

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

   3. Taylor expanded in 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

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

      4. *-commutative N/A

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

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

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

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

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

      7. +-commutative N/A

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

      8. accelerator-lowering-fma.f64 90.4

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

   5. Simplified 90.4%

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

   if -1.34999999999999997e-78 < z < 8.4000000000000001e-135

   1. Initial program 99.4%

      \[\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b \]
   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. accelerator-lowering-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. accelerator-lowering-fma.f64 93.1

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

   5. Simplified 93.1%

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

   if 8.4000000000000001e-135 < z

   1. Initial program 86.4%

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

   3. Taylor expanded in z around inf

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. associate-+r+ N/A

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

      4. distribute-lft-in N/A

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

      5. *-commutative N/A

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

      6. associate-*l/ N/A

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

      7. associate-/l* N/A

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

      8. *-inverses N/A

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

      9. *-rgt-identity N/A

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

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

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

      11. associate-+l+ N/A

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

      12. +-commutative N/A

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

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

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

      14. associate-/l* N/A

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

      15. distribute-lft-out N/A

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

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

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

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

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

      18. /-lowering-/.f64 98.0

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

   5. Simplified 98.0%

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

   6. Taylor expanded in t around 0

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

      1. *-lowering-*.f64 92.2

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

   8. Simplified 92.2%

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

Alternative 4: 87.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -6.5 \cdot 10^{-29}:\\ \;\;\;\;\mathsf{fma}\left(z, \mathsf{fma}\left(a, b, y\right), x\right)\\ \mathbf{elif}\;b \leq 3800000000000:\\ \;\;\;\;\mathsf{fma}\left(a, t, \mathsf{fma}\left(z, y, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, y + a \cdot b, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= b -6.5e-29)
   (fma z (fma a b y) x)
   (if (<= b 3800000000000.0) (fma a t (fma z y x)) (fma z (+ y (* a b)) x))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (b <= -6.5e-29) {
		tmp = fma(z, fma(a, b, y), x);
	} else if (b <= 3800000000000.0) {
		tmp = fma(a, t, fma(z, y, x));
	} else {
		tmp = fma(z, (y + (a * b)), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (b <= -6.5e-29)
		tmp = fma(z, fma(a, b, y), x);
	elseif (b <= 3800000000000.0)
		tmp = fma(a, t, fma(z, y, x));
	else
		tmp = fma(z, Float64(y + Float64(a * b)), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[b, -6.5e-29], N[(z * N[(a * b + y), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[b, 3800000000000.0], N[(a * t + N[(z * y + x), $MachinePrecision]), $MachinePrecision], N[(z * N[(y + N[(a * b), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -6.5e-29

   1. Initial program 92.2%

      \[\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b \]
   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. accelerator-lowering-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. accelerator-lowering-fma.f64 82.2

