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

Percentage Accurate: 99.8% → 99.8%

Time: 9.0s

Alternatives: 11

Speedup: 1.0×

• 26.4% 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: 99.8% 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: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z, double t) {
	return (x * ((y + (z + (y + z))) + 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 + (y + z))) + t)) + (y * 5.0d0)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

Alternative 2: 88.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \mathsf{fma}\left(2, y + z, t\right)\\ \mathbf{if}\;x \leq -1.05 \cdot 10^{-54}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 9 \cdot 10^{-95}:\\ \;\;\;\;\mathsf{fma}\left(x, z \cdot 2, y \cdot 5\right)\\ \mathbf{elif}\;x \leq 5 \cdot 10^{-18}:\\ \;\;\;\;\mathsf{fma}\left(y, 5, x \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* x (fma 2.0 (+ y z) t))))
   (if (<= x -1.05e-54)
     t_1
     (if (<= x 9e-95)
       (fma x (* z 2.0) (* y 5.0))
       (if (<= x 5e-18) (fma y 5.0 (* x t)) t_1)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x * fma(2.0, (y + z), t);
	double tmp;
	if (x <= -1.05e-54) {
		tmp = t_1;
	} else if (x <= 9e-95) {
		tmp = fma(x, (z * 2.0), (y * 5.0));
	} else if (x <= 5e-18) {
		tmp = fma(y, 5.0, (x * t));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(x * fma(2.0, Float64(y + z), t))
	tmp = 0.0
	if (x <= -1.05e-54)
		tmp = t_1;
	elseif (x <= 9e-95)
		tmp = fma(x, Float64(z * 2.0), Float64(y * 5.0));
	elseif (x <= 5e-18)
		tmp = fma(y, 5.0, Float64(x * t));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x * N[(2.0 * N[(y + z), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.05e-54], t$95$1, If[LessEqual[x, 9e-95], N[(x * N[(z * 2.0), $MachinePrecision] + N[(y * 5.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 5e-18], N[(y * 5.0 + N[(x * t), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.05e-54 or 5.00000000000000036e-18 < 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. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. distribute-lft-out N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-+.f64 98.2

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

   5. Applied rewrites 98.2%

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

   if -1.05e-54 < x < 9e-95

   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

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

      6. lift-+.f64 N/A

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

      7. associate-+l+ N/A

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

      8. +-commutative N/A

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

      9. lift-+.f64 N/A

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

      10. flip-+ N/A

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

      11. +-inverses N/A

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

      12. +-inverses N/A

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

      13. +-inverses N/A

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

      14. +-inverses N/A

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

      15. flip-+ N/A

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

      16. lower-+.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in t around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 84.5

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

   7. Applied rewrites 84.5%

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

   if 9e-95 < x < 5.00000000000000036e-18

   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. 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

