Numeric.Signal.Multichannel:$cput from hsignal-0.2.7.1

Percentage Accurate: 96.7% → 93.6%

Time: 10.9s

Alternatives: 11

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 96.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 93.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x - y}{z - y}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+27}:\\ \;\;\;\;\left(x - y\right) \cdot \frac{t}{z - y}\\ \mathbf{elif}\;t\_1 \leq 3 \cdot 10^{-13}:\\ \;\;\;\;t \cdot \frac{x - y}{z}\\ \mathbf{elif}\;t\_1 \leq 2:\\ \;\;\;\;t \cdot \frac{y}{y - z}\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{x}{z - y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (- x y) (- z y))))
   (if (<= t_1 -1e+27)
     (* (- x y) (/ t (- z y)))
     (if (<= t_1 3e-13)
       (* t (/ (- x y) z))
       (if (<= t_1 2.0) (* t (/ y (- y z))) (* t (/ x (- z y))))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x - y) / (z - y);
	double tmp;
	if (t_1 <= -1e+27) {
		tmp = (x - y) * (t / (z - y));
	} else if (t_1 <= 3e-13) {
		tmp = t * ((x - y) / z);
	} else if (t_1 <= 2.0) {
		tmp = t * (y / (y - z));
	} else {
		tmp = t * (x / (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) :: t_1
    real(8) :: tmp
    t_1 = (x - y) / (z - y)
    if (t_1 <= (-1d+27)) then
        tmp = (x - y) * (t / (z - y))
    else if (t_1 <= 3d-13) then
        tmp = t * ((x - y) / z)
    else if (t_1 <= 2.0d0) then
        tmp = t * (y / (y - z))
    else
        tmp = t * (x / (z - y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (x - y) / (z - y);
	double tmp;
	if (t_1 <= -1e+27) {
		tmp = (x - y) * (t / (z - y));
	} else if (t_1 <= 3e-13) {
		tmp = t * ((x - y) / z);
	} else if (t_1 <= 2.0) {
		tmp = t * (y / (y - z));
	} else {
		tmp = t * (x / (z - y));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (x - y) / (z - y)
	tmp = 0
	if t_1 <= -1e+27:
		tmp = (x - y) * (t / (z - y))
	elif t_1 <= 3e-13:
		tmp = t * ((x - y) / z)
	elif t_1 <= 2.0:
		tmp = t * (y / (y - z))
	else:
		tmp = t * (x / (z - y))
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x - y) / Float64(z - y))
	tmp = 0.0
	if (t_1 <= -1e+27)
		tmp = Float64(Float64(x - y) * Float64(t / Float64(z - y)));
	elseif (t_1 <= 3e-13)
		tmp = Float64(t * Float64(Float64(x - y) / z));
	elseif (t_1 <= 2.0)
		tmp = Float64(t * Float64(y / Float64(y - z)));
	else
		tmp = Float64(t * Float64(x / Float64(z - y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (x - y) / (z - y);
	tmp = 0.0;
	if (t_1 <= -1e+27)
		tmp = (x - y) * (t / (z - y));
	elseif (t_1 <= 3e-13)
		tmp = t * ((x - y) / z);
	elseif (t_1 <= 2.0)
		tmp = t * (y / (y - z));
	else
		tmp = t * (x / (z - y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x - y), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e+27], N[(N[(x - y), $MachinePrecision] * N[(t / N[(z - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 3e-13], N[(t * N[(N[(x - y), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2.0], N[(t * N[(y / N[(y - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t * N[(x / N[(z - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (-.f64 x y) (-.f64 z y)) < -1e27

   1. Initial program 90.1%

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

      1. associate-*l/ N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

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

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

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

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

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

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

      7. --lowering--.f64 98.1

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

   4. Applied egg-rr 98.1%

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

   if -1e27 < (/.f64 (-.f64 x y) (-.f64 z y)) < 2.99999999999999984e-13

   1. Initial program 96.0%

      \[\frac{x - y}{z - y} \cdot t \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