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

   5. Simplified 82.2%

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

   if -6.5e-29 < b < 3.8e12

   1. Initial program 89.8%

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

   3. Taylor expanded in b around 0

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

   5. Simplified 93.6%

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

   if 3.8e12 < b

   1. Initial program 92.3%

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

   3. Taylor expanded in z around inf

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. associate-+r+ N/A

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

      4. distribute-lft-in N/A

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

      5. *-commutative N/A

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

      6. associate-*l/ N/A

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

      7. associate-/l* N/A

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

      8. *-inverses N/A

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

      9. *-rgt-identity N/A

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

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

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

      11. associate-+l+ N/A

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

      12. +-commutative N/A

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

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

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

      14. associate-/l* N/A

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

      15. distribute-lft-out N/A

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

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

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

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

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

      18. /-lowering-/.f64 90.9

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

   5. Simplified 90.9%

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

   6. Taylor expanded in t around 0

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

      1. *-lowering-*.f64 91.4

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

   8. Simplified 91.4%

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

Alternative 5: 61.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -0.019:\\ \;\;\;\;\mathsf{fma}\left(a, t, x\right)\\ \mathbf{elif}\;t \leq -8.2 \cdot 10^{-55}:\\ \;\;\;\;\left(z \cdot a\right) \cdot b\\ \mathbf{elif}\;t \leq 1.9 \cdot 10^{+91}:\\ \;\;\;\;\mathsf{fma}\left(z, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, t, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= t -0.019)
   (fma a t x)
   (if (<= t -8.2e-55)
     (* (* z a) b)
     (if (<= t 1.9e+91) (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 <= -0.019) {
		tmp = fma(a, t, x);
	} else if (t <= -8.2e-55) {
		tmp = (z * a) * b;
	} else if (t <= 1.9e+91) {
		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 <= -0.019)
		tmp = fma(a, t, x);
	elseif (t <= -8.2e-55)
		tmp = Float64(Float64(z * a) * b);
	elseif (t <= 1.9e+91)
		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, -0.019], N[(a * t + x), $MachinePrecision], If[LessEqual[t, -8.2e-55], N[(N[(z * a), $MachinePrecision] * b), $MachinePrecision], If[LessEqual[t, 1.9e+91], N[(z * y + x), $MachinePrecision], N[(a * t + x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if t < -0.0189999999999999995 or 1.8999999999999999e91 < 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 z around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 70.5

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

   5. Simplified 70.5%

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

   if -0.0189999999999999995 < t < -8.1999999999999996e-55

   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 z around inf

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. associate-+r+ N/A

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

      4. distribute-lft-in N/A

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

      5. *-commutative N/A

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

      6. associate-*l/ N/A

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

      7. associate-/l* N/A

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

      8. *-inverses N/A

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

      9. *-rgt-identity N/A

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

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

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

      11. associate-+l+ N/A

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

      12. +-commutative N/A

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

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

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

      14. associate-/l* N/A

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

      15. distribute-lft-out N/A

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

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

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

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

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

      18. /-lowering-/.f64 82.9

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

   5. Simplified 82.9%

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

   6. Taylor expanded in a around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

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

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

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

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

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

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

      6. *-lowering-*.f64 73.8

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

   8. Simplified 73.8%

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

   9. Taylor expanded in b around inf

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

   11. Simplified 73.8%

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

   if -8.1999999999999996e-55 < t < 1.8999999999999999e91

   1. Initial program 91.1%

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

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{x + y \cdot z} \]
   4. 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. accelerator-lowering-fma.f64 64.1

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

   5. Simplified 64.1%

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

4. Final simplification 66.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.019:\\ \;\;\;\;\mathsf{fma}\left(a, t, x\right)\\ \mathbf{elif}\;t \leq -8.2 \cdot 10^{-55}:\\ \;\;\;\;\left(z \cdot a\right) \cdot b\\ \mathbf{elif}\;t \leq 1.9 \cdot 10^{+91}:\\ \;\;\;\;\mathsf{fma}\left(z, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, t, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 87.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(z, \mathsf{fma}\left(a, b, y\right), x\right)\\ \mathbf{if}\;b \leq -1.15 \cdot 10^{-20}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;b \leq 120000000000:\\ \;\;\;\;\mathsf{fma}\left(a, t, \mathsf{fma}\left(z, y, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (fma z (fma a b y) x)))
   (if (<= b -1.15e-20)
     t_1
     (if (<= b 120000000000.0) (fma a t (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(z, fma(a, b, y), x);
	double tmp;
	if (b <= -1.15e-20) {
		tmp = t_1;
	} else if (b <= 120000000000.0) {
		tmp = fma(a, t, fma(z, y, x));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = fma(z, fma(a, b, y), x)
	tmp = 0.0
	if (b <= -1.15e-20)
		tmp = t_1;
	elseif (b <= 120000000000.0)
		tmp = fma(a, t, 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[(z * N[(a * b + y), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[b, -1.15e-20], t$95$1, If[LessEqual[b, 120000000000.0], N[(a * t + N[(z * y + x), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if b < -1.15e-20 or 1.2e11 < b

   1. Initial program 92.2%

      \[\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b \]
   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. accelerator-lowering-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. accelerator-lowering-fma.f64 86.1

