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

Percentage Accurate: 95.4% → 97.4%

Time: 12.8s

Alternatives: 20

Speedup: 1.4×

• 27.4% 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: 95.4% 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: 97.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot 3 \leq 5 \cdot 10^{-78}:\\ \;\;\;\;x + \frac{\frac{t}{y} - y}{z \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{3 \cdot \left(z \cdot y\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* z 3.0) 5e-78)
   (+ x (/ (- (/ t y) y) (* z 3.0)))
   (+ (- x (/ y (* z 3.0))) (/ t (* 3.0 (* z y))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * 3.0) <= 5e-78) {
		tmp = x + (((t / y) - y) / (z * 3.0));
	} else {
		tmp = (x - (y / (z * 3.0))) + (t / (3.0 * (z * 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) :: tmp
    if ((z * 3.0d0) <= 5d-78) then
        tmp = x + (((t / y) - y) / (z * 3.0d0))
    else
        tmp = (x - (y / (z * 3.0d0))) + (t / (3.0d0 * (z * y)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * 3.0) <= 5e-78) {
		tmp = x + (((t / y) - y) / (z * 3.0));
	} else {
		tmp = (x - (y / (z * 3.0))) + (t / (3.0 * (z * y)));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z * 3.0) <= 5e-78:
		tmp = x + (((t / y) - y) / (z * 3.0))
	else:
		tmp = (x - (y / (z * 3.0))) + (t / (3.0 * (z * y)))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z * 3.0) <= 5e-78)
		tmp = Float64(x + Float64(Float64(Float64(t / y) - y) / Float64(z * 3.0)));
	else
		tmp = Float64(Float64(x - Float64(y / Float64(z * 3.0))) + Float64(t / Float64(3.0 * Float64(z * y))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((z * 3.0) <= 5e-78)
		tmp = x + (((t / y) - y) / (z * 3.0));
	else
		tmp = (x - (y / (z * 3.0))) + (t / (3.0 * (z * y)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(z * 3.0), $MachinePrecision], 5e-78], N[(x + N[(N[(N[(t / y), $MachinePrecision] - y), $MachinePrecision] / N[(z * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x - N[(y / N[(z * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t / N[(3.0 * N[(z * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z #s(literal 3 binary64)) < 4.9999999999999996e-78

   1. Initial program 89.8%

      \[\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \]
   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 99.2

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

   4. Applied egg-rr 99.2%

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

   if 4.9999999999999996e-78 < (*.f64 z #s(literal 3 binary64))

   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

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

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

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

      5. *-lowering-*.f64 99.8

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

   4. Applied egg-rr 99.8%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot 3 \leq 5 \cdot 10^{-78}:\\ \;\;\;\;x + \frac{\frac{t}{y} - y}{z \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{3 \cdot \left(z \cdot y\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 97.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot 3 \leq 4 \cdot 10^{-95}:\\ \;\;\;\;x + \frac{\frac{t}{y} - y}{z \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* z 3.0) 4e-95)
   (+ x (/ (- (/ t y) y) (* z 3.0)))
   (+ (- x (/ y (* z 3.0))) (/ t (* (* z 3.0) y)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * 3.0) <= 4e-95) {
		tmp = x + (((t / y) - y) / (z * 3.0));
	} else {
		tmp = (x - (y / (z * 3.0))) + (t / ((z * 3.0) * 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) :: tmp
    if ((z * 3.0d0) <= 4d-95) then
        tmp = x + (((t / y) - y) / (z * 3.0d0))
    else
        tmp = (x - (y / (z * 3.0d0))) + (t / ((z * 3.0d0) * y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * 3.0) <= 4e-95) {
		tmp = x + (((t / y) - y) / (z * 3.0));
	} else {
		tmp = (x - (y / (z * 3.0))) + (t / ((z * 3.0) * y));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z * 3.0) <= 4e-95:
		tmp = x + (((t / y) - y) / (z * 3.0))
	else:
		tmp = (x - (y / (z * 3.0))) + (t / ((z * 3.0) * y))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z * 3.0) <= 4e-95)
		tmp = Float64(x + Float64(Float64(Float64(t / y) - y) / Float64(z * 3.0)));
	else
		tmp = Float64(Float64(x - Float64(y / Float64(z * 3.0))) + Float64(t / Float64(Float64(z * 3.0) * y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((z * 3.0) <= 4e-95)
		tmp = x + (((t / y) - y) / (z * 3.0));
	else
		tmp = (x - (y / (z * 3.0))) + (t / ((z * 3.0) * y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(z * 3.0), $MachinePrecision], 4e-95], N[(x + N[(N[(N[(t / y), $MachinePrecision] - y), $MachinePrecision] / N[(z * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x - N[(y / N[(z * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t / N[(N[(z * 3.0), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z #s(literal 3 binary64)) < 3.99999999999999996e-95

   1. Initial program 89.5%

      \[\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \]
   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 99.2

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

   4. Applied egg-rr 99.2%

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

   if 3.99999999999999996e-95 < (*.f64 z #s(literal 3 binary64))

   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. Recombined 2 regimes into one program.