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

      6. lift-+.f64 N/A

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

      7. associate-+l+ N/A

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

      8. +-commutative N/A

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

      9. lift-+.f64 N/A

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

      10. flip-+ N/A

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

      11. +-inverses N/A

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

      12. +-inverses N/A

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

      13. +-inverses N/A

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

      14. +-inverses N/A

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

      15. flip-+ N/A

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

      16. lower-+.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 100.0

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

   7. Applied rewrites 100.0%

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

4. Final simplification 93.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.05 \cdot 10^{-54}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(2, y + z, t\right)\\ \mathbf{elif}\;x \leq 9 \cdot 10^{-95}:\\ \;\;\;\;\mathsf{fma}\left(x, z \cdot 2, y \cdot 5\right)\\ \mathbf{elif}\;x \leq 5 \cdot 10^{-18}:\\ \;\;\;\;\mathsf{fma}\left(y, 5, x \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(2, y + z, t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 78.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y \cdot \mathsf{fma}\left(x, 2, 5\right)\\ \mathbf{if}\;y \leq -1.05 \cdot 10^{+83}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq -5.2 \cdot 10^{-12}:\\ \;\;\;\;\mathsf{fma}\left(y, 5, x \cdot t\right)\\ \mathbf{elif}\;y \leq 1.56 \cdot 10^{+133}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(2, z, t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* y (fma x 2.0 5.0))))
   (if (<= y -1.05e+83)
     t_1
     (if (<= y -5.2e-12)
       (fma y 5.0 (* x t))
       (if (<= y 1.56e+133) (* x (fma 2.0 z t)) t_1)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = y * fma(x, 2.0, 5.0);
	double tmp;
	if (y <= -1.05e+83) {
		tmp = t_1;
	} else if (y <= -5.2e-12) {
		tmp = fma(y, 5.0, (x * t));
	} else if (y <= 1.56e+133) {
		tmp = x * fma(2.0, z, t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(y * fma(x, 2.0, 5.0))
	tmp = 0.0
	if (y <= -1.05e+83)
		tmp = t_1;
	elseif (y <= -5.2e-12)
		tmp = fma(y, 5.0, Float64(x * t));
	elseif (y <= 1.56e+133)
		tmp = Float64(x * fma(2.0, z, t));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(y * N[(x * 2.0 + 5.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.05e+83], t$95$1, If[LessEqual[y, -5.2e-12], N[(y * 5.0 + N[(x * t), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.56e+133], N[(x * N[(2.0 * z + t), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.05000000000000001e83 or 1.56e133 < y

   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. lower-*.f64 N/A

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

      8. neg-sub0 N/A

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

      9. associate--r- N/A

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

      10. neg-sub0 N/A

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

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

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

      12. metadata-eval N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 78.1

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

   5. Applied rewrites 78.1%

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

   if -1.05000000000000001e83 < y < -5.19999999999999965e-12

   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 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

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

      6. lift-+.f64 N/A

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

      7. associate-+l+ N/A

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

      8. +-commutative N/A

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

      9. lift-+.f64 N/A

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

      10. flip-+ N/A

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

      11. +-inverses N/A

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

      12. +-inverses N/A

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

      13. +-inverses N/A

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

      14. +-inverses N/A

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

      15. flip-+ N/A

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

      16. lower-+.f64 94.2

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

   4. Applied rewrites 94.2%

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

   5. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 92.0

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

   7. Applied rewrites 92.0%

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

   if -5.19999999999999965e-12 < y < 1.56e133

   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 y around 0

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 79.6

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

   5. Applied rewrites 79.6%

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

Alternative 4: 98.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \mathsf{fma}\left(2, y + z, t\right)\\ \mathbf{if}\;x \leq -34000000000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 6.4 \cdot 10^{-18}:\\ \;\;\;\;\mathsf{fma}\left(y, 5, x \cdot \left(z + \left(z + t\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* x (fma 2.0 (+ y z) t))))
   (if (<= x -34000000000000.0)
     t_1
     (if (<= x 6.4e-18) (fma y 5.0 (* x (+ z (+ z t)))) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x * fma(2.0, (y + z), t);
	double tmp;
	if (x <= -34000000000000.0) {
		tmp = t_1;
	} else if (x <= 6.4e-18) {
		tmp = fma(y, 5.0, (x * (z + (z + t))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(x * fma(2.0, Float64(y + z), t))
	tmp = 0.0
	if (x <= -34000000000000.0)
		tmp = t_1;
	elseif (x <= 6.4e-18)
		tmp = fma(y, 5.0, Float64(x * Float64(z + Float64(z + t))));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x * N[(2.0 * N[(y + z), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -34000000000000.0], t$95$1, If[LessEqual[x, 6.4e-18], N[(y * 5.0 + N[(x * N[(z + N[(z + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -3.4e13 or 6.3999999999999998e-18 < 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. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. distribute-lft-out N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-+.f64 99.4

