Graphics.Rendering.Plot.Render.Plot.Legend:renderLegendOutside from plot-0.2.3.4, B

Percentage Accurate: 99.9% → 99.9%

Time: 6.8s

Alternatives: 13

Speedup: 1.0×

• 26.2% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5 \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (+ (* x (+ (+ (+ (+ y z) z) y) t)) (* y 5.0)))

C

double code(double x, double y, double z, double t) {
	return (x * ((((y + z) + z) + y) + t)) + (y * 5.0);
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = (x * ((((y + z) + z) + y) + t)) + (y * 5.0d0)
end function

Java

public static double code(double x, double y, double z, double t) {
	return (x * ((((y + z) + z) + y) + t)) + (y * 5.0);
}

Python

def code(x, y, z, t):
	return (x * ((((y + z) + z) + y) + t)) + (y * 5.0)

Julia

function code(x, y, z, t)
	return Float64(Float64(x * Float64(Float64(Float64(Float64(y + z) + z) + y) + t)) + Float64(y * 5.0))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = (x * ((((y + z) + z) + y) + t)) + (y * 5.0);
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(x * N[(N[(N[(N[(y + z), $MachinePrecision] + z), $MachinePrecision] + y), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision] + N[(y * 5.0), $MachinePrecision]), $MachinePrecision]

Initial Program: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5 \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (+ (* x (+ (+ (+ (+ y z) z) y) t)) (* y 5.0)))

C

double code(double x, double y, double z, double t) {
	return (x * ((((y + z) + z) + y) + t)) + (y * 5.0);
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = (x * ((((y + z) + z) + y) + t)) + (y * 5.0d0)
end function

Java

public static double code(double x, double y, double z, double t) {
	return (x * ((((y + z) + z) + y) + t)) + (y * 5.0);
}

Python

def code(x, y, z, t):
	return (x * ((((y + z) + z) + y) + t)) + (y * 5.0)

Julia

function code(x, y, z, t)
	return Float64(Float64(x * Float64(Float64(Float64(Float64(y + z) + z) + y) + t)) + Float64(y * 5.0))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = (x * ((((y + z) + z) + y) + t)) + (y * 5.0);
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(x * N[(N[(N[(N[(y + z), $MachinePrecision] + z), $MachinePrecision] + y), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision] + N[(y * 5.0), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ 5 \cdot y + \left(\left(t + y\right) + \mathsf{fma}\left(2, z, y\right)\right) \cdot x \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (+ (* 5.0 y) (* (+ (+ t y) (fma 2.0 z y)) x)))

C

double code(double x, double y, double z, double t) {
	return (5.0 * y) + (((t + y) + fma(2.0, z, y)) * x);
}

Julia

function code(x, y, z, t)
	return Float64(Float64(5.0 * y) + Float64(Float64(Float64(t + y) + fma(2.0, z, y)) * x))
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(5.0 * y), $MachinePrecision] + N[(N[(N[(t + y), $MachinePrecision] + N[(2.0 * z + y), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.9%

   \[x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5 \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-+.f64 N/A

      \[\leadsto x \cdot \color{blue}{\left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right)} + y \cdot 5 \]

   2. lift-+.f64 N/A

      \[\leadsto x \cdot \left(\color{blue}{\left(\left(\left(y + z\right) + z\right) + y\right)} + t\right) + y \cdot 5 \]

   3. associate-+l+ N/A

      \[\leadsto x \cdot \color{blue}{\left(\left(\left(y + z\right) + z\right) + \left(y + t\right)\right)} + y \cdot 5 \]

   4. lower-+.f64 N/A

      \[\leadsto x \cdot \color{blue}{\left(\left(\left(y + z\right) + z\right) + \left(y + t\right)\right)} + y \cdot 5 \]

   5. lift-+.f64 N/A

      \[\leadsto x \cdot \left(\color{blue}{\left(\left(y + z\right) + z\right)} + \left(y + t\right)\right) + y \cdot 5 \]

   6. lift-+.f64 N/A

      \[\leadsto x \cdot \left(\left(\color{blue}{\left(y + z\right)} + z\right) + \left(y + t\right)\right) + y \cdot 5 \]

   7. associate-+l+ N/A

      \[\leadsto x \cdot \left(\color{blue}{\left(y + \left(z + z\right)\right)} + \left(y + t\right)\right) + y \cdot 5 \]

   8. +-commutative N/A

      \[\leadsto x \cdot \left(\color{blue}{\left(\left(z + z\right) + y\right)} + \left(y + t\right)\right) + y \cdot 5 \]

   9. count-2 N/A

      \[\leadsto x \cdot \left(\left(\color{blue}{2 \cdot z} + y\right) + \left(y + t\right)\right) + y \cdot 5 \]

   10. lower-fma.f64 N/A

       \[\leadsto x \cdot \left(\color{blue}{\mathsf{fma}\left(2, z, y\right)} + \left(y + t\right)\right) + y \cdot 5 \]

   11. +-commutative N/A

       \[\leadsto x \cdot \left(\mathsf{fma}\left(2, z, y\right) + \color{blue}{\left(t + y\right)}\right) + y \cdot 5 \]

   12. lower-+.f64 99.9

       \[\leadsto x \cdot \left(\mathsf{fma}\left(2, z, y\right) + \color{blue}{\left(t + y\right)}\right) + y \cdot 5 \]

4. Applied rewrites 99.9%

   \[\leadsto x \cdot \color{blue}{\left(\mathsf{fma}\left(2, z, y\right) + \left(t + y\right)\right)} + y \cdot 5 \]

5. Final simplification 99.9%

   \[\leadsto 5 \cdot y + \left(\left(t + y\right) + \mathsf{fma}\left(2, z, y\right)\right) \cdot x \]
6. Add Preprocessing

Alternative 2: 87.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -9.8 \cdot 10^{-30}:\\ \;\;\;\;\mathsf{fma}\left(y + z, 2, t\right) \cdot x\\ \mathbf{elif}\;x \leq -6 \cdot 10^{-168}:\\ \;\;\;\;\mathsf{fma}\left(t, x, 5 \cdot y\right)\\ \mathbf{elif}\;x \leq 1.55 \cdot 10^{-59}:\\ \;\;\;\;\mathsf{fma}\left(y, 5, \left(z + z\right) \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, 2, \left(t + y\right) + y\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= x -9.8e-30)
   (* (fma (+ y z) 2.0 t) x)
   (if (<= x -6e-168)
     (fma t x (* 5.0 y))
     (if (<= x 1.55e-59)
       (fma y 5.0 (* (+ z z) x))
       (* (fma z 2.0 (+ (+ t y) y)) x)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -9.8e-30) {
		tmp = fma((y + z), 2.0, t) * x;
	} else if (x <= -6e-168) {
		tmp = fma(t, x, (5.0 * y));
	} else if (x <= 1.55e-59) {
		tmp = fma(y, 5.0, ((z + z) * x));
	} else {
		tmp = fma(z, 2.0, ((t + y) + y)) * x;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x <= -9.8e-30)
		tmp = Float64(fma(Float64(y + z), 2.0, t) * x);
	elseif (x <= -6e-168)
		tmp = fma(t, x, Float64(5.0 * y));
	elseif (x <= 1.55e-59)
		tmp = fma(y, 5.0, Float64(Float64(z + z) * x));
	else
		tmp = Float64(fma(z, 2.0, Float64(Float64(t + y) + y)) * x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[x, -9.8e-30], N[(N[(N[(y + z), $MachinePrecision] * 2.0 + t), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[x, -6e-168], N[(t * x + N[(5.0 * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.55e-59], N[(y * 5.0 + N[(N[(z + z), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision], N[(N[(z * 2.0 + N[(N[(t + y), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -9.79999999999999942e-30

   1. Initial program 100.0%

      \[x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5 \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(t + \left(2 \cdot y + 2 \cdot z\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(t + \left(2 \cdot y + 2 \cdot z\right)\right) \cdot x} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(t + \left(2 \cdot y + 2 \cdot z\right)\right) \cdot x} \]

      3. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(2 \cdot y + 2 \cdot z\right) + t\right)} \cdot x \]

      4. distribute-lft-out N/A

         \[\leadsto \left(\color{blue}{2 \cdot \left(y + z\right)} + t\right) \cdot x \]

      5. *-commutative N/A

         \[\leadsto \left(\color{blue}{\left(y + z\right) \cdot 2} + t\right) \cdot x \]

      6. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(y + z, 2, t\right)} \cdot x \]

      7. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{z + y}, 2, t\right) \cdot x \]

      8. lower-+.f64 97.9

         \[\leadsto \mathsf{fma}\left(\color{blue}{z + y}, 2, t\right) \cdot x \]

   5. Applied rewrites 97.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z + y, 2, t\right) \cdot x} \]

   if -9.79999999999999942e-30 < x < -5.99999999999999983e-168

   1. Initial program 99.8%

      \[x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5 \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{y \cdot 5 + x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \color{blue}{y \cdot 5} + x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) \]

      4. lower-fma.f64 99.9

         \[\leadsto \color{blue}{\mathsf{fma}\left(y, 5, x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right)\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, 5, \color{blue}{x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right)}\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(y, 5, \color{blue}{\left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) \cdot x}\right) \]

      7. lower-*.f64 99.9

         \[\leadsto \mathsf{fma}\left(y, 5, \color{blue}{\left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) \cdot x}\right) \]