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

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

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

      2. --lowering--.f64 95.1

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

   5. Simplified 95.1%

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

   if 2.99999999999999984e-13 < (/.f64 (-.f64 x y) (-.f64 z y)) < 2

   1. Initial program 99.9%

      \[\frac{x - y}{z - y} \cdot t \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. neg-lowering-neg.f64 99.3

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

   5. Simplified 99.3%

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

      1. distribute-frac-neg N/A

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

      2. distribute-frac-neg2 N/A

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

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

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

      4. neg-sub0 N/A

         \[\leadsto \frac{y}{\color{blue}{0 - \left(z - y\right)}} \cdot t \]

      5. sub-neg N/A

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

      6. +-commutative N/A

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

      7. associate--r+ N/A

         \[\leadsto \frac{y}{\color{blue}{\left(0 - \left(\mathsf{neg}\left(y\right)\right)\right) - z}} \cdot t \]

      8. neg-sub0 N/A

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

      9. remove-double-neg N/A

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

      10. --lowering--.f64 99.3

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

   7. Applied egg-rr 99.3%

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

   if 2 < (/.f64 (-.f64 x y) (-.f64 z y))

   1. Initial program 99.7%

      \[\frac{x - y}{z - y} \cdot t \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

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

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

      2. --lowering--.f64 98.9

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

   5. Simplified 98.9%

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

4. Final simplification 97.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{z - y} \leq -1 \cdot 10^{+27}:\\ \;\;\;\;\left(x - y\right) \cdot \frac{t}{z - y}\\ \mathbf{elif}\;\frac{x - y}{z - y} \leq 3 \cdot 10^{-13}:\\ \;\;\;\;t \cdot \frac{x - y}{z}\\ \mathbf{elif}\;\frac{x - y}{z - y} \leq 2:\\ \;\;\;\;t \cdot \frac{y}{y - z}\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{x}{z - y}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 93.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x - y}{z - y}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+27}:\\ \;\;\;\;\frac{x \cdot t}{z - y}\\ \mathbf{elif}\;t\_1 \leq 3 \cdot 10^{-13}:\\ \;\;\;\;t \cdot \frac{x - y}{z}\\ \mathbf{elif}\;t\_1 \leq 2:\\ \;\;\;\;t \cdot \frac{y}{y - z}\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{x}{z - y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (- x y) (- z y))))
   (if (<= t_1 -1e+27)
     (/ (* x t) (- z y))
     (if (<= t_1 3e-13)
       (* t (/ (- x y) z))
       (if (<= t_1 2.0) (* t (/ y (- y z))) (* t (/ x (- z y))))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x - y) / (z - y);
	double tmp;
	if (t_1 <= -1e+27) {
		tmp = (x * t) / (z - y);
	} else if (t_1 <= 3e-13) {
		tmp = t * ((x - y) / z);
	} else if (t_1 <= 2.0) {
		tmp = t * (y / (y - z));
	} else {
		tmp = t * (x / (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) :: t_1
    real(8) :: tmp
    t_1 = (x - y) / (z - y)
    if (t_1 <= (-1d+27)) then
        tmp = (x * t) / (z - y)
    else if (t_1 <= 3d-13) then
        tmp = t * ((x - y) / z)
    else if (t_1 <= 2.0d0) then
        tmp = t * (y / (y - z))
    else
        tmp = t * (x / (z - y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (x - y) / (z - y);
	double tmp;
	if (t_1 <= -1e+27) {
		tmp = (x * t) / (z - y);
	} else if (t_1 <= 3e-13) {
		tmp = t * ((x - y) / z);
	} else if (t_1 <= 2.0) {
		tmp = t * (y / (y - z));
	} else {
		tmp = t * (x / (z - y));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (x - y) / (z - y)
	tmp = 0
	if t_1 <= -1e+27:
		tmp = (x * t) / (z - y)
	elif t_1 <= 3e-13:
		tmp = t * ((x - y) / z)
	elif t_1 <= 2.0:
		tmp = t * (y / (y - z))
	else:
		tmp = t * (x / (z - y))
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x - y) / Float64(z - y))
	tmp = 0.0
	if (t_1 <= -1e+27)
		tmp = Float64(Float64(x * t) / Float64(z - y));
	elseif (t_1 <= 3e-13)
		tmp = Float64(t * Float64(Float64(x - y) / z));
	elseif (t_1 <= 2.0)
		tmp = Float64(t * Float64(y / Float64(y - z)));
	else
		tmp = Float64(t * Float64(x / Float64(z - y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (x - y) / (z - y);
	tmp = 0.0;
	if (t_1 <= -1e+27)
		tmp = (x * t) / (z - y);
	elseif (t_1 <= 3e-13)
		tmp = t * ((x - y) / z);
	elseif (t_1 <= 2.0)
		tmp = t * (y / (y - z));
	else
		tmp = t * (x / (z - y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x - y), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e+27], N[(N[(x * t), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 3e-13], N[(t * N[(N[(x - y), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2.0], N[(t * N[(y / N[(y - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t * N[(x / N[(z - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (-.f64 x y) (-.f64 z y)) < -1e27

