Diagrams.Solve.Polynomial:cubForm from diagrams-solve-0.1, H

Percentage Accurate: 95.7% → 97.9%

Time: 11.9s

Alternatives: 18

Speedup: 1.0×

• 27.9% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (+ (- x (/ y (* z 3.0))) (/ t (* (* z 3.0) y))))

C

double code(double x, double y, double z, double t) {
	return (x - (y / (z * 3.0))) + (t / ((z * 3.0) * 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 = (x - (y / (z * 3.0d0))) + (t / ((z * 3.0d0) * y))
end function

Java

public static double code(double x, double y, double z, double t) {
	return (x - (y / (z * 3.0))) + (t / ((z * 3.0) * y));
}

Python

def code(x, y, z, t):
	return (x - (y / (z * 3.0))) + (t / ((z * 3.0) * y))

Julia

function code(x, y, z, t)
	return Float64(Float64(x - Float64(y / Float64(z * 3.0))) + Float64(t / Float64(Float64(z * 3.0) * y)))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = (x - (y / (z * 3.0))) + (t / ((z * 3.0) * y));
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(x - N[(y / N[(z * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t / N[(N[(z * 3.0), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 95.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (+ (- x (/ y (* z 3.0))) (/ t (* (* z 3.0) y))))

C

double code(double x, double y, double z, double t) {
	return (x - (y / (z * 3.0))) + (t / ((z * 3.0) * 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 = (x - (y / (z * 3.0d0))) + (t / ((z * 3.0d0) * y))
end function

Java

public static double code(double x, double y, double z, double t) {
	return (x - (y / (z * 3.0))) + (t / ((z * 3.0) * y));
}

Python

def code(x, y, z, t):
	return (x - (y / (z * 3.0))) + (t / ((z * 3.0) * y))

Julia

function code(x, y, z, t)
	return Float64(Float64(x - Float64(y / Float64(z * 3.0))) + Float64(t / Float64(Float64(z * 3.0) * y)))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = (x - (y / (z * 3.0))) + (t / ((z * 3.0) * y));
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(x - N[(y / N[(z * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t / N[(N[(z * 3.0), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 97.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t}{y} - y\\ \mathbf{if}\;y \leq -6.8 \cdot 10^{-54}:\\ \;\;\;\;\mathsf{fma}\left(\frac{0.3333333333333333}{z}, t\_1, x\right)\\ \mathbf{elif}\;y \leq 1.3 \cdot 10^{-53}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{z}, \frac{0.3333333333333333}{y}, x\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{t\_1}{z \cdot 3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (/ t y) y)))
   (if (<= y -6.8e-54)
     (fma (/ 0.3333333333333333 z) t_1 x)
     (if (<= y 1.3e-53)
       (fma (/ t z) (/ 0.3333333333333333 y) x)
       (+ x (/ t_1 (* z 3.0)))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (t / y) - y;
	double tmp;
	if (y <= -6.8e-54) {
		tmp = fma((0.3333333333333333 / z), t_1, x);
	} else if (y <= 1.3e-53) {
		tmp = fma((t / z), (0.3333333333333333 / y), x);
	} else {
		tmp = x + (t_1 / (z * 3.0));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(t / y) - y)
	tmp = 0.0
	if (y <= -6.8e-54)
		tmp = fma(Float64(0.3333333333333333 / z), t_1, x);
	elseif (y <= 1.3e-53)
		tmp = fma(Float64(t / z), Float64(0.3333333333333333 / y), x);
	else
		tmp = Float64(x + Float64(t_1 / Float64(z * 3.0)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(t / y), $MachinePrecision] - y), $MachinePrecision]}, If[LessEqual[y, -6.8e-54], N[(N[(0.3333333333333333 / z), $MachinePrecision] * t$95$1 + x), $MachinePrecision], If[LessEqual[y, 1.3e-53], N[(N[(t / z), $MachinePrecision] * N[(0.3333333333333333 / y), $MachinePrecision] + x), $MachinePrecision], N[(x + N[(t$95$1 / N[(z * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < -6.79999999999999975e-54

   1. Initial program 97.2%

      \[\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(x + \frac{1}{3} \cdot \frac{t}{y \cdot z}\right) - \frac{1}{3} \cdot \frac{y}{z}} \]
   4. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \color{blue}{x + \left(\frac{1}{3} \cdot \frac{t}{y \cdot z} - \frac{1}{3} \cdot \frac{y}{z}\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{1}{3} \cdot \frac{t}{y \cdot z} - \frac{1}{3} \cdot \frac{y}{z}\right) + x} \]

      3. associate-*r/ N/A

         \[\leadsto \left(\color{blue}{\frac{\frac{1}{3} \cdot t}{y \cdot z}} - \frac{1}{3} \cdot \frac{y}{z}\right) + x \]

      4. *-commutative N/A

         \[\leadsto \left(\frac{\frac{1}{3} \cdot t}{\color{blue}{z \cdot y}} - \frac{1}{3} \cdot \frac{y}{z}\right) + x \]

      5. times-frac N/A

         \[\leadsto \left(\color{blue}{\frac{\frac{1}{3}}{z} \cdot \frac{t}{y}} - \frac{1}{3} \cdot \frac{y}{z}\right) + x \]

      6. metadata-eval N/A

         \[\leadsto \left(\frac{\color{blue}{\frac{1}{3} \cdot 1}}{z} \cdot \frac{t}{y} - \frac{1}{3} \cdot \frac{y}{z}\right) + x \]

      7. associate-*r/ N/A

         \[\leadsto \left(\color{blue}{\left(\frac{1}{3} \cdot \frac{1}{z}\right)} \cdot \frac{t}{y} - \frac{1}{3} \cdot \frac{y}{z}\right) + x \]

      8. associate-*r/ N/A

         \[\leadsto \left(\left(\frac{1}{3} \cdot \frac{1}{z}\right) \cdot \frac{t}{y} - \color{blue}{\frac{\frac{1}{3} \cdot y}{z}}\right) + x \]

      9. associate-*l/ N/A

         \[\leadsto \left(\left(\frac{1}{3} \cdot \frac{1}{z}\right) \cdot \frac{t}{y} - \color{blue}{\frac{\frac{1}{3}}{z} \cdot y}\right) + x \]

      10. metadata-eval N/A

          \[\leadsto \left(\left(\frac{1}{3} \cdot \frac{1}{z}\right) \cdot \frac{t}{y} - \frac{\color{blue}{\frac{1}{3} \cdot 1}}{z} \cdot y\right) + x \]

      11. associate-*r/ N/A

          \[\leadsto \left(\left(\frac{1}{3} \cdot \frac{1}{z}\right) \cdot \frac{t}{y} - \color{blue}{\left(\frac{1}{3} \cdot \frac{1}{z}\right)} \cdot y\right) + x \]

      12. distribute-lft-out-- N/A

          \[\leadsto \color{blue}{\left(\frac{1}{3} \cdot \frac{1}{z}\right) \cdot \left(\frac{t}{y} - y\right)} + x \]

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

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{3} \cdot \frac{1}{z}, \frac{t}{y} - y, x\right)} \]

      14. associate-*r/ N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\frac{1}{3} \cdot 1}{z}}, \frac{t}{y} - y, x\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{\frac{1}{3}}}{z}, \frac{t}{y} - y, x\right) \]

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

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\frac{1}{3}}{z}}, \frac{t}{y} - y, x\right) \]

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

          \[\leadsto \mathsf{fma}\left(\frac{\frac{1}{3}}{z}, \color{blue}{\frac{t}{y} - y}, x\right) \]

      18. /-lowering-/.f64 99.7

          \[\leadsto \mathsf{fma}\left(\frac{0.3333333333333333}{z}, \color{blue}{\frac{t}{y}} - y, x\right) \]

   5. Simplified 99.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{0.3333333333333333}{z}, \frac{t}{y} - y, x\right)} \]

   if -6.79999999999999975e-54 < y < 1.29999999999999998e-53

   1. Initial program 94.7%

      \[\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x} + \frac{t}{\left(z \cdot 3\right) \cdot y} \]
   4. Step-by-step derivation

   5. Simplified 94.7%

      \[\leadsto \color{blue}{x} + \frac{t}{\left(z \cdot 3\right) \cdot y} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{t}{\left(z \cdot 3\right) \cdot y} + x} \]

      2. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{t}{z \cdot 3}}{y}} + x \]

      3. div-inv N/A

         \[\leadsto \frac{\color{blue}{t \cdot \frac{1}{z \cdot 3}}}{y} + x \]

      4. *-commutative N/A

         \[\leadsto \frac{t \cdot \frac{1}{\color{blue}{3 \cdot z}}}{y} + x \]

      5. associate-/r* N/A

         \[\leadsto \frac{t \cdot \color{blue}{\frac{\frac{1}{3}}{z}}}{y} + x \]

      6. metadata-eval N/A

         \[\leadsto \frac{t \cdot \frac{\color{blue}{\frac{1}{3}}}{z}}{y} + x \]

      7. associate-/l* N/A

         \[\leadsto \frac{\color{blue}{\frac{t \cdot \frac{1}{3}}{z}}}{y} + x \]

      8. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{t \cdot \frac{1}{3}}{z \cdot y}} + x \]

      9. un-div-inv N/A

         \[\leadsto \color{blue}{\left(t \cdot \frac{1}{3}\right) \cdot \frac{1}{z \cdot y}} + x \]

      10. un-div-inv N/A

          \[\leadsto \color{blue}{\frac{t \cdot \frac{1}{3}}{z \cdot y}} + x \]

      11. times-frac N/A

          \[\leadsto \color{blue}{\frac{t}{z} \cdot \frac{\frac{1}{3}}{y}} + x \]

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

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{t}{z}, \frac{\frac{1}{3}}{y}, x\right)} \]

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

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{t}{z}}, \frac{\frac{1}{3}}{y}, x\right) \]

      14. /-lowering-/.f64 98.3

          \[\leadsto \mathsf{fma}\left(\frac{t}{z}, \color{blue}{\frac{0.3333333333333333}{y}}, x\right) \]

   7. Applied egg-rr 98.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{t}{z}, \frac{0.3333333333333333}{y}, x\right)} \]

   if 1.29999999999999998e-53 < y

   1. Initial program 99.2%

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

      1. associate-+l- N/A

         \[\leadsto \color{blue}{x - \left(\frac{y}{z \cdot 3} - \frac{t}{\left(z \cdot 3\right) \cdot y}\right)} \]

      2. --lowering--.f64 N/A

         \[\leadsto \color{blue}{x - \left(\frac{y}{z \cdot 3} - \frac{t}{\left(z \cdot 3\right) \cdot y}\right)} \]

      3. *-commutative N/A

         \[\leadsto x - \left(\frac{y}{z \cdot 3} - \frac{t}{\color{blue}{y \cdot \left(z \cdot 3\right)}}\right) \]

      4. associate-/r* N/A

         \[\leadsto x - \left(\frac{y}{z \cdot 3} - \color{blue}{\frac{\frac{t}{y}}{z \cdot 3}}\right) \]

      5. sub-div N/A

         \[\leadsto x - \color{blue}{\frac{y - \frac{t}{y}}{z \cdot 3}} \]

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

         \[\leadsto x - \color{blue}{\frac{y - \frac{t}{y}}{z \cdot 3}} \]

      7. --lowering--.f64 N/A

         \[\leadsto x - \frac{\color{blue}{y - \frac{t}{y}}}{z \cdot 3} \]

      8. /-lowering-/.f64 N/A

         \[\leadsto x - \frac{y - \color{blue}{\frac{t}{y}}}{z \cdot 3} \]

      9. *-lowering-*.f64 99.7

         \[\leadsto x - \frac{y - \frac{t}{y}}{\color{blue}{z \cdot 3}} \]

   4. Applied egg-rr 99.7%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -6.8 \cdot 10^{-54}:\\ \;\;\;\;\mathsf{fma}\left(\frac{0.3333333333333333}{z}, \frac{t}{y} - y, x\right)\\ \mathbf{elif}\;y \leq 1.3 \cdot 10^{-53}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{z}, \frac{0.3333333333333333}{y}, x\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{\frac{t}{y} - y}{z \cdot 3}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 97.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x - \frac{y}{z \cdot 3}\\ \mathbf{if}\;t\_1 + \frac{t}{y \cdot \left(z \cdot 3\right)} \leq 10^{+287}:\\ \;\;\;\;t\_1 + \frac{t}{z \cdot \left(y \cdot 3\right)}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{z \cdot \frac{3}{\frac{t}{y} - y}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- x (/ y (* z 3.0)))))
   (if (<= (+ t_1 (/ t (* y (* z 3.0)))) 1e+287)
     (+ t_1 (/ t (* z (* y 3.0))))
     (+ x (/ 1.0 (* z (/ 3.0 (- (/ t y) y))))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x - (y / (z * 3.0));
	double tmp;
	if ((t_1 + (t / (y * (z * 3.0)))) <= 1e+287) {
		tmp = t_1 + (t / (z * (y * 3.0)));
	} else {
		tmp = x + (1.0 / (z * (3.0 / ((t / y) - y))));
	}
	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 - (y / (z * 3.0d0))
    if ((t_1 + (t / (y * (z * 3.0d0)))) <= 1d+287) then
        tmp = t_1 + (t / (z * (y * 3.0d0)))
    else
        tmp = x + (1.0d0 / (z * (3.0d0 / ((t / y) - y))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = x - (y / (z * 3.0));
	double tmp;
	if ((t_1 + (t / (y * (z * 3.0)))) <= 1e+287) {
		tmp = t_1 + (t / (z * (y * 3.0)));
	} else {
		tmp = x + (1.0 / (z * (3.0 / ((t / y) - y))));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = x - (y / (z * 3.0))
	tmp = 0
	if (t_1 + (t / (y * (z * 3.0)))) <= 1e+287:
		tmp = t_1 + (t / (z * (y * 3.0)))
	else:
		tmp = x + (1.0 / (z * (3.0 / ((t / y) - y))))
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(x - Float64(y / Float64(z * 3.0)))
	tmp = 0.0
	if (Float64(t_1 + Float64(t / Float64(y * Float64(z * 3.0)))) <= 1e+287)
		tmp = Float64(t_1 + Float64(t / Float64(z * Float64(y * 3.0))));
	else
		tmp = Float64(x + Float64(1.0 / Float64(z * Float64(3.0 / Float64(Float64(t / y) - y)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = x - (y / (z * 3.0));
	tmp = 0.0;
	if ((t_1 + (t / (y * (z * 3.0)))) <= 1e+287)
		tmp = t_1 + (t / (z * (y * 3.0)));
	else
		tmp = x + (1.0 / (z * (3.0 / ((t / y) - y))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x - N[(y / N[(z * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$1 + N[(t / N[(y * N[(z * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1e+287], N[(t$95$1 + N[(t / N[(z * N[(y * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(1.0 / N[(z * N[(3.0 / N[(N[(t / y), $MachinePrecision] - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (-.f64 x (/.f64 y (*.f64 z #s(literal 3 binary64)))) (/.f64 t (*.f64 (*.f64 z #s(literal 3 binary64)) y))) < 1.0000000000000001e287

   1. Initial program 98.1%

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

      1. associate-*l* N/A

         \[\leadsto \left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\color{blue}{z \cdot \left(3 \cdot y\right)}} \]

      2. *-commutative N/A

         \[\leadsto \left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\color{blue}{\left(3 \cdot y\right) \cdot z}} \]

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

         \[\leadsto \left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\color{blue}{\left(3 \cdot y\right) \cdot z}} \]

      4. *-commutative N/A

         \[\leadsto \left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\color{blue}{\left(y \cdot 3\right)} \cdot z} \]

      5. *-lowering-*.f64 98.2

         \[\leadsto \left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\color{blue}{\left(y \cdot 3\right)} \cdot z} \]

   4. Applied egg-rr 98.2%

      \[\leadsto \left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\color{blue}{\left(y \cdot 3\right) \cdot z}} \]

   if 1.0000000000000001e287 < (+.f64 (-.f64 x (/.f64 y (*.f64 z #s(literal 3 binary64)))) (/.f64 t (*.f64 (*.f64 z #s(literal 3 binary64)) y)))

   1. Initial program 88.8%

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

      1. associate-+l- N/A

         \[\leadsto \color{blue}{x - \left(\frac{y}{z \cdot 3} - \frac{t}{\left(z \cdot 3\right) \cdot y}\right)} \]

      2. --lowering--.f64 N/A

         \[\leadsto \color{blue}{x - \left(\frac{y}{z \cdot 3} - \frac{t}{\left(z \cdot 3\right) \cdot y}\right)} \]

      3. *-commutative N/A

         \[\leadsto x - \left(\frac{y}{z \cdot 3} - \frac{t}{\color{blue}{y \cdot \left(z \cdot 3\right)}}\right) \]

      4. associate-/r* N/A

         \[\leadsto x - \left(\frac{y}{z \cdot 3} - \color{blue}{\frac{\frac{t}{y}}{z \cdot 3}}\right) \]

      5. sub-div N/A

         \[\leadsto x - \color{blue}{\frac{y - \frac{t}{y}}{z \cdot 3}} \]

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

         \[\leadsto x - \color{blue}{\frac{y - \frac{t}{y}}{z \cdot 3}} \]

      7. --lowering--.f64 N/A

         \[\leadsto x - \frac{\color{blue}{y - \frac{t}{y}}}{z \cdot 3} \]

      8. /-lowering-/.f64 N/A

         \[\leadsto x - \frac{y - \color{blue}{\frac{t}{y}}}{z \cdot 3} \]