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot 3 \leq 4 \cdot 10^{-95}:\\ \;\;\;\;x + \frac{\frac{t}{y} - y}{z \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot 3 \leq 0.1:\\ \;\;\;\;x + \frac{\frac{t}{y} - y}{z \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{z \cdot y}, 0.3333333333333333, \mathsf{fma}\left(\frac{y}{z}, -0.3333333333333333, x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* z 3.0) 0.1)
   (+ x (/ (- (/ t y) y) (* z 3.0)))
   (fma (/ t (* z y)) 0.3333333333333333 (fma (/ y z) -0.3333333333333333 x))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * 3.0) <= 0.1) {
		tmp = x + (((t / y) - y) / (z * 3.0));
	} else {
		tmp = fma((t / (z * y)), 0.3333333333333333, fma((y / z), -0.3333333333333333, x));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z * 3.0) <= 0.1)
		tmp = Float64(x + Float64(Float64(Float64(t / y) - y) / Float64(z * 3.0)));
	else
		tmp = fma(Float64(t / Float64(z * y)), 0.3333333333333333, fma(Float64(y / z), -0.3333333333333333, x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(z * 3.0), $MachinePrecision], 0.1], N[(x + N[(N[(N[(t / y), $MachinePrecision] - y), $MachinePrecision] / N[(z * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t / N[(z * y), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333 + N[(N[(y / z), $MachinePrecision] * -0.3333333333333333 + x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z #s(literal 3 binary64)) < 0.10000000000000001

   1. Initial program 90.6%

      \[\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \]
   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 99.3

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

   4. Applied egg-rr 99.3%

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

   if 0.10000000000000001 < (*.f64 z #s(literal 3 binary64))

   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. +-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 99.6

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

   4. Applied egg-rr 99.6%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot 3 \leq 0.1:\\ \;\;\;\;x + \frac{\frac{t}{y} - y}{z \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{z \cdot y}, 0.3333333333333333, \mathsf{fma}\left(\frac{y}{z}, -0.3333333333333333, x\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 92.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -6.8 \cdot 10^{-6}:\\ \;\;\;\;x - \frac{y}{z \cdot 3}\\ \mathbf{elif}\;y \leq 780:\\ \;\;\;\;x + \frac{\frac{t}{z \cdot 3}}{y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{-0.3333333333333333}{z}, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y -6.8e-6)
   (- x (/ y (* z 3.0)))
   (if (<= y 780.0)
     (+ x (/ (/ t (* z 3.0)) y))
     (fma y (/ -0.3333333333333333 z) x))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -6.8e-6) {
		tmp = x - (y / (z * 3.0));
	} else if (y <= 780.0) {
		tmp = x + ((t / (z * 3.0)) / y);
	} else {
		tmp = fma(y, (-0.3333333333333333 / z), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= -6.8e-6)
		tmp = Float64(x - Float64(y / Float64(z * 3.0)));
	elseif (y <= 780.0)
		tmp = Float64(x + Float64(Float64(t / Float64(z * 3.0)) / y));
	else
		tmp = fma(y, Float64(-0.3333333333333333 / z), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, -6.8e-6], N[(x - N[(y / N[(z * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 780.0], N[(x + N[(N[(t / N[(z * 3.0), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(y * N[(-0.3333333333333333 / z), $MachinePrecision] + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -6.80000000000000012e-6

   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. 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 99.8

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in y around inf

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

   7. Simplified 93.8%

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

   if -6.80000000000000012e-6 < y < 780

   1. Initial program 88.1%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 82.0%

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

      1. associate-/r* N/A

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

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

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

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

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

      4. *-lowering-*.f64 90.4

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

   7. Applied egg-rr 90.4%

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

   if 780 < 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. 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 92.0%