   1. Initial program 90.1%

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

      1. associate-*l/ N/A

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

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

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

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

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

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

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

      5. --lowering--.f64 93.0

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

   4. Applied egg-rr 93.0%

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

   5. Taylor expanded in x around inf

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

   7. Simplified 93.0%

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

   if -1e27 < (/.f64 (-.f64 x y) (-.f64 z y)) < 2.99999999999999984e-13

   1. Initial program 96.0%

      \[\frac{x - y}{z - y} \cdot t \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

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

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

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

      2. --lowering--.f64 95.1

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

   5. Simplified 95.1%

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

   if 2.99999999999999984e-13 < (/.f64 (-.f64 x y) (-.f64 z y)) < 2

   1. Initial program 99.9%

      \[\frac{x - y}{z - y} \cdot t \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. neg-lowering-neg.f64 99.3

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

   5. Simplified 99.3%

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

      1. distribute-frac-neg N/A

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

      2. distribute-frac-neg2 N/A

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

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

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

      4. neg-sub0 N/A

         \[\leadsto \frac{y}{\color{blue}{0 - \left(z - y\right)}} \cdot t \]

      5. sub-neg N/A

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

      6. +-commutative N/A

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

      7. associate--r+ N/A

         \[\leadsto \frac{y}{\color{blue}{\left(0 - \left(\mathsf{neg}\left(y\right)\right)\right) - z}} \cdot t \]

      8. neg-sub0 N/A

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

      9. remove-double-neg N/A

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

      10. --lowering--.f64 99.3

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

   7. Applied egg-rr 99.3%

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

   if 2 < (/.f64 (-.f64 x y) (-.f64 z y))

   1. Initial program 99.7%

      \[\frac{x - y}{z - y} \cdot t \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