      9. *-lowering-*.f64 99.9

         \[\leadsto x - \frac{y - \frac{t}{y}}{\color{blue}{z \cdot 3}} \]

   4. Applied egg-rr 99.9%

      \[\leadsto \color{blue}{x - \frac{y - \frac{t}{y}}{z \cdot 3}} \]
   5. Step-by-step derivation

      1. clear-num N/A

         \[\leadsto x - \color{blue}{\frac{1}{\frac{z \cdot 3}{y - \frac{t}{y}}}} \]

      2. /-lowering-/.f64 N/A

         \[\leadsto x - \color{blue}{\frac{1}{\frac{z \cdot 3}{y - \frac{t}{y}}}} \]

      3. associate-/l* N/A

         \[\leadsto x - \frac{1}{\color{blue}{z \cdot \frac{3}{y - \frac{t}{y}}}} \]

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

         \[\leadsto x - \frac{1}{\color{blue}{z \cdot \frac{3}{y - \frac{t}{y}}}} \]

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

         \[\leadsto x - \frac{1}{z \cdot \color{blue}{\frac{3}{y - \frac{t}{y}}}} \]

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

         \[\leadsto x - \frac{1}{z \cdot \frac{3}{\color{blue}{y - \frac{t}{y}}}} \]

      7. /-lowering-/.f64 99.9

         \[\leadsto x - \frac{1}{z \cdot \frac{3}{y - \color{blue}{\frac{t}{y}}}} \]

   6. Applied egg-rr 99.9%

      \[\leadsto x - \color{blue}{\frac{1}{z \cdot \frac{3}{y - \frac{t}{y}}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{y \cdot \left(z \cdot 3\right)} \leq 10^{+287}:\\ \;\;\;\;\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{z \cdot \left(y \cdot 3\right)}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{z \cdot \frac{3}{\frac{t}{y} - y}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x - \frac{y}{z \cdot 3}\\ \mathbf{if}\;t\_1 + \frac{t}{y \cdot \left(z \cdot 3\right)} \leq 2 \cdot 10^{+286}:\\ \;\;\;\;t\_1 + \frac{t}{z \cdot \left(y \cdot 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{0.3333333333333333}{z}, \frac{t}{y} - y, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- x (/ y (* z 3.0)))))
   (if (<= (+ t_1 (/ t (* y (* z 3.0)))) 2e+286)
     (+ t_1 (/ t (* z (* y 3.0))))
     (fma (/ 0.3333333333333333 z) (- (/ t y) y) x))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x - (y / (z * 3.0));
	double tmp;
	if ((t_1 + (t / (y * (z * 3.0)))) <= 2e+286) {
		tmp = t_1 + (t / (z * (y * 3.0)));
	} else {
		tmp = fma((0.3333333333333333 / z), ((t / y) - y), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(x - Float64(y / Float64(z * 3.0)))
	tmp = 0.0
	if (Float64(t_1 + Float64(t / Float64(y * Float64(z * 3.0)))) <= 2e+286)
		tmp = Float64(t_1 + Float64(t / Float64(z * Float64(y * 3.0))));
	else
		tmp = fma(Float64(0.3333333333333333 / z), Float64(Float64(t / y) - y), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x - N[(y / N[(z * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$1 + N[(t / N[(y * N[(z * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2e+286], N[(t$95$1 + N[(t / N[(z * N[(y * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.3333333333333333 / z), $MachinePrecision] * N[(N[(t / y), $MachinePrecision] - y), $MachinePrecision] + x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (-.f64 x (/.f64 y (*.f64 z #s(literal 3 binary64)))) (/.f64 t (*.f64 (*.f64 z #s(literal 3 binary64)) y))) < 2.00000000000000007e286

   1. Initial program 98.1%

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

      1. associate-*l* N/A

         \[\leadsto \left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\color{blue}{z \cdot \left(3 \cdot y\right)}} \]

      2. *-commutative N/A

         \[\leadsto \left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\color{blue}{\left(3 \cdot y\right) \cdot z}} \]

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

         \[\leadsto \left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\color{blue}{\left(3 \cdot y\right) \cdot z}} \]

      4. *-commutative N/A

         \[\leadsto \left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\color{blue}{\left(y \cdot 3\right)} \cdot z} \]

      5. *-lowering-*.f64 98.2

         \[\leadsto \left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\color{blue}{\left(y \cdot 3\right)} \cdot z} \]

   4. Applied egg-rr 98.2%

      \[\leadsto \left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\color{blue}{\left(y \cdot 3\right) \cdot z}} \]

   if 2.00000000000000007e286 < (+.f64 (-.f64 x (/.f64 y (*.f64 z #s(literal 3 binary64)))) (/.f64 t (*.f64 (*.f64 z #s(literal 3 binary64)) y)))

   1. Initial program 89.1%

      \[\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(x + \frac{1}{3} \cdot \frac{t}{y \cdot z}\right) - \frac{1}{3} \cdot \frac{y}{z}} \]
   4. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \color{blue}{x + \left(\frac{1}{3} \cdot \frac{t}{y \cdot z} - \frac{1}{3} \cdot \frac{y}{z}\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{1}{3} \cdot \frac{t}{y \cdot z} - \frac{1}{3} \cdot \frac{y}{z}\right) + x} \]

      3. associate-*r/ N/A

         \[\leadsto \left(\color{blue}{\frac{\frac{1}{3} \cdot t}{y \cdot z}} - \frac{1}{3} \cdot \frac{y}{z}\right) + x \]

      4. *-commutative N/A

         \[\leadsto \left(\frac{\frac{1}{3} \cdot t}{\color{blue}{z \cdot y}} - \frac{1}{3} \cdot \frac{y}{z}\right) + x \]

      5. times-frac N/A

         \[\leadsto \left(\color{blue}{\frac{\frac{1}{3}}{z} \cdot \frac{t}{y}} - \frac{1}{3} \cdot \frac{y}{z}\right) + x \]

      6. metadata-eval N/A

         \[\leadsto \left(\frac{\color{blue}{\frac{1}{3} \cdot 1}}{z} \cdot \frac{t}{y} - \frac{1}{3} \cdot \frac{y}{z}\right) + x \]

      7. associate-*r/ N/A

         \[\leadsto \left(\color{blue}{\left(\frac{1}{3} \cdot \frac{1}{z}\right)} \cdot \frac{t}{y} - \frac{1}{3} \cdot \frac{y}{z}\right) + x \]

      8. associate-*r/ N/A

         \[\leadsto \left(\left(\frac{1}{3} \cdot \frac{1}{z}\right) \cdot \frac{t}{y} - \color{blue}{\frac{\frac{1}{3} \cdot y}{z}}\right) + x \]

      9. associate-*l/ N/A

         \[\leadsto \left(\left(\frac{1}{3} \cdot \frac{1}{z}\right) \cdot \frac{t}{y} - \color{blue}{\frac{\frac{1}{3}}{z} \cdot y}\right) + x \]

      10. metadata-eval N/A

          \[\leadsto \left(\left(\frac{1}{3} \cdot \frac{1}{z}\right) \cdot \frac{t}{y} - \frac{\color{blue}{\frac{1}{3} \cdot 1}}{z} \cdot y\right) + x \]

      11. associate-*r/ N/A

          \[\leadsto \left(\left(\frac{1}{3} \cdot \frac{1}{z}\right) \cdot \frac{t}{y} - \color{blue}{\left(\frac{1}{3} \cdot \frac{1}{z}\right)} \cdot y\right) + x \]

      12. distribute-lft-out-- N/A

          \[\leadsto \color{blue}{\left(\frac{1}{3} \cdot \frac{1}{z}\right) \cdot \left(\frac{t}{y} - y\right)} + x \]

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

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{3} \cdot \frac{1}{z}, \frac{t}{y} - y, x\right)} \]

      14. associate-*r/ N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\frac{1}{3} \cdot 1}{z}}, \frac{t}{y} - y, x\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{\frac{1}{3}}}{z}, \frac{t}{y} - y, x\right) \]

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

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\frac{1}{3}}{z}}, \frac{t}{y} - y, x\right) \]

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

          \[\leadsto \mathsf{fma}\left(\frac{\frac{1}{3}}{z}, \color{blue}{\frac{t}{y} - y}, x\right) \]

      18. /-lowering-/.f64 99.9

          \[\leadsto \mathsf{fma}\left(\frac{0.3333333333333333}{z}, \color{blue}{\frac{t}{y}} - y, x\right) \]

   5. Simplified 99.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{0.3333333333333333}{z}, \frac{t}{y} - y, x\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{y \cdot \left(z \cdot 3\right)} \leq 2 \cdot 10^{+286}:\\ \;\;\;\;\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{z \cdot \left(y \cdot 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{0.3333333333333333}{z}, \frac{t}{y} - y, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 97.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{y \cdot \left(z \cdot 3\right)} \leq 2 \cdot 10^{+286}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{y \cdot z}, 0.3333333333333333, \mathsf{fma}\left(\frac{y}{z}, -0.3333333333333333, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{0.3333333333333333}{z}, \frac{t}{y} - y, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (+ (- x (/ y (* z 3.0))) (/ t (* y (* z 3.0)))) 2e+286)
   (fma (/ t (* y z)) 0.3333333333333333 (fma (/ y z) -0.3333333333333333 x))
   (fma (/ 0.3333333333333333 z) (- (/ t y) y) x)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (((x - (y / (z * 3.0))) + (t / (y * (z * 3.0)))) <= 2e+286) {
		tmp = fma((t / (y * z)), 0.3333333333333333, fma((y / z), -0.3333333333333333, x));
	} else {
		tmp = fma((0.3333333333333333 / z), ((t / y) - y), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(Float64(x - Float64(y / Float64(z * 3.0))) + Float64(t / Float64(y * Float64(z * 3.0)))) <= 2e+286)
		tmp = fma(Float64(t / Float64(y * z)), 0.3333333333333333, fma(Float64(y / z), -0.3333333333333333, x));
	else
		tmp = fma(Float64(0.3333333333333333 / z), Float64(Float64(t / y) - y), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(N[(x - N[(y / N[(z * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t / N[(y * N[(z * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2e+286], N[(N[(t / N[(y * z), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333 + N[(N[(y / z), $MachinePrecision] * -0.3333333333333333 + x), $MachinePrecision]), $MachinePrecision], N[(N[(0.3333333333333333 / z), $MachinePrecision] * N[(N[(t / y), $MachinePrecision] - y), $MachinePrecision] + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (-.f64 x (/.f64 y (*.f64 z #s(literal 3 binary64)))) (/.f64 t (*.f64 (*.f64 z #s(literal 3 binary64)) y))) < 2.00000000000000007e286

   1. Initial program 98.1%

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{t}{\left(z \cdot 3\right) \cdot y} + \left(x - \frac{y}{z \cdot 3}\right)} \]

      2. *-rgt-identity N/A

         \[\leadsto \frac{\color{blue}{t \cdot 1}}{\left(z \cdot 3\right) \cdot y} + \left(x - \frac{y}{z \cdot 3}\right) \]

      3. *-commutative N/A

         \[\leadsto \frac{t \cdot 1}{\color{blue}{y \cdot \left(z \cdot 3\right)}} + \left(x - \frac{y}{z \cdot 3}\right) \]

      4. associate-*r* N/A

         \[\leadsto \frac{t \cdot 1}{\color{blue}{\left(y \cdot z\right) \cdot 3}} + \left(x - \frac{y}{z \cdot 3}\right) \]

      5. times-frac N/A

         \[\leadsto \color{blue}{\frac{t}{y \cdot z} \cdot \frac{1}{3}} + \left(x - \frac{y}{z \cdot 3}\right) \]

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

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{t}{y \cdot z}, \frac{1}{3}, x - \frac{y}{z \cdot 3}\right)} \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{t}{y \cdot z}}, \frac{1}{3}, x - \frac{y}{z \cdot 3}\right) \]

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

         \[\leadsto \mathsf{fma}\left(\frac{t}{\color{blue}{y \cdot z}}, \frac{1}{3}, x - \frac{y}{z \cdot 3}\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\frac{t}{y \cdot z}, \color{blue}{\frac{1}{3}}, x - \frac{y}{z \cdot 3}\right) \]

      10. sub-neg N/A

          \[\leadsto \mathsf{fma}\left(\frac{t}{y \cdot z}, \frac{1}{3}, \color{blue}{x + \left(\mathsf{neg}\left(\frac{y}{z \cdot 3}\right)\right)}\right) \]

      11. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{t}{y \cdot z}, \frac{1}{3}, \color{blue}{\left(\mathsf{neg}\left(\frac{y}{z \cdot 3}\right)\right) + x}\right) \]

      12. associate-/r* N/A

          \[\leadsto \mathsf{fma}\left(\frac{t}{y \cdot z}, \frac{1}{3}, \left(\mathsf{neg}\left(\color{blue}{\frac{\frac{y}{z}}{3}}\right)\right) + x\right) \]

      13. div-inv N/A

          \[\leadsto \mathsf{fma}\left(\frac{t}{y \cdot z}, \frac{1}{3}, \left(\mathsf{neg}\left(\color{blue}{\frac{y}{z} \cdot \frac{1}{3}}\right)\right) + x\right) \]

      14. distribute-rgt-neg-in N/A

          \[\leadsto \mathsf{fma}\left(\frac{t}{y \cdot z}, \frac{1}{3}, \color{blue}{\frac{y}{z} \cdot \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)} + x\right) \]

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

          \[\leadsto \mathsf{fma}\left(\frac{t}{y \cdot z}, \frac{1}{3}, \color{blue}{\mathsf{fma}\left(\frac{y}{z}, \mathsf{neg}\left(\frac{1}{3}\right), x\right)}\right) \]

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

          \[\leadsto \mathsf{fma}\left(\frac{t}{y \cdot z}, \frac{1}{3}, \mathsf{fma}\left(\color{blue}{\frac{y}{z}}, \mathsf{neg}\left(\frac{1}{3}\right), x\right)\right) \]

      17. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\frac{t}{y \cdot z}, \frac{1}{3}, \mathsf{fma}\left(\frac{y}{z}, \mathsf{neg}\left(\color{blue}{\frac{1}{3}}\right), x\right)\right) \]

      18. metadata-eval 97.7

          \[\leadsto \mathsf{fma}\left(\frac{t}{y \cdot z}, 0.3333333333333333, \mathsf{fma}\left(\frac{y}{z}, \color{blue}{-0.3333333333333333}, x\right)\right) \]

   4. Applied egg-rr 97.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{t}{y \cdot z}, 0.3333333333333333, \mathsf{fma}\left(\frac{y}{z}, -0.3333333333333333, x\right)\right)} \]

   if 2.00000000000000007e286 < (+.f64 (-.f64 x (/.f64 y (*.f64 z #s(literal 3 binary64)))) (/.f64 t (*.f64 (*.f64 z #s(literal 3 binary64)) y)))

   1. Initial program 89.1%

      \[\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(x + \frac{1}{3} \cdot \frac{t}{y \cdot z}\right) - \frac{1}{3} \cdot \frac{y}{z}} \]
   4. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \color{blue}{x + \left(\frac{1}{3} \cdot \frac{t}{y \cdot z} - \frac{1}{3} \cdot \frac{y}{z}\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{1}{3} \cdot \frac{t}{y \cdot z} - \frac{1}{3} \cdot \frac{y}{z}\right) + x} \]

      3. associate-*r/ N/A

         \[\leadsto \left(\color{blue}{\frac{\frac{1}{3} \cdot t}{y \cdot z}} - \frac{1}{3} \cdot \frac{y}{z}\right) + x \]

      4. *-commutative N/A

         \[\leadsto \left(\frac{\frac{1}{3} \cdot t}{\color{blue}{z \cdot y}} - \frac{1}{3} \cdot \frac{y}{z}\right) + x \]

      5. times-frac N/A

         \[\leadsto \left(\color{blue}{\frac{\frac{1}{3}}{z} \cdot \frac{t}{y}} - \frac{1}{3} \cdot \frac{y}{z}\right) + x \]

      6. metadata-eval N/A

         \[\leadsto \left(\frac{\color{blue}{\frac{1}{3} \cdot 1}}{z} \cdot \frac{t}{y} - \frac{1}{3} \cdot \frac{y}{z}\right) + x \]

      7. associate-*r/ N/A

         \[\leadsto \left(\color{blue}{\left(\frac{1}{3} \cdot \frac{1}{z}\right)} \cdot \frac{t}{y} - \frac{1}{3} \cdot \frac{y}{z}\right) + x \]

      8. associate-*r/ N/A

         \[\leadsto \left(\left(\frac{1}{3} \cdot \frac{1}{z}\right) \cdot \frac{t}{y} - \color{blue}{\frac{\frac{1}{3} \cdot y}{z}}\right) + x \]

      9. associate-*l/ N/A

         \[\leadsto \left(\left(\frac{1}{3} \cdot \frac{1}{z}\right) \cdot \frac{t}{y} - \color{blue}{\frac{\frac{1}{3}}{z} \cdot y}\right) + x \]

      10. metadata-eval N/A

          \[\leadsto \left(\left(\frac{1}{3} \cdot \frac{1}{z}\right) \cdot \frac{t}{y} - \frac{\color{blue}{\frac{1}{3} \cdot 1}}{z} \cdot y\right) + x \]

      11. associate-*r/ N/A

          \[\leadsto \left(\left(\frac{1}{3} \cdot \frac{1}{z}\right) \cdot \frac{t}{y} - \color{blue}{\left(\frac{1}{3} \cdot \frac{1}{z}\right)} \cdot y\right) + x \]