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

Alternative 5: 92.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4.9 \cdot 10^{-5}:\\ \;\;\;\;x - \frac{y}{z \cdot 3}\\ \mathbf{elif}\;y \leq 840:\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{z}, \frac{0.3333333333333333}{y}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{-0.3333333333333333}{z}, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y -4.9e-5)
   (- x (/ y (* z 3.0)))
   (if (<= y 840.0)
     (fma (/ t z) (/ 0.3333333333333333 y) x)
     (fma y (/ -0.3333333333333333 z) x))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -4.9e-5) {
		tmp = x - (y / (z * 3.0));
	} else if (y <= 840.0) {
		tmp = fma((t / z), (0.3333333333333333 / y), x);
	} else {
		tmp = fma(y, (-0.3333333333333333 / z), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= -4.9e-5)
		tmp = Float64(x - Float64(y / Float64(z * 3.0)));
	elseif (y <= 840.0)
		tmp = fma(Float64(t / z), Float64(0.3333333333333333 / y), x);
	else
		tmp = fma(y, Float64(-0.3333333333333333 / z), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, -4.9e-5], N[(x - N[(y / N[(z * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 840.0], N[(N[(t / z), $MachinePrecision] * N[(0.3333333333333333 / y), $MachinePrecision] + x), $MachinePrecision], N[(y * N[(-0.3333333333333333 / z), $MachinePrecision] + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -4.9e-5

   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. 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 99.8

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in y around inf

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

   7. Simplified 93.8%

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

   if -4.9e-5 < y < 840

   1. Initial program 88.1%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 82.0%

      \[\leadsto \color{blue}{x} + \frac{t}{\left(z \cdot 3\right) \cdot y} \]
   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. associate-/r* N/A

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

      5. div-inv N/A

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

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

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

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

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

      8. *-commutative N/A

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

      9. associate-/r* N/A

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

      10. metadata-eval N/A

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

      11. /-lowering-/.f64 90.3

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

   7. Applied egg-rr 90.3%

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

   if 840 < 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. 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 92.0%

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

Alternative 6: 92.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -8.5 \cdot 10^{-6}:\\ \;\;\;\;x - \frac{y}{z \cdot 3}\\ \mathbf{elif}\;y \leq 800:\\ \;\;\;\;\mathsf{fma}\left(0.3333333333333333, \frac{\frac{t}{z}}{y}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{-0.3333333333333333}{z}, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y -8.5e-6)
   (- x (/ y (* z 3.0)))
   (if (<= y 800.0)
     (fma 0.3333333333333333 (/ (/ t z) y) x)
     (fma y (/ -0.3333333333333333 z) x))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -8.5e-6) {
		tmp = x - (y / (z * 3.0));
	} else if (y <= 800.0) {
		tmp = fma(0.3333333333333333, ((t / z) / y), x);
	} else {
		tmp = fma(y, (-0.3333333333333333 / z), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= -8.5e-6)
		tmp = Float64(x - Float64(y / Float64(z * 3.0)));
	elseif (y <= 800.0)
		tmp = fma(0.3333333333333333, Float64(Float64(t / z) / y), x);
	else
		tmp = fma(y, Float64(-0.3333333333333333 / z), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, -8.5e-6], N[(x - N[(y / N[(z * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 800.0], N[(0.3333333333333333 * N[(N[(t / z), $MachinePrecision] / y), $MachinePrecision] + x), $MachinePrecision], N[(y * N[(-0.3333333333333333 / z), $MachinePrecision] + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -8.4999999999999999e-6

   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. 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 99.8

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in y around inf

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

   7. Simplified 93.8%

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

   if -8.4999999999999999e-6 < y < 800

   1. Initial program 88.1%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 82.0%

      \[\leadsto \color{blue}{x} + \frac{t}{\left(z \cdot 3\right) \cdot y} \]
   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. associate-/r* N/A

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

      5. div-inv N/A

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

      6. *-commutative N/A

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

      7. associate-/r* N/A

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

      8. times-frac N/A

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

      9. associate-/r/ N/A

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

      10. div-inv N/A

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

      11. clear-num N/A

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

      12. times-frac N/A

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

      13. metadata-eval N/A

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

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

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

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

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

      16. /-lowering-/.f64 86.9

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

   7. Applied egg-rr 86.9%

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

      1. associate-/l/ N/A

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

      2. associate-/r* N/A

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

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

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

      4. /-lowering-/.f64 90.3

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

   9. Applied egg-rr 90.3%

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

   if 800 < 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. 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 92.0%