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

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

      2. --lowering--.f64 98.9

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

   5. Simplified 98.9%

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

4. Final simplification 96.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{z - y} \leq -1 \cdot 10^{+27}:\\ \;\;\;\;\frac{x \cdot t}{z - y}\\ \mathbf{elif}\;\frac{x - y}{z - y} \leq 3 \cdot 10^{-13}:\\ \;\;\;\;t \cdot \frac{x - y}{z}\\ \mathbf{elif}\;\frac{x - y}{z - y} \leq 2:\\ \;\;\;\;t \cdot \frac{y}{y - z}\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{x}{z - y}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 94.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x - y}{z - y}\\ t_2 := t \cdot \frac{x}{z - y}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+27}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 3 \cdot 10^{-13}:\\ \;\;\;\;t \cdot \frac{x - y}{z}\\ \mathbf{elif}\;t\_1 \leq 2:\\ \;\;\;\;t \cdot \frac{y}{y - z}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (- x y) (- z y))) (t_2 (* t (/ x (- z y)))))
   (if (<= t_1 -1e+27)
     t_2
     (if (<= t_1 3e-13)
       (* t (/ (- x y) z))
       (if (<= t_1 2.0) (* t (/ y (- y z))) t_2)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x - y) / (z - y);
	double t_2 = t * (x / (z - y));
	double tmp;
	if (t_1 <= -1e+27) {
		tmp = t_2;
	} else if (t_1 <= 3e-13) {
		tmp = t * ((x - y) / z);
	} else if (t_1 <= 2.0) {
		tmp = t * (y / (y - z));
	} else {
		tmp = t_2;
	}
	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) :: t_2
    real(8) :: tmp
    t_1 = (x - y) / (z - y)
    t_2 = t * (x / (z - y))
    if (t_1 <= (-1d+27)) then
        tmp = t_2
    else if (t_1 <= 3d-13) then
        tmp = t * ((x - y) / z)
    else if (t_1 <= 2.0d0) then
        tmp = t * (y / (y - z))
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (x - y) / (z - y);
	double t_2 = t * (x / (z - y));
	double tmp;
	if (t_1 <= -1e+27) {
		tmp = t_2;
	} else if (t_1 <= 3e-13) {
		tmp = t * ((x - y) / z);
	} else if (t_1 <= 2.0) {
		tmp = t * (y / (y - z));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (x - y) / (z - y)
	t_2 = t * (x / (z - y))
	tmp = 0
	if t_1 <= -1e+27:
		tmp = t_2
	elif t_1 <= 3e-13:
		tmp = t * ((x - y) / z)
	elif t_1 <= 2.0:
		tmp = t * (y / (y - z))
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x - y) / Float64(z - y))
	t_2 = Float64(t * Float64(x / Float64(z - y)))
	tmp = 0.0
	if (t_1 <= -1e+27)
		tmp = t_2;
	elseif (t_1 <= 3e-13)
		tmp = Float64(t * Float64(Float64(x - y) / z));
	elseif (t_1 <= 2.0)
		tmp = Float64(t * Float64(y / Float64(y - z)));
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (x - y) / (z - y);
	t_2 = t * (x / (z - y));
	tmp = 0.0;
	if (t_1 <= -1e+27)
		tmp = t_2;
	elseif (t_1 <= 3e-13)
		tmp = t * ((x - y) / z);
	elseif (t_1 <= 2.0)
		tmp = t * (y / (y - z));
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x - y), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t * N[(x / N[(z - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e+27], t$95$2, If[LessEqual[t$95$1, 3e-13], N[(t * N[(N[(x - y), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2.0], N[(t * N[(y / N[(y - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 x y) (-.f64 z y)) < -1e27 or 2 < (/.f64 (-.f64 x y) (-.f64 z y))

   1. Initial program 95.6%

      \[\frac{x - y}{z - y} \cdot t \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

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

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

      2. --lowering--.f64 95.2

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

   5. Simplified 95.2%

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

   if -1e27 < (/.f64 (-.f64 x y) (-.f64 z y)) < 2.99999999999999984e-13

   1. Initial program 96.0%

      \[\frac{x - y}{z - y} \cdot t \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

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

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

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

      2. --lowering--.f64 95.1

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

   5. Simplified 95.1%

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

   if 2.99999999999999984e-13 < (/.f64 (-.f64 x y) (-.f64 z y)) < 2

   1. Initial program 99.9%

      \[\frac{x - y}{z - y} \cdot t \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. neg-lowering-neg.f64 99.3

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

   5. Simplified 99.3%

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

      1. distribute-frac-neg N/A

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

      2. distribute-frac-neg2 N/A

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

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

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

      4. neg-sub0 N/A

         \[\leadsto \frac{y}{\color{blue}{0 - \left(z - y\right)}} \cdot t \]

      5. sub-neg N/A

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

      6. +-commutative N/A

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

      7. associate--r+ N/A

         \[\leadsto \frac{y}{\color{blue}{\left(0 - \left(\mathsf{neg}\left(y\right)\right)\right) - z}} \cdot t \]