      8. lift-+.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, 5, \color{blue}{\left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right)} \cdot x\right) \]

      9. lift-+.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, 5, \left(\color{blue}{\left(\left(\left(y + z\right) + z\right) + y\right)} + t\right) \cdot x\right) \]

      10. lift-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(y, 5, \left(\left(\color{blue}{\left(\left(y + z\right) + z\right)} + y\right) + t\right) \cdot x\right) \]

      11. associate-+l+ N/A

          \[\leadsto \mathsf{fma}\left(y, 5, \left(\color{blue}{\left(\left(y + z\right) + \left(z + y\right)\right)} + t\right) \cdot x\right) \]

      12. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(y, 5, \left(\left(\left(y + z\right) + \color{blue}{\left(y + z\right)}\right) + t\right) \cdot x\right) \]

      13. lift-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(y, 5, \left(\left(\left(y + z\right) + \color{blue}{\left(y + z\right)}\right) + t\right) \cdot x\right) \]

      14. flip-+ N/A

          \[\leadsto \mathsf{fma}\left(y, 5, \left(\color{blue}{\frac{\left(y + z\right) \cdot \left(y + z\right) - \left(y + z\right) \cdot \left(y + z\right)}{\left(y + z\right) - \left(y + z\right)}} + t\right) \cdot x\right) \]

      15. +-inverses N/A

          \[\leadsto \mathsf{fma}\left(y, 5, \left(\frac{\color{blue}{0}}{\left(y + z\right) - \left(y + z\right)} + t\right) \cdot x\right) \]

      16. +-inverses N/A

          \[\leadsto \mathsf{fma}\left(y, 5, \left(\frac{\color{blue}{z \cdot z - z \cdot z}}{\left(y + z\right) - \left(y + z\right)} + t\right) \cdot x\right) \]

      17. +-inverses N/A

          \[\leadsto \mathsf{fma}\left(y, 5, \left(\frac{z \cdot z - z \cdot z}{\color{blue}{0}} + t\right) \cdot x\right) \]

      18. +-inverses N/A

          \[\leadsto \mathsf{fma}\left(y, 5, \left(\frac{z \cdot z - z \cdot z}{\color{blue}{z - z}} + t\right) \cdot x\right) \]

      19. flip-+ N/A

          \[\leadsto \mathsf{fma}\left(y, 5, \left(\color{blue}{\left(z + z\right)} + t\right) \cdot x\right) \]

      20. count-2 N/A

          \[\leadsto \mathsf{fma}\left(y, 5, \left(\color{blue}{2 \cdot z} + t\right) \cdot x\right) \]

      21. lower-fma.f64 99.8

          \[\leadsto \mathsf{fma}\left(y, 5, \color{blue}{\mathsf{fma}\left(2, z, t\right)} \cdot x\right) \]

   4. Applied rewrites 99.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, 5, \mathsf{fma}\left(2, z, t\right) \cdot x\right)} \]

   5. Taylor expanded in z around 0

      \[\leadsto \color{blue}{5 \cdot y + t \cdot x} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{t \cdot x + 5 \cdot y} \]

      2. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(t, x, 5 \cdot y\right)} \]

      3. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(t, x, \color{blue}{y \cdot 5}\right) \]

      4. lower-*.f64 94.8

         \[\leadsto \mathsf{fma}\left(t, x, \color{blue}{y \cdot 5}\right) \]

   7. Applied rewrites 94.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(t, x, y \cdot 5\right)} \]

   if -5.99999999999999983e-168 < x < 1.55e-59

   1. Initial program 99.9%

      \[x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5 \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{y \cdot 5 + x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \color{blue}{y \cdot 5} + x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) \]

      4. lower-fma.f64 100.0

         \[\leadsto \color{blue}{\mathsf{fma}\left(y, 5, x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right)\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, 5, \color{blue}{x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right)}\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(y, 5, \color{blue}{\left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) \cdot x}\right) \]

      7. lower-*.f64 100.0

         \[\leadsto \mathsf{fma}\left(y, 5, \color{blue}{\left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) \cdot x}\right) \]

      8. lift-+.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, 5, \color{blue}{\left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right)} \cdot x\right) \]

      9. lift-+.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, 5, \left(\color{blue}{\left(\left(\left(y + z\right) + z\right) + y\right)} + t\right) \cdot x\right) \]

      10. lift-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(y, 5, \left(\left(\color{blue}{\left(\left(y + z\right) + z\right)} + y\right) + t\right) \cdot x\right) \]

      11. associate-+l+ N/A

          \[\leadsto \mathsf{fma}\left(y, 5, \left(\color{blue}{\left(\left(y + z\right) + \left(z + y\right)\right)} + t\right) \cdot x\right) \]

      12. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(y, 5, \left(\left(\left(y + z\right) + \color{blue}{\left(y + z\right)}\right) + t\right) \cdot x\right) \]

      13. lift-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(y, 5, \left(\left(\left(y + z\right) + \color{blue}{\left(y + z\right)}\right) + t\right) \cdot x\right) \]

      14. flip-+ N/A

          \[\leadsto \mathsf{fma}\left(y, 5, \left(\color{blue}{\frac{\left(y + z\right) \cdot \left(y + z\right) - \left(y + z\right) \cdot \left(y + z\right)}{\left(y + z\right) - \left(y + z\right)}} + t\right) \cdot x\right) \]

      15. +-inverses N/A

          \[\leadsto \mathsf{fma}\left(y, 5, \left(\frac{\color{blue}{0}}{\left(y + z\right) - \left(y + z\right)} + t\right) \cdot x\right) \]

      16. +-inverses N/A

          \[\leadsto \mathsf{fma}\left(y, 5, \left(\frac{\color{blue}{z \cdot z - z \cdot z}}{\left(y + z\right) - \left(y + z\right)} + t\right) \cdot x\right) \]

      17. +-inverses N/A

          \[\leadsto \mathsf{fma}\left(y, 5, \left(\frac{z \cdot z - z \cdot z}{\color{blue}{0}} + t\right) \cdot x\right) \]

      18. +-inverses N/A

          \[\leadsto \mathsf{fma}\left(y, 5, \left(\frac{z \cdot z - z \cdot z}{\color{blue}{z - z}} + t\right) \cdot x\right) \]

      19. flip-+ N/A

          \[\leadsto \mathsf{fma}\left(y, 5, \left(\color{blue}{\left(z + z\right)} + t\right) \cdot x\right) \]

      20. count-2 N/A

          \[\leadsto \mathsf{fma}\left(y, 5, \left(\color{blue}{2 \cdot z} + t\right) \cdot x\right) \]

      21. lower-fma.f64 99.9

          \[\leadsto \mathsf{fma}\left(y, 5, \color{blue}{\mathsf{fma}\left(2, z, t\right)} \cdot x\right) \]

   4. Applied rewrites 99.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, 5, \mathsf{fma}\left(2, z, t\right) \cdot x\right)} \]

   5. Taylor expanded in t around 0

      \[\leadsto \mathsf{fma}\left(y, 5, \color{blue}{\left(2 \cdot z\right)} \cdot x\right) \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(y, 5, \color{blue}{\left(z \cdot 2\right)} \cdot x\right) \]

      2. lower-*.f64 87.9

         \[\leadsto \mathsf{fma}\left(y, 5, \color{blue}{\left(z \cdot 2\right)} \cdot x\right) \]

   7. Applied rewrites 87.9%

      \[\leadsto \mathsf{fma}\left(y, 5, \color{blue}{\left(z \cdot 2\right)} \cdot x\right) \]
   8. Step-by-step derivation

   9. Applied rewrites 87.9%

      \[\leadsto \mathsf{fma}\left(y, 5, \left(z + \color{blue}{z}\right) \cdot x\right) \]

   if 1.55e-59 < x

   1. Initial program 99.9%

      \[x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5 \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(t + \left(2 \cdot y + 2 \cdot z\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(t + \left(2 \cdot y + 2 \cdot z\right)\right) \cdot x} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(t + \left(2 \cdot y + 2 \cdot z\right)\right) \cdot x} \]

      3. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(2 \cdot y + 2 \cdot z\right) + t\right)} \cdot x \]

      4. distribute-lft-out N/A

         \[\leadsto \left(\color{blue}{2 \cdot \left(y + z\right)} + t\right) \cdot x \]

      5. *-commutative N/A

         \[\leadsto \left(\color{blue}{\left(y + z\right) \cdot 2} + t\right) \cdot x \]

      6. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(y + z, 2, t\right)} \cdot x \]

      7. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{z + y}, 2, t\right) \cdot x \]

      8. lower-+.f64 98.6

         \[\leadsto \mathsf{fma}\left(\color{blue}{z + y}, 2, t\right) \cdot x \]

   5. Applied rewrites 98.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z + y, 2, t\right) \cdot x} \]
   6. Step-by-step derivation

   7. Applied rewrites 98.7%

      \[\leadsto \mathsf{fma}\left(z, 2, \left(t + y\right) + y\right) \cdot x \]
3. Recombined 4 regimes into one program.