      12. distribute-lft-out-- N/A

          \[\leadsto \color{blue}{\left(\frac{1}{3} \cdot \frac{1}{z}\right) \cdot \left(\frac{t}{y} - y\right)} + x \]

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

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{3} \cdot \frac{1}{z}, \frac{t}{y} - y, x\right)} \]

      14. associate-*r/ N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\frac{1}{3} \cdot 1}{z}}, \frac{t}{y} - y, x\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{\frac{1}{3}}}{z}, \frac{t}{y} - y, x\right) \]

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

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\frac{1}{3}}{z}}, \frac{t}{y} - y, x\right) \]

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

          \[\leadsto \mathsf{fma}\left(\frac{\frac{1}{3}}{z}, \color{blue}{\frac{t}{y} - y}, x\right) \]

      18. /-lowering-/.f64 99.9

          \[\leadsto \mathsf{fma}\left(\frac{0.3333333333333333}{z}, \color{blue}{\frac{t}{y}} - y, x\right) \]

   5. Simplified 99.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{0.3333333333333333}{z}, \frac{t}{y} - y, x\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{y \cdot \left(z \cdot 3\right)} \leq 2 \cdot 10^{+286}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{y \cdot z}, 0.3333333333333333, \mathsf{fma}\left(\frac{y}{z}, -0.3333333333333333, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{0.3333333333333333}{z}, \frac{t}{y} - y, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 97.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{0.3333333333333333}{z}, \frac{t}{y} - y, x\right)\\ \mathbf{if}\;y \leq -5.5 \cdot 10^{-55}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 1.3 \cdot 10^{-53}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{z}, \frac{0.3333333333333333}{y}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (fma (/ 0.3333333333333333 z) (- (/ t y) y) x)))
   (if (<= y -5.5e-55)
     t_1
     (if (<= y 1.3e-53) (fma (/ t z) (/ 0.3333333333333333 y) x) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fma((0.3333333333333333 / z), ((t / y) - y), x);
	double tmp;
	if (y <= -5.5e-55) {
		tmp = t_1;
	} else if (y <= 1.3e-53) {
		tmp = fma((t / z), (0.3333333333333333 / y), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = fma(Float64(0.3333333333333333 / z), Float64(Float64(t / y) - y), x)
	tmp = 0.0
	if (y <= -5.5e-55)
		tmp = t_1;
	elseif (y <= 1.3e-53)
		tmp = fma(Float64(t / z), Float64(0.3333333333333333 / y), x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(0.3333333333333333 / z), $MachinePrecision] * N[(N[(t / y), $MachinePrecision] - y), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[y, -5.5e-55], t$95$1, If[LessEqual[y, 1.3e-53], N[(N[(t / z), $MachinePrecision] * N[(0.3333333333333333 / y), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -5.4999999999999999e-55 or 1.29999999999999998e-53 < y

   1. Initial program 98.2%

      \[\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(x + \frac{1}{3} \cdot \frac{t}{y \cdot z}\right) - \frac{1}{3} \cdot \frac{y}{z}} \]
   4. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \color{blue}{x + \left(\frac{1}{3} \cdot \frac{t}{y \cdot z} - \frac{1}{3} \cdot \frac{y}{z}\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{1}{3} \cdot \frac{t}{y \cdot z} - \frac{1}{3} \cdot \frac{y}{z}\right) + x} \]

      3. associate-*r/ N/A

         \[\leadsto \left(\color{blue}{\frac{\frac{1}{3} \cdot t}{y \cdot z}} - \frac{1}{3} \cdot \frac{y}{z}\right) + x \]

      4. *-commutative N/A

         \[\leadsto \left(\frac{\frac{1}{3} \cdot t}{\color{blue}{z \cdot y}} - \frac{1}{3} \cdot \frac{y}{z}\right) + x \]

      5. times-frac N/A

         \[\leadsto \left(\color{blue}{\frac{\frac{1}{3}}{z} \cdot \frac{t}{y}} - \frac{1}{3} \cdot \frac{y}{z}\right) + x \]

      6. metadata-eval N/A

         \[\leadsto \left(\frac{\color{blue}{\frac{1}{3} \cdot 1}}{z} \cdot \frac{t}{y} - \frac{1}{3} \cdot \frac{y}{z}\right) + x \]

      7. associate-*r/ N/A

         \[\leadsto \left(\color{blue}{\left(\frac{1}{3} \cdot \frac{1}{z}\right)} \cdot \frac{t}{y} - \frac{1}{3} \cdot \frac{y}{z}\right) + x \]

      8. associate-*r/ N/A

         \[\leadsto \left(\left(\frac{1}{3} \cdot \frac{1}{z}\right) \cdot \frac{t}{y} - \color{blue}{\frac{\frac{1}{3} \cdot y}{z}}\right) + x \]

      9. associate-*l/ N/A

         \[\leadsto \left(\left(\frac{1}{3} \cdot \frac{1}{z}\right) \cdot \frac{t}{y} - \color{blue}{\frac{\frac{1}{3}}{z} \cdot y}\right) + x \]

      10. metadata-eval N/A

          \[\leadsto \left(\left(\frac{1}{3} \cdot \frac{1}{z}\right) \cdot \frac{t}{y} - \frac{\color{blue}{\frac{1}{3} \cdot 1}}{z} \cdot y\right) + x \]

      11. associate-*r/ N/A

          \[\leadsto \left(\left(\frac{1}{3} \cdot \frac{1}{z}\right) \cdot \frac{t}{y} - \color{blue}{\left(\frac{1}{3} \cdot \frac{1}{z}\right)} \cdot y\right) + x \]

      12. distribute-lft-out-- N/A

          \[\leadsto \color{blue}{\left(\frac{1}{3} \cdot \frac{1}{z}\right) \cdot \left(\frac{t}{y} - y\right)} + x \]

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

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{3} \cdot \frac{1}{z}, \frac{t}{y} - y, x\right)} \]

      14. associate-*r/ N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\frac{1}{3} \cdot 1}{z}}, \frac{t}{y} - y, x\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{\frac{1}{3}}}{z}, \frac{t}{y} - y, x\right) \]

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

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\frac{1}{3}}{z}}, \frac{t}{y} - y, x\right) \]

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

          \[\leadsto \mathsf{fma}\left(\frac{\frac{1}{3}}{z}, \color{blue}{\frac{t}{y} - y}, x\right) \]

      18. /-lowering-/.f64 99.7

          \[\leadsto \mathsf{fma}\left(\frac{0.3333333333333333}{z}, \color{blue}{\frac{t}{y}} - y, x\right) \]

   5. Simplified 99.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{0.3333333333333333}{z}, \frac{t}{y} - y, x\right)} \]

   if -5.4999999999999999e-55 < y < 1.29999999999999998e-53

   1. Initial program 94.7%

      \[\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x} + \frac{t}{\left(z \cdot 3\right) \cdot y} \]
   4. Step-by-step derivation

   5. Simplified 94.7%

      \[\leadsto \color{blue}{x} + \frac{t}{\left(z \cdot 3\right) \cdot y} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{t}{\left(z \cdot 3\right) \cdot y} + x} \]

      2. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{t}{z \cdot 3}}{y}} + x \]

      3. div-inv N/A

         \[\leadsto \frac{\color{blue}{t \cdot \frac{1}{z \cdot 3}}}{y} + x \]

      4. *-commutative N/A

         \[\leadsto \frac{t \cdot \frac{1}{\color{blue}{3 \cdot z}}}{y} + x \]

      5. associate-/r* N/A

         \[\leadsto \frac{t \cdot \color{blue}{\frac{\frac{1}{3}}{z}}}{y} + x \]

      6. metadata-eval N/A

         \[\leadsto \frac{t \cdot \frac{\color{blue}{\frac{1}{3}}}{z}}{y} + x \]

      7. associate-/l* N/A

         \[\leadsto \frac{\color{blue}{\frac{t \cdot \frac{1}{3}}{z}}}{y} + x \]

      8. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{t \cdot \frac{1}{3}}{z \cdot y}} + x \]

      9. un-div-inv N/A

         \[\leadsto \color{blue}{\left(t \cdot \frac{1}{3}\right) \cdot \frac{1}{z \cdot y}} + x \]

      10. un-div-inv N/A

          \[\leadsto \color{blue}{\frac{t \cdot \frac{1}{3}}{z \cdot y}} + x \]

      11. times-frac N/A

          \[\leadsto \color{blue}{\frac{t}{z} \cdot \frac{\frac{1}{3}}{y}} + x \]

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

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{t}{z}, \frac{\frac{1}{3}}{y}, x\right)} \]

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

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{t}{z}}, \frac{\frac{1}{3}}{y}, x\right) \]

      14. /-lowering-/.f64 98.3

          \[\leadsto \mathsf{fma}\left(\frac{t}{z}, \color{blue}{\frac{0.3333333333333333}{y}}, x\right) \]

   7. Applied egg-rr 98.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{t}{z}, \frac{0.3333333333333333}{y}, x\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 90.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.2 \cdot 10^{-51}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{\frac{1}{z}}{-3}, x\right)\\ \mathbf{elif}\;y \leq 9.5 \cdot 10^{-18}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{z}, \frac{0.3333333333333333}{y}, x\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{-1}{z \cdot \frac{3}{y}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y -1.2e-51)
   (fma y (/ (/ 1.0 z) -3.0) x)
   (if (<= y 9.5e-18)
     (fma (/ t z) (/ 0.3333333333333333 y) x)
     (+ x (/ -1.0 (* z (/ 3.0 y)))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -1.2e-51) {
		tmp = fma(y, ((1.0 / z) / -3.0), x);
	} else if (y <= 9.5e-18) {
		tmp = fma((t / z), (0.3333333333333333 / y), x);
	} else {
		tmp = x + (-1.0 / (z * (3.0 / y)));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= -1.2e-51)
		tmp = fma(y, Float64(Float64(1.0 / z) / -3.0), x);
	elseif (y <= 9.5e-18)
		tmp = fma(Float64(t / z), Float64(0.3333333333333333 / y), x);
	else
		tmp = Float64(x + Float64(-1.0 / Float64(z * Float64(3.0 / y))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, -1.2e-51], N[(y * N[(N[(1.0 / z), $MachinePrecision] / -3.0), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[y, 9.5e-18], N[(N[(t / z), $MachinePrecision] * N[(0.3333333333333333 / y), $MachinePrecision] + x), $MachinePrecision], N[(x + N[(-1.0 / N[(z * N[(3.0 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.2e-51

   1. Initial program 97.2%

      \[\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{y \cdot \left(\frac{x}{y} - \frac{1}{3} \cdot \frac{1}{z}\right)} \]
   4. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto y \cdot \color{blue}{\left(\frac{x}{y} + \left(\mathsf{neg}\left(\frac{1}{3} \cdot \frac{1}{z}\right)\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto y \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\frac{1}{3} \cdot \frac{1}{z}\right)\right) + \frac{x}{y}\right)} \]

      3. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{1}{3} \cdot \frac{1}{z}\right)\right) \cdot y + \frac{x}{y} \cdot y} \]

      4. associate-*r/ N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\frac{\frac{1}{3} \cdot 1}{z}}\right)\right) \cdot y + \frac{x}{y} \cdot y \]

      5. metadata-eval N/A

         \[\leadsto \left(\mathsf{neg}\left(\frac{\color{blue}{\frac{1}{3}}}{z}\right)\right) \cdot y + \frac{x}{y} \cdot y \]

      6. distribute-neg-frac N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(\frac{1}{3}\right)}{z}} \cdot y + \frac{x}{y} \cdot y \]

      7. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{3}}}{z} \cdot y + \frac{x}{y} \cdot y \]

      8. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\frac{-1}{3} \cdot y}{z}} + \frac{x}{y} \cdot y \]

      9. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{-1}{3} \cdot \frac{y}{z}} + \frac{x}{y} \cdot y \]

      10. cancel-sign-sub N/A

          \[\leadsto \color{blue}{\frac{-1}{3} \cdot \frac{y}{z} - \left(\mathsf{neg}\left(\frac{x}{y}\right)\right) \cdot y} \]

      11. mul-1-neg N/A

          \[\leadsto \frac{-1}{3} \cdot \frac{y}{z} - \color{blue}{\left(-1 \cdot \frac{x}{y}\right)} \cdot y \]

      12. associate-*r/ N/A

          \[\leadsto \frac{-1}{3} \cdot \frac{y}{z} - \color{blue}{\frac{-1 \cdot x}{y}} \cdot y \]

      13. associate-*l/ N/A

          \[\leadsto \frac{-1}{3} \cdot \frac{y}{z} - \color{blue}{\frac{\left(-1 \cdot x\right) \cdot y}{y}} \]

      14. associate-/l* N/A

          \[\leadsto \frac{-1}{3} \cdot \frac{y}{z} - \color{blue}{\left(-1 \cdot x\right) \cdot \frac{y}{y}} \]

      15. mul-1-neg N/A

          \[\leadsto \frac{-1}{3} \cdot \frac{y}{z} - \color{blue}{\left(\mathsf{neg}\left(x\right)\right)} \cdot \frac{y}{y} \]

      16. *-inverses N/A

          \[\leadsto \frac{-1}{3} \cdot \frac{y}{z} - \left(\mathsf{neg}\left(x\right)\right) \cdot \color{blue}{1} \]

      17. cancel-sign-sub N/A

          \[\leadsto \color{blue}{\frac{-1}{3} \cdot \frac{y}{z} + x \cdot 1} \]

      18. *-rgt-identity N/A

          \[\leadsto \frac{-1}{3} \cdot \frac{y}{z} + \color{blue}{x} \]

   5. Simplified 89.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, \frac{-0.3333333333333333}{z}, x\right)} \]
   6. Step-by-step derivation

      1. clear-num N/A

         \[\leadsto \mathsf{fma}\left(y, \color{blue}{\frac{1}{\frac{z}{\frac{-1}{3}}}}, x\right) \]

      2. associate-/r/ N/A

         \[\leadsto \mathsf{fma}\left(y, \color{blue}{\frac{1}{z} \cdot \frac{-1}{3}}, x\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(y, \frac{1}{z} \cdot \color{blue}{\frac{1}{-3}}, x\right) \]

      4. div-inv N/A

         \[\leadsto \mathsf{fma}\left(y, \color{blue}{\frac{\frac{1}{z}}{-3}}, x\right) \]

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

         \[\leadsto \mathsf{fma}\left(y, \color{blue}{\frac{\frac{1}{z}}{-3}}, x\right) \]

      6. /-lowering-/.f64 89.3

         \[\leadsto \mathsf{fma}\left(y, \frac{\color{blue}{\frac{1}{z}}}{-3}, x\right) \]

   7. Applied egg-rr 89.3%

      \[\leadsto \mathsf{fma}\left(y, \color{blue}{\frac{\frac{1}{z}}{-3}}, x\right) \]

   if -1.2e-51 < y < 9.5000000000000003e-18

   1. Initial program 94.7%

      \[\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x} + \frac{t}{\left(z \cdot 3\right) \cdot y} \]
   4. Step-by-step derivation

   5. Simplified 94.3%

      \[\leadsto \color{blue}{x} + \frac{t}{\left(z \cdot 3\right) \cdot y} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{t}{\left(z \cdot 3\right) \cdot y} + x} \]

      2. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{t}{z \cdot 3}}{y}} + x \]

      3. div-inv N/A

         \[\leadsto \frac{\color{blue}{t \cdot \frac{1}{z \cdot 3}}}{y} + x \]

      4. *-commutative N/A

         \[\leadsto \frac{t \cdot \frac{1}{\color{blue}{3 \cdot z}}}{y} + x \]

      5. associate-/r* N/A

         \[\leadsto \frac{t \cdot \color{blue}{\frac{\frac{1}{3}}{z}}}{y} + x \]

      6. metadata-eval N/A

         \[\leadsto \frac{t \cdot \frac{\color{blue}{\frac{1}{3}}}{z}}{y} + x \]

      7. associate-/l* N/A

         \[\leadsto \frac{\color{blue}{\frac{t \cdot \frac{1}{3}}{z}}}{y} + x \]

      8. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{t \cdot \frac{1}{3}}{z \cdot y}} + x \]

      9. un-div-inv N/A

         \[\leadsto \color{blue}{\left(t \cdot \frac{1}{3}\right) \cdot \frac{1}{z \cdot y}} + x \]

      10. un-div-inv N/A

          \[\leadsto \color{blue}{\frac{t \cdot \frac{1}{3}}{z \cdot y}} + x \]

      11. times-frac N/A

          \[\leadsto \color{blue}{\frac{t}{z} \cdot \frac{\frac{1}{3}}{y}} + x \]

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

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{t}{z}, \frac{\frac{1}{3}}{y}, x\right)} \]

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

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{t}{z}}, \frac{\frac{1}{3}}{y}, x\right) \]

      14. /-lowering-/.f64 97.7

          \[\leadsto \mathsf{fma}\left(\frac{t}{z}, \color{blue}{\frac{0.3333333333333333}{y}}, x\right) \]

   7. Applied egg-rr 97.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{t}{z}, \frac{0.3333333333333333}{y}, x\right)} \]

   if 9.5000000000000003e-18 < y

   1. Initial program 99.7%

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

      1. associate-+l- N/A

         \[\leadsto \color{blue}{x - \left(\frac{y}{z \cdot 3} - \frac{t}{\left(z \cdot 3\right) \cdot y}\right)} \]

      2. --lowering--.f64 N/A

         \[\leadsto \color{blue}{x - \left(\frac{y}{z \cdot 3} - \frac{t}{\left(z \cdot 3\right) \cdot y}\right)} \]