      8. neg-sub0 N/A

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

      9. remove-double-neg N/A

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

      10. --lowering--.f64 99.3

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

   7. Applied egg-rr 99.3%

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

4. Final simplification 96.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{z - y} \leq -1 \cdot 10^{+27}:\\ \;\;\;\;t \cdot \frac{x}{z - y}\\ \mathbf{elif}\;\frac{x - y}{z - y} \leq 3 \cdot 10^{-13}:\\ \;\;\;\;t \cdot \frac{x - y}{z}\\ \mathbf{elif}\;\frac{x - y}{z - y} \leq 2:\\ \;\;\;\;t \cdot \frac{y}{y - z}\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{x}{z - y}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 93.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x - y}{z - y}\\ t_2 := t \cdot \frac{x}{z - y}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{-15}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 3 \cdot 10^{-13}:\\ \;\;\;\;\left(x - y\right) \cdot \frac{t}{z}\\ \mathbf{elif}\;t\_1 \leq 2:\\ \;\;\;\;t \cdot \frac{y}{y - z}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (- x y) (- z y))) (t_2 (* t (/ x (- z y)))))
   (if (<= t_1 -2e-15)
     t_2
     (if (<= t_1 3e-13)
       (* (- x y) (/ t z))
       (if (<= t_1 2.0) (* t (/ y (- y z))) t_2)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x - y) / (z - y);
	double t_2 = t * (x / (z - y));
	double tmp;
	if (t_1 <= -2e-15) {
		tmp = t_2;
	} else if (t_1 <= 3e-13) {
		tmp = (x - y) * (t / z);
	} else if (t_1 <= 2.0) {
		tmp = t * (y / (y - z));
	} else {
		tmp = t_2;
	}
	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) :: t_2
    real(8) :: tmp
    t_1 = (x - y) / (z - y)
    t_2 = t * (x / (z - y))
    if (t_1 <= (-2d-15)) then
        tmp = t_2
    else if (t_1 <= 3d-13) then
        tmp = (x - y) * (t / z)
    else if (t_1 <= 2.0d0) then
        tmp = t * (y / (y - z))
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (x - y) / (z - y);
	double t_2 = t * (x / (z - y));
	double tmp;
	if (t_1 <= -2e-15) {
		tmp = t_2;
	} else if (t_1 <= 3e-13) {
		tmp = (x - y) * (t / z);
	} else if (t_1 <= 2.0) {
		tmp = t * (y / (y - z));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (x - y) / (z - y)
	t_2 = t * (x / (z - y))
	tmp = 0
	if t_1 <= -2e-15:
		tmp = t_2
	elif t_1 <= 3e-13:
		tmp = (x - y) * (t / z)
	elif t_1 <= 2.0:
		tmp = t * (y / (y - z))
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x - y) / Float64(z - y))
	t_2 = Float64(t * Float64(x / Float64(z - y)))
	tmp = 0.0
	if (t_1 <= -2e-15)
		tmp = t_2;
	elseif (t_1 <= 3e-13)
		tmp = Float64(Float64(x - y) * Float64(t / z));
	elseif (t_1 <= 2.0)
		tmp = Float64(t * Float64(y / Float64(y - z)));
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (x - y) / (z - y);
	t_2 = t * (x / (z - y));
	tmp = 0.0;
	if (t_1 <= -2e-15)
		tmp = t_2;
	elseif (t_1 <= 3e-13)
		tmp = (x - y) * (t / z);
	elseif (t_1 <= 2.0)
		tmp = t * (y / (y - z));
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x - y), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t * N[(x / N[(z - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e-15], t$95$2, If[LessEqual[t$95$1, 3e-13], N[(N[(x - y), $MachinePrecision] * N[(t / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2.0], N[(t * N[(y / N[(y - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 x y) (-.f64 z y)) < -2.0000000000000002e-15 or 2 < (/.f64 (-.f64 x y) (-.f64 z y))

   1. Initial program 95.9%

      \[\frac{x - y}{z - y} \cdot t \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

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

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

      2. --lowering--.f64 94.1

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

   5. Simplified 94.1%

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

   if -2.0000000000000002e-15 < (/.f64 (-.f64 x y) (-.f64 z y)) < 2.99999999999999984e-13