4. Final simplification 95.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -9.8 \cdot 10^{-30}:\\ \;\;\;\;\mathsf{fma}\left(y + z, 2, t\right) \cdot x\\ \mathbf{elif}\;x \leq -6 \cdot 10^{-168}:\\ \;\;\;\;\mathsf{fma}\left(t, x, 5 \cdot y\right)\\ \mathbf{elif}\;x \leq 1.55 \cdot 10^{-59}:\\ \;\;\;\;\mathsf{fma}\left(y, 5, \left(z + z\right) \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, 2, \left(t + y\right) + y\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 46.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(z \cdot x\right) \cdot 2\\ \mathbf{if}\;x \leq -3 \cdot 10^{+105}:\\ \;\;\;\;\left(y \cdot 2\right) \cdot x\\ \mathbf{elif}\;x \leq -9.8 \cdot 10^{-30}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 7.5 \cdot 10^{-56}:\\ \;\;\;\;5 \cdot y\\ \mathbf{elif}\;x \leq 7 \cdot 10^{+255}:\\ \;\;\;\;t \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (* z x) 2.0)))
   (if (<= x -3e+105)
     (* (* y 2.0) x)
     (if (<= x -9.8e-30)
       t_1
       (if (<= x 7.5e-56) (* 5.0 y) (if (<= x 7e+255) (* t x) t_1))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (z * x) * 2.0;
	double tmp;
	if (x <= -3e+105) {
		tmp = (y * 2.0) * x;
	} else if (x <= -9.8e-30) {
		tmp = t_1;
	} else if (x <= 7.5e-56) {
		tmp = 5.0 * y;
	} else if (x <= 7e+255) {
		tmp = t * x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (z * x) * 2.0d0
    if (x <= (-3d+105)) then
        tmp = (y * 2.0d0) * x
    else if (x <= (-9.8d-30)) then
        tmp = t_1
    else if (x <= 7.5d-56) then
        tmp = 5.0d0 * y
    else if (x <= 7d+255) then
        tmp = t * 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 t_1 = (z * x) * 2.0;
	double tmp;
	if (x <= -3e+105) {
		tmp = (y * 2.0) * x;
	} else if (x <= -9.8e-30) {
		tmp = t_1;
	} else if (x <= 7.5e-56) {
		tmp = 5.0 * y;
	} else if (x <= 7e+255) {
		tmp = t * x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (z * x) * 2.0
	tmp = 0
	if x <= -3e+105:
		tmp = (y * 2.0) * x
	elif x <= -9.8e-30:
		tmp = t_1
	elif x <= 7.5e-56:
		tmp = 5.0 * y
	elif x <= 7e+255:
		tmp = t * x
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(z * x) * 2.0)
	tmp = 0.0
	if (x <= -3e+105)
		tmp = Float64(Float64(y * 2.0) * x);
	elseif (x <= -9.8e-30)
		tmp = t_1;
	elseif (x <= 7.5e-56)
		tmp = Float64(5.0 * y);
	elseif (x <= 7e+255)
		tmp = Float64(t * x);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (z * x) * 2.0;
	tmp = 0.0;
	if (x <= -3e+105)
		tmp = (y * 2.0) * x;
	elseif (x <= -9.8e-30)
		tmp = t_1;
	elseif (x <= 7.5e-56)
		tmp = 5.0 * y;
	elseif (x <= 7e+255)
		tmp = t * x;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(z * x), $MachinePrecision] * 2.0), $MachinePrecision]}, If[LessEqual[x, -3e+105], N[(N[(y * 2.0), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[x, -9.8e-30], t$95$1, If[LessEqual[x, 7.5e-56], N[(5.0 * y), $MachinePrecision], If[LessEqual[x, 7e+255], N[(t * x), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -3.0000000000000001e105

   1. Initial program 100.0%

      \[x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5 \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(t + \left(2 \cdot y + 2 \cdot z\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(t + \left(2 \cdot y + 2 \cdot z\right)\right) \cdot x} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(t + \left(2 \cdot y + 2 \cdot z\right)\right) \cdot x} \]

      3. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(2 \cdot y + 2 \cdot z\right) + t\right)} \cdot x \]

      4. distribute-lft-out N/A

         \[\leadsto \left(\color{blue}{2 \cdot \left(y + z\right)} + t\right) \cdot x \]

      5. *-commutative N/A

         \[\leadsto \left(\color{blue}{\left(y + z\right) \cdot 2} + t\right) \cdot x \]

      6. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(y + z, 2, t\right)} \cdot x \]

      7. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{z + y}, 2, t\right) \cdot x \]

      8. lower-+.f64 100.0

         \[\leadsto \mathsf{fma}\left(\color{blue}{z + y}, 2, t\right) \cdot x \]

   5. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z + y, 2, t\right) \cdot x} \]

   6. Taylor expanded in y around inf

      \[\leadsto \left(2 \cdot y\right) \cdot x \]
   7. Step-by-step derivation

   8. Applied rewrites 46.8%

      \[\leadsto \left(y \cdot 2\right) \cdot x \]

   if -3.0000000000000001e105 < x < -9.79999999999999942e-30 or 6.99999999999999971e255 < x

   1. Initial program 100.0%

      \[x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5 \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{2 \cdot \left(x \cdot z\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(x \cdot z\right) \cdot 2} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(x \cdot z\right) \cdot 2} \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\left(z \cdot x\right)} \cdot 2 \]

      4. lower-*.f64 55.1

         \[\leadsto \color{blue}{\left(z \cdot x\right)} \cdot 2 \]

   5. Applied rewrites 55.1%

      \[\leadsto \color{blue}{\left(z \cdot x\right) \cdot 2} \]

   if -9.79999999999999942e-30 < x < 7.50000000000000041e-56

   1. Initial program 99.8%

      \[x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5 \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

         \[\leadsto \color{blue}{y \cdot 5} \]

      2. lower-*.f64 65.0

         \[\leadsto \color{blue}{y \cdot 5} \]

   5. Applied rewrites 65.0%

      \[\leadsto \color{blue}{y \cdot 5} \]

   if 7.50000000000000041e-56 < x < 6.99999999999999971e255

   1. Initial program 99.9%

      \[x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5 \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

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

      1. lower-*.f64 45.4

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

   5. Applied rewrites 45.4%

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

4. Final simplification 54.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3 \cdot 10^{+105}:\\ \;\;\;\;\left(y \cdot 2\right) \cdot x\\ \mathbf{elif}\;x \leq -9.8 \cdot 10^{-30}:\\ \;\;\;\;\left(z \cdot x\right) \cdot 2\\ \mathbf{elif}\;x \leq 7.5 \cdot 10^{-56}:\\ \;\;\;\;5 \cdot y\\ \mathbf{elif}\;x \leq 7 \cdot 10^{+255}:\\ \;\;\;\;t \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot x\right) \cdot 2\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 87.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(y + z, 2, t\right) \cdot x\\ \mathbf{if}\;x \leq -9.8 \cdot 10^{-30}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq -6 \cdot 10^{-168}:\\ \;\;\;\;\mathsf{fma}\left(t, x, 5 \cdot y\right)\\ \mathbf{elif}\;x \leq 1.55 \cdot 10^{-59}:\\ \;\;\;\;\mathsf{fma}\left(y, 5, \left(z + z\right) \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (fma (+ y z) 2.0 t) x)))
   (if (<= x -9.8e-30)
     t_1
     (if (<= x -6e-168)
       (fma t x (* 5.0 y))
       (if (<= x 1.55e-59) (fma y 5.0 (* (+ z z) x)) t_1)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fma((y + z), 2.0, t) * x;
	double tmp;
	if (x <= -9.8e-30) {
		tmp = t_1;
	} else if (x <= -6e-168) {
		tmp = fma(t, x, (5.0 * y));
	} else if (x <= 1.55e-59) {
		tmp = fma(y, 5.0, ((z + z) * x));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(fma(Float64(y + z), 2.0, t) * x)
	tmp = 0.0
	if (x <= -9.8e-30)
		tmp = t_1;
	elseif (x <= -6e-168)
		tmp = fma(t, x, Float64(5.0 * y));
	elseif (x <= 1.55e-59)
		tmp = fma(y, 5.0, Float64(Float64(z + z) * x));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(N[(y + z), $MachinePrecision] * 2.0 + t), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[x, -9.8e-30], t$95$1, If[LessEqual[x, -6e-168], N[(t * x + N[(5.0 * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.55e-59], N[(y * 5.0 + N[(N[(z + z), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if x < -9.79999999999999942e-30 or 1.55e-59 < x

   1. Initial program 100.0%

      \[x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5 \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(t + \left(2 \cdot y + 2 \cdot z\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(t + \left(2 \cdot y + 2 \cdot z\right)\right) \cdot x} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(t + \left(2 \cdot y + 2 \cdot z\right)\right) \cdot x} \]

      3. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(2 \cdot y + 2 \cdot z\right) + t\right)} \cdot x \]

      4. distribute-lft-out N/A

         \[\leadsto \left(\color{blue}{2 \cdot \left(y + z\right)} + t\right) \cdot x \]

      5. *-commutative N/A

         \[\leadsto \left(\color{blue}{\left(y + z\right) \cdot 2} + t\right) \cdot x \]

      6. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(y + z, 2, t\right)} \cdot x \]

      7. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{z + y}, 2, t\right) \cdot x \]

      8. lower-+.f64 98.3

         \[\leadsto \mathsf{fma}\left(\color{blue}{z + y}, 2, t\right) \cdot x \]

   5. Applied rewrites 98.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z + y, 2, t\right) \cdot x} \]

   if -9.79999999999999942e-30 < x < -5.99999999999999983e-168

   1. Initial program 99.8%

      \[x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5 \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{y \cdot 5 + x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \color{blue}{y \cdot 5} + x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) \]