      3. *-commutative N/A

         \[\leadsto x - \left(\frac{y}{z \cdot 3} - \frac{t}{\color{blue}{y \cdot \left(z \cdot 3\right)}}\right) \]

      4. associate-/r* N/A

         \[\leadsto x - \left(\frac{y}{z \cdot 3} - \color{blue}{\frac{\frac{t}{y}}{z \cdot 3}}\right) \]

      5. sub-div N/A

         \[\leadsto x - \color{blue}{\frac{y - \frac{t}{y}}{z \cdot 3}} \]

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

         \[\leadsto x - \color{blue}{\frac{y - \frac{t}{y}}{z \cdot 3}} \]

      7. --lowering--.f64 N/A

         \[\leadsto x - \frac{\color{blue}{y - \frac{t}{y}}}{z \cdot 3} \]

      8. /-lowering-/.f64 N/A

         \[\leadsto x - \frac{y - \color{blue}{\frac{t}{y}}}{z \cdot 3} \]

      9. *-lowering-*.f64 99.7

         \[\leadsto x - \frac{y - \frac{t}{y}}{\color{blue}{z \cdot 3}} \]

   4. Applied egg-rr 99.7%

      \[\leadsto \color{blue}{x - \frac{y - \frac{t}{y}}{z \cdot 3}} \]
   5. Step-by-step derivation

      1. clear-num N/A

         \[\leadsto x - \color{blue}{\frac{1}{\frac{z \cdot 3}{y - \frac{t}{y}}}} \]

      2. /-lowering-/.f64 N/A

         \[\leadsto x - \color{blue}{\frac{1}{\frac{z \cdot 3}{y - \frac{t}{y}}}} \]

      3. associate-/l* N/A

         \[\leadsto x - \frac{1}{\color{blue}{z \cdot \frac{3}{y - \frac{t}{y}}}} \]

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

         \[\leadsto x - \frac{1}{\color{blue}{z \cdot \frac{3}{y - \frac{t}{y}}}} \]

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

         \[\leadsto x - \frac{1}{z \cdot \color{blue}{\frac{3}{y - \frac{t}{y}}}} \]

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

         \[\leadsto x - \frac{1}{z \cdot \frac{3}{\color{blue}{y - \frac{t}{y}}}} \]

      7. /-lowering-/.f64 99.8

         \[\leadsto x - \frac{1}{z \cdot \frac{3}{y - \color{blue}{\frac{t}{y}}}} \]

   6. Applied egg-rr 99.8%

      \[\leadsto x - \color{blue}{\frac{1}{z \cdot \frac{3}{y - \frac{t}{y}}}} \]

   7. Taylor expanded in y around inf

      \[\leadsto x - \frac{1}{z \cdot \color{blue}{\frac{3}{y}}} \]
   8. Step-by-step derivation

      1. /-lowering-/.f64 91.4

         \[\leadsto x - \frac{1}{z \cdot \color{blue}{\frac{3}{y}}} \]

   9. Simplified 91.4%

      \[\leadsto x - \frac{1}{z \cdot \color{blue}{\frac{3}{y}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 93.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.2 \cdot 10^{-51}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{\frac{1}{z}}{-3}, x\right)\\ \mathbf{elif}\;y \leq 9.5 \cdot 10^{-18}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{z}, \frac{0.3333333333333333}{y}, x\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{-1}{z \cdot \frac{3}{y}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 90.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.2 \cdot 10^{-51}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{\frac{1}{z}}{-3}, x\right)\\ \mathbf{elif}\;y \leq 3.8 \cdot 10^{-17}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{z}, \frac{0.3333333333333333}{y}, x\right)\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{z \cdot 3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y -1.2e-51)
   (fma y (/ (/ 1.0 z) -3.0) x)
   (if (<= y 3.8e-17)
     (fma (/ t z) (/ 0.3333333333333333 y) x)
     (- x (/ y (* z 3.0))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -1.2e-51) {
		tmp = fma(y, ((1.0 / z) / -3.0), x);
	} else if (y <= 3.8e-17) {
		tmp = fma((t / z), (0.3333333333333333 / y), x);
	} else {
		tmp = x - (y / (z * 3.0));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= -1.2e-51)
		tmp = fma(y, Float64(Float64(1.0 / z) / -3.0), x);
	elseif (y <= 3.8e-17)
		tmp = fma(Float64(t / z), Float64(0.3333333333333333 / y), x);
	else
		tmp = Float64(x - Float64(y / Float64(z * 3.0)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, -1.2e-51], N[(y * N[(N[(1.0 / z), $MachinePrecision] / -3.0), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[y, 3.8e-17], N[(N[(t / z), $MachinePrecision] * N[(0.3333333333333333 / y), $MachinePrecision] + x), $MachinePrecision], N[(x - N[(y / N[(z * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.2e-51

   1. Initial program 97.2%

      \[\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{y \cdot \left(\frac{x}{y} - \frac{1}{3} \cdot \frac{1}{z}\right)} \]
   4. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto y \cdot \color{blue}{\left(\frac{x}{y} + \left(\mathsf{neg}\left(\frac{1}{3} \cdot \frac{1}{z}\right)\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto y \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\frac{1}{3} \cdot \frac{1}{z}\right)\right) + \frac{x}{y}\right)} \]

      3. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{1}{3} \cdot \frac{1}{z}\right)\right) \cdot y + \frac{x}{y} \cdot y} \]

      4. associate-*r/ N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\frac{\frac{1}{3} \cdot 1}{z}}\right)\right) \cdot y + \frac{x}{y} \cdot y \]

      5. metadata-eval N/A

         \[\leadsto \left(\mathsf{neg}\left(\frac{\color{blue}{\frac{1}{3}}}{z}\right)\right) \cdot y + \frac{x}{y} \cdot y \]

      6. distribute-neg-frac N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(\frac{1}{3}\right)}{z}} \cdot y + \frac{x}{y} \cdot y \]

      7. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{3}}}{z} \cdot y + \frac{x}{y} \cdot y \]

      8. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\frac{-1}{3} \cdot y}{z}} + \frac{x}{y} \cdot y \]

      9. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{-1}{3} \cdot \frac{y}{z}} + \frac{x}{y} \cdot y \]

      10. cancel-sign-sub N/A

          \[\leadsto \color{blue}{\frac{-1}{3} \cdot \frac{y}{z} - \left(\mathsf{neg}\left(\frac{x}{y}\right)\right) \cdot y} \]

      11. mul-1-neg N/A

          \[\leadsto \frac{-1}{3} \cdot \frac{y}{z} - \color{blue}{\left(-1 \cdot \frac{x}{y}\right)} \cdot y \]

      12. associate-*r/ N/A

          \[\leadsto \frac{-1}{3} \cdot \frac{y}{z} - \color{blue}{\frac{-1 \cdot x}{y}} \cdot y \]

      13. associate-*l/ N/A

          \[\leadsto \frac{-1}{3} \cdot \frac{y}{z} - \color{blue}{\frac{\left(-1 \cdot x\right) \cdot y}{y}} \]

      14. associate-/l* N/A

          \[\leadsto \frac{-1}{3} \cdot \frac{y}{z} - \color{blue}{\left(-1 \cdot x\right) \cdot \frac{y}{y}} \]

      15. mul-1-neg N/A

          \[\leadsto \frac{-1}{3} \cdot \frac{y}{z} - \color{blue}{\left(\mathsf{neg}\left(x\right)\right)} \cdot \frac{y}{y} \]

      16. *-inverses N/A

          \[\leadsto \frac{-1}{3} \cdot \frac{y}{z} - \left(\mathsf{neg}\left(x\right)\right) \cdot \color{blue}{1} \]

      17. cancel-sign-sub N/A

          \[\leadsto \color{blue}{\frac{-1}{3} \cdot \frac{y}{z} + x \cdot 1} \]

      18. *-rgt-identity N/A

          \[\leadsto \frac{-1}{3} \cdot \frac{y}{z} + \color{blue}{x} \]

   5. Simplified 89.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, \frac{-0.3333333333333333}{z}, x\right)} \]
   6. Step-by-step derivation

      1. clear-num N/A

         \[\leadsto \mathsf{fma}\left(y, \color{blue}{\frac{1}{\frac{z}{\frac{-1}{3}}}}, x\right) \]

      2. associate-/r/ N/A

         \[\leadsto \mathsf{fma}\left(y, \color{blue}{\frac{1}{z} \cdot \frac{-1}{3}}, x\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(y, \frac{1}{z} \cdot \color{blue}{\frac{1}{-3}}, x\right) \]

      4. div-inv N/A

         \[\leadsto \mathsf{fma}\left(y, \color{blue}{\frac{\frac{1}{z}}{-3}}, x\right) \]

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

         \[\leadsto \mathsf{fma}\left(y, \color{blue}{\frac{\frac{1}{z}}{-3}}, x\right) \]

      6. /-lowering-/.f64 89.3

         \[\leadsto \mathsf{fma}\left(y, \frac{\color{blue}{\frac{1}{z}}}{-3}, x\right) \]

   7. Applied egg-rr 89.3%

      \[\leadsto \mathsf{fma}\left(y, \color{blue}{\frac{\frac{1}{z}}{-3}}, x\right) \]

   if -1.2e-51 < y < 3.8000000000000001e-17

   1. Initial program 94.7%

      \[\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x} + \frac{t}{\left(z \cdot 3\right) \cdot y} \]
   4. Step-by-step derivation

   5. Simplified 94.3%

      \[\leadsto \color{blue}{x} + \frac{t}{\left(z \cdot 3\right) \cdot y} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{t}{\left(z \cdot 3\right) \cdot y} + x} \]

      2. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{t}{z \cdot 3}}{y}} + x \]

      3. div-inv N/A

         \[\leadsto \frac{\color{blue}{t \cdot \frac{1}{z \cdot 3}}}{y} + x \]

      4. *-commutative N/A

         \[\leadsto \frac{t \cdot \frac{1}{\color{blue}{3 \cdot z}}}{y} + x \]

      5. associate-/r* N/A

         \[\leadsto \frac{t \cdot \color{blue}{\frac{\frac{1}{3}}{z}}}{y} + x \]

      6. metadata-eval N/A

         \[\leadsto \frac{t \cdot \frac{\color{blue}{\frac{1}{3}}}{z}}{y} + x \]

      7. associate-/l* N/A

         \[\leadsto \frac{\color{blue}{\frac{t \cdot \frac{1}{3}}{z}}}{y} + x \]

      8. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{t \cdot \frac{1}{3}}{z \cdot y}} + x \]

      9. un-div-inv N/A

         \[\leadsto \color{blue}{\left(t \cdot \frac{1}{3}\right) \cdot \frac{1}{z \cdot y}} + x \]

      10. un-div-inv N/A

          \[\leadsto \color{blue}{\frac{t \cdot \frac{1}{3}}{z \cdot y}} + x \]

      11. times-frac N/A

          \[\leadsto \color{blue}{\frac{t}{z} \cdot \frac{\frac{1}{3}}{y}} + x \]

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

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{t}{z}, \frac{\frac{1}{3}}{y}, x\right)} \]

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

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{t}{z}}, \frac{\frac{1}{3}}{y}, x\right) \]

      14. /-lowering-/.f64 97.7

          \[\leadsto \mathsf{fma}\left(\frac{t}{z}, \color{blue}{\frac{0.3333333333333333}{y}}, x\right) \]

   7. Applied egg-rr 97.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{t}{z}, \frac{0.3333333333333333}{y}, x\right)} \]

   if 3.8000000000000001e-17 < y

   1. Initial program 99.7%

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

      1. associate-+l- N/A

         \[\leadsto \color{blue}{x - \left(\frac{y}{z \cdot 3} - \frac{t}{\left(z \cdot 3\right) \cdot y}\right)} \]

      2. --lowering--.f64 N/A

         \[\leadsto \color{blue}{x - \left(\frac{y}{z \cdot 3} - \frac{t}{\left(z \cdot 3\right) \cdot y}\right)} \]

      3. *-commutative N/A

         \[\leadsto x - \left(\frac{y}{z \cdot 3} - \frac{t}{\color{blue}{y \cdot \left(z \cdot 3\right)}}\right) \]

      4. associate-/r* N/A

         \[\leadsto x - \left(\frac{y}{z \cdot 3} - \color{blue}{\frac{\frac{t}{y}}{z \cdot 3}}\right) \]

      5. sub-div N/A

         \[\leadsto x - \color{blue}{\frac{y - \frac{t}{y}}{z \cdot 3}} \]

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

         \[\leadsto x - \color{blue}{\frac{y - \frac{t}{y}}{z \cdot 3}} \]

      7. --lowering--.f64 N/A

         \[\leadsto x - \frac{\color{blue}{y - \frac{t}{y}}}{z \cdot 3} \]

      8. /-lowering-/.f64 N/A

         \[\leadsto x - \frac{y - \color{blue}{\frac{t}{y}}}{z \cdot 3} \]

      9. *-lowering-*.f64 99.7

         \[\leadsto x - \frac{y - \frac{t}{y}}{\color{blue}{z \cdot 3}} \]

   4. Applied egg-rr 99.7%

      \[\leadsto \color{blue}{x - \frac{y - \frac{t}{y}}{z \cdot 3}} \]

   5. Taylor expanded in y around inf

      \[\leadsto x - \frac{\color{blue}{y}}{z \cdot 3} \]
   6. Step-by-step derivation

   7. Simplified 91.3%

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

Alternative 8: 88.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.2 \cdot 10^{-51}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{\frac{1}{z}}{-3}, x\right)\\ \mathbf{elif}\;y \leq 3.55 \cdot 10^{-17}:\\ \;\;\;\;x + \frac{t}{z \cdot \left(y \cdot 3\right)}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{z \cdot 3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y -1.2e-51)
   (fma y (/ (/ 1.0 z) -3.0) x)
   (if (<= y 3.55e-17) (+ x (/ t (* z (* y 3.0)))) (- x (/ y (* z 3.0))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -1.2e-51) {
		tmp = fma(y, ((1.0 / z) / -3.0), x);
	} else if (y <= 3.55e-17) {
		tmp = x + (t / (z * (y * 3.0)));
	} else {
		tmp = x - (y / (z * 3.0));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= -1.2e-51)
		tmp = fma(y, Float64(Float64(1.0 / z) / -3.0), x);
	elseif (y <= 3.55e-17)
		tmp = Float64(x + Float64(t / Float64(z * Float64(y * 3.0))));
	else
		tmp = Float64(x - Float64(y / Float64(z * 3.0)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, -1.2e-51], N[(y * N[(N[(1.0 / z), $MachinePrecision] / -3.0), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[y, 3.55e-17], N[(x + N[(t / N[(z * N[(y * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x - N[(y / N[(z * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.2e-51

   1. Initial program 97.2%

      \[\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{y \cdot \left(\frac{x}{y} - \frac{1}{3} \cdot \frac{1}{z}\right)} \]
   4. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto y \cdot \color{blue}{\left(\frac{x}{y} + \left(\mathsf{neg}\left(\frac{1}{3} \cdot \frac{1}{z}\right)\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto y \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\frac{1}{3} \cdot \frac{1}{z}\right)\right) + \frac{x}{y}\right)} \]

      3. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{1}{3} \cdot \frac{1}{z}\right)\right) \cdot y + \frac{x}{y} \cdot y} \]

      4. associate-*r/ N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\frac{\frac{1}{3} \cdot 1}{z}}\right)\right) \cdot y + \frac{x}{y} \cdot y \]

      5. metadata-eval N/A

         \[\leadsto \left(\mathsf{neg}\left(\frac{\color{blue}{\frac{1}{3}}}{z}\right)\right) \cdot y + \frac{x}{y} \cdot y \]

      6. distribute-neg-frac N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(\frac{1}{3}\right)}{z}} \cdot y + \frac{x}{y} \cdot y \]

      7. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{3}}}{z} \cdot y + \frac{x}{y} \cdot y \]

      8. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\frac{-1}{3} \cdot y}{z}} + \frac{x}{y} \cdot y \]

      9. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{-1}{3} \cdot \frac{y}{z}} + \frac{x}{y} \cdot y \]

      10. cancel-sign-sub N/A

          \[\leadsto \color{blue}{\frac{-1}{3} \cdot \frac{y}{z} - \left(\mathsf{neg}\left(\frac{x}{y}\right)\right) \cdot y} \]

      11. mul-1-neg N/A

          \[\leadsto \frac{-1}{3} \cdot \frac{y}{z} - \color{blue}{\left(-1 \cdot \frac{x}{y}\right)} \cdot y \]

      12. associate-*r/ N/A

          \[\leadsto \frac{-1}{3} \cdot \frac{y}{z} - \color{blue}{\frac{-1 \cdot x}{y}} \cdot y \]

      13. associate-*l/ N/A

          \[\leadsto \frac{-1}{3} \cdot \frac{y}{z} - \color{blue}{\frac{\left(-1 \cdot x\right) \cdot y}{y}} \]

      14. associate-/l* N/A

          \[\leadsto \frac{-1}{3} \cdot \frac{y}{z} - \color{blue}{\left(-1 \cdot x\right) \cdot \frac{y}{y}} \]

      15. mul-1-neg N/A

          \[\leadsto \frac{-1}{3} \cdot \frac{y}{z} - \color{blue}{\left(\mathsf{neg}\left(x\right)\right)} \cdot \frac{y}{y} \]

      16. *-inverses N/A

          \[\leadsto \frac{-1}{3} \cdot \frac{y}{z} - \left(\mathsf{neg}\left(x\right)\right) \cdot \color{blue}{1} \]