   1. Initial program 95.8%

      \[\frac{x - y}{z - y} \cdot t \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

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

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

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

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

      5. /-lowering-/.f64 86.1

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

   5. Simplified 86.1%

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

   if 2.99999999999999984e-13 < (/.f64 (-.f64 x y) (-.f64 z y)) < 2

   1. Initial program 99.9%

      \[\frac{x - y}{z - y} \cdot t \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. neg-lowering-neg.f64 99.3

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

   5. Simplified 99.3%

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

      1. distribute-frac-neg N/A

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

      2. distribute-frac-neg2 N/A

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

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

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

      4. neg-sub0 N/A

         \[\leadsto \frac{y}{\color{blue}{0 - \left(z - y\right)}} \cdot t \]

      5. sub-neg N/A

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

      6. +-commutative N/A

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

      7. associate--r+ N/A

         \[\leadsto \frac{y}{\color{blue}{\left(0 - \left(\mathsf{neg}\left(y\right)\right)\right) - z}} \cdot t \]

      8. neg-sub0 N/A

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

      9. remove-double-neg N/A

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

      10. --lowering--.f64 99.3

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

   7. Applied egg-rr 99.3%

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

4. Final simplification 92.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{z - y} \leq -2 \cdot 10^{-15}:\\ \;\;\;\;t \cdot \frac{x}{z - y}\\ \mathbf{elif}\;\frac{x - y}{z - y} \leq 3 \cdot 10^{-13}:\\ \;\;\;\;\left(x - y\right) \cdot \frac{t}{z}\\ \mathbf{elif}\;\frac{x - y}{z - y} \leq 2:\\ \;\;\;\;t \cdot \frac{y}{y - z}\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{x}{z - y}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 93.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x - y}{z - y}\\ t_2 := t \cdot \frac{x}{z - y}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{-15}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 0.005:\\ \;\;\;\;\left(x - y\right) \cdot \frac{t}{z}\\ \mathbf{elif}\;t\_1 \leq 2:\\ \;\;\;\;\mathsf{fma}\left(t, \frac{z}{y}, t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (- x y) (- z y))) (t_2 (* t (/ x (- z y)))))
   (if (<= t_1 -2e-15)
     t_2
     (if (<= t_1 0.005)
       (* (- x y) (/ t z))
       (if (<= t_1 2.0) (fma t (/ z y) t) t_2)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x - y) / (z - y);
	double t_2 = t * (x / (z - y));
	double tmp;
	if (t_1 <= -2e-15) {
		tmp = t_2;
	} else if (t_1 <= 0.005) {
		tmp = (x - y) * (t / z);
	} else if (t_1 <= 2.0) {
		tmp = fma(t, (z / y), t);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x - y) / Float64(z - y))
	t_2 = Float64(t * Float64(x / Float64(z - y)))
	tmp = 0.0
	if (t_1 <= -2e-15)
		tmp = t_2;
	elseif (t_1 <= 0.005)
		tmp = Float64(Float64(x - y) * Float64(t / z));
	elseif (t_1 <= 2.0)
		tmp = fma(t, Float64(z / y), t);
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x - y), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t * N[(x / N[(z - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e-15], t$95$2, If[LessEqual[t$95$1, 0.005], N[(N[(x - y), $MachinePrecision] * N[(t / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2.0], N[(t * N[(z / y), $MachinePrecision] + t), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 x y) (-.f64 z y)) < -2.0000000000000002e-15 or 2 < (/.f64 (-.f64 x y) (-.f64 z y))

   1. Initial program 95.9%

      \[\frac{x - y}{z - y} \cdot t \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

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

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

      2. --lowering--.f64 94.1

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

   5. Simplified 94.1%

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

   if -2.0000000000000002e-15 < (/.f64 (-.f64 x y) (-.f64 z y)) < 0.0050000000000000001

   1. Initial program 95.9%

      \[\frac{x - y}{z - y} \cdot t \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