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

   5. Simplified 86.1%

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

   if -1.15e-20 < b < 1.2e11

   1. Initial program 89.8%

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

   3. Taylor expanded in b around 0

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

   5. Simplified 93.6%

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

Alternative 7: 86.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(a, \mathsf{fma}\left(b, z, t\right), x\right)\\ \mathbf{if}\;a \leq -3.75 \cdot 10^{-25}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 3.8 \cdot 10^{+104}:\\ \;\;\;\;\mathsf{fma}\left(a, t, \mathsf{fma}\left(z, y, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (fma a (fma b z t) x)))
   (if (<= a -3.75e-25) t_1 (if (<= a 3.8e+104) (fma a t (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, fma(b, z, t), x);
	double tmp;
	if (a <= -3.75e-25) {
		tmp = t_1;
	} else if (a <= 3.8e+104) {
		tmp = fma(a, t, fma(z, y, x));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = fma(a, fma(b, z, t), x)
	tmp = 0.0
	if (a <= -3.75e-25)
		tmp = t_1;
	elseif (a <= 3.8e+104)
		tmp = fma(a, t, 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[(a * N[(b * z + t), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[a, -3.75e-25], t$95$1, If[LessEqual[a, 3.8e+104], N[(a * t + N[(z * y + x), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if a < -3.74999999999999994e-25 or 3.79999999999999969e104 < a

   1. Initial program 83.5%

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

   3. Taylor expanded in y around 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. accelerator-lowering-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. accelerator-lowering-fma.f64 89.2

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

   5. Simplified 89.2%

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

   if -3.74999999999999994e-25 < a < 3.79999999999999969e104

   1. Initial program 96.9%

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

   3. Taylor expanded in b around 0

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

   5. Simplified 85.4%

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

Alternative 8: 82.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := z \cdot \mathsf{fma}\left(a, b, y\right)\\ \mathbf{if}\;z \leq -2.65 \cdot 10^{+85}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1.05 \cdot 10^{+117}:\\ \;\;\;\;\mathsf{fma}\left(a, t, \mathsf{fma}\left(z, y, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* z (fma a b y))))
   (if (<= z -2.65e+85) t_1 (if (<= z 1.05e+117) (fma a t (fma z y x)) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = z * fma(a, b, y);
	double tmp;
	if (z <= -2.65e+85) {
		tmp = t_1;
	} else if (z <= 1.05e+117) {
		tmp = fma(a, t, fma(z, y, x));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(z * fma(a, b, y))
	tmp = 0.0
	if (z <= -2.65e+85)
		tmp = t_1;
	elseif (z <= 1.05e+117)
		tmp = fma(a, t, 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[(z * N[(a * b + y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -2.65e+85], t$95$1, If[LessEqual[z, 1.05e+117], N[(a * t + N[(z * y + x), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -2.65e85 or 1.0500000000000001e117 < z

   1. Initial program 80.8%

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

   3. Taylor expanded in z around inf

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

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

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

      2. +-commutative N/A

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

      3. accelerator-lowering-fma.f64 86.5

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

   5. Simplified 86.5%

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

   if -2.65e85 < z < 1.0500000000000001e117

   1. Initial program 97.7%

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

   3. Taylor expanded in b around 0

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

   5. Simplified 84.4%

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

Alternative 9: 38.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.1 \cdot 10^{+40}:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;z \leq -8.2 \cdot 10^{-190}:\\ \;\;\;\;t \cdot a\\ \mathbf{elif}\;z \leq 1.6 \cdot 10^{-81}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= z -2.1e+40)
   (* y z)
   (if (<= z -8.2e-190) (* t a) (if (<= z 1.6e-81) x (* y z)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (z <= -2.1e+40) {
		tmp = y * z;
	} else if (z <= -8.2e-190) {
		tmp = t * a;
	} else if (z <= 1.6e-81) {
		tmp = x;
	} else {
		tmp = y * z;
	}
	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 <= (-2.1d+40)) then
        tmp = y * z
    else if (z <= (-8.2d-190)) then
        tmp = t * a
    else if (z <= 1.6d-81) then
        tmp = x
    else
        tmp = y * z
    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 <= -2.1e+40) {
		tmp = y * z;
	} else if (z <= -8.2e-190) {
		tmp = t * a;
	} else if (z <= 1.6e-81) {
		tmp = x;
	} else {
		tmp = y * z;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if z <= -2.1e+40:
		tmp = y * z
	elif z <= -8.2e-190:
		tmp = t * a
	elif z <= 1.6e-81:
		tmp = x
	else:
		tmp = y * z
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (z <= -2.1e+40)
		tmp = Float64(y * z);
	elseif (z <= -8.2e-190)
		tmp = Float64(t * a);
	elseif (z <= 1.6e-81)
		tmp = x;
	else
		tmp = Float64(y * z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (z <= -2.1e+40)
		tmp = y * z;
	elseif (z <= -8.2e-190)
		tmp = t * a;
	elseif (z <= 1.6e-81)
		tmp = x;
	else
		tmp = y * z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[z, -2.1e+40], N[(y * z), $MachinePrecision], If[LessEqual[z, -8.2e-190], N[(t * a), $MachinePrecision], If[LessEqual[z, 1.6e-81], x, N[(y * z), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if z < -2.1000000000000001e40 or 1.6e-81 < z