      4. lower-fma.f64 99.9

         \[\leadsto \color{blue}{\mathsf{fma}\left(y, 5, x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right)\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, 5, \color{blue}{x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right)}\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(y, 5, \color{blue}{\left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) \cdot x}\right) \]

      7. lower-*.f64 99.9

         \[\leadsto \mathsf{fma}\left(y, 5, \color{blue}{\left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) \cdot x}\right) \]

      8. lift-+.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, 5, \color{blue}{\left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right)} \cdot x\right) \]

      9. lift-+.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, 5, \left(\color{blue}{\left(\left(\left(y + z\right) + z\right) + y\right)} + t\right) \cdot x\right) \]

      10. lift-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(y, 5, \left(\left(\color{blue}{\left(\left(y + z\right) + z\right)} + y\right) + t\right) \cdot x\right) \]

      11. associate-+l+ N/A

          \[\leadsto \mathsf{fma}\left(y, 5, \left(\color{blue}{\left(\left(y + z\right) + \left(z + y\right)\right)} + t\right) \cdot x\right) \]

      12. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(y, 5, \left(\left(\left(y + z\right) + \color{blue}{\left(y + z\right)}\right) + t\right) \cdot x\right) \]

      13. lift-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(y, 5, \left(\left(\left(y + z\right) + \color{blue}{\left(y + z\right)}\right) + t\right) \cdot x\right) \]

      14. flip-+ N/A

          \[\leadsto \mathsf{fma}\left(y, 5, \left(\color{blue}{\frac{\left(y + z\right) \cdot \left(y + z\right) - \left(y + z\right) \cdot \left(y + z\right)}{\left(y + z\right) - \left(y + z\right)}} + t\right) \cdot x\right) \]

      15. +-inverses N/A

          \[\leadsto \mathsf{fma}\left(y, 5, \left(\frac{\color{blue}{0}}{\left(y + z\right) - \left(y + z\right)} + t\right) \cdot x\right) \]

      16. +-inverses N/A

          \[\leadsto \mathsf{fma}\left(y, 5, \left(\frac{\color{blue}{z \cdot z - z \cdot z}}{\left(y + z\right) - \left(y + z\right)} + t\right) \cdot x\right) \]

      17. +-inverses N/A

          \[\leadsto \mathsf{fma}\left(y, 5, \left(\frac{z \cdot z - z \cdot z}{\color{blue}{0}} + t\right) \cdot x\right) \]

      18. +-inverses N/A

          \[\leadsto \mathsf{fma}\left(y, 5, \left(\frac{z \cdot z - z \cdot z}{\color{blue}{z - z}} + t\right) \cdot x\right) \]

      19. flip-+ N/A

          \[\leadsto \mathsf{fma}\left(y, 5, \left(\color{blue}{\left(z + z\right)} + t\right) \cdot x\right) \]

      20. count-2 N/A

          \[\leadsto \mathsf{fma}\left(y, 5, \left(\color{blue}{2 \cdot z} + t\right) \cdot x\right) \]

      21. lower-fma.f64 99.8

          \[\leadsto \mathsf{fma}\left(y, 5, \color{blue}{\mathsf{fma}\left(2, z, t\right)} \cdot x\right) \]

   4. Applied rewrites 99.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, 5, \mathsf{fma}\left(2, z, t\right) \cdot x\right)} \]

   5. Taylor expanded in z around 0

      \[\leadsto \color{blue}{5 \cdot y + t \cdot x} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{t \cdot x + 5 \cdot y} \]

      2. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(t, x, 5 \cdot y\right)} \]

      3. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(t, x, \color{blue}{y \cdot 5}\right) \]

      4. lower-*.f64 94.8

         \[\leadsto \mathsf{fma}\left(t, x, \color{blue}{y \cdot 5}\right) \]

   7. Applied rewrites 94.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(t, x, y \cdot 5\right)} \]

   if -5.99999999999999983e-168 < x < 1.55e-59

   1. Initial program 99.9%

      \[x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5 \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{y \cdot 5 + x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \color{blue}{y \cdot 5} + x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) \]

      4. lower-fma.f64 100.0

         \[\leadsto \color{blue}{\mathsf{fma}\left(y, 5, x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right)\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, 5, \color{blue}{x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right)}\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(y, 5, \color{blue}{\left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) \cdot x}\right) \]

      7. lower-*.f64 100.0

         \[\leadsto \mathsf{fma}\left(y, 5, \color{blue}{\left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) \cdot x}\right) \]

      8. lift-+.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, 5, \color{blue}{\left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right)} \cdot x\right) \]

      9. lift-+.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, 5, \left(\color{blue}{\left(\left(\left(y + z\right) + z\right) + y\right)} + t\right) \cdot x\right) \]

      10. lift-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(y, 5, \left(\left(\color{blue}{\left(\left(y + z\right) + z\right)} + y\right) + t\right) \cdot x\right) \]

      11. associate-+l+ N/A

          \[\leadsto \mathsf{fma}\left(y, 5, \left(\color{blue}{\left(\left(y + z\right) + \left(z + y\right)\right)} + t\right) \cdot x\right) \]

      12. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(y, 5, \left(\left(\left(y + z\right) + \color{blue}{\left(y + z\right)}\right) + t\right) \cdot x\right) \]

      13. lift-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(y, 5, \left(\left(\left(y + z\right) + \color{blue}{\left(y + z\right)}\right) + t\right) \cdot x\right) \]

      14. flip-+ N/A

          \[\leadsto \mathsf{fma}\left(y, 5, \left(\color{blue}{\frac{\left(y + z\right) \cdot \left(y + z\right) - \left(y + z\right) \cdot \left(y + z\right)}{\left(y + z\right) - \left(y + z\right)}} + t\right) \cdot x\right) \]

      15. +-inverses N/A

          \[\leadsto \mathsf{fma}\left(y, 5, \left(\frac{\color{blue}{0}}{\left(y + z\right) - \left(y + z\right)} + t\right) \cdot x\right) \]

      16. +-inverses N/A

          \[\leadsto \mathsf{fma}\left(y, 5, \left(\frac{\color{blue}{z \cdot z - z \cdot z}}{\left(y + z\right) - \left(y + z\right)} + t\right) \cdot x\right) \]

      17. +-inverses N/A

          \[\leadsto \mathsf{fma}\left(y, 5, \left(\frac{z \cdot z - z \cdot z}{\color{blue}{0}} + t\right) \cdot x\right) \]

      18. +-inverses N/A

          \[\leadsto \mathsf{fma}\left(y, 5, \left(\frac{z \cdot z - z \cdot z}{\color{blue}{z - z}} + t\right) \cdot x\right) \]

      19. flip-+ N/A

          \[\leadsto \mathsf{fma}\left(y, 5, \left(\color{blue}{\left(z + z\right)} + t\right) \cdot x\right) \]

      20. count-2 N/A

          \[\leadsto \mathsf{fma}\left(y, 5, \left(\color{blue}{2 \cdot z} + t\right) \cdot x\right) \]

      21. lower-fma.f64 99.9

          \[\leadsto \mathsf{fma}\left(y, 5, \color{blue}{\mathsf{fma}\left(2, z, t\right)} \cdot x\right) \]

   4. Applied rewrites 99.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, 5, \mathsf{fma}\left(2, z, t\right) \cdot x\right)} \]

   5. Taylor expanded in t around 0

      \[\leadsto \mathsf{fma}\left(y, 5, \color{blue}{\left(2 \cdot z\right)} \cdot x\right) \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(y, 5, \color{blue}{\left(z \cdot 2\right)} \cdot x\right) \]

      2. lower-*.f64 87.9

         \[\leadsto \mathsf{fma}\left(y, 5, \color{blue}{\left(z \cdot 2\right)} \cdot x\right) \]

   7. Applied rewrites 87.9%

      \[\leadsto \mathsf{fma}\left(y, 5, \color{blue}{\left(z \cdot 2\right)} \cdot x\right) \]
   8. Step-by-step derivation

   9. Applied rewrites 87.9%

      \[\leadsto \mathsf{fma}\left(y, 5, \left(z + \color{blue}{z}\right) \cdot x\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 95.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -9.8 \cdot 10^{-30}:\\ \;\;\;\;\mathsf{fma}\left(y + z, 2, t\right) \cdot x\\ \mathbf{elif}\;x \leq -6 \cdot 10^{-168}:\\ \;\;\;\;\mathsf{fma}\left(t, x, 5 \cdot y\right)\\ \mathbf{elif}\;x \leq 1.55 \cdot 10^{-59}:\\ \;\;\;\;\mathsf{fma}\left(y, 5, \left(z + z\right) \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y + z, 2, t\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 62.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\left(t + y\right) + y\right) \cdot x\\ \mathbf{if}\;x \leq -4.5 \cdot 10^{+21}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq -9.8 \cdot 10^{-30}:\\ \;\;\;\;\left(z \cdot x\right) \cdot 2\\ \mathbf{elif}\;x \leq 6.8 \cdot 10^{-56}:\\ \;\;\;\;\mathsf{fma}\left(2, x, 5\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (+ (+ t y) y) x)))
   (if (<= x -4.5e+21)
     t_1
     (if (<= x -9.8e-30)
       (* (* z x) 2.0)
       (if (<= x 6.8e-56) (* (fma 2.0 x 5.0) y) t_1)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = ((t + y) + y) * x;
	double tmp;
	if (x <= -4.5e+21) {
		tmp = t_1;
	} else if (x <= -9.8e-30) {
		tmp = (z * x) * 2.0;
	} else if (x <= 6.8e-56) {
		tmp = fma(2.0, x, 5.0) * y;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(Float64(t + y) + y) * x)
	tmp = 0.0
	if (x <= -4.5e+21)
		tmp = t_1;
	elseif (x <= -9.8e-30)
		tmp = Float64(Float64(z * x) * 2.0);
	elseif (x <= 6.8e-56)
		tmp = Float64(fma(2.0, x, 5.0) * y);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(N[(t + y), $MachinePrecision] + y), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[x, -4.5e+21], t$95$1, If[LessEqual[x, -9.8e-30], N[(N[(z * x), $MachinePrecision] * 2.0), $MachinePrecision], If[LessEqual[x, 6.8e-56], N[(N[(2.0 * x + 5.0), $MachinePrecision] * y), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.5e21 or 6.79999999999999964e-56 < x