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

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

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

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

      5. /-lowering-/.f64 84.7

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

   5. Simplified 84.7%

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

   if 0.0050000000000000001 < (/.f64 (-.f64 x y) (-.f64 z y)) < 2

   1. Initial program 100.0%

      \[\frac{x - y}{z - y} \cdot t \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. neg-lowering-neg.f64 99.3

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

   5. Simplified 99.3%

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

   6. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

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

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

      4. /-lowering-/.f64 99.3

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

   8. Simplified 99.3%

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

4. Final simplification 92.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{z - y} \leq -2 \cdot 10^{-15}:\\ \;\;\;\;t \cdot \frac{x}{z - y}\\ \mathbf{elif}\;\frac{x - y}{z - y} \leq 0.005:\\ \;\;\;\;\left(x - y\right) \cdot \frac{t}{z}\\ \mathbf{elif}\;\frac{x - y}{z - y} \leq 2:\\ \;\;\;\;\mathsf{fma}\left(t, \frac{z}{y}, t\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{x}{z - y}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 79.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x - y}{z - y}\\ \mathbf{if}\;t\_1 \leq -4 \cdot 10^{+41}:\\ \;\;\;\;-x \cdot \frac{t}{y}\\ \mathbf{elif}\;t\_1 \leq 0.005:\\ \;\;\;\;\left(x - y\right) \cdot \frac{t}{z}\\ \mathbf{elif}\;t\_1 \leq 2:\\ \;\;\;\;\mathsf{fma}\left(t, \frac{z}{y}, t\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (- x y) (- z y))))
   (if (<= t_1 -4e+41)
     (- (* x (/ t y)))
     (if (<= t_1 0.005)
       (* (- x y) (/ t z))
       (if (<= t_1 2.0) (fma t (/ z y) t) (* t (/ x z)))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x - y) / (z - y);
	double tmp;
	if (t_1 <= -4e+41) {
		tmp = -(x * (t / y));
	} else if (t_1 <= 0.005) {
		tmp = (x - y) * (t / z);
	} else if (t_1 <= 2.0) {
		tmp = fma(t, (z / y), t);
	} else {
		tmp = t * (x / z);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x - y) / Float64(z - y))
	tmp = 0.0
	if (t_1 <= -4e+41)
		tmp = Float64(-Float64(x * Float64(t / y)));
	elseif (t_1 <= 0.005)
		tmp = Float64(Float64(x - y) * Float64(t / z));
	elseif (t_1 <= 2.0)
		tmp = fma(t, Float64(z / y), t);
	else
		tmp = Float64(t * Float64(x / z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x - y), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -4e+41], (-N[(x * N[(t / y), $MachinePrecision]), $MachinePrecision]), If[LessEqual[t$95$1, 0.005], N[(N[(x - y), $MachinePrecision] * N[(t / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2.0], N[(t * N[(z / y), $MachinePrecision] + t), $MachinePrecision], N[(t * N[(x / z), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (-.f64 x y) (-.f64 z y)) < -4.00000000000000002e41

   1. Initial program 88.9%

      \[\frac{x - y}{z - y} \cdot t \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. div-sub N/A

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

      4. sub-neg N/A

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

      5. *-inverses N/A

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

      6. metadata-eval N/A

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

      7. distribute-lft-in N/A

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

      8. neg-mul-1 N/A

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

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

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

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

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

      11. mul-1-neg N/A

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

      12. neg-mul-1 N/A

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

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

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

      14. *-commutative N/A

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

      15. neg-mul-1 N/A

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

      16. remove-double-neg N/A

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

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

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

      18. mul-1-neg N/A

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

      19. distribute-neg-frac2 N/A

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

      20. /-lowering-/.f64 N/A

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

      21. neg-lowering-neg.f64 69.6

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

   5. Simplified 69.6%

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

   6. Taylor expanded in x around inf

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

      1. associate-*r/ N/A

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

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

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

      3. mul-1-neg N/A

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

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

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

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

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

      6. neg-lowering-neg.f64 72.5

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

   8. Simplified 72.5%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

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

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

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

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

      6. neg-lowering-neg.f64 74.9

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

   10. Applied egg-rr 74.9%

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

   if -4.00000000000000002e41 < (/.f64 (-.f64 x y) (-.f64 z y)) < 0.0050000000000000001