      17. cancel-sign-sub N/A

          \[\leadsto \color{blue}{\frac{-1}{3} \cdot \frac{y}{z} + x \cdot 1} \]

      18. *-rgt-identity N/A

          \[\leadsto \frac{-1}{3} \cdot \frac{y}{z} + \color{blue}{x} \]

   5. Simplified 89.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, \frac{-0.3333333333333333}{z}, x\right)} \]
   6. Step-by-step derivation

      1. clear-num N/A

         \[\leadsto \mathsf{fma}\left(y, \color{blue}{\frac{1}{\frac{z}{\frac{-1}{3}}}}, x\right) \]

      2. associate-/r/ N/A

         \[\leadsto \mathsf{fma}\left(y, \color{blue}{\frac{1}{z} \cdot \frac{-1}{3}}, x\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(y, \frac{1}{z} \cdot \color{blue}{\frac{1}{-3}}, x\right) \]

      4. div-inv N/A

         \[\leadsto \mathsf{fma}\left(y, \color{blue}{\frac{\frac{1}{z}}{-3}}, x\right) \]

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

         \[\leadsto \mathsf{fma}\left(y, \color{blue}{\frac{\frac{1}{z}}{-3}}, x\right) \]

      6. /-lowering-/.f64 89.3

         \[\leadsto \mathsf{fma}\left(y, \frac{\color{blue}{\frac{1}{z}}}{-3}, x\right) \]

   7. Applied egg-rr 89.3%

      \[\leadsto \mathsf{fma}\left(y, \color{blue}{\frac{\frac{1}{z}}{-3}}, x\right) \]

   if -1.2e-51 < y < 3.5499999999999998e-17

   1. Initial program 94.7%

      \[\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x} + \frac{t}{\left(z \cdot 3\right) \cdot y} \]
   4. Step-by-step derivation

   5. Simplified 94.3%

      \[\leadsto \color{blue}{x} + \frac{t}{\left(z \cdot 3\right) \cdot y} \]
   6. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto x + \frac{t}{\color{blue}{z \cdot \left(3 \cdot y\right)}} \]

      2. *-commutative N/A

         \[\leadsto x + \frac{t}{z \cdot \color{blue}{\left(y \cdot 3\right)}} \]

      3. *-commutative N/A

         \[\leadsto x + \frac{t}{\color{blue}{\left(y \cdot 3\right) \cdot z}} \]

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

         \[\leadsto x + \frac{t}{\color{blue}{\left(y \cdot 3\right) \cdot z}} \]

      5. *-lowering-*.f64 94.4

         \[\leadsto x + \frac{t}{\color{blue}{\left(y \cdot 3\right)} \cdot z} \]

   7. Applied egg-rr 94.4%

      \[\leadsto x + \frac{t}{\color{blue}{\left(y \cdot 3\right) \cdot z}} \]

   if 3.5499999999999998e-17 < y

   1. Initial program 99.7%

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

      1. associate-+l- N/A

         \[\leadsto \color{blue}{x - \left(\frac{y}{z \cdot 3} - \frac{t}{\left(z \cdot 3\right) \cdot y}\right)} \]

      2. --lowering--.f64 N/A

         \[\leadsto \color{blue}{x - \left(\frac{y}{z \cdot 3} - \frac{t}{\left(z \cdot 3\right) \cdot y}\right)} \]

      3. *-commutative N/A

         \[\leadsto x - \left(\frac{y}{z \cdot 3} - \frac{t}{\color{blue}{y \cdot \left(z \cdot 3\right)}}\right) \]

      4. associate-/r* N/A

         \[\leadsto x - \left(\frac{y}{z \cdot 3} - \color{blue}{\frac{\frac{t}{y}}{z \cdot 3}}\right) \]

      5. sub-div N/A

         \[\leadsto x - \color{blue}{\frac{y - \frac{t}{y}}{z \cdot 3}} \]

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

         \[\leadsto x - \color{blue}{\frac{y - \frac{t}{y}}{z \cdot 3}} \]

      7. --lowering--.f64 N/A

         \[\leadsto x - \frac{\color{blue}{y - \frac{t}{y}}}{z \cdot 3} \]

      8. /-lowering-/.f64 N/A

         \[\leadsto x - \frac{y - \color{blue}{\frac{t}{y}}}{z \cdot 3} \]

      9. *-lowering-*.f64 99.7

         \[\leadsto x - \frac{y - \frac{t}{y}}{\color{blue}{z \cdot 3}} \]

   4. Applied egg-rr 99.7%

      \[\leadsto \color{blue}{x - \frac{y - \frac{t}{y}}{z \cdot 3}} \]

   5. Taylor expanded in y around inf

      \[\leadsto x - \frac{\color{blue}{y}}{z \cdot 3} \]
   6. Step-by-step derivation

   7. Simplified 91.3%

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

4. Final simplification 92.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.2 \cdot 10^{-51}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{\frac{1}{z}}{-3}, x\right)\\ \mathbf{elif}\;y \leq 3.55 \cdot 10^{-17}:\\ \;\;\;\;x + \frac{t}{z \cdot \left(y \cdot 3\right)}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{z \cdot 3}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 88.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.08 \cdot 10^{-51}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{-0.3333333333333333}{z}, x\right)\\ \mathbf{elif}\;y \leq 3.6 \cdot 10^{-17}:\\ \;\;\;\;x + \frac{t}{z \cdot \left(y \cdot 3\right)}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{z \cdot 3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y -1.08e-51)
   (fma y (/ -0.3333333333333333 z) x)
   (if (<= y 3.6e-17) (+ x (/ t (* z (* y 3.0)))) (- x (/ y (* z 3.0))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -1.08e-51) {
		tmp = fma(y, (-0.3333333333333333 / z), x);
	} else if (y <= 3.6e-17) {
		tmp = x + (t / (z * (y * 3.0)));
	} else {
		tmp = x - (y / (z * 3.0));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= -1.08e-51)
		tmp = fma(y, Float64(-0.3333333333333333 / z), x);
	elseif (y <= 3.6e-17)
		tmp = Float64(x + Float64(t / Float64(z * Float64(y * 3.0))));
	else
		tmp = Float64(x - Float64(y / Float64(z * 3.0)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, -1.08e-51], N[(y * N[(-0.3333333333333333 / z), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[y, 3.6e-17], N[(x + N[(t / N[(z * N[(y * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x - N[(y / N[(z * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.08000000000000004e-51

   1. Initial program 97.2%

      \[\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{y \cdot \left(\frac{x}{y} - \frac{1}{3} \cdot \frac{1}{z}\right)} \]
   4. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto y \cdot \color{blue}{\left(\frac{x}{y} + \left(\mathsf{neg}\left(\frac{1}{3} \cdot \frac{1}{z}\right)\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto y \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\frac{1}{3} \cdot \frac{1}{z}\right)\right) + \frac{x}{y}\right)} \]

      3. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{1}{3} \cdot \frac{1}{z}\right)\right) \cdot y + \frac{x}{y} \cdot y} \]

      4. associate-*r/ N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\frac{\frac{1}{3} \cdot 1}{z}}\right)\right) \cdot y + \frac{x}{y} \cdot y \]

      5. metadata-eval N/A

         \[\leadsto \left(\mathsf{neg}\left(\frac{\color{blue}{\frac{1}{3}}}{z}\right)\right) \cdot y + \frac{x}{y} \cdot y \]

      6. distribute-neg-frac N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(\frac{1}{3}\right)}{z}} \cdot y + \frac{x}{y} \cdot y \]

      7. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{3}}}{z} \cdot y + \frac{x}{y} \cdot y \]

      8. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\frac{-1}{3} \cdot y}{z}} + \frac{x}{y} \cdot y \]

      9. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{-1}{3} \cdot \frac{y}{z}} + \frac{x}{y} \cdot y \]

      10. cancel-sign-sub N/A

          \[\leadsto \color{blue}{\frac{-1}{3} \cdot \frac{y}{z} - \left(\mathsf{neg}\left(\frac{x}{y}\right)\right) \cdot y} \]

      11. mul-1-neg N/A

          \[\leadsto \frac{-1}{3} \cdot \frac{y}{z} - \color{blue}{\left(-1 \cdot \frac{x}{y}\right)} \cdot y \]

      12. associate-*r/ N/A

          \[\leadsto \frac{-1}{3} \cdot \frac{y}{z} - \color{blue}{\frac{-1 \cdot x}{y}} \cdot y \]

      13. associate-*l/ N/A

          \[\leadsto \frac{-1}{3} \cdot \frac{y}{z} - \color{blue}{\frac{\left(-1 \cdot x\right) \cdot y}{y}} \]

      14. associate-/l* N/A

          \[\leadsto \frac{-1}{3} \cdot \frac{y}{z} - \color{blue}{\left(-1 \cdot x\right) \cdot \frac{y}{y}} \]

      15. mul-1-neg N/A

          \[\leadsto \frac{-1}{3} \cdot \frac{y}{z} - \color{blue}{\left(\mathsf{neg}\left(x\right)\right)} \cdot \frac{y}{y} \]

      16. *-inverses N/A

          \[\leadsto \frac{-1}{3} \cdot \frac{y}{z} - \left(\mathsf{neg}\left(x\right)\right) \cdot \color{blue}{1} \]

      17. cancel-sign-sub N/A

          \[\leadsto \color{blue}{\frac{-1}{3} \cdot \frac{y}{z} + x \cdot 1} \]

      18. *-rgt-identity N/A

          \[\leadsto \frac{-1}{3} \cdot \frac{y}{z} + \color{blue}{x} \]

   5. Simplified 89.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, \frac{-0.3333333333333333}{z}, x\right)} \]

   if -1.08000000000000004e-51 < y < 3.59999999999999995e-17

   1. Initial program 94.7%

      \[\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x} + \frac{t}{\left(z \cdot 3\right) \cdot y} \]
   4. Step-by-step derivation

   5. Simplified 94.3%

      \[\leadsto \color{blue}{x} + \frac{t}{\left(z \cdot 3\right) \cdot y} \]
   6. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto x + \frac{t}{\color{blue}{z \cdot \left(3 \cdot y\right)}} \]

      2. *-commutative N/A

         \[\leadsto x + \frac{t}{z \cdot \color{blue}{\left(y \cdot 3\right)}} \]

      3. *-commutative N/A

         \[\leadsto x + \frac{t}{\color{blue}{\left(y \cdot 3\right) \cdot z}} \]

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

         \[\leadsto x + \frac{t}{\color{blue}{\left(y \cdot 3\right) \cdot z}} \]

      5. *-lowering-*.f64 94.4

         \[\leadsto x + \frac{t}{\color{blue}{\left(y \cdot 3\right)} \cdot z} \]

   7. Applied egg-rr 94.4%

      \[\leadsto x + \frac{t}{\color{blue}{\left(y \cdot 3\right) \cdot z}} \]

   if 3.59999999999999995e-17 < y

   1. Initial program 99.7%

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

      1. associate-+l- N/A

         \[\leadsto \color{blue}{x - \left(\frac{y}{z \cdot 3} - \frac{t}{\left(z \cdot 3\right) \cdot y}\right)} \]

      2. --lowering--.f64 N/A

         \[\leadsto \color{blue}{x - \left(\frac{y}{z \cdot 3} - \frac{t}{\left(z \cdot 3\right) \cdot y}\right)} \]

      3. *-commutative N/A

         \[\leadsto x - \left(\frac{y}{z \cdot 3} - \frac{t}{\color{blue}{y \cdot \left(z \cdot 3\right)}}\right) \]

      4. associate-/r* N/A

         \[\leadsto x - \left(\frac{y}{z \cdot 3} - \color{blue}{\frac{\frac{t}{y}}{z \cdot 3}}\right) \]

      5. sub-div N/A

         \[\leadsto x - \color{blue}{\frac{y - \frac{t}{y}}{z \cdot 3}} \]

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

         \[\leadsto x - \color{blue}{\frac{y - \frac{t}{y}}{z \cdot 3}} \]

      7. --lowering--.f64 N/A

         \[\leadsto x - \frac{\color{blue}{y - \frac{t}{y}}}{z \cdot 3} \]

      8. /-lowering-/.f64 N/A

         \[\leadsto x - \frac{y - \color{blue}{\frac{t}{y}}}{z \cdot 3} \]

      9. *-lowering-*.f64 99.7

         \[\leadsto x - \frac{y - \frac{t}{y}}{\color{blue}{z \cdot 3}} \]

   4. Applied egg-rr 99.7%

      \[\leadsto \color{blue}{x - \frac{y - \frac{t}{y}}{z \cdot 3}} \]

   5. Taylor expanded in y around inf

      \[\leadsto x - \frac{\color{blue}{y}}{z \cdot 3} \]
   6. Step-by-step derivation

   7. Simplified 91.3%

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

4. Final simplification 92.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.08 \cdot 10^{-51}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{-0.3333333333333333}{z}, x\right)\\ \mathbf{elif}\;y \leq 3.6 \cdot 10^{-17}:\\ \;\;\;\;x + \frac{t}{z \cdot \left(y \cdot 3\right)}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{z \cdot 3}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 88.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.2 \cdot 10^{-51}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{-0.3333333333333333}{z}, x\right)\\ \mathbf{elif}\;y \leq 2.25 \cdot 10^{-20}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{y \cdot z}, 0.3333333333333333, x\right)\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{z \cdot 3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y -1.2e-51)
   (fma y (/ -0.3333333333333333 z) x)
   (if (<= y 2.25e-20)
     (fma (/ t (* y z)) 0.3333333333333333 x)
     (- x (/ y (* z 3.0))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -1.2e-51) {
		tmp = fma(y, (-0.3333333333333333 / z), x);
	} else if (y <= 2.25e-20) {
		tmp = fma((t / (y * z)), 0.3333333333333333, x);
	} else {
		tmp = x - (y / (z * 3.0));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= -1.2e-51)
		tmp = fma(y, Float64(-0.3333333333333333 / z), x);
	elseif (y <= 2.25e-20)
		tmp = fma(Float64(t / Float64(y * z)), 0.3333333333333333, x);
	else
		tmp = Float64(x - Float64(y / Float64(z * 3.0)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, -1.2e-51], N[(y * N[(-0.3333333333333333 / z), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[y, 2.25e-20], N[(N[(t / N[(y * z), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333 + x), $MachinePrecision], N[(x - N[(y / N[(z * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.2e-51

   1. Initial program 97.2%

      \[\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{y \cdot \left(\frac{x}{y} - \frac{1}{3} \cdot \frac{1}{z}\right)} \]
   4. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto y \cdot \color{blue}{\left(\frac{x}{y} + \left(\mathsf{neg}\left(\frac{1}{3} \cdot \frac{1}{z}\right)\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto y \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\frac{1}{3} \cdot \frac{1}{z}\right)\right) + \frac{x}{y}\right)} \]

      3. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{1}{3} \cdot \frac{1}{z}\right)\right) \cdot y + \frac{x}{y} \cdot y} \]

      4. associate-*r/ N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\frac{\frac{1}{3} \cdot 1}{z}}\right)\right) \cdot y + \frac{x}{y} \cdot y \]

      5. metadata-eval N/A

         \[\leadsto \left(\mathsf{neg}\left(\frac{\color{blue}{\frac{1}{3}}}{z}\right)\right) \cdot y + \frac{x}{y} \cdot y \]

      6. distribute-neg-frac N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(\frac{1}{3}\right)}{z}} \cdot y + \frac{x}{y} \cdot y \]

      7. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{3}}}{z} \cdot y + \frac{x}{y} \cdot y \]

      8. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\frac{-1}{3} \cdot y}{z}} + \frac{x}{y} \cdot y \]

      9. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{-1}{3} \cdot \frac{y}{z}} + \frac{x}{y} \cdot y \]

      10. cancel-sign-sub N/A

          \[\leadsto \color{blue}{\frac{-1}{3} \cdot \frac{y}{z} - \left(\mathsf{neg}\left(\frac{x}{y}\right)\right) \cdot y} \]

      11. mul-1-neg N/A

          \[\leadsto \frac{-1}{3} \cdot \frac{y}{z} - \color{blue}{\left(-1 \cdot \frac{x}{y}\right)} \cdot y \]

      12. associate-*r/ N/A

          \[\leadsto \frac{-1}{3} \cdot \frac{y}{z} - \color{blue}{\frac{-1 \cdot x}{y}} \cdot y \]

      13. associate-*l/ N/A

          \[\leadsto \frac{-1}{3} \cdot \frac{y}{z} - \color{blue}{\frac{\left(-1 \cdot x\right) \cdot y}{y}} \]

      14. associate-/l* N/A

          \[\leadsto \frac{-1}{3} \cdot \frac{y}{z} - \color{blue}{\left(-1 \cdot x\right) \cdot \frac{y}{y}} \]

      15. mul-1-neg N/A

          \[\leadsto \frac{-1}{3} \cdot \frac{y}{z} - \color{blue}{\left(\mathsf{neg}\left(x\right)\right)} \cdot \frac{y}{y} \]

      16. *-inverses N/A

          \[\leadsto \frac{-1}{3} \cdot \frac{y}{z} - \left(\mathsf{neg}\left(x\right)\right) \cdot \color{blue}{1} \]

      17. cancel-sign-sub N/A

          \[\leadsto \color{blue}{\frac{-1}{3} \cdot \frac{y}{z} + x \cdot 1} \]

      18. *-rgt-identity N/A

          \[\leadsto \frac{-1}{3} \cdot \frac{y}{z} + \color{blue}{x} \]