   1. Initial program 100.0%

      \[x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5 \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(t + \left(2 \cdot y + 2 \cdot z\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(t + \left(2 \cdot y + 2 \cdot z\right)\right) \cdot x} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(t + \left(2 \cdot y + 2 \cdot z\right)\right) \cdot x} \]

      3. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(2 \cdot y + 2 \cdot z\right) + t\right)} \cdot x \]

      4. distribute-lft-out N/A

         \[\leadsto \left(\color{blue}{2 \cdot \left(y + z\right)} + t\right) \cdot x \]

      5. *-commutative N/A

         \[\leadsto \left(\color{blue}{\left(y + z\right) \cdot 2} + t\right) \cdot x \]

      6. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(y + z, 2, t\right)} \cdot x \]

      7. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{z + y}, 2, t\right) \cdot x \]

      8. lower-+.f64 99.2

         \[\leadsto \mathsf{fma}\left(\color{blue}{z + y}, 2, t\right) \cdot x \]

   5. Applied rewrites 99.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z + y, 2, t\right) \cdot x} \]

   6. Taylor expanded in z around 0

      \[\leadsto \left(t + 2 \cdot y\right) \cdot x \]
   7. Step-by-step derivation

   8. Applied rewrites 66.7%

      \[\leadsto \mathsf{fma}\left(y, 2, t\right) \cdot x \]
   9. Step-by-step derivation

   10. Applied rewrites 66.7%

       \[\leadsto \left(\left(t + y\right) + y\right) \cdot x \]

   if -4.5e21 < x < -9.79999999999999942e-30

   1. Initial program 99.9%

      \[x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5 \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{2 \cdot \left(x \cdot z\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(x \cdot z\right) \cdot 2} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(x \cdot z\right) \cdot 2} \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\left(z \cdot x\right)} \cdot 2 \]

      4. lower-*.f64 65.6

         \[\leadsto \color{blue}{\left(z \cdot x\right)} \cdot 2 \]

   5. Applied rewrites 65.6%

      \[\leadsto \color{blue}{\left(z \cdot x\right) \cdot 2} \]

   if -9.79999999999999942e-30 < x < 6.79999999999999964e-56

   1. Initial program 99.8%

      \[x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5 \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{y \cdot \left(5 + 2 \cdot x\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto y \cdot \color{blue}{\left(2 \cdot x + 5\right)} \]

      2. metadata-eval N/A

         \[\leadsto y \cdot \left(\color{blue}{\left(\mathsf{neg}\left(-2\right)\right)} \cdot x + 5\right) \]

      3. distribute-lft-neg-in N/A

         \[\leadsto y \cdot \left(\color{blue}{\left(\mathsf{neg}\left(-2 \cdot x\right)\right)} + 5\right) \]

      4. neg-sub0 N/A

         \[\leadsto y \cdot \left(\color{blue}{\left(0 - -2 \cdot x\right)} + 5\right) \]

      5. associate--r- N/A

         \[\leadsto y \cdot \color{blue}{\left(0 - \left(-2 \cdot x - 5\right)\right)} \]

      6. neg-sub0 N/A

         \[\leadsto y \cdot \color{blue}{\left(\mathsf{neg}\left(\left(-2 \cdot x - 5\right)\right)\right)} \]

      7. *-commutative N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(-2 \cdot x - 5\right)\right)\right) \cdot y} \]

      8. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(-2 \cdot x - 5\right)\right)\right) \cdot y} \]

      9. neg-sub0 N/A

         \[\leadsto \color{blue}{\left(0 - \left(-2 \cdot x - 5\right)\right)} \cdot y \]

      10. associate--r- N/A

          \[\leadsto \color{blue}{\left(\left(0 - -2 \cdot x\right) + 5\right)} \cdot y \]

      11. neg-sub0 N/A

          \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(-2 \cdot x\right)\right)} + 5\right) \cdot y \]

      12. distribute-lft-neg-in N/A

          \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(-2\right)\right) \cdot x} + 5\right) \cdot y \]

      13. metadata-eval N/A

          \[\leadsto \left(\color{blue}{2} \cdot x + 5\right) \cdot y \]

      14. lower-fma.f64 65.0

          \[\leadsto \color{blue}{\mathsf{fma}\left(2, x, 5\right)} \cdot y \]

   5. Applied rewrites 65.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(2, x, 5\right) \cdot y} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 6: 62.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\left(t + y\right) + y\right) \cdot x\\ \mathbf{if}\;x \leq -4.5 \cdot 10^{+21}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq -9.8 \cdot 10^{-30}:\\ \;\;\;\;\left(z \cdot x\right) \cdot 2\\ \mathbf{elif}\;x \leq 6.8 \cdot 10^{-56}:\\ \;\;\;\;5 \cdot y\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (+ (+ t y) y) x)))
   (if (<= x -4.5e+21)
     t_1
     (if (<= x -9.8e-30) (* (* z x) 2.0) (if (<= x 6.8e-56) (* 5.0 y) t_1)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = ((t + y) + y) * x;
	double tmp;
	if (x <= -4.5e+21) {
		tmp = t_1;
	} else if (x <= -9.8e-30) {
		tmp = (z * x) * 2.0;
	} else if (x <= 6.8e-56) {
		tmp = 5.0 * y;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = ((t + y) + y) * x
    if (x <= (-4.5d+21)) then
        tmp = t_1
    else if (x <= (-9.8d-30)) then
        tmp = (z * x) * 2.0d0
    else if (x <= 6.8d-56) then
        tmp = 5.0d0 * y
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = ((t + y) + y) * x;
	double tmp;
	if (x <= -4.5e+21) {
		tmp = t_1;
	} else if (x <= -9.8e-30) {
		tmp = (z * x) * 2.0;
	} else if (x <= 6.8e-56) {
		tmp = 5.0 * y;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = ((t + y) + y) * x
	tmp = 0
	if x <= -4.5e+21:
		tmp = t_1
	elif x <= -9.8e-30:
		tmp = (z * x) * 2.0
	elif x <= 6.8e-56:
		tmp = 5.0 * y
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(Float64(t + y) + y) * x)
	tmp = 0.0
	if (x <= -4.5e+21)
		tmp = t_1;
	elseif (x <= -9.8e-30)
		tmp = Float64(Float64(z * x) * 2.0);
	elseif (x <= 6.8e-56)
		tmp = Float64(5.0 * y);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = ((t + y) + y) * x;
	tmp = 0.0;
	if (x <= -4.5e+21)
		tmp = t_1;
	elseif (x <= -9.8e-30)
		tmp = (z * x) * 2.0;
	elseif (x <= 6.8e-56)
		tmp = 5.0 * y;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(N[(t + y), $MachinePrecision] + y), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[x, -4.5e+21], t$95$1, If[LessEqual[x, -9.8e-30], N[(N[(z * x), $MachinePrecision] * 2.0), $MachinePrecision], If[LessEqual[x, 6.8e-56], N[(5.0 * y), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.5e21 or 6.79999999999999964e-56 < x

   1. Initial program 100.0%

      \[x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5 \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(t + \left(2 \cdot y + 2 \cdot z\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(t + \left(2 \cdot y + 2 \cdot z\right)\right) \cdot x} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(t + \left(2 \cdot y + 2 \cdot z\right)\right) \cdot x} \]

      3. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(2 \cdot y + 2 \cdot z\right) + t\right)} \cdot x \]

      4. distribute-lft-out N/A

         \[\leadsto \left(\color{blue}{2 \cdot \left(y + z\right)} + t\right) \cdot x \]

      5. *-commutative N/A

         \[\leadsto \left(\color{blue}{\left(y + z\right) \cdot 2} + t\right) \cdot x \]

      6. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(y + z, 2, t\right)} \cdot x \]

      7. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{z + y}, 2, t\right) \cdot x \]

      8. lower-+.f64 99.2

         \[\leadsto \mathsf{fma}\left(\color{blue}{z + y}, 2, t\right) \cdot x \]

   5. Applied rewrites 99.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z + y, 2, t\right) \cdot x} \]

   6. Taylor expanded in z around 0

      \[\leadsto \left(t + 2 \cdot y\right) \cdot x \]
   7. Step-by-step derivation

   8. Applied rewrites 66.7%

      \[\leadsto \mathsf{fma}\left(y, 2, t\right) \cdot x \]
   9. Step-by-step derivation

   10. Applied rewrites 66.7%

       \[\leadsto \left(\left(t + y\right) + y\right) \cdot x \]

   if -4.5e21 < x < -9.79999999999999942e-30

   1. Initial program 99.9%

      \[x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5 \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{2 \cdot \left(x \cdot z\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(x \cdot z\right) \cdot 2} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(x \cdot z\right) \cdot 2} \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\left(z \cdot x\right)} \cdot 2 \]

      4. lower-*.f64 65.6

         \[\leadsto \color{blue}{\left(z \cdot x\right)} \cdot 2 \]

   5. Applied rewrites 65.6%

      \[\leadsto \color{blue}{\left(z \cdot x\right) \cdot 2} \]

   if -9.79999999999999942e-30 < x < 6.79999999999999964e-56

   1. Initial program 99.8%

      \[x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5 \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

         \[\leadsto \color{blue}{y \cdot 5} \]

      2. lower-*.f64 65.0

         \[\leadsto \color{blue}{y \cdot 5} \]

   5. Applied rewrites 65.0%

      \[\leadsto \color{blue}{y \cdot 5} \]
3. Recombined 3 regimes into one program.