   1. Initial program 85.3%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 46.0

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

   5. Simplified 46.0%

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

   if -2.1000000000000001e40 < z < -8.2000000000000004e-190

   1. Initial program 97.1%

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

   3. Taylor expanded in t around inf

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

      1. *-lowering-*.f64 41.2

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

   5. Simplified 41.2%

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

   if -8.2000000000000004e-190 < z < 1.6e-81

   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 x around inf

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

   5. Simplified 52.2%

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

4. Final simplification 46.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.1 \cdot 10^{+40}:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;z \leq -8.2 \cdot 10^{-190}:\\ \;\;\;\;t \cdot a\\ \mathbf{elif}\;z \leq 1.6 \cdot 10^{-81}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 72.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := z \cdot \mathsf{fma}\left(a, b, y\right)\\ \mathbf{if}\;z \leq -1.7 \cdot 10^{+40}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 9 \cdot 10^{-73}:\\ \;\;\;\;\mathsf{fma}\left(a, t, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* z (fma a b y))))
   (if (<= z -1.7e+40) t_1 (if (<= z 9e-73) (fma a t x) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = z * fma(a, b, y);
	double tmp;
	if (z <= -1.7e+40) {
		tmp = t_1;
	} else if (z <= 9e-73) {
		tmp = fma(a, t, x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(z * fma(a, b, y))
	tmp = 0.0
	if (z <= -1.7e+40)
		tmp = t_1;
	elseif (z <= 9e-73)
		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[(z * N[(a * b + y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -1.7e+40], t$95$1, If[LessEqual[z, 9e-73], N[(a * t + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.69999999999999994e40 or 9e-73 < z

   1. Initial program 85.1%

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

   3. Taylor expanded in z around inf

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

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

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

      2. +-commutative N/A

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

      3. accelerator-lowering-fma.f64 79.8

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

   5. Simplified 79.8%

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

   if -1.69999999999999994e40 < z < 9e-73

   1. Initial program 98.7%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 74.7

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

   5. Simplified 74.7%

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

Alternative 11: 75.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := a \cdot \mathsf{fma}\left(b, z, t\right)\\ \mathbf{if}\;a \leq -3.4 \cdot 10^{-17}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 2.7 \cdot 10^{+77}:\\ \;\;\;\;\mathsf{fma}\left(z, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* a (fma b z t))))
   (if (<= a -3.4e-17) t_1 (if (<= a 2.7e+77) (fma z y x) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = a * fma(b, z, t);
	double tmp;
	if (a <= -3.4e-17) {
		tmp = t_1;
	} else if (a <= 2.7e+77) {
		tmp = fma(z, y, x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(a * fma(b, z, t))
	tmp = 0.0
	if (a <= -3.4e-17)
		tmp = t_1;
	elseif (a <= 2.7e+77)
		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[(a * N[(b * z + t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -3.4e-17], t$95$1, If[LessEqual[a, 2.7e+77], N[(z * y + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if a < -3.3999999999999998e-17 or 2.6999999999999998e77 < a

   1. Initial program 83.0%

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

   3. Taylor expanded in a around inf

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

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

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

      2. +-commutative N/A

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

      3. accelerator-lowering-fma.f64 77.5

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

   5. Simplified 77.5%

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

   if -3.3999999999999998e-17 < a < 2.6999999999999998e77

   1. Initial program 98.2%

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

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{x + y \cdot z} \]
   4. 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. accelerator-lowering-fma.f64 75.1