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

   5. Applied rewrites 99.4%

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

   if -3.4e13 < x < 6.3999999999999998e-18

   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

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

      6. lift-+.f64 N/A

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

      7. associate-+l+ N/A

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

      8. +-commutative N/A

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

      9. lift-+.f64 N/A

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

      10. flip-+ N/A

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

      11. +-inverses N/A

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

      12. +-inverses N/A

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

      13. +-inverses N/A

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

      14. +-inverses N/A

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

      15. flip-+ N/A

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

      16. lower-+.f64 99.9

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

   4. Applied rewrites 99.9%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. associate-+l+ N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. lower-+.f64 99.9

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

   6. Applied rewrites 99.9%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -34000000000000:\\ \;\;\;\;x \cdot \mathsf{fma}\left(2, y + z, t\right)\\ \mathbf{elif}\;x \leq 6.4 \cdot 10^{-18}:\\ \;\;\;\;\mathsf{fma}\left(y, 5, x \cdot \left(z + \left(z + t\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(2, y + z, t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 88.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \mathsf{fma}\left(2, y + z, t\right)\\ \mathbf{if}\;x \leq -2.5 \cdot 10^{-56}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 5 \cdot 10^{-18}:\\ \;\;\;\;\mathsf{fma}\left(y, 5, x \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* x (fma 2.0 (+ y z) t))))
   (if (<= x -2.5e-56) t_1 (if (<= x 5e-18) (fma y 5.0 (* x t)) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x * fma(2.0, (y + z), t);
	double tmp;
	if (x <= -2.5e-56) {
		tmp = t_1;
	} else if (x <= 5e-18) {
		tmp = fma(y, 5.0, (x * t));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(x * fma(2.0, Float64(y + z), t))
	tmp = 0.0
	if (x <= -2.5e-56)
		tmp = t_1;
	elseif (x <= 5e-18)
		tmp = fma(y, 5.0, Float64(x * t));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x * N[(2.0 * N[(y + z), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -2.5e-56], t$95$1, If[LessEqual[x, 5e-18], N[(y * 5.0 + N[(x * t), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -2.49999999999999999e-56 or 5.00000000000000036e-18 < 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. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. distribute-lft-out N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-+.f64 98.2

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

   5. Applied rewrites 98.2%

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

   if -2.49999999999999999e-56 < x < 5.00000000000000036e-18

   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

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

      6. lift-+.f64 N/A

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

      7. associate-+l+ N/A

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

      8. +-commutative N/A

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

      9. lift-+.f64 N/A

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

      10. flip-+ N/A

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

      11. +-inverses N/A

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

      12. +-inverses N/A

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

      13. +-inverses N/A

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

      14. +-inverses N/A

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

      15. flip-+ N/A

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

      16. lower-+.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 76.7

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

   7. Applied rewrites 76.7%

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

Alternative 6: 77.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y \cdot \mathsf{fma}\left(x, 2, 5\right)\\ \mathbf{if}\;y \leq -1 \cdot 10^{-23}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 1.56 \cdot 10^{+133}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(2, z, t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* y (fma x 2.0 5.0))))
   (if (<= y -1e-23) t_1 (if (<= y 1.56e+133) (* x (fma 2.0 z t)) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = y * fma(x, 2.0, 5.0);
	double tmp;
	if (y <= -1e-23) {
		tmp = t_1;
	} else if (y <= 1.56e+133) {
		tmp = x * fma(2.0, z, t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(y * fma(x, 2.0, 5.0))
	tmp = 0.0
	if (y <= -1e-23)
		tmp = t_1;
	elseif (y <= 1.56e+133)
		tmp = Float64(x * fma(2.0, z, t));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(y * N[(x * 2.0 + 5.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1e-23], t$95$1, If[LessEqual[y, 1.56e+133], N[(x * N[(2.0 * z + t), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -9.9999999999999996e-24 or 1.56e133 < y