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

Alternative 7: 89.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.62 \cdot 10^{+38}:\\ \;\;\;\;x - \frac{y}{z \cdot 3}\\ \mathbf{elif}\;y \leq 1150:\\ \;\;\;\;\mathsf{fma}\left(0.3333333333333333, \frac{\frac{t}{y}}{z}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{-0.3333333333333333}{z}, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y -1.62e+38)
   (- x (/ y (* z 3.0)))
   (if (<= y 1150.0)
     (fma 0.3333333333333333 (/ (/ t y) z) x)
     (fma y (/ -0.3333333333333333 z) x))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -1.62e+38) {
		tmp = x - (y / (z * 3.0));
	} else if (y <= 1150.0) {
		tmp = fma(0.3333333333333333, ((t / y) / z), x);
	} else {
		tmp = fma(y, (-0.3333333333333333 / z), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= -1.62e+38)
		tmp = Float64(x - Float64(y / Float64(z * 3.0)));
	elseif (y <= 1150.0)
		tmp = fma(0.3333333333333333, Float64(Float64(t / y) / z), x);
	else
		tmp = fma(y, Float64(-0.3333333333333333 / z), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, -1.62e+38], N[(x - N[(y / N[(z * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1150.0], N[(0.3333333333333333 * N[(N[(t / y), $MachinePrecision] / z), $MachinePrecision] + x), $MachinePrecision], N[(y * N[(-0.3333333333333333 / z), $MachinePrecision] + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.62000000000000001e38

   1. Initial program 98.0%

      \[\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \]
   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. Taylor expanded in y around inf

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

   7. Simplified 98.1%

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

   if -1.62000000000000001e38 < y < 1150

   1. Initial program 89.0%

      \[\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \]
   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 81.4%

      \[\leadsto \color{blue}{x} + \frac{t}{\left(z \cdot 3\right) \cdot y} \]
   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. associate-/r* N/A

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

      5. div-inv N/A

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

      6. *-commutative N/A

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

      7. associate-/r* N/A

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

      8. times-frac N/A

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

      9. associate-/r/ N/A

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

      10. div-inv N/A

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

      11. clear-num N/A

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

      12. times-frac N/A

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

      13. metadata-eval N/A

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

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

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

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

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

      16. /-lowering-/.f64 86.0

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

   7. Applied egg-rr 86.0%

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

   if 1150 < 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. 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 92.0%

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

Alternative 8: 89.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.1 \cdot 10^{+38}:\\ \;\;\;\;x - \frac{y}{z \cdot 3}\\ \mathbf{elif}\;y \leq 1020:\\ \;\;\;\;x + \frac{t}{\left(z \cdot 3\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{-0.3333333333333333}{z}, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y -2.1e+38)
   (- x (/ y (* z 3.0)))
   (if (<= y 1020.0)
     (+ x (/ t (* (* z 3.0) y)))
     (fma y (/ -0.3333333333333333 z) x))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -2.1e+38) {
		tmp = x - (y / (z * 3.0));
	} else if (y <= 1020.0) {
		tmp = x + (t / ((z * 3.0) * y));
	} else {
		tmp = fma(y, (-0.3333333333333333 / z), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= -2.1e+38)
		tmp = Float64(x - Float64(y / Float64(z * 3.0)));
	elseif (y <= 1020.0)
		tmp = Float64(x + Float64(t / Float64(Float64(z * 3.0) * y)));
	else
		tmp = fma(y, Float64(-0.3333333333333333 / z), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, -2.1e+38], N[(x - N[(y / N[(z * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1020.0], N[(x + N[(t / N[(N[(z * 3.0), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * N[(-0.3333333333333333 / z), $MachinePrecision] + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.1e38

   1. Initial program 98.0%

      \[\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \]
   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. Taylor expanded in y around inf

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

   7. Simplified 98.1%

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

   if -2.1e38 < y < 1020

   1. Initial program 89.0%

      \[\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \]
   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 81.4%

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

   if 1020 < 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. 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 92.0%