   5. Simplified 89.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, \frac{-0.3333333333333333}{z}, x\right)} \]

   if -1.2e-51 < y < 2.2500000000000001e-20

   1. Initial program 94.7%

      \[\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x} + \frac{t}{\left(z \cdot 3\right) \cdot y} \]
   4. Step-by-step derivation

   5. Simplified 94.3%

      \[\leadsto \color{blue}{x} + \frac{t}{\left(z \cdot 3\right) \cdot y} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{t}{\left(z \cdot 3\right) \cdot y} + x} \]

      2. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{t}{z \cdot 3}}{y}} + x \]

      3. div-inv N/A

         \[\leadsto \frac{\color{blue}{t \cdot \frac{1}{z \cdot 3}}}{y} + x \]

      4. *-commutative N/A

         \[\leadsto \frac{t \cdot \frac{1}{\color{blue}{3 \cdot z}}}{y} + x \]

      5. associate-/r* N/A

         \[\leadsto \frac{t \cdot \color{blue}{\frac{\frac{1}{3}}{z}}}{y} + x \]

      6. metadata-eval N/A

         \[\leadsto \frac{t \cdot \frac{\color{blue}{\frac{1}{3}}}{z}}{y} + x \]

      7. associate-/l* N/A

         \[\leadsto \frac{\color{blue}{\frac{t \cdot \frac{1}{3}}{z}}}{y} + x \]

      8. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{t \cdot \frac{1}{3}}{z \cdot y}} + x \]

      9. un-div-inv N/A

         \[\leadsto \color{blue}{\left(t \cdot \frac{1}{3}\right) \cdot \frac{1}{z \cdot y}} + x \]

      10. clear-num N/A

          \[\leadsto \left(t \cdot \frac{1}{3}\right) \cdot \color{blue}{\frac{1}{\frac{z \cdot y}{1}}} + x \]

      11. un-div-inv N/A

          \[\leadsto \color{blue}{\frac{t \cdot \frac{1}{3}}{\frac{z \cdot y}{1}}} + x \]

      12. div-inv N/A

          \[\leadsto \frac{t \cdot \frac{1}{3}}{\color{blue}{\left(z \cdot y\right) \cdot \frac{1}{1}}} + x \]

      13. metadata-eval N/A

          \[\leadsto \frac{t \cdot \frac{1}{3}}{\left(z \cdot y\right) \cdot \color{blue}{1}} + x \]

      14. times-frac N/A

          \[\leadsto \color{blue}{\frac{t}{z \cdot y} \cdot \frac{\frac{1}{3}}{1}} + x \]

      15. metadata-eval N/A

          \[\leadsto \frac{t}{z \cdot y} \cdot \color{blue}{\frac{1}{3}} + x \]

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

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{t}{z \cdot y}, \frac{1}{3}, x\right)} \]

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

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{t}{z \cdot y}}, \frac{1}{3}, x\right) \]

      18. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{t}{\color{blue}{y \cdot z}}, \frac{1}{3}, x\right) \]

      19. *-lowering-*.f64 94.3

          \[\leadsto \mathsf{fma}\left(\frac{t}{\color{blue}{y \cdot z}}, 0.3333333333333333, x\right) \]

   7. Applied egg-rr 94.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{t}{y \cdot z}, 0.3333333333333333, x\right)} \]

   if 2.2500000000000001e-20 < y

   1. Initial program 99.7%

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

      1. associate-+l- N/A

         \[\leadsto \color{blue}{x - \left(\frac{y}{z \cdot 3} - \frac{t}{\left(z \cdot 3\right) \cdot y}\right)} \]

      2. --lowering--.f64 N/A

         \[\leadsto \color{blue}{x - \left(\frac{y}{z \cdot 3} - \frac{t}{\left(z \cdot 3\right) \cdot y}\right)} \]

      3. *-commutative N/A

         \[\leadsto x - \left(\frac{y}{z \cdot 3} - \frac{t}{\color{blue}{y \cdot \left(z \cdot 3\right)}}\right) \]

      4. associate-/r* N/A

         \[\leadsto x - \left(\frac{y}{z \cdot 3} - \color{blue}{\frac{\frac{t}{y}}{z \cdot 3}}\right) \]

      5. sub-div N/A

         \[\leadsto x - \color{blue}{\frac{y - \frac{t}{y}}{z \cdot 3}} \]

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

         \[\leadsto x - \color{blue}{\frac{y - \frac{t}{y}}{z \cdot 3}} \]

      7. --lowering--.f64 N/A

         \[\leadsto x - \frac{\color{blue}{y - \frac{t}{y}}}{z \cdot 3} \]

      8. /-lowering-/.f64 N/A

         \[\leadsto x - \frac{y - \color{blue}{\frac{t}{y}}}{z \cdot 3} \]

      9. *-lowering-*.f64 99.7

         \[\leadsto x - \frac{y - \frac{t}{y}}{\color{blue}{z \cdot 3}} \]

   4. Applied egg-rr 99.7%

      \[\leadsto \color{blue}{x - \frac{y - \frac{t}{y}}{z \cdot 3}} \]

   5. Taylor expanded in y around inf

      \[\leadsto x - \frac{\color{blue}{y}}{z \cdot 3} \]
   6. Step-by-step derivation

   7. Simplified 91.3%

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

Alternative 11: 88.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.2 \cdot 10^{-51}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{-0.3333333333333333}{z}, x\right)\\ \mathbf{elif}\;y \leq 2.52 \cdot 10^{-20}:\\ \;\;\;\;\mathsf{fma}\left(\frac{0.3333333333333333}{y \cdot z}, t, x\right)\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{z \cdot 3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y -1.2e-51)
   (fma y (/ -0.3333333333333333 z) x)
   (if (<= y 2.52e-20)
     (fma (/ 0.3333333333333333 (* y z)) t x)
     (- x (/ y (* z 3.0))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -1.2e-51) {
		tmp = fma(y, (-0.3333333333333333 / z), x);
	} else if (y <= 2.52e-20) {
		tmp = fma((0.3333333333333333 / (y * z)), t, x);
	} else {
		tmp = x - (y / (z * 3.0));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= -1.2e-51)
		tmp = fma(y, Float64(-0.3333333333333333 / z), x);
	elseif (y <= 2.52e-20)
		tmp = fma(Float64(0.3333333333333333 / Float64(y * z)), t, x);
	else
		tmp = Float64(x - Float64(y / Float64(z * 3.0)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, -1.2e-51], N[(y * N[(-0.3333333333333333 / z), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[y, 2.52e-20], N[(N[(0.3333333333333333 / N[(y * z), $MachinePrecision]), $MachinePrecision] * t + x), $MachinePrecision], N[(x - N[(y / N[(z * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.2e-51

   1. Initial program 97.2%

      \[\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{y \cdot \left(\frac{x}{y} - \frac{1}{3} \cdot \frac{1}{z}\right)} \]
   4. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto y \cdot \color{blue}{\left(\frac{x}{y} + \left(\mathsf{neg}\left(\frac{1}{3} \cdot \frac{1}{z}\right)\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto y \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\frac{1}{3} \cdot \frac{1}{z}\right)\right) + \frac{x}{y}\right)} \]

      3. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{1}{3} \cdot \frac{1}{z}\right)\right) \cdot y + \frac{x}{y} \cdot y} \]

      4. associate-*r/ N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\frac{\frac{1}{3} \cdot 1}{z}}\right)\right) \cdot y + \frac{x}{y} \cdot y \]

      5. metadata-eval N/A

         \[\leadsto \left(\mathsf{neg}\left(\frac{\color{blue}{\frac{1}{3}}}{z}\right)\right) \cdot y + \frac{x}{y} \cdot y \]

      6. distribute-neg-frac N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(\frac{1}{3}\right)}{z}} \cdot y + \frac{x}{y} \cdot y \]

      7. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{3}}}{z} \cdot y + \frac{x}{y} \cdot y \]

      8. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\frac{-1}{3} \cdot y}{z}} + \frac{x}{y} \cdot y \]

      9. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{-1}{3} \cdot \frac{y}{z}} + \frac{x}{y} \cdot y \]

      10. cancel-sign-sub N/A

          \[\leadsto \color{blue}{\frac{-1}{3} \cdot \frac{y}{z} - \left(\mathsf{neg}\left(\frac{x}{y}\right)\right) \cdot y} \]

      11. mul-1-neg N/A

          \[\leadsto \frac{-1}{3} \cdot \frac{y}{z} - \color{blue}{\left(-1 \cdot \frac{x}{y}\right)} \cdot y \]

      12. associate-*r/ N/A

          \[\leadsto \frac{-1}{3} \cdot \frac{y}{z} - \color{blue}{\frac{-1 \cdot x}{y}} \cdot y \]

      13. associate-*l/ N/A

          \[\leadsto \frac{-1}{3} \cdot \frac{y}{z} - \color{blue}{\frac{\left(-1 \cdot x\right) \cdot y}{y}} \]

      14. associate-/l* N/A

          \[\leadsto \frac{-1}{3} \cdot \frac{y}{z} - \color{blue}{\left(-1 \cdot x\right) \cdot \frac{y}{y}} \]

      15. mul-1-neg N/A

          \[\leadsto \frac{-1}{3} \cdot \frac{y}{z} - \color{blue}{\left(\mathsf{neg}\left(x\right)\right)} \cdot \frac{y}{y} \]

      16. *-inverses N/A

          \[\leadsto \frac{-1}{3} \cdot \frac{y}{z} - \left(\mathsf{neg}\left(x\right)\right) \cdot \color{blue}{1} \]

      17. cancel-sign-sub N/A

          \[\leadsto \color{blue}{\frac{-1}{3} \cdot \frac{y}{z} + x \cdot 1} \]

      18. *-rgt-identity N/A

          \[\leadsto \frac{-1}{3} \cdot \frac{y}{z} + \color{blue}{x} \]

   5. Simplified 89.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, \frac{-0.3333333333333333}{z}, x\right)} \]

   if -1.2e-51 < y < 2.52e-20

   1. Initial program 94.7%

      \[\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x} + \frac{t}{\left(z \cdot 3\right) \cdot y} \]
   4. Step-by-step derivation

   5. Simplified 94.3%

      \[\leadsto \color{blue}{x} + \frac{t}{\left(z \cdot 3\right) \cdot y} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{t}{\left(z \cdot 3\right) \cdot y} + x} \]

      2. associate-*l* N/A

         \[\leadsto \frac{t}{\color{blue}{z \cdot \left(3 \cdot y\right)}} + x \]

      3. *-commutative N/A

         \[\leadsto \frac{t}{z \cdot \color{blue}{\left(y \cdot 3\right)}} + x \]

      4. *-commutative N/A

         \[\leadsto \frac{t}{\color{blue}{\left(y \cdot 3\right) \cdot z}} + x \]

      5. div-inv N/A

         \[\leadsto \color{blue}{t \cdot \frac{1}{\left(y \cdot 3\right) \cdot z}} + x \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\frac{1}{\left(y \cdot 3\right) \cdot z} \cdot t} + x \]

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

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{\left(y \cdot 3\right) \cdot z}, t, x\right)} \]

      8. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{\color{blue}{z \cdot \left(y \cdot 3\right)}}, t, x\right) \]

      9. associate-/r* N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\frac{1}{z}}{y \cdot 3}}, t, x\right) \]

      10. div-inv N/A

          \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{1 \cdot \frac{1}{z}}}{y \cdot 3}, t, x\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{1 \cdot \frac{1}{z}}{\color{blue}{3 \cdot y}}, t, x\right) \]

      12. times-frac N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{3} \cdot \frac{\frac{1}{z}}{y}}, t, x\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{3}} \cdot \frac{\frac{1}{z}}{y}, t, x\right) \]

      14. associate-/r* N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{3} \cdot \color{blue}{\frac{1}{z \cdot y}}, t, x\right) \]

      15. un-div-inv N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\frac{1}{3}}{z \cdot y}}, t, x\right) \]

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

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\frac{1}{3}}{z \cdot y}}, t, x\right) \]

      17. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{\frac{1}{3}}{\color{blue}{y \cdot z}}, t, x\right) \]

      18. *-lowering-*.f64 93.2

          \[\leadsto \mathsf{fma}\left(\frac{0.3333333333333333}{\color{blue}{y \cdot z}}, t, x\right) \]

   7. Applied egg-rr 93.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{0.3333333333333333}{y \cdot z}, t, x\right)} \]

   if 2.52e-20 < y

   1. Initial program 99.7%

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

      1. associate-+l- N/A

         \[\leadsto \color{blue}{x - \left(\frac{y}{z \cdot 3} - \frac{t}{\left(z \cdot 3\right) \cdot y}\right)} \]

      2. --lowering--.f64 N/A

         \[\leadsto \color{blue}{x - \left(\frac{y}{z \cdot 3} - \frac{t}{\left(z \cdot 3\right) \cdot y}\right)} \]

      3. *-commutative N/A

         \[\leadsto x - \left(\frac{y}{z \cdot 3} - \frac{t}{\color{blue}{y \cdot \left(z \cdot 3\right)}}\right) \]

      4. associate-/r* N/A

         \[\leadsto x - \left(\frac{y}{z \cdot 3} - \color{blue}{\frac{\frac{t}{y}}{z \cdot 3}}\right) \]

      5. sub-div N/A

         \[\leadsto x - \color{blue}{\frac{y - \frac{t}{y}}{z \cdot 3}} \]

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

         \[\leadsto x - \color{blue}{\frac{y - \frac{t}{y}}{z \cdot 3}} \]

      7. --lowering--.f64 N/A

         \[\leadsto x - \frac{\color{blue}{y - \frac{t}{y}}}{z \cdot 3} \]

      8. /-lowering-/.f64 N/A

         \[\leadsto x - \frac{y - \color{blue}{\frac{t}{y}}}{z \cdot 3} \]

      9. *-lowering-*.f64 99.7

         \[\leadsto x - \frac{y - \frac{t}{y}}{\color{blue}{z \cdot 3}} \]

   4. Applied egg-rr 99.7%

      \[\leadsto \color{blue}{x - \frac{y - \frac{t}{y}}{z \cdot 3}} \]

   5. Taylor expanded in y around inf

      \[\leadsto x - \frac{\color{blue}{y}}{z \cdot 3} \]
   6. Step-by-step derivation

   7. Simplified 91.3%

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

Alternative 12: 76.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.2 \cdot 10^{-53}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{-0.3333333333333333}{z}, x\right)\\ \mathbf{elif}\;y \leq 1.12 \cdot 10^{-41}:\\ \;\;\;\;\frac{t}{y \cdot \left(z \cdot 3\right)}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{z \cdot 3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y -3.2e-53)
   (fma y (/ -0.3333333333333333 z) x)
   (if (<= y 1.12e-41) (/ t (* y (* z 3.0))) (- x (/ y (* z 3.0))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -3.2e-53) {
		tmp = fma(y, (-0.3333333333333333 / z), x);
	} else if (y <= 1.12e-41) {
		tmp = t / (y * (z * 3.0));
	} else {
		tmp = x - (y / (z * 3.0));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= -3.2e-53)
		tmp = fma(y, Float64(-0.3333333333333333 / z), x);
	elseif (y <= 1.12e-41)
		tmp = Float64(t / Float64(y * Float64(z * 3.0)));
	else
		tmp = Float64(x - Float64(y / Float64(z * 3.0)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, -3.2e-53], N[(y * N[(-0.3333333333333333 / z), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[y, 1.12e-41], N[(t / N[(y * N[(z * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x - N[(y / N[(z * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -3.2000000000000001e-53

   1. Initial program 97.2%

      \[\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{y \cdot \left(\frac{x}{y} - \frac{1}{3} \cdot \frac{1}{z}\right)} \]
   4. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto y \cdot \color{blue}{\left(\frac{x}{y} + \left(\mathsf{neg}\left(\frac{1}{3} \cdot \frac{1}{z}\right)\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto y \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\frac{1}{3} \cdot \frac{1}{z}\right)\right) + \frac{x}{y}\right)} \]

      3. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{1}{3} \cdot \frac{1}{z}\right)\right) \cdot y + \frac{x}{y} \cdot y} \]

      4. associate-*r/ N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\frac{\frac{1}{3} \cdot 1}{z}}\right)\right) \cdot y + \frac{x}{y} \cdot y \]

      5. metadata-eval N/A

         \[\leadsto \left(\mathsf{neg}\left(\frac{\color{blue}{\frac{1}{3}}}{z}\right)\right) \cdot y + \frac{x}{y} \cdot y \]

      6. distribute-neg-frac N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(\frac{1}{3}\right)}{z}} \cdot y + \frac{x}{y} \cdot y \]

      7. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{3}}}{z} \cdot y + \frac{x}{y} \cdot y \]

      8. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\frac{-1}{3} \cdot y}{z}} + \frac{x}{y} \cdot y \]

      9. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{-1}{3} \cdot \frac{y}{z}} + \frac{x}{y} \cdot y \]

      10. cancel-sign-sub N/A

          \[\leadsto \color{blue}{\frac{-1}{3} \cdot \frac{y}{z} - \left(\mathsf{neg}\left(\frac{x}{y}\right)\right) \cdot y} \]

      11. mul-1-neg N/A

          \[\leadsto \frac{-1}{3} \cdot \frac{y}{z} - \color{blue}{\left(-1 \cdot \frac{x}{y}\right)} \cdot y \]

      12. associate-*r/ N/A

          \[\leadsto \frac{-1}{3} \cdot \frac{y}{z} - \color{blue}{\frac{-1 \cdot x}{y}} \cdot y \]

      13. associate-*l/ N/A

          \[\leadsto \frac{-1}{3} \cdot \frac{y}{z} - \color{blue}{\frac{\left(-1 \cdot x\right) \cdot y}{y}} \]