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

   5. Simplified 75.1%

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

Alternative 12: 63.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -3.4 \cdot 10^{+15}:\\ \;\;\;\;\mathsf{fma}\left(a, t, x\right)\\ \mathbf{elif}\;t \leq 4.9 \cdot 10^{+90}:\\ \;\;\;\;\mathsf{fma}\left(z, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, t, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= t -3.4e+15) (fma a t x) (if (<= t 4.9e+90) (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 <= -3.4e+15) {
		tmp = fma(a, t, x);
	} else if (t <= 4.9e+90) {
		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 <= -3.4e+15)
		tmp = fma(a, t, x);
	elseif (t <= 4.9e+90)
		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, -3.4e+15], N[(a * t + x), $MachinePrecision], If[LessEqual[t, 4.9e+90], N[(z * y + x), $MachinePrecision], N[(a * t + x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t < -3.4e15 or 4.9000000000000003e90 < t

   1. Initial program 89.4%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 70.6

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

   5. Simplified 70.6%

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

   if -3.4e15 < t < 4.9000000000000003e90

   1. Initial program 91.8%

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

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{x + y \cdot z} \]
   4. 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. accelerator-lowering-fma.f64 61.8

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

   5. Simplified 61.8%

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

Alternative 13: 58.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -9 \cdot 10^{+147}:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;z \leq 1.9 \cdot 10^{+122}:\\ \;\;\;\;\mathsf{fma}\left(a, t, x\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= z -9e+147) (* y z) (if (<= z 1.9e+122) (fma a t x) (* y z))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (z <= -9e+147) {
		tmp = y * z;
	} else if (z <= 1.9e+122) {
		tmp = fma(a, t, x);
	} else {
		tmp = y * z;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (z <= -9e+147)
		tmp = Float64(y * z);
	elseif (z <= 1.9e+122)
		tmp = fma(a, t, x);
	else
		tmp = Float64(y * z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[z, -9e+147], N[(y * z), $MachinePrecision], If[LessEqual[z, 1.9e+122], N[(a * t + x), $MachinePrecision], N[(y * z), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -9.00000000000000016e147 or 1.8999999999999999e122 < z

   1. Initial program 80.1%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 54.9

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

   5. Simplified 54.9%

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

   if -9.00000000000000016e147 < z < 1.8999999999999999e122

   1. Initial program 96.2%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 62.5

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

   5. Simplified 62.5%

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

4. Final simplification 60.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -9 \cdot 10^{+147}:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;z \leq 1.9 \cdot 10^{+122}:\\ \;\;\;\;\mathsf{fma}\left(a, t, x\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 39.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -240000000:\\ \;\;\;\;t \cdot a\\ \mathbf{elif}\;a \leq 1.26 \cdot 10^{+82}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= a -240000000.0) (* t a) (if (<= a 1.26e+82) x (* t a))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (a <= -240000000.0) {
		tmp = t * a;
	} else if (a <= 1.26e+82) {
		tmp = x;
	} else {
		tmp = t * 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) :: tmp
    if (a <= (-240000000.0d0)) then
        tmp = t * a
    else if (a <= 1.26d+82) then
        tmp = x
    else
        tmp = t * 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 tmp;
	if (a <= -240000000.0) {
		tmp = t * a;
	} else if (a <= 1.26e+82) {
		tmp = x;
	} else {
		tmp = t * a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if a <= -240000000.0:
		tmp = t * a
	elif a <= 1.26e+82:
		tmp = x
	else:
		tmp = t * a
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (a <= -240000000.0)
		tmp = Float64(t * a);
	elseif (a <= 1.26e+82)
		tmp = x;
	else
		tmp = Float64(t * a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (a <= -240000000.0)
		tmp = t * a;
	elseif (a <= 1.26e+82)
		tmp = x;
	else
		tmp = t * a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[a, -240000000.0], N[(t * a), $MachinePrecision], If[LessEqual[a, 1.26e+82], x, N[(t * a), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if a < -2.4e8 or 1.2600000000000001e82 < a

   1. Initial program 82.4%

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

   3. Taylor expanded in t around inf

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

      1. *-lowering-*.f64 38.9

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

   5. Simplified 38.9%

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

   if -2.4e8 < a < 1.2600000000000001e82

   1. Initial program 98.2%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 34.7%

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

4. Final simplification 36.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -240000000:\\ \;\;\;\;t \cdot a\\ \mathbf{elif}\;a \leq 1.26 \cdot 10^{+82}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t \cdot a\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 26.2% accurate, 30.0× 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 90.9%

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

3. Taylor expanded in x around inf

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

5. Simplified 24.3%

   \[\leadsto \color{blue}{x} \]
6. 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 2024201 
(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)))