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

Alternative 9: 75.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3 \cdot 10^{-27}:\\ \;\;\;\;x - \frac{y}{z \cdot 3}\\ \mathbf{elif}\;y \leq 1.22 \cdot 10^{-104}:\\ \;\;\;\;\frac{t \cdot 0.3333333333333333}{z \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{-0.3333333333333333}{z}, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y -3e-27)
   (- x (/ y (* z 3.0)))
   (if (<= y 1.22e-104)
     (/ (* t 0.3333333333333333) (* z y))
     (fma y (/ -0.3333333333333333 z) x))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -3e-27) {
		tmp = x - (y / (z * 3.0));
	} else if (y <= 1.22e-104) {
		tmp = (t * 0.3333333333333333) / (z * y);
	} else {
		tmp = fma(y, (-0.3333333333333333 / z), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= -3e-27)
		tmp = Float64(x - Float64(y / Float64(z * 3.0)));
	elseif (y <= 1.22e-104)
		tmp = Float64(Float64(t * 0.3333333333333333) / Float64(z * y));
	else
		tmp = fma(y, Float64(-0.3333333333333333 / z), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, -3e-27], N[(x - N[(y / N[(z * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.22e-104], N[(N[(t * 0.3333333333333333), $MachinePrecision] / N[(z * y), $MachinePrecision]), $MachinePrecision], N[(y * N[(-0.3333333333333333 / z), $MachinePrecision] + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -3.0000000000000001e-27

   1. Initial program 98.5%

      \[\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \]
   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 99.8

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in y around inf

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

   7. Simplified 92.0%

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

   if -3.0000000000000001e-27 < y < 1.21999999999999997e-104

   1. Initial program 85.8%

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

   3. Taylor expanded in y around 0

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

      1. associate-*r/ N/A

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

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

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

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

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

      4. *-lowering-*.f64 64.0

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

   5. Simplified 64.0%

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

   if 1.21999999999999997e-104 < y

   1. Initial program 98.4%

      \[\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \]
   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 78.9%

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

4. Final simplification 76.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3 \cdot 10^{-27}:\\ \;\;\;\;x - \frac{y}{z \cdot 3}\\ \mathbf{elif}\;y \leq 1.22 \cdot 10^{-104}:\\ \;\;\;\;\frac{t \cdot 0.3333333333333333}{z \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{-0.3333333333333333}{z}, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 75.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.9 \cdot 10^{-27}:\\ \;\;\;\;x - \frac{y}{z \cdot 3}\\ \mathbf{elif}\;y \leq 2.6 \cdot 10^{-104}:\\ \;\;\;\;t \cdot \frac{0.3333333333333333}{z \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{-0.3333333333333333}{z}, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y -1.9e-27)
   (- x (/ y (* z 3.0)))
   (if (<= y 2.6e-104)
     (* t (/ 0.3333333333333333 (* z y)))
     (fma y (/ -0.3333333333333333 z) x))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -1.9e-27) {
		tmp = x - (y / (z * 3.0));
	} else if (y <= 2.6e-104) {
		tmp = t * (0.3333333333333333 / (z * y));
	} else {
		tmp = fma(y, (-0.3333333333333333 / z), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= -1.9e-27)
		tmp = Float64(x - Float64(y / Float64(z * 3.0)));
	elseif (y <= 2.6e-104)
		tmp = Float64(t * Float64(0.3333333333333333 / Float64(z * y)));
	else
		tmp = fma(y, Float64(-0.3333333333333333 / z), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, -1.9e-27], N[(x - N[(y / N[(z * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.6e-104], N[(t * N[(0.3333333333333333 / N[(z * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * N[(-0.3333333333333333 / z), $MachinePrecision] + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.9e-27

   1. Initial program 98.5%

      \[\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \]
   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 99.8

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in y around inf

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

   7. Simplified 92.0%

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

   if -1.9e-27 < y < 2.60000000000000003e-104

   1. Initial program 85.8%

      \[\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \]
   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 92.3

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

   4. Applied egg-rr 92.3%

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

      1. frac-2neg N/A

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

      2. div-inv N/A

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

      3. sub-neg N/A

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

      4. distribute-neg-in N/A

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

      5. distribute-neg-frac2 N/A

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

      6. distribute-frac-neg N/A

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

      7. frac-2neg N/A

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

      8. +-commutative N/A

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

      9. sub-neg N/A

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

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

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

      11. --lowering--.f64 N/A

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

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

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

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

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

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

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

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

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

      16. metadata-eval 92.2

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

   6. Applied egg-rr 92.2%

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

   7. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{t}{y \cdot z}} \]
   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. *-commutative N/A

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

      4. *-lowering-*.f64 63.9

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

   9. Simplified 63.9%

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

       1. clear-num N/A

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

       2. associate-*r/ N/A

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

       3. div-inv N/A

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

       4. associate-*r/ N/A

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

       5. associate-/r/ N/A

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

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

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

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

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

       8. *-commutative N/A

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

       9. *-lowering-*.f64 64.0

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

   11. Applied egg-rr 64.0%

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

   if 2.60000000000000003e-104 < y

   1. Initial program 98.4%

      \[\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \]
   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 78.9%