      14. associate-/l* N/A

          \[\leadsto \frac{-1}{3} \cdot \frac{y}{z} - \color{blue}{\left(-1 \cdot x\right) \cdot \frac{y}{y}} \]

      15. mul-1-neg N/A

          \[\leadsto \frac{-1}{3} \cdot \frac{y}{z} - \color{blue}{\left(\mathsf{neg}\left(x\right)\right)} \cdot \frac{y}{y} \]

      16. *-inverses N/A

          \[\leadsto \frac{-1}{3} \cdot \frac{y}{z} - \left(\mathsf{neg}\left(x\right)\right) \cdot \color{blue}{1} \]

      17. cancel-sign-sub N/A

          \[\leadsto \color{blue}{\frac{-1}{3} \cdot \frac{y}{z} + x \cdot 1} \]

      18. *-rgt-identity N/A

          \[\leadsto \frac{-1}{3} \cdot \frac{y}{z} + \color{blue}{x} \]

   5. Simplified 89.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, \frac{-0.3333333333333333}{z}, x\right)} \]

   if -3.2000000000000001e-53 < y < 1.11999999999999999e-41

   1. Initial program 94.5%

      \[\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{\frac{1}{3} \cdot \frac{t}{z} + x \cdot y}{y}} \]
   4. Step-by-step derivation

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

         \[\leadsto \color{blue}{\frac{\frac{1}{3} \cdot \frac{t}{z} + x \cdot y}{y}} \]

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

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{1}{3}, \frac{t}{z}, x \cdot y\right)}}{y} \]

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

         \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{3}, \color{blue}{\frac{t}{z}}, x \cdot y\right)}{y} \]

      4. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{3}, \frac{t}{z}, \color{blue}{y \cdot x}\right)}{y} \]

      5. *-lowering-*.f64 97.5

         \[\leadsto \frac{\mathsf{fma}\left(0.3333333333333333, \frac{t}{z}, \color{blue}{y \cdot x}\right)}{y} \]

   5. Simplified 97.5%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(0.3333333333333333, \frac{t}{z}, y \cdot x\right)}{y}} \]

   6. Taylor expanded in t around inf

      \[\leadsto \frac{\color{blue}{\frac{1}{3} \cdot \frac{t}{z}}}{y} \]
   7. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \frac{\color{blue}{\frac{\frac{1}{3} \cdot t}{z}}}{y} \]

      2. *-commutative N/A

         \[\leadsto \frac{\frac{\color{blue}{t \cdot \frac{1}{3}}}{z}}{y} \]

      3. associate-*r/ N/A

         \[\leadsto \frac{\color{blue}{t \cdot \frac{\frac{1}{3}}{z}}}{y} \]

      4. metadata-eval N/A

         \[\leadsto \frac{t \cdot \frac{\color{blue}{\frac{1}{3} \cdot 1}}{z}}{y} \]

      5. associate-*r/ N/A

         \[\leadsto \frac{t \cdot \color{blue}{\left(\frac{1}{3} \cdot \frac{1}{z}\right)}}{y} \]

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

         \[\leadsto \frac{\color{blue}{t \cdot \left(\frac{1}{3} \cdot \frac{1}{z}\right)}}{y} \]

      7. associate-*r/ N/A

         \[\leadsto \frac{t \cdot \color{blue}{\frac{\frac{1}{3} \cdot 1}{z}}}{y} \]

      8. metadata-eval N/A

         \[\leadsto \frac{t \cdot \frac{\color{blue}{\frac{1}{3}}}{z}}{y} \]

      9. /-lowering-/.f64 69.2

         \[\leadsto \frac{t \cdot \color{blue}{\frac{0.3333333333333333}{z}}}{y} \]

   8. Simplified 69.2%

      \[\leadsto \frac{\color{blue}{t \cdot \frac{0.3333333333333333}{z}}}{y} \]
   9. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \color{blue}{t \cdot \frac{\frac{\frac{1}{3}}{z}}{y}} \]

      2. clear-num N/A

         \[\leadsto t \cdot \color{blue}{\frac{1}{\frac{y}{\frac{\frac{1}{3}}{z}}}} \]

      3. un-div-inv N/A

         \[\leadsto \color{blue}{\frac{t}{\frac{y}{\frac{\frac{1}{3}}{z}}}} \]

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

         \[\leadsto \color{blue}{\frac{t}{\frac{y}{\frac{\frac{1}{3}}{z}}}} \]

      5. div-inv N/A

         \[\leadsto \frac{t}{\color{blue}{y \cdot \frac{1}{\frac{\frac{1}{3}}{z}}}} \]

      6. clear-num N/A

         \[\leadsto \frac{t}{y \cdot \color{blue}{\frac{z}{\frac{1}{3}}}} \]

      7. div-inv N/A

         \[\leadsto \frac{t}{y \cdot \color{blue}{\left(z \cdot \frac{1}{\frac{1}{3}}\right)}} \]

      8. metadata-eval N/A

         \[\leadsto \frac{t}{y \cdot \left(z \cdot \color{blue}{3}\right)} \]

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

         \[\leadsto \frac{t}{\color{blue}{y \cdot \left(z \cdot 3\right)}} \]

      10. *-lowering-*.f64 65.9

          \[\leadsto \frac{t}{y \cdot \color{blue}{\left(z \cdot 3\right)}} \]

   10. Applied egg-rr 65.9%

       \[\leadsto \color{blue}{\frac{t}{y \cdot \left(z \cdot 3\right)}} \]

   if 1.11999999999999999e-41 < y

   1. Initial program 99.7%

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

      1. associate-+l- N/A

         \[\leadsto \color{blue}{x - \left(\frac{y}{z \cdot 3} - \frac{t}{\left(z \cdot 3\right) \cdot y}\right)} \]

      2. --lowering--.f64 N/A

         \[\leadsto \color{blue}{x - \left(\frac{y}{z \cdot 3} - \frac{t}{\left(z \cdot 3\right) \cdot y}\right)} \]

      3. *-commutative N/A

         \[\leadsto x - \left(\frac{y}{z \cdot 3} - \frac{t}{\color{blue}{y \cdot \left(z \cdot 3\right)}}\right) \]

      4. associate-/r* N/A

         \[\leadsto x - \left(\frac{y}{z \cdot 3} - \color{blue}{\frac{\frac{t}{y}}{z \cdot 3}}\right) \]

      5. sub-div N/A

         \[\leadsto x - \color{blue}{\frac{y - \frac{t}{y}}{z \cdot 3}} \]

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

         \[\leadsto x - \color{blue}{\frac{y - \frac{t}{y}}{z \cdot 3}} \]

      7. --lowering--.f64 N/A

         \[\leadsto x - \frac{\color{blue}{y - \frac{t}{y}}}{z \cdot 3} \]

      8. /-lowering-/.f64 N/A

         \[\leadsto x - \frac{y - \color{blue}{\frac{t}{y}}}{z \cdot 3} \]

      9. *-lowering-*.f64 99.7

         \[\leadsto x - \frac{y - \frac{t}{y}}{\color{blue}{z \cdot 3}} \]

   4. Applied egg-rr 99.7%

      \[\leadsto \color{blue}{x - \frac{y - \frac{t}{y}}{z \cdot 3}} \]

   5. Taylor expanded in y around inf

      \[\leadsto x - \frac{\color{blue}{y}}{z \cdot 3} \]
   6. Step-by-step derivation

   7. Simplified 90.3%

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

Alternative 13: 76.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -7.2 \cdot 10^{-53}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{-0.3333333333333333}{z}, x\right)\\ \mathbf{elif}\;y \leq 7 \cdot 10^{-41}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{t}{y \cdot z}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{z \cdot 3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y -7.2e-53)
   (fma y (/ -0.3333333333333333 z) x)
   (if (<= y 7e-41)
     (* 0.3333333333333333 (/ t (* y z)))
     (- x (/ y (* z 3.0))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -7.2e-53) {
		tmp = fma(y, (-0.3333333333333333 / z), x);
	} else if (y <= 7e-41) {
		tmp = 0.3333333333333333 * (t / (y * z));
	} else {
		tmp = x - (y / (z * 3.0));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= -7.2e-53)
		tmp = fma(y, Float64(-0.3333333333333333 / z), x);
	elseif (y <= 7e-41)
		tmp = Float64(0.3333333333333333 * Float64(t / Float64(y * z)));
	else
		tmp = Float64(x - Float64(y / Float64(z * 3.0)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, -7.2e-53], N[(y * N[(-0.3333333333333333 / z), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[y, 7e-41], N[(0.3333333333333333 * N[(t / N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x - N[(y / N[(z * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -7.1999999999999998e-53

   1. Initial program 97.2%

      \[\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{y \cdot \left(\frac{x}{y} - \frac{1}{3} \cdot \frac{1}{z}\right)} \]
   4. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto y \cdot \color{blue}{\left(\frac{x}{y} + \left(\mathsf{neg}\left(\frac{1}{3} \cdot \frac{1}{z}\right)\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto y \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\frac{1}{3} \cdot \frac{1}{z}\right)\right) + \frac{x}{y}\right)} \]

      3. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{1}{3} \cdot \frac{1}{z}\right)\right) \cdot y + \frac{x}{y} \cdot y} \]

      4. associate-*r/ N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\frac{\frac{1}{3} \cdot 1}{z}}\right)\right) \cdot y + \frac{x}{y} \cdot y \]

      5. metadata-eval N/A

         \[\leadsto \left(\mathsf{neg}\left(\frac{\color{blue}{\frac{1}{3}}}{z}\right)\right) \cdot y + \frac{x}{y} \cdot y \]

      6. distribute-neg-frac N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(\frac{1}{3}\right)}{z}} \cdot y + \frac{x}{y} \cdot y \]

      7. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{3}}}{z} \cdot y + \frac{x}{y} \cdot y \]

      8. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\frac{-1}{3} \cdot y}{z}} + \frac{x}{y} \cdot y \]

      9. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{-1}{3} \cdot \frac{y}{z}} + \frac{x}{y} \cdot y \]

      10. cancel-sign-sub N/A

          \[\leadsto \color{blue}{\frac{-1}{3} \cdot \frac{y}{z} - \left(\mathsf{neg}\left(\frac{x}{y}\right)\right) \cdot y} \]

      11. mul-1-neg N/A

          \[\leadsto \frac{-1}{3} \cdot \frac{y}{z} - \color{blue}{\left(-1 \cdot \frac{x}{y}\right)} \cdot y \]

      12. associate-*r/ N/A

          \[\leadsto \frac{-1}{3} \cdot \frac{y}{z} - \color{blue}{\frac{-1 \cdot x}{y}} \cdot y \]

      13. associate-*l/ N/A

          \[\leadsto \frac{-1}{3} \cdot \frac{y}{z} - \color{blue}{\frac{\left(-1 \cdot x\right) \cdot y}{y}} \]

      14. associate-/l* N/A

          \[\leadsto \frac{-1}{3} \cdot \frac{y}{z} - \color{blue}{\left(-1 \cdot x\right) \cdot \frac{y}{y}} \]

      15. mul-1-neg N/A

          \[\leadsto \frac{-1}{3} \cdot \frac{y}{z} - \color{blue}{\left(\mathsf{neg}\left(x\right)\right)} \cdot \frac{y}{y} \]

      16. *-inverses N/A

          \[\leadsto \frac{-1}{3} \cdot \frac{y}{z} - \left(\mathsf{neg}\left(x\right)\right) \cdot \color{blue}{1} \]

      17. cancel-sign-sub N/A

          \[\leadsto \color{blue}{\frac{-1}{3} \cdot \frac{y}{z} + x \cdot 1} \]

      18. *-rgt-identity N/A

          \[\leadsto \frac{-1}{3} \cdot \frac{y}{z} + \color{blue}{x} \]

   5. Simplified 89.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, \frac{-0.3333333333333333}{z}, x\right)} \]

   if -7.1999999999999998e-53 < y < 6.9999999999999999e-41

   1. Initial program 94.5%

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

      1. associate-+l- N/A

         \[\leadsto \color{blue}{x - \left(\frac{y}{z \cdot 3} - \frac{t}{\left(z \cdot 3\right) \cdot y}\right)} \]

      2. --lowering--.f64 N/A

         \[\leadsto \color{blue}{x - \left(\frac{y}{z \cdot 3} - \frac{t}{\left(z \cdot 3\right) \cdot y}\right)} \]

      3. *-commutative N/A

         \[\leadsto x - \left(\frac{y}{z \cdot 3} - \frac{t}{\color{blue}{y \cdot \left(z \cdot 3\right)}}\right) \]

      4. associate-/r* N/A

         \[\leadsto x - \left(\frac{y}{z \cdot 3} - \color{blue}{\frac{\frac{t}{y}}{z \cdot 3}}\right) \]

      5. sub-div N/A

         \[\leadsto x - \color{blue}{\frac{y - \frac{t}{y}}{z \cdot 3}} \]

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

         \[\leadsto x - \color{blue}{\frac{y - \frac{t}{y}}{z \cdot 3}} \]

      7. --lowering--.f64 N/A

         \[\leadsto x - \frac{\color{blue}{y - \frac{t}{y}}}{z \cdot 3} \]

      8. /-lowering-/.f64 N/A

         \[\leadsto x - \frac{y - \color{blue}{\frac{t}{y}}}{z \cdot 3} \]

      9. *-lowering-*.f64 87.8

         \[\leadsto x - \frac{y - \frac{t}{y}}{\color{blue}{z \cdot 3}} \]

   4. Applied egg-rr 87.8%

      \[\leadsto \color{blue}{x - \frac{y - \frac{t}{y}}{z \cdot 3}} \]
   5. Step-by-step derivation

      1. clear-num N/A

         \[\leadsto x - \color{blue}{\frac{1}{\frac{z \cdot 3}{y - \frac{t}{y}}}} \]

      2. /-lowering-/.f64 N/A

         \[\leadsto x - \color{blue}{\frac{1}{\frac{z \cdot 3}{y - \frac{t}{y}}}} \]

      3. associate-/l* N/A

         \[\leadsto x - \frac{1}{\color{blue}{z \cdot \frac{3}{y - \frac{t}{y}}}} \]

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

         \[\leadsto x - \frac{1}{\color{blue}{z \cdot \frac{3}{y - \frac{t}{y}}}} \]

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

         \[\leadsto x - \frac{1}{z \cdot \color{blue}{\frac{3}{y - \frac{t}{y}}}} \]

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

         \[\leadsto x - \frac{1}{z \cdot \frac{3}{\color{blue}{y - \frac{t}{y}}}} \]

      7. /-lowering-/.f64 87.8

         \[\leadsto x - \frac{1}{z \cdot \frac{3}{y - \color{blue}{\frac{t}{y}}}} \]

   6. Applied egg-rr 87.8%

      \[\leadsto x - \color{blue}{\frac{1}{z \cdot \frac{3}{y - \frac{t}{y}}}} \]

   7. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{t}{y \cdot z}} \]
   8. Step-by-step derivation

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

         \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{t}{y \cdot z}} \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \color{blue}{\frac{t}{y \cdot z}} \]

      3. *-lowering-*.f64 65.8

         \[\leadsto 0.3333333333333333 \cdot \frac{t}{\color{blue}{y \cdot z}} \]

   9. Simplified 65.8%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \frac{t}{y \cdot z}} \]

   if 6.9999999999999999e-41 < y

   1. Initial program 99.7%

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

      1. associate-+l- N/A

         \[\leadsto \color{blue}{x - \left(\frac{y}{z \cdot 3} - \frac{t}{\left(z \cdot 3\right) \cdot y}\right)} \]

      2. --lowering--.f64 N/A

         \[\leadsto \color{blue}{x - \left(\frac{y}{z \cdot 3} - \frac{t}{\left(z \cdot 3\right) \cdot y}\right)} \]

      3. *-commutative N/A

         \[\leadsto x - \left(\frac{y}{z \cdot 3} - \frac{t}{\color{blue}{y \cdot \left(z \cdot 3\right)}}\right) \]

      4. associate-/r* N/A

         \[\leadsto x - \left(\frac{y}{z \cdot 3} - \color{blue}{\frac{\frac{t}{y}}{z \cdot 3}}\right) \]

      5. sub-div N/A

         \[\leadsto x - \color{blue}{\frac{y - \frac{t}{y}}{z \cdot 3}} \]

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

         \[\leadsto x - \color{blue}{\frac{y - \frac{t}{y}}{z \cdot 3}} \]

      7. --lowering--.f64 N/A

         \[\leadsto x - \frac{\color{blue}{y - \frac{t}{y}}}{z \cdot 3} \]

      8. /-lowering-/.f64 N/A

         \[\leadsto x - \frac{y - \color{blue}{\frac{t}{y}}}{z \cdot 3} \]

      9. *-lowering-*.f64 99.7

         \[\leadsto x - \frac{y - \frac{t}{y}}{\color{blue}{z \cdot 3}} \]

   4. Applied egg-rr 99.7%

      \[\leadsto \color{blue}{x - \frac{y - \frac{t}{y}}{z \cdot 3}} \]

   5. Taylor expanded in y around inf

      \[\leadsto x - \frac{\color{blue}{y}}{z \cdot 3} \]
   6. Step-by-step derivation