4. Final simplification 66.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.5 \cdot 10^{+21}:\\ \;\;\;\;\left(\left(t + y\right) + y\right) \cdot x\\ \mathbf{elif}\;x \leq -9.8 \cdot 10^{-30}:\\ \;\;\;\;\left(z \cdot x\right) \cdot 2\\ \mathbf{elif}\;x \leq 6.8 \cdot 10^{-56}:\\ \;\;\;\;5 \cdot y\\ \mathbf{else}:\\ \;\;\;\;\left(\left(t + y\right) + y\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 99.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -175000:\\ \;\;\;\;\mathsf{fma}\left(y + z, 2, t\right) \cdot x\\ \mathbf{elif}\;x \leq 0.7:\\ \;\;\;\;\mathsf{fma}\left(y, 5, \mathsf{fma}\left(2, z, t\right) \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, 2, \left(t + y\right) + y\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= x -175000.0)
   (* (fma (+ y z) 2.0 t) x)
   (if (<= x 0.7)
     (fma y 5.0 (* (fma 2.0 z t) x))
     (* (fma z 2.0 (+ (+ t y) y)) x))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -175000.0) {
		tmp = fma((y + z), 2.0, t) * x;
	} else if (x <= 0.7) {
		tmp = fma(y, 5.0, (fma(2.0, z, t) * x));
	} else {
		tmp = fma(z, 2.0, ((t + y) + y)) * x;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x <= -175000.0)
		tmp = Float64(fma(Float64(y + z), 2.0, t) * x);
	elseif (x <= 0.7)
		tmp = fma(y, 5.0, Float64(fma(2.0, z, t) * x));
	else
		tmp = Float64(fma(z, 2.0, Float64(Float64(t + y) + y)) * x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[x, -175000.0], N[(N[(N[(y + z), $MachinePrecision] * 2.0 + t), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[x, 0.7], N[(y * 5.0 + N[(N[(2.0 * z + t), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision], N[(N[(z * 2.0 + N[(N[(t + y), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -175000

   1. Initial program 100.0%

      \[x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5 \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(t + \left(2 \cdot y + 2 \cdot z\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(t + \left(2 \cdot y + 2 \cdot z\right)\right) \cdot x} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(t + \left(2 \cdot y + 2 \cdot z\right)\right) \cdot x} \]

      3. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(2 \cdot y + 2 \cdot z\right) + t\right)} \cdot x \]

      4. distribute-lft-out N/A

         \[\leadsto \left(\color{blue}{2 \cdot \left(y + z\right)} + t\right) \cdot x \]

      5. *-commutative N/A

         \[\leadsto \left(\color{blue}{\left(y + z\right) \cdot 2} + t\right) \cdot x \]

      6. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(y + z, 2, t\right)} \cdot x \]

      7. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{z + y}, 2, t\right) \cdot x \]

      8. lower-+.f64 99.2

         \[\leadsto \mathsf{fma}\left(\color{blue}{z + y}, 2, t\right) \cdot x \]

   5. Applied rewrites 99.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z + y, 2, t\right) \cdot x} \]

   if -175000 < x < 0.69999999999999996

   1. Initial program 99.8%

      \[x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5 \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{y \cdot 5 + x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \color{blue}{y \cdot 5} + x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) \]

      4. lower-fma.f64 100.0

         \[\leadsto \color{blue}{\mathsf{fma}\left(y, 5, x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right)\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, 5, \color{blue}{x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right)}\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(y, 5, \color{blue}{\left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) \cdot x}\right) \]

      7. lower-*.f64 100.0

         \[\leadsto \mathsf{fma}\left(y, 5, \color{blue}{\left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) \cdot x}\right) \]

      8. lift-+.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, 5, \color{blue}{\left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right)} \cdot x\right) \]

      9. lift-+.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, 5, \left(\color{blue}{\left(\left(\left(y + z\right) + z\right) + y\right)} + t\right) \cdot x\right) \]

      10. lift-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(y, 5, \left(\left(\color{blue}{\left(\left(y + z\right) + z\right)} + y\right) + t\right) \cdot x\right) \]

      11. associate-+l+ N/A

          \[\leadsto \mathsf{fma}\left(y, 5, \left(\color{blue}{\left(\left(y + z\right) + \left(z + y\right)\right)} + t\right) \cdot x\right) \]

      12. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(y, 5, \left(\left(\left(y + z\right) + \color{blue}{\left(y + z\right)}\right) + t\right) \cdot x\right) \]

      13. lift-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(y, 5, \left(\left(\left(y + z\right) + \color{blue}{\left(y + z\right)}\right) + t\right) \cdot x\right) \]

      14. flip-+ N/A

          \[\leadsto \mathsf{fma}\left(y, 5, \left(\color{blue}{\frac{\left(y + z\right) \cdot \left(y + z\right) - \left(y + z\right) \cdot \left(y + z\right)}{\left(y + z\right) - \left(y + z\right)}} + t\right) \cdot x\right) \]

      15. +-inverses N/A

          \[\leadsto \mathsf{fma}\left(y, 5, \left(\frac{\color{blue}{0}}{\left(y + z\right) - \left(y + z\right)} + t\right) \cdot x\right) \]

      16. +-inverses N/A

          \[\leadsto \mathsf{fma}\left(y, 5, \left(\frac{\color{blue}{z \cdot z - z \cdot z}}{\left(y + z\right) - \left(y + z\right)} + t\right) \cdot x\right) \]

      17. +-inverses N/A

          \[\leadsto \mathsf{fma}\left(y, 5, \left(\frac{z \cdot z - z \cdot z}{\color{blue}{0}} + t\right) \cdot x\right) \]

      18. +-inverses N/A

          \[\leadsto \mathsf{fma}\left(y, 5, \left(\frac{z \cdot z - z \cdot z}{\color{blue}{z - z}} + t\right) \cdot x\right) \]

      19. flip-+ N/A

          \[\leadsto \mathsf{fma}\left(y, 5, \left(\color{blue}{\left(z + z\right)} + t\right) \cdot x\right) \]

      20. count-2 N/A

          \[\leadsto \mathsf{fma}\left(y, 5, \left(\color{blue}{2 \cdot z} + t\right) \cdot x\right) \]

      21. lower-fma.f64 99.1

          \[\leadsto \mathsf{fma}\left(y, 5, \color{blue}{\mathsf{fma}\left(2, z, t\right)} \cdot x\right) \]

   4. Applied rewrites 99.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, 5, \mathsf{fma}\left(2, z, t\right) \cdot x\right)} \]

   if 0.69999999999999996 < x

   1. Initial program 99.9%

      \[x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5 \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(t + \left(2 \cdot y + 2 \cdot z\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(t + \left(2 \cdot y + 2 \cdot z\right)\right) \cdot x} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(t + \left(2 \cdot y + 2 \cdot z\right)\right) \cdot x} \]

      3. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(2 \cdot y + 2 \cdot z\right) + t\right)} \cdot x \]

      4. distribute-lft-out N/A

         \[\leadsto \left(\color{blue}{2 \cdot \left(y + z\right)} + t\right) \cdot x \]

      5. *-commutative N/A

         \[\leadsto \left(\color{blue}{\left(y + z\right) \cdot 2} + t\right) \cdot x \]

      6. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(y + z, 2, t\right)} \cdot x \]

      7. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{z + y}, 2, t\right) \cdot x \]

      8. lower-+.f64 99.6

         \[\leadsto \mathsf{fma}\left(\color{blue}{z + y}, 2, t\right) \cdot x \]

   5. Applied rewrites 99.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z + y, 2, t\right) \cdot x} \]
   6. Step-by-step derivation