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

4. Final simplification 76.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.9 \cdot 10^{-27}:\\ \;\;\;\;x - \frac{y}{z \cdot 3}\\ \mathbf{elif}\;y \leq 2.6 \cdot 10^{-104}:\\ \;\;\;\;t \cdot \frac{0.3333333333333333}{z \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{-0.3333333333333333}{z}, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 75.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.9 \cdot 10^{-27}:\\ \;\;\;\;x - \frac{y}{z \cdot 3}\\ \mathbf{elif}\;y \leq 1.3 \cdot 10^{-104}:\\ \;\;\;\;\frac{t}{z \cdot y} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{-0.3333333333333333}{z}, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y -1.9e-27)
   (- x (/ y (* z 3.0)))
   (if (<= y 1.3e-104)
     (* (/ t (* z y)) 0.3333333333333333)
     (fma y (/ -0.3333333333333333 z) x))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -1.9e-27) {
		tmp = x - (y / (z * 3.0));
	} else if (y <= 1.3e-104) {
		tmp = (t / (z * y)) * 0.3333333333333333;
	} else {
		tmp = fma(y, (-0.3333333333333333 / z), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= -1.9e-27)
		tmp = Float64(x - Float64(y / Float64(z * 3.0)));
	elseif (y <= 1.3e-104)
		tmp = Float64(Float64(t / Float64(z * y)) * 0.3333333333333333);
	else
		tmp = fma(y, Float64(-0.3333333333333333 / z), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, -1.9e-27], N[(x - N[(y / N[(z * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.3e-104], N[(N[(t / N[(z * y), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision], N[(y * N[(-0.3333333333333333 / z), $MachinePrecision] + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.9e-27

   1. Initial program 98.5%

      \[\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \]
   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 99.8

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in y around inf

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

   7. Simplified 92.0%

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

   if -1.9e-27 < y < 1.30000000000000001e-104

   1. Initial program 85.8%

      \[\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \]
   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 92.3

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

   4. Applied egg-rr 92.3%

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

   5. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{t}{y \cdot z}} \]
   6. 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 63.9

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

   7. Simplified 63.9%

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

   if 1.30000000000000001e-104 < y

   1. Initial program 98.4%

      \[\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \]
   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 78.9%

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

4. Final simplification 76.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.9 \cdot 10^{-27}:\\ \;\;\;\;x - \frac{y}{z \cdot 3}\\ \mathbf{elif}\;y \leq 1.3 \cdot 10^{-104}:\\ \;\;\;\;\frac{t}{z \cdot y} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{-0.3333333333333333}{z}, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 95.6% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 92.9%

   \[\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \]
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 96.5

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

4. Applied egg-rr 96.5%

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

5. Final simplification 96.5%

   \[\leadsto x + \frac{\frac{t}{y} - y}{z \cdot 3} \]
6. Add Preprocessing

Alternative 13: 95.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(-0.3333333333333333, \frac{y - \frac{t}{y}}{z}, x\right) \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (fma -0.3333333333333333 (/ (- y (/ t y)) z) x))

C

double code(double x, double y, double z, double t) {
	return fma(-0.3333333333333333, ((y - (t / y)) / z), x);
}

Julia

function code(x, y, z, t)
	return fma(-0.3333333333333333, Float64(Float64(y - Float64(t / y)) / z), x)
end

Wolfram

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

Derivation

1. Initial program 92.9%

   \[\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \]
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 96.5