   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. lower-*.f64 N/A

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

      8. neg-sub0 N/A

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

      9. associate--r- N/A

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

      10. neg-sub0 N/A

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

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

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

      12. metadata-eval N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 75.0

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

   5. Applied rewrites 75.0%

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

   if -9.9999999999999996e-24 < y < 1.56e133

   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 y around 0

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 80.0

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

   5. Applied rewrites 80.0%

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

Alternative 7: 63.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \mathsf{fma}\left(2, z, t\right)\\ \mathbf{if}\;z \leq -1.3 \cdot 10^{-43}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 6.6 \cdot 10^{+96}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(2, y, t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* x (fma 2.0 z t))))
   (if (<= z -1.3e-43) t_1 (if (<= z 6.6e+96) (* x (fma 2.0 y t)) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x * fma(2.0, z, t);
	double tmp;
	if (z <= -1.3e-43) {
		tmp = t_1;
	} else if (z <= 6.6e+96) {
		tmp = x * fma(2.0, y, t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(x * fma(2.0, z, t))
	tmp = 0.0
	if (z <= -1.3e-43)
		tmp = t_1;
	elseif (z <= 6.6e+96)
		tmp = Float64(x * fma(2.0, y, t));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x * N[(2.0 * z + t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -1.3e-43], t$95$1, If[LessEqual[z, 6.6e+96], N[(x * N[(2.0 * y + t), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.3e-43 or 6.59999999999999969e96 < z

   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 y around 0

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 76.1

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

   5. Applied rewrites 76.1%

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

   if -1.3e-43 < z < 6.59999999999999969e96

   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. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. distribute-lft-out N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-+.f64 71.4

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

   5. Applied rewrites 71.4%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 65.9%

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

Alternative 8: 58.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \left(z + z\right)\\ \mathbf{if}\;z \leq -2 \cdot 10^{+89}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 2.25 \cdot 10^{+112}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(2, y, t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* x (+ z z))))
   (if (<= z -2e+89) t_1 (if (<= z 2.25e+112) (* x (fma 2.0 y t)) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x * (z + z);
	double tmp;
	if (z <= -2e+89) {
		tmp = t_1;
	} else if (z <= 2.25e+112) {
		tmp = x * fma(2.0, y, t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(x * Float64(z + z))
	tmp = 0.0
	if (z <= -2e+89)
		tmp = t_1;
	elseif (z <= 2.25e+112)
		tmp = Float64(x * fma(2.0, y, t));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x * N[(z + z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -2e+89], t$95$1, If[LessEqual[z, 2.25e+112], N[(x * N[(2.0 * y + t), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.99999999999999999e89 or 2.24999999999999995e112 < z

   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. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 65.9

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

   5. Applied rewrites 65.9%

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

   7. Applied rewrites 65.9%

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

   if -1.99999999999999999e89 < z < 2.24999999999999995e112

   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. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. distribute-lft-out N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-+.f64 74.1

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

   5. Applied rewrites 74.1%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 64.2%

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

4. Final simplification 64.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2 \cdot 10^{+89}:\\ \;\;\;\;x \cdot \left(z + z\right)\\ \mathbf{elif}\;z \leq 2.25 \cdot 10^{+112}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(2, y, t\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(z + z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 45.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \left(z + z\right)\\ \mathbf{if}\;z \leq -3.9 \cdot 10^{-29}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 2.25 \cdot 10^{+112}:\\ \;\;\;\;x \cdot t\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* x (+ z z))))
   (if (<= z -3.9e-29) t_1 (if (<= z 2.25e+112) (* x t) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x * (z + z);
	double tmp;
	if (z <= -3.9e-29) {
		tmp = t_1;
	} else if (z <= 2.25e+112) {
		tmp = x * t;
	} 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 = x * (z + z)
    if (z <= (-3.9d-29)) then
        tmp = t_1
    else if (z <= 2.25d+112) then
        tmp = x * t
    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 = x * (z + z);
	double tmp;
	if (z <= -3.9e-29) {
		tmp = t_1;
	} else if (z <= 2.25e+112) {
		tmp = x * t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = x * (z + z)
	tmp = 0
	if z <= -3.9e-29:
		tmp = t_1
	elif z <= 2.25e+112:
		tmp = x * t
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(x * Float64(z + z))
	tmp = 0.0
	if (z <= -3.9e-29)
		tmp = t_1;
	elseif (z <= 2.25e+112)
		tmp = Float64(x * t);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = x * (z + z);
	tmp = 0.0;
	if (z <= -3.9e-29)
		tmp = t_1;
	elseif (z <= 2.25e+112)
		tmp = x * t;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x * N[(z + z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -3.9e-29], t$95$1, If[LessEqual[z, 2.25e+112], N[(x * t), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -3.8999999999999998e-29 or 2.24999999999999995e112 < z