   7. Applied rewrites 99.6%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -175000:\\ \;\;\;\;\mathsf{fma}\left(y + z, 2, t\right) \cdot x\\ \mathbf{elif}\;x \leq 0.7:\\ \;\;\;\;\mathsf{fma}\left(y, 5, \mathsf{fma}\left(2, z, t\right) \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, 2, \left(t + y\right) + y\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 88.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(y + z, 2, t\right) \cdot x\\ \mathbf{if}\;x \leq -9.8 \cdot 10^{-30}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 3.25 \cdot 10^{-105}:\\ \;\;\;\;\mathsf{fma}\left(y, 5, t \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (fma (+ y z) 2.0 t) x)))
   (if (<= x -9.8e-30) t_1 (if (<= x 3.25e-105) (fma y 5.0 (* t x)) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fma((y + z), 2.0, t) * x;
	double tmp;
	if (x <= -9.8e-30) {
		tmp = t_1;
	} else if (x <= 3.25e-105) {
		tmp = fma(y, 5.0, (t * x));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(fma(Float64(y + z), 2.0, t) * x)
	tmp = 0.0
	if (x <= -9.8e-30)
		tmp = t_1;
	elseif (x <= 3.25e-105)
		tmp = fma(y, 5.0, Float64(t * x));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(N[(y + z), $MachinePrecision] * 2.0 + t), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[x, -9.8e-30], t$95$1, If[LessEqual[x, 3.25e-105], N[(y * 5.0 + N[(t * x), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -9.79999999999999942e-30 or 3.25000000000000003e-105 < x

   1. Initial program 100.0%

      \[x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5 \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(t + \left(2 \cdot y + 2 \cdot z\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(t + \left(2 \cdot y + 2 \cdot z\right)\right) \cdot x} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(t + \left(2 \cdot y + 2 \cdot z\right)\right) \cdot x} \]

      3. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(2 \cdot y + 2 \cdot z\right) + t\right)} \cdot x \]

      4. distribute-lft-out N/A

         \[\leadsto \left(\color{blue}{2 \cdot \left(y + z\right)} + t\right) \cdot x \]

      5. *-commutative N/A

         \[\leadsto \left(\color{blue}{\left(y + z\right) \cdot 2} + t\right) \cdot x \]

      6. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(y + z, 2, t\right)} \cdot x \]

      7. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{z + y}, 2, t\right) \cdot x \]

      8. lower-+.f64 97.2

         \[\leadsto \mathsf{fma}\left(\color{blue}{z + y}, 2, t\right) \cdot x \]

   5. Applied rewrites 97.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z + y, 2, t\right) \cdot x} \]

   if -9.79999999999999942e-30 < x < 3.25000000000000003e-105

   1. Initial program 99.8%

      \[x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5 \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{y \cdot 5 + x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \color{blue}{y \cdot 5} + x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) \]

      4. lower-fma.f64 100.0

         \[\leadsto \color{blue}{\mathsf{fma}\left(y, 5, x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right)\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, 5, \color{blue}{x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right)}\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(y, 5, \color{blue}{\left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) \cdot x}\right) \]

      7. lower-*.f64 100.0

         \[\leadsto \mathsf{fma}\left(y, 5, \color{blue}{\left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) \cdot x}\right) \]

      8. lift-+.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, 5, \color{blue}{\left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right)} \cdot x\right) \]

      9. lift-+.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, 5, \left(\color{blue}{\left(\left(\left(y + z\right) + z\right) + y\right)} + t\right) \cdot x\right) \]

      10. lift-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(y, 5, \left(\left(\color{blue}{\left(\left(y + z\right) + z\right)} + y\right) + t\right) \cdot x\right) \]

      11. associate-+l+ N/A

          \[\leadsto \mathsf{fma}\left(y, 5, \left(\color{blue}{\left(\left(y + z\right) + \left(z + y\right)\right)} + t\right) \cdot x\right) \]

      12. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(y, 5, \left(\left(\left(y + z\right) + \color{blue}{\left(y + z\right)}\right) + t\right) \cdot x\right) \]

      13. lift-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(y, 5, \left(\left(\left(y + z\right) + \color{blue}{\left(y + z\right)}\right) + t\right) \cdot x\right) \]

      14. flip-+ N/A

          \[\leadsto \mathsf{fma}\left(y, 5, \left(\color{blue}{\frac{\left(y + z\right) \cdot \left(y + z\right) - \left(y + z\right) \cdot \left(y + z\right)}{\left(y + z\right) - \left(y + z\right)}} + t\right) \cdot x\right) \]

      15. +-inverses N/A

          \[\leadsto \mathsf{fma}\left(y, 5, \left(\frac{\color{blue}{0}}{\left(y + z\right) - \left(y + z\right)} + t\right) \cdot x\right) \]

      16. +-inverses N/A

          \[\leadsto \mathsf{fma}\left(y, 5, \left(\frac{\color{blue}{z \cdot z - z \cdot z}}{\left(y + z\right) - \left(y + z\right)} + t\right) \cdot x\right) \]

      17. +-inverses N/A

          \[\leadsto \mathsf{fma}\left(y, 5, \left(\frac{z \cdot z - z \cdot z}{\color{blue}{0}} + t\right) \cdot x\right) \]

      18. +-inverses N/A

          \[\leadsto \mathsf{fma}\left(y, 5, \left(\frac{z \cdot z - z \cdot z}{\color{blue}{z - z}} + t\right) \cdot x\right) \]

      19. flip-+ N/A

          \[\leadsto \mathsf{fma}\left(y, 5, \left(\color{blue}{\left(z + z\right)} + t\right) \cdot x\right) \]

      20. count-2 N/A

          \[\leadsto \mathsf{fma}\left(y, 5, \left(\color{blue}{2 \cdot z} + t\right) \cdot x\right) \]

      21. lower-fma.f64 99.9

          \[\leadsto \mathsf{fma}\left(y, 5, \color{blue}{\mathsf{fma}\left(2, z, t\right)} \cdot x\right) \]

   4. Applied rewrites 99.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, 5, \mathsf{fma}\left(2, z, t\right) \cdot x\right)} \]

   5. Taylor expanded in t around inf

      \[\leadsto \mathsf{fma}\left(y, 5, \color{blue}{t \cdot x}\right) \]
   6. Step-by-step derivation

      1. lower-*.f64 83.5

         \[\leadsto \mathsf{fma}\left(y, 5, \color{blue}{t \cdot x}\right) \]

   7. Applied rewrites 83.5%

      \[\leadsto \mathsf{fma}\left(y, 5, \color{blue}{t \cdot x}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 92.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -9.8 \cdot 10^{-30}:\\ \;\;\;\;\mathsf{fma}\left(y + z, 2, t\right) \cdot x\\ \mathbf{elif}\;x \leq 3.25 \cdot 10^{-105}:\\ \;\;\;\;\mathsf{fma}\left(y, 5, t \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y + z, 2, t\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) \cdot x + 5 \cdot y \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (+ (* (+ (+ (+ (+ y z) z) y) t) x) (* 5.0 y)))

C

double code(double x, double y, double z, double t) {
	return (((((y + z) + z) + y) + t) * x) + (5.0 * y);
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = (((((y + z) + z) + y) + t) * x) + (5.0d0 * y)
end function

Java

public static double code(double x, double y, double z, double t) {
	return (((((y + z) + z) + y) + t) * x) + (5.0 * y);
}

Python

def code(x, y, z, t):
	return (((((y + z) + z) + y) + t) * x) + (5.0 * y)

Julia

function code(x, y, z, t)
	return Float64(Float64(Float64(Float64(Float64(Float64(y + z) + z) + y) + t) * x) + Float64(5.0 * y))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = (((((y + z) + z) + y) + t) * x) + (5.0 * y);
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(N[(N[(N[(N[(y + z), $MachinePrecision] + z), $MachinePrecision] + y), $MachinePrecision] + t), $MachinePrecision] * x), $MachinePrecision] + N[(5.0 * y), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.9%

   \[x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5 \]
2. Add Preprocessing

3. Final simplification 99.9%

   \[\leadsto \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) \cdot x + 5 \cdot y \]
4. Add Preprocessing

Alternative 10: 79.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(2, x, 5\right) \cdot y\\ \mathbf{if}\;y \leq -1.3 \cdot 10^{+57}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 3.7 \cdot 10^{+52}:\\ \;\;\;\;\mathsf{fma}\left(z, 2, t\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (fma 2.0 x 5.0) y)))
   (if (<= y -1.3e+57) t_1 (if (<= y 3.7e+52) (* (fma z 2.0 t) x) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fma(2.0, x, 5.0) * y;
	double tmp;
	if (y <= -1.3e+57) {
		tmp = t_1;
	} else if (y <= 3.7e+52) {
		tmp = fma(z, 2.0, t) * x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(fma(2.0, x, 5.0) * y)
	tmp = 0.0
	if (y <= -1.3e+57)
		tmp = t_1;
	elseif (y <= 3.7e+52)
		tmp = Float64(fma(z, 2.0, t) * x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(2.0 * x + 5.0), $MachinePrecision] * y), $MachinePrecision]}, If[LessEqual[y, -1.3e+57], t$95$1, If[LessEqual[y, 3.7e+52], N[(N[(z * 2.0 + t), $MachinePrecision] * x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.3e57 or 3.7e52 < y

   1. Initial program 99.9%

      \[x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5 \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{y \cdot \left(5 + 2 \cdot x\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto y \cdot \color{blue}{\left(2 \cdot x + 5\right)} \]

      2. metadata-eval N/A

         \[\leadsto y \cdot \left(\color{blue}{\left(\mathsf{neg}\left(-2\right)\right)} \cdot x + 5\right) \]

      3. distribute-lft-neg-in N/A

         \[\leadsto y \cdot \left(\color{blue}{\left(\mathsf{neg}\left(-2 \cdot x\right)\right)} + 5\right) \]