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

4. Applied egg-rr 96.5%

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

   1. sub-neg N/A

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

   2. +-commutative N/A

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

   3. distribute-neg-frac N/A

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

   4. neg-mul-1 N/A

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

   5. *-commutative N/A

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

   6. times-frac N/A

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

   7. metadata-eval N/A

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

   8. metadata-eval N/A

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

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

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

   10. metadata-eval N/A

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

   11. /-lowering-/.f64 N/A

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

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

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

   13. /-lowering-/.f64 96.4

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

6. Applied egg-rr 96.4%

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

Alternative 14: 48.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.5 \cdot 10^{+74}:\\ \;\;\;\;\frac{y}{z \cdot -3}\\ \mathbf{elif}\;y \leq 4.7 \cdot 10^{+88}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot -0.3333333333333333}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y -3.5e+74)
   (/ y (* z -3.0))
   (if (<= y 4.7e+88) x (/ (* y -0.3333333333333333) z))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -3.5e+74) {
		tmp = y / (z * -3.0);
	} else if (y <= 4.7e+88) {
		tmp = x;
	} else {
		tmp = (y * -0.3333333333333333) / z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (y <= (-3.5d+74)) then
        tmp = y / (z * (-3.0d0))
    else if (y <= 4.7d+88) then
        tmp = x
    else
        tmp = (y * (-0.3333333333333333d0)) / z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -3.5e+74) {
		tmp = y / (z * -3.0);
	} else if (y <= 4.7e+88) {
		tmp = x;
	} else {
		tmp = (y * -0.3333333333333333) / z;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if y <= -3.5e+74:
		tmp = y / (z * -3.0)
	elif y <= 4.7e+88:
		tmp = x
	else:
		tmp = (y * -0.3333333333333333) / z
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= -3.5e+74)
		tmp = Float64(y / Float64(z * -3.0));
	elseif (y <= 4.7e+88)
		tmp = x;
	else
		tmp = Float64(Float64(y * -0.3333333333333333) / z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (y <= -3.5e+74)
		tmp = y / (z * -3.0);
	elseif (y <= 4.7e+88)
		tmp = x;
	else
		tmp = (y * -0.3333333333333333) / z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, -3.5e+74], N[(y / N[(z * -3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 4.7e+88], x, N[(N[(y * -0.3333333333333333), $MachinePrecision] / z), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -3.50000000000000014e74

   1. Initial program 97.7%

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

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{-1}{3} \cdot \frac{y}{z}} \]
   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 80.2

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

   5. Simplified 80.2%

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

      1. div-inv N/A

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

      2. div-inv N/A

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

      3. clear-num N/A

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

      4. div-inv N/A

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

      5. metadata-eval N/A

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

      6. un-div-inv N/A

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

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

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

      8. *-lowering-*.f64 80.3

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

   7. Applied egg-rr 80.3%

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

   if -3.50000000000000014e74 < y < 4.70000000000000007e88

   1. Initial program 90.3%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 31.1%

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

   if 4.70000000000000007e88 < y

   1. Initial program 99.6%

      \[\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \]
   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 99.6

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

   4. Applied egg-rr 99.6%

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

   5. Taylor expanded in y around inf

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

      1. associate-*r/ N/A

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

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

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

      3. *-lowering-*.f64 88.4

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

   7. Simplified 88.4%

      \[\leadsto \color{blue}{\frac{-0.3333333333333333 \cdot y}{z}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 47.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.5 \cdot 10^{+74}:\\ \;\;\;\;\frac{y}{z \cdot -3}\\ \mathbf{elif}\;y \leq 4.7 \cdot 10^{+88}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot -0.3333333333333333}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 48.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4.2 \cdot 10^{+74}:\\ \;\;\;\;\frac{y}{z \cdot -3}\\ \mathbf{elif}\;y \leq 4.7 \cdot 10^{+88}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{-0.3333333333333333}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y -4.2e+74)
   (/ y (* z -3.0))
   (if (<= y 4.7e+88) x (* y (/ -0.3333333333333333 z)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -4.2e+74) {
		tmp = y / (z * -3.0);
	} else if (y <= 4.7e+88) {
		tmp = x;
	} else {
		tmp = y * (-0.3333333333333333 / z);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (y <= (-4.2d+74)) then
        tmp = y / (z * (-3.0d0))
    else if (y <= 4.7d+88) then
        tmp = x
    else
        tmp = y * ((-0.3333333333333333d0) / z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -4.2e+74) {
		tmp = y / (z * -3.0);
	} else if (y <= 4.7e+88) {
		tmp = x;
	} else {
		tmp = y * (-0.3333333333333333 / z);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if y <= -4.2e+74:
		tmp = y / (z * -3.0)
	elif y <= 4.7e+88:
		tmp = x
	else:
		tmp = y * (-0.3333333333333333 / z)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= -4.2e+74)
		tmp = Float64(y / Float64(z * -3.0));
	elseif (y <= 4.7e+88)
		tmp = x;
	else
		tmp = Float64(y * Float64(-0.3333333333333333 / z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (y <= -4.2e+74)
		tmp = y / (z * -3.0);
	elseif (y <= 4.7e+88)
		tmp = x;
	else
		tmp = y * (-0.3333333333333333 / z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, -4.2e+74], N[(y / N[(z * -3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 4.7e+88], x, N[(y * N[(-0.3333333333333333 / z), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -4.1999999999999998e74