   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. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 61.9

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

   5. Applied rewrites 61.9%

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

   7. Applied rewrites 61.9%

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

   if -3.8999999999999998e-29 < z < 2.24999999999999995e112

   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. *-commutative N/A

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

      2. lower-*.f64 46.2

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

   5. Applied rewrites 46.2%

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

4. Final simplification 53.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.9 \cdot 10^{-29}:\\ \;\;\;\;x \cdot \left(z + z\right)\\ \mathbf{elif}\;z \leq 2.25 \cdot 10^{+112}:\\ \;\;\;\;x \cdot t\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(z + z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 48.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.05 \cdot 10^{-54}:\\ \;\;\;\;x \cdot t\\ \mathbf{elif}\;x \leq 9 \cdot 10^{-68}:\\ \;\;\;\;y \cdot 5\\ \mathbf{else}:\\ \;\;\;\;x \cdot t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= x -1.05e-54) (* x t) (if (<= x 9e-68) (* y 5.0) (* x t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -1.05e-54) {
		tmp = x * t;
	} else if (x <= 9e-68) {
		tmp = y * 5.0;
	} else {
		tmp = x * t;
	}
	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 <= (-1.05d-54)) then
        tmp = x * t
    else if (x <= 9d-68) then
        tmp = y * 5.0d0
    else
        tmp = x * t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -1.05e-54) {
		tmp = x * t;
	} else if (x <= 9e-68) {
		tmp = y * 5.0;
	} else {
		tmp = x * t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if x <= -1.05e-54:
		tmp = x * t
	elif x <= 9e-68:
		tmp = y * 5.0
	else:
		tmp = x * t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x <= -1.05e-54)
		tmp = Float64(x * t);
	elseif (x <= 9e-68)
		tmp = Float64(y * 5.0);
	else
		tmp = Float64(x * t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (x <= -1.05e-54)
		tmp = x * t;
	elseif (x <= 9e-68)
		tmp = y * 5.0;
	else
		tmp = x * t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[x, -1.05e-54], N[(x * t), $MachinePrecision], If[LessEqual[x, 9e-68], N[(y * 5.0), $MachinePrecision], N[(x * t), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.05e-54 or 8.99999999999999998e-68 < 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 t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 44.2

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

   5. Applied rewrites 44.2%

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

   if -1.05e-54 < x < 8.99999999999999998e-68

   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. lower-*.f64 58.3

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

   5. Applied rewrites 58.3%

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

4. Final simplification 49.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.05 \cdot 10^{-54}:\\ \;\;\;\;x \cdot t\\ \mathbf{elif}\;x \leq 9 \cdot 10^{-68}:\\ \;\;\;\;y \cdot 5\\ \mathbf{else}:\\ \;\;\;\;x \cdot t\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 30.5% accurate, 4.3× speedup

Math

\[\begin{array}{l} \\ y \cdot 5 \end{array} \]

FPCore

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

C

double code(double x, double y, double z, double t) {
	return 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 = y * 5.0d0
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_] := N[(y * 5.0), $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 x around 0

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

   1. lower-*.f64 25.3

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

5. Applied rewrites 25.3%

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

6. Final simplification 25.3%

   \[\leadsto y \cdot 5 \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024220 
(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)))