   7. Simplified 90.3%

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

Alternative 14: 46.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{y}{z \cdot -3}\\ \mathbf{if}\;y \leq -1.15 \cdot 10^{-61}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 1.85 \cdot 10^{+34}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ y (* z -3.0))))
   (if (<= y -1.15e-61) t_1 (if (<= y 1.85e+34) x t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = y / (z * -3.0);
	double tmp;
	if (y <= -1.15e-61) {
		tmp = t_1;
	} else if (y <= 1.85e+34) {
		tmp = 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 = y / (z * (-3.0d0))
    if (y <= (-1.15d-61)) then
        tmp = t_1
    else if (y <= 1.85d+34) then
        tmp = 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 = y / (z * -3.0);
	double tmp;
	if (y <= -1.15e-61) {
		tmp = t_1;
	} else if (y <= 1.85e+34) {
		tmp = x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = y / (z * -3.0)
	tmp = 0
	if y <= -1.15e-61:
		tmp = t_1
	elif y <= 1.85e+34:
		tmp = x
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(y / Float64(z * -3.0))
	tmp = 0.0
	if (y <= -1.15e-61)
		tmp = t_1;
	elseif (y <= 1.85e+34)
		tmp = x;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = y / (z * -3.0);
	tmp = 0.0;
	if (y <= -1.15e-61)
		tmp = t_1;
	elseif (y <= 1.85e+34)
		tmp = x;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(y / N[(z * -3.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.15e-61], t$95$1, If[LessEqual[y, 1.85e+34], x, t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.14999999999999996e-61 or 1.85000000000000004e34 < y

   1. Initial program 98.3%

      \[\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{-1}{3} \cdot \frac{y}{z}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{-1}{3} \cdot y}{z}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\color{blue}{y \cdot \frac{-1}{3}}}{z} \]

      3. associate-/l* N/A

         \[\leadsto \color{blue}{y \cdot \frac{\frac{-1}{3}}{z}} \]

      4. metadata-eval N/A

         \[\leadsto y \cdot \frac{\color{blue}{\mathsf{neg}\left(\frac{1}{3}\right)}}{z} \]

      5. distribute-neg-frac N/A

         \[\leadsto y \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{\frac{1}{3}}{z}\right)\right)} \]

      6. metadata-eval N/A

         \[\leadsto y \cdot \left(\mathsf{neg}\left(\frac{\color{blue}{\frac{1}{3} \cdot 1}}{z}\right)\right) \]

      7. associate-*r/ N/A

         \[\leadsto y \cdot \left(\mathsf{neg}\left(\color{blue}{\frac{1}{3} \cdot \frac{1}{z}}\right)\right) \]

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

         \[\leadsto \color{blue}{y \cdot \left(\mathsf{neg}\left(\frac{1}{3} \cdot \frac{1}{z}\right)\right)} \]

      9. associate-*r/ N/A

         \[\leadsto y \cdot \left(\mathsf{neg}\left(\color{blue}{\frac{\frac{1}{3} \cdot 1}{z}}\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto y \cdot \left(\mathsf{neg}\left(\frac{\color{blue}{\frac{1}{3}}}{z}\right)\right) \]

      11. distribute-neg-frac N/A

          \[\leadsto y \cdot \color{blue}{\frac{\mathsf{neg}\left(\frac{1}{3}\right)}{z}} \]

      12. metadata-eval N/A

          \[\leadsto y \cdot \frac{\color{blue}{\frac{-1}{3}}}{z} \]

      13. /-lowering-/.f64 65.8

          \[\leadsto y \cdot \color{blue}{\frac{-0.3333333333333333}{z}} \]

   5. Simplified 65.8%

      \[\leadsto \color{blue}{y \cdot \frac{-0.3333333333333333}{z}} \]
   6. Step-by-step derivation

      1. clear-num N/A

         \[\leadsto y \cdot \color{blue}{\frac{1}{\frac{z}{\frac{-1}{3}}}} \]

      2. un-div-inv N/A

         \[\leadsto \color{blue}{\frac{y}{\frac{z}{\frac{-1}{3}}}} \]

      3. div-inv N/A

         \[\leadsto \frac{y}{\color{blue}{z \cdot \frac{1}{\frac{-1}{3}}}} \]

      4. metadata-eval N/A

         \[\leadsto \frac{y}{z \cdot \color{blue}{-3}} \]

      5. metadata-eval N/A

         \[\leadsto \frac{y}{z \cdot \color{blue}{\left(\mathsf{neg}\left(3\right)\right)}} \]

      6. distribute-rgt-neg-in N/A

         \[\leadsto \frac{y}{\color{blue}{\mathsf{neg}\left(z \cdot 3\right)}} \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{y}{\mathsf{neg}\left(z \cdot 3\right)}} \]

      8. distribute-rgt-neg-in N/A

         \[\leadsto \frac{y}{\color{blue}{z \cdot \left(\mathsf{neg}\left(3\right)\right)}} \]

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

         \[\leadsto \frac{y}{\color{blue}{z \cdot \left(\mathsf{neg}\left(3\right)\right)}} \]

      10. metadata-eval 65.8

          \[\leadsto \frac{y}{z \cdot \color{blue}{-3}} \]

   7. Applied egg-rr 65.8%

      \[\leadsto \color{blue}{\frac{y}{z \cdot -3}} \]

   if -1.14999999999999996e-61 < y < 1.85000000000000004e34

   1. Initial program 95.1%

      \[\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

   5. Simplified 32.8%

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

Alternative 15: 46.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y \cdot \frac{-0.3333333333333333}{z}\\ \mathbf{if}\;y \leq -1.15 \cdot 10^{-61}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 1.6 \cdot 10^{+31}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* y (/ -0.3333333333333333 z))))
   (if (<= y -1.15e-61) t_1 (if (<= y 1.6e+31) x t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = y * (-0.3333333333333333 / z);
	double tmp;
	if (y <= -1.15e-61) {
		tmp = t_1;
	} else if (y <= 1.6e+31) {
		tmp = 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 = y * ((-0.3333333333333333d0) / z)
    if (y <= (-1.15d-61)) then
        tmp = t_1
    else if (y <= 1.6d+31) then
        tmp = 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 = y * (-0.3333333333333333 / z);
	double tmp;
	if (y <= -1.15e-61) {
		tmp = t_1;
	} else if (y <= 1.6e+31) {
		tmp = x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = y * (-0.3333333333333333 / z)
	tmp = 0
	if y <= -1.15e-61:
		tmp = t_1
	elif y <= 1.6e+31:
		tmp = x
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(y * Float64(-0.3333333333333333 / z))
	tmp = 0.0
	if (y <= -1.15e-61)
		tmp = t_1;
	elseif (y <= 1.6e+31)
		tmp = x;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = y * (-0.3333333333333333 / z);
	tmp = 0.0;
	if (y <= -1.15e-61)
		tmp = t_1;
	elseif (y <= 1.6e+31)
		tmp = x;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(y * N[(-0.3333333333333333 / z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.15e-61], t$95$1, If[LessEqual[y, 1.6e+31], x, t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.14999999999999996e-61 or 1.6e31 < y

   1. Initial program 98.3%

      \[\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{-1}{3} \cdot \frac{y}{z}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{-1}{3} \cdot y}{z}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\color{blue}{y \cdot \frac{-1}{3}}}{z} \]

      3. associate-/l* N/A

         \[\leadsto \color{blue}{y \cdot \frac{\frac{-1}{3}}{z}} \]

      4. metadata-eval N/A

         \[\leadsto y \cdot \frac{\color{blue}{\mathsf{neg}\left(\frac{1}{3}\right)}}{z} \]

      5. distribute-neg-frac N/A

         \[\leadsto y \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{\frac{1}{3}}{z}\right)\right)} \]

      6. metadata-eval N/A

         \[\leadsto y \cdot \left(\mathsf{neg}\left(\frac{\color{blue}{\frac{1}{3} \cdot 1}}{z}\right)\right) \]

      7. associate-*r/ N/A

         \[\leadsto y \cdot \left(\mathsf{neg}\left(\color{blue}{\frac{1}{3} \cdot \frac{1}{z}}\right)\right) \]

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

         \[\leadsto \color{blue}{y \cdot \left(\mathsf{neg}\left(\frac{1}{3} \cdot \frac{1}{z}\right)\right)} \]

      9. associate-*r/ N/A

         \[\leadsto y \cdot \left(\mathsf{neg}\left(\color{blue}{\frac{\frac{1}{3} \cdot 1}{z}}\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto y \cdot \left(\mathsf{neg}\left(\frac{\color{blue}{\frac{1}{3}}}{z}\right)\right) \]

      11. distribute-neg-frac N/A

          \[\leadsto y \cdot \color{blue}{\frac{\mathsf{neg}\left(\frac{1}{3}\right)}{z}} \]

      12. metadata-eval N/A

          \[\leadsto y \cdot \frac{\color{blue}{\frac{-1}{3}}}{z} \]

      13. /-lowering-/.f64 65.8

          \[\leadsto y \cdot \color{blue}{\frac{-0.3333333333333333}{z}} \]

   5. Simplified 65.8%

      \[\leadsto \color{blue}{y \cdot \frac{-0.3333333333333333}{z}} \]

   if -1.14999999999999996e-61 < y < 1.6e31

   1. Initial program 95.1%

      \[\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

   5. Simplified 32.8%

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

Alternative 16: 63.6% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ x - \frac{y}{z \cdot 3} \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (- x (/ y (* z 3.0))))

C

double code(double x, double y, double z, double t) {
	return x - (y / (z * 3.0));
}

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 * 3.0d0))
end function

Java

public static double code(double x, double y, double z, double t) {
	return x - (y / (z * 3.0));
}

Python

def code(x, y, z, t):
	return x - (y / (z * 3.0))

Julia

function code(x, y, z, t)
	return Float64(x - Float64(y / Float64(z * 3.0)))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = x - (y / (z * 3.0));
end

Wolfram

code[x_, y_, z_, t_] := N[(x - N[(y / N[(z * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 96.7%

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

   1. associate-+l- N/A

      \[\leadsto \color{blue}{x - \left(\frac{y}{z \cdot 3} - \frac{t}{\left(z \cdot 3\right) \cdot y}\right)} \]

   2. --lowering--.f64 N/A

      \[\leadsto \color{blue}{x - \left(\frac{y}{z \cdot 3} - \frac{t}{\left(z \cdot 3\right) \cdot y}\right)} \]

   3. *-commutative N/A

      \[\leadsto x - \left(\frac{y}{z \cdot 3} - \frac{t}{\color{blue}{y \cdot \left(z \cdot 3\right)}}\right) \]

   4. associate-/r* N/A

      \[\leadsto x - \left(\frac{y}{z \cdot 3} - \color{blue}{\frac{\frac{t}{y}}{z \cdot 3}}\right) \]

   5. sub-div N/A

      \[\leadsto x - \color{blue}{\frac{y - \frac{t}{y}}{z \cdot 3}} \]

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

      \[\leadsto x - \color{blue}{\frac{y - \frac{t}{y}}{z \cdot 3}} \]

   7. --lowering--.f64 N/A

      \[\leadsto x - \frac{\color{blue}{y - \frac{t}{y}}}{z \cdot 3} \]

   8. /-lowering-/.f64 N/A

      \[\leadsto x - \frac{y - \color{blue}{\frac{t}{y}}}{z \cdot 3} \]

   9. *-lowering-*.f64 94.6

      \[\leadsto x - \frac{y - \frac{t}{y}}{\color{blue}{z \cdot 3}} \]

4. Applied egg-rr 94.6%

   \[\leadsto \color{blue}{x - \frac{y - \frac{t}{y}}{z \cdot 3}} \]

5. Taylor expanded in y around inf

   \[\leadsto x - \frac{\color{blue}{y}}{z \cdot 3} \]
6. Step-by-step derivation

7. Simplified 64.1%

   \[\leadsto x - \frac{\color{blue}{y}}{z \cdot 3} \]
8. Add Preprocessing

Alternative 17: 63.5% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(y, \frac{-0.3333333333333333}{z}, x\right) \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (fma y (/ -0.3333333333333333 z) x))

C

double code(double x, double y, double z, double t) {
	return fma(y, (-0.3333333333333333 / z), x);
}

Julia

function code(x, y, z, t)
	return fma(y, Float64(-0.3333333333333333 / z), x)
end

Wolfram

code[x_, y_, z_, t_] := N[(y * N[(-0.3333333333333333 / z), $MachinePrecision] + x), $MachinePrecision]

Derivation

1. Initial program 96.7%

   \[\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \]
2. Add Preprocessing

3. Taylor expanded in y around inf

   \[\leadsto \color{blue}{y \cdot \left(\frac{x}{y} - \frac{1}{3} \cdot \frac{1}{z}\right)} \]
4. Step-by-step derivation

   1. sub-neg N/A

      \[\leadsto y \cdot \color{blue}{\left(\frac{x}{y} + \left(\mathsf{neg}\left(\frac{1}{3} \cdot \frac{1}{z}\right)\right)\right)} \]

   2. +-commutative N/A

      \[\leadsto y \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\frac{1}{3} \cdot \frac{1}{z}\right)\right) + \frac{x}{y}\right)} \]

   3. distribute-rgt-in N/A

      \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{1}{3} \cdot \frac{1}{z}\right)\right) \cdot y + \frac{x}{y} \cdot y} \]

   4. associate-*r/ N/A

      \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\frac{\frac{1}{3} \cdot 1}{z}}\right)\right) \cdot y + \frac{x}{y} \cdot y \]

   5. metadata-eval N/A

      \[\leadsto \left(\mathsf{neg}\left(\frac{\color{blue}{\frac{1}{3}}}{z}\right)\right) \cdot y + \frac{x}{y} \cdot y \]

   6. distribute-neg-frac N/A

      \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(\frac{1}{3}\right)}{z}} \cdot y + \frac{x}{y} \cdot y \]

   7. metadata-eval N/A

      \[\leadsto \frac{\color{blue}{\frac{-1}{3}}}{z} \cdot y + \frac{x}{y} \cdot y \]

   8. associate-*l/ N/A

      \[\leadsto \color{blue}{\frac{\frac{-1}{3} \cdot y}{z}} + \frac{x}{y} \cdot y \]

   9. associate-*r/ N/A

      \[\leadsto \color{blue}{\frac{-1}{3} \cdot \frac{y}{z}} + \frac{x}{y} \cdot y \]

   10. cancel-sign-sub N/A

       \[\leadsto \color{blue}{\frac{-1}{3} \cdot \frac{y}{z} - \left(\mathsf{neg}\left(\frac{x}{y}\right)\right) \cdot y} \]

   11. mul-1-neg N/A

       \[\leadsto \frac{-1}{3} \cdot \frac{y}{z} - \color{blue}{\left(-1 \cdot \frac{x}{y}\right)} \cdot y \]

   12. associate-*r/ N/A

       \[\leadsto \frac{-1}{3} \cdot \frac{y}{z} - \color{blue}{\frac{-1 \cdot x}{y}} \cdot y \]

   13. associate-*l/ N/A

       \[\leadsto \frac{-1}{3} \cdot \frac{y}{z} - \color{blue}{\frac{\left(-1 \cdot x\right) \cdot y}{y}} \]

   14. associate-/l* N/A

       \[\leadsto \frac{-1}{3} \cdot \frac{y}{z} - \color{blue}{\left(-1 \cdot x\right) \cdot \frac{y}{y}} \]

   15. mul-1-neg N/A

       \[\leadsto \frac{-1}{3} \cdot \frac{y}{z} - \color{blue}{\left(\mathsf{neg}\left(x\right)\right)} \cdot \frac{y}{y} \]

   16. *-inverses N/A

       \[\leadsto \frac{-1}{3} \cdot \frac{y}{z} - \left(\mathsf{neg}\left(x\right)\right) \cdot \color{blue}{1} \]

   17. cancel-sign-sub N/A

       \[\leadsto \color{blue}{\frac{-1}{3} \cdot \frac{y}{z} + x \cdot 1} \]

   18. *-rgt-identity N/A

       \[\leadsto \frac{-1}{3} \cdot \frac{y}{z} + \color{blue}{x} \]

5. Simplified 64.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(y, \frac{-0.3333333333333333}{z}, x\right)} \]
6. Add Preprocessing

Alternative 18: 30.5% accurate, 44.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 96.7%

   \[\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \]
2. Add Preprocessing

3. Taylor expanded in x around inf

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

5. Simplified 28.8%

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

Developer Target 1: 96.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \left(x - \frac{y}{z \cdot 3}\right) + \frac{\frac{t}{z \cdot 3}}{y} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (+ (- x (/ y (* z 3.0))) (/ (/ t (* z 3.0)) y)))

C

double code(double x, double y, double z, double t) {
	return (x - (y / (z * 3.0))) + ((t / (z * 3.0)) / 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 = (x - (y / (z * 3.0d0))) + ((t / (z * 3.0d0)) / y)
end function

Java

public static double code(double x, double y, double z, double t) {
	return (x - (y / (z * 3.0))) + ((t / (z * 3.0)) / y);
}

Python

def code(x, y, z, t):
	return (x - (y / (z * 3.0))) + ((t / (z * 3.0)) / y)

Julia

function code(x, y, z, t)
	return Float64(Float64(x - Float64(y / Float64(z * 3.0))) + Float64(Float64(t / Float64(z * 3.0)) / y))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = (x - (y / (z * 3.0))) + ((t / (z * 3.0)) / y);
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(x - N[(y / N[(z * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(t / N[(z * 3.0), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2024198 
(FPCore (x y z t)
  :name "Diagrams.Solve.Polynomial:cubForm  from diagrams-solve-0.1, H"
  :precision binary64

  :alt
  (! :herbie-platform default (+ (- x (/ y (* z 3))) (/ (/ t (* z 3)) y)))

  (+ (- x (/ y (* z 3.0))) (/ t (* (* z 3.0) y))))