   1. Initial program 97.7%

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

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{-1}{3} \cdot \frac{y}{z}} \]
   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 80.2

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

   5. Simplified 80.2%

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

      1. div-inv N/A

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

      2. div-inv N/A

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

      3. clear-num N/A

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

      4. div-inv N/A

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

      5. metadata-eval N/A

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

      6. un-div-inv N/A

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

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

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

      8. *-lowering-*.f64 80.3

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

   7. Applied egg-rr 80.3%

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

   if -4.1999999999999998e74 < y < 4.70000000000000007e88

   1. Initial program 90.3%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 31.1%

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

   if 4.70000000000000007e88 < y

   1. Initial program 99.6%

      \[\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \]
   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 88.2

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

   5. Simplified 88.2%

      \[\leadsto \color{blue}{y \cdot \frac{-0.3333333333333333}{z}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 16: 48.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y \cdot \frac{-0.3333333333333333}{z}\\ \mathbf{if}\;y \leq -1.6 \cdot 10^{+75}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 4.8 \cdot 10^{+88}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* y (/ -0.3333333333333333 z))))
   (if (<= y -1.6e+75) t_1 (if (<= y 4.8e+88) 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 <= -1.6e+75) {
		tmp = t_1;
	} else if (y <= 4.8e+88) {
		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 <= (-1.6d+75)) then
        tmp = t_1
    else if (y <= 4.8d+88) 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 <= -1.6e+75) {
		tmp = t_1;
	} else if (y <= 4.8e+88) {
		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 <= -1.6e+75:
		tmp = t_1
	elif y <= 4.8e+88:
		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 <= -1.6e+75)
		tmp = t_1;
	elseif (y <= 4.8e+88)
		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 <= -1.6e+75)
		tmp = t_1;
	elseif (y <= 4.8e+88)
		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, -1.6e+75], t$95$1, If[LessEqual[y, 4.8e+88], x, t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.59999999999999992e75 or 4.7999999999999998e88 < y

   1. Initial program 98.5%

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

   3. Taylor expanded in y around 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 83.6

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

   5. Simplified 83.6%

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

   if -1.59999999999999992e75 < y < 4.7999999999999998e88

   1. Initial program 90.3%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 31.1%

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

Alternative 17: 64.0% 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 92.9%

   \[\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \]
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 96.5

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

4. Applied egg-rr 96.5%

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

5. Taylor expanded in y around inf

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

7. Simplified 59.4%

   \[\leadsto x - \frac{\color{blue}{y}}{z \cdot 3} \]
8. Add Preprocessing

Alternative 18: 64.0% 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 92.9%

   \[\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \]
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 59.4%

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

Alternative 19: 64.0% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(-0.3333333333333333, \frac{y}{z}, x\right) \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (fma -0.3333333333333333 (/ y z) x))

C

double code(double x, double y, double z, double t) {
	return fma(-0.3333333333333333, (y / z), x);
}

Julia

function code(x, y, z, t)
	return fma(-0.3333333333333333, Float64(y / z), x)
end

Wolfram

code[x_, y_, z_, t_] := N[(-0.3333333333333333 * N[(y / z), $MachinePrecision] + x), $MachinePrecision]

Derivation

1. Initial program 92.9%

   \[\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \]
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 96.5

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

4. Applied egg-rr 96.5%

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

5. Taylor expanded in y around inf

   \[\leadsto \color{blue}{y \cdot \left(\frac{x}{y} - \frac{1}{3} \cdot \frac{1}{z}\right)} \]
6. 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. associate-*r/ N/A

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

   3. metadata-eval N/A

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

   4. distribute-neg-frac N/A

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

   5. metadata-eval N/A

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

   6. +-commutative N/A

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

   7. distribute-lft-in N/A

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

   8. associate-/l* N/A

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

   9. *-commutative N/A

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

   10. associate-*r/ N/A

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

   11. *-commutative N/A

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

   12. associate-*l/ N/A

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

   13. associate-/l* N/A

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

   14. *-inverses N/A

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

   15. *-rgt-identity N/A

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

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

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

   17. /-lowering-/.f64 59.3

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

7. Simplified 59.3%

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

Alternative 20: 30.7% 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 92.9%

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

3. Taylor expanded in x around inf

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

5. Simplified 26.4%

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

Developer Target 1: 95.9% 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 2024199 
(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))))