      4. neg-sub0 N/A

         \[\leadsto y \cdot \left(\color{blue}{\left(0 - -2 \cdot x\right)} + 5\right) \]

      5. associate--r- N/A

         \[\leadsto y \cdot \color{blue}{\left(0 - \left(-2 \cdot x - 5\right)\right)} \]

      6. neg-sub0 N/A

         \[\leadsto y \cdot \color{blue}{\left(\mathsf{neg}\left(\left(-2 \cdot x - 5\right)\right)\right)} \]

      7. *-commutative N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(-2 \cdot x - 5\right)\right)\right) \cdot y} \]

      8. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(-2 \cdot x - 5\right)\right)\right) \cdot y} \]

      9. neg-sub0 N/A

         \[\leadsto \color{blue}{\left(0 - \left(-2 \cdot x - 5\right)\right)} \cdot y \]

      10. associate--r- N/A

          \[\leadsto \color{blue}{\left(\left(0 - -2 \cdot x\right) + 5\right)} \cdot y \]

      11. neg-sub0 N/A

          \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(-2 \cdot x\right)\right)} + 5\right) \cdot y \]

      12. distribute-lft-neg-in N/A

          \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(-2\right)\right) \cdot x} + 5\right) \cdot y \]

      13. metadata-eval N/A

          \[\leadsto \left(\color{blue}{2} \cdot x + 5\right) \cdot y \]

      14. lower-fma.f64 82.5

          \[\leadsto \color{blue}{\mathsf{fma}\left(2, x, 5\right)} \cdot y \]

   5. Applied rewrites 82.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(2, x, 5\right) \cdot y} \]

   if -1.3e57 < y < 3.7e52

   1. Initial program 99.9%

      \[x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5 \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{x \cdot \left(t + 2 \cdot z\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(t + 2 \cdot z\right) \cdot x} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(t + 2 \cdot z\right) \cdot x} \]

      3. +-commutative N/A

         \[\leadsto \color{blue}{\left(2 \cdot z + t\right)} \cdot x \]

      4. *-commutative N/A

         \[\leadsto \left(\color{blue}{z \cdot 2} + t\right) \cdot x \]

      5. lower-fma.f64 79.9

         \[\leadsto \color{blue}{\mathsf{fma}\left(z, 2, t\right)} \cdot x \]

   5. Applied rewrites 79.9%

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

Alternative 11: 46.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.5:\\ \;\;\;\;\left(y \cdot 2\right) \cdot x\\ \mathbf{elif}\;x \leq 7.5 \cdot 10^{-56}:\\ \;\;\;\;5 \cdot y\\ \mathbf{else}:\\ \;\;\;\;t \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= x -2.5) (* (* y 2.0) x) (if (<= x 7.5e-56) (* 5.0 y) (* t x))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -2.5) {
		tmp = (y * 2.0) * x;
	} else if (x <= 7.5e-56) {
		tmp = 5.0 * y;
	} else {
		tmp = t * x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (x <= (-2.5d0)) then
        tmp = (y * 2.0d0) * x
    else if (x <= 7.5d-56) then
        tmp = 5.0d0 * y
    else
        tmp = t * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -2.5) {
		tmp = (y * 2.0) * x;
	} else if (x <= 7.5e-56) {
		tmp = 5.0 * y;
	} else {
		tmp = t * x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if x <= -2.5:
		tmp = (y * 2.0) * x
	elif x <= 7.5e-56:
		tmp = 5.0 * y
	else:
		tmp = t * x
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x <= -2.5)
		tmp = Float64(Float64(y * 2.0) * x);
	elseif (x <= 7.5e-56)
		tmp = Float64(5.0 * y);
	else
		tmp = Float64(t * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (x <= -2.5)
		tmp = (y * 2.0) * x;
	elseif (x <= 7.5e-56)
		tmp = 5.0 * y;
	else
		tmp = t * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[x, -2.5], N[(N[(y * 2.0), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[x, 7.5e-56], N[(5.0 * y), $MachinePrecision], N[(t * x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.5

   1. Initial program 100.0%

      \[x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5 \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(t + \left(2 \cdot y + 2 \cdot z\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(t + \left(2 \cdot y + 2 \cdot z\right)\right) \cdot x} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(t + \left(2 \cdot y + 2 \cdot z\right)\right) \cdot x} \]

      3. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(2 \cdot y + 2 \cdot z\right) + t\right)} \cdot x \]

      4. distribute-lft-out N/A

         \[\leadsto \left(\color{blue}{2 \cdot \left(y + z\right)} + t\right) \cdot x \]

      5. *-commutative N/A

         \[\leadsto \left(\color{blue}{\left(y + z\right) \cdot 2} + t\right) \cdot x \]

      6. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(y + z, 2, t\right)} \cdot x \]

      7. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{z + y}, 2, t\right) \cdot x \]

      8. lower-+.f64 99.2

         \[\leadsto \mathsf{fma}\left(\color{blue}{z + y}, 2, t\right) \cdot x \]

   5. Applied rewrites 99.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z + y, 2, t\right) \cdot x} \]

   6. Taylor expanded in y around inf

      \[\leadsto \left(2 \cdot y\right) \cdot x \]
   7. Step-by-step derivation

   8. Applied rewrites 40.4%

      \[\leadsto \left(y \cdot 2\right) \cdot x \]

   if -2.5 < x < 7.50000000000000041e-56

   1. Initial program 99.8%

      \[x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5 \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

         \[\leadsto \color{blue}{y \cdot 5} \]

      2. lower-*.f64 61.1

         \[\leadsto \color{blue}{y \cdot 5} \]

   5. Applied rewrites 61.1%

      \[\leadsto \color{blue}{y \cdot 5} \]

   if 7.50000000000000041e-56 < x

   1. Initial program 99.9%

      \[x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5 \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

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

      1. lower-*.f64 42.5

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

   5. Applied rewrites 42.5%

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

4. Final simplification 49.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.5:\\ \;\;\;\;\left(y \cdot 2\right) \cdot x\\ \mathbf{elif}\;x \leq 7.5 \cdot 10^{-56}:\\ \;\;\;\;5 \cdot y\\ \mathbf{else}:\\ \;\;\;\;t \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 46.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.8 \cdot 10^{-63}:\\ \;\;\;\;t \cdot x\\ \mathbf{elif}\;x \leq 7.5 \cdot 10^{-56}:\\ \;\;\;\;5 \cdot y\\ \mathbf{else}:\\ \;\;\;\;t \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= x -1.8e-63) (* t x) (if (<= x 7.5e-56) (* 5.0 y) (* t x))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -1.8e-63) {
		tmp = t * x;
	} else if (x <= 7.5e-56) {
		tmp = 5.0 * y;
	} else {
		tmp = t * x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (x <= (-1.8d-63)) then
        tmp = t * x
    else if (x <= 7.5d-56) then
        tmp = 5.0d0 * y
    else
        tmp = t * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -1.8e-63) {
		tmp = t * x;
	} else if (x <= 7.5e-56) {
		tmp = 5.0 * y;
	} else {
		tmp = t * x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if x <= -1.8e-63:
		tmp = t * x
	elif x <= 7.5e-56:
		tmp = 5.0 * y
	else:
		tmp = t * x
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x <= -1.8e-63)
		tmp = Float64(t * x);
	elseif (x <= 7.5e-56)
		tmp = Float64(5.0 * y);
	else
		tmp = Float64(t * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (x <= -1.8e-63)
		tmp = t * x;
	elseif (x <= 7.5e-56)
		tmp = 5.0 * y;
	else
		tmp = t * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[x, -1.8e-63], N[(t * x), $MachinePrecision], If[LessEqual[x, 7.5e-56], N[(5.0 * y), $MachinePrecision], N[(t * x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.80000000000000004e-63 or 7.50000000000000041e-56 < x

   1. Initial program 99.9%

      \[x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5 \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

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

      1. lower-*.f64 36.8

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

   5. Applied rewrites 36.8%

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

   if -1.80000000000000004e-63 < x < 7.50000000000000041e-56

   1. Initial program 99.8%

      \[x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5 \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

         \[\leadsto \color{blue}{y \cdot 5} \]

      2. lower-*.f64 66.8

         \[\leadsto \color{blue}{y \cdot 5} \]

   5. Applied rewrites 66.8%

      \[\leadsto \color{blue}{y \cdot 5} \]
3. Recombined 2 regimes into one program.

4. Final simplification 47.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.8 \cdot 10^{-63}:\\ \;\;\;\;t \cdot x\\ \mathbf{elif}\;x \leq 7.5 \cdot 10^{-56}:\\ \;\;\;\;5 \cdot y\\ \mathbf{else}:\\ \;\;\;\;t \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 30.3% accurate, 4.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_] := N[(t * x), $MachinePrecision]

Derivation

1. Initial program 99.9%

   \[x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5 \]
2. Add Preprocessing

3. Taylor expanded in t around inf

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

   1. lower-*.f64 29.3

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

5. Applied rewrites 29.3%

   \[\leadsto \color{blue}{t \cdot x} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024254 
(FPCore (x y z t)
  :name "Graphics.Rendering.Plot.Render.Plot.Legend:renderLegendOutside from plot-0.2.3.4, B"
  :precision binary64
  (+ (* x (+ (+ (+ (+ y z) z) y) t)) (* y 5.0)))