   1. Initial program 96.1%

      \[\frac{x - y}{z - y} \cdot t \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

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

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

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

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

      5. /-lowering-/.f64 82.3

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

   5. Simplified 82.3%

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

   if 0.0050000000000000001 < (/.f64 (-.f64 x y) (-.f64 z y)) < 2

   1. Initial program 100.0%

      \[\frac{x - y}{z - y} \cdot t \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. neg-lowering-neg.f64 99.3

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

   5. Simplified 99.3%

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

   6. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

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

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

      4. /-lowering-/.f64 99.3

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

   8. Simplified 99.3%

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

   if 2 < (/.f64 (-.f64 x y) (-.f64 z y))

   1. Initial program 99.7%

      \[\frac{x - y}{z - y} \cdot t \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. /-lowering-/.f64 62.1

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

   5. Simplified 62.1%

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

4. Final simplification 84.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{z - y} \leq -4 \cdot 10^{+41}:\\ \;\;\;\;-x \cdot \frac{t}{y}\\ \mathbf{elif}\;\frac{x - y}{z - y} \leq 0.005:\\ \;\;\;\;\left(x - y\right) \cdot \frac{t}{z}\\ \mathbf{elif}\;\frac{x - y}{z - y} \leq 2:\\ \;\;\;\;\mathsf{fma}\left(t, \frac{z}{y}, t\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 69.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := t \cdot \frac{x}{z}\\ t_2 := \frac{x - y}{z - y}\\ \mathbf{if}\;t\_2 \leq -4 \cdot 10^{+41}:\\ \;\;\;\;-x \cdot \frac{t}{y}\\ \mathbf{elif}\;t\_2 \leq 3 \cdot 10^{-13}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 2:\\ \;\;\;\;t\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* t (/ x z))) (t_2 (/ (- x y) (- z y))))
   (if (<= t_2 -4e+41)
     (- (* x (/ t y)))
     (if (<= t_2 3e-13) t_1 (if (<= t_2 2.0) t t_1)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = t * (x / z);
	double t_2 = (x - y) / (z - y);
	double tmp;
	if (t_2 <= -4e+41) {
		tmp = -(x * (t / y));
	} else if (t_2 <= 3e-13) {
		tmp = t_1;
	} else if (t_2 <= 2.0) {
		tmp = t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = t * (x / z)
    t_2 = (x - y) / (z - y)
    if (t_2 <= (-4d+41)) then
        tmp = -(x * (t / y))
    else if (t_2 <= 3d-13) then
        tmp = t_1
    else if (t_2 <= 2.0d0) then
        tmp = t
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = t * (x / z);
	double t_2 = (x - y) / (z - y);
	double tmp;
	if (t_2 <= -4e+41) {
		tmp = -(x * (t / y));
	} else if (t_2 <= 3e-13) {
		tmp = t_1;
	} else if (t_2 <= 2.0) {
		tmp = t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = t * (x / z)
	t_2 = (x - y) / (z - y)
	tmp = 0
	if t_2 <= -4e+41:
		tmp = -(x * (t / y))
	elif t_2 <= 3e-13:
		tmp = t_1
	elif t_2 <= 2.0:
		tmp = t
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(t * Float64(x / z))
	t_2 = Float64(Float64(x - y) / Float64(z - y))
	tmp = 0.0
	if (t_2 <= -4e+41)
		tmp = Float64(-Float64(x * Float64(t / y)));
	elseif (t_2 <= 3e-13)
		tmp = t_1;
	elseif (t_2 <= 2.0)
		tmp = t;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = t * (x / z);
	t_2 = (x - y) / (z - y);
	tmp = 0.0;
	if (t_2 <= -4e+41)
		tmp = -(x * (t / y));
	elseif (t_2 <= 3e-13)
		tmp = t_1;
	elseif (t_2 <= 2.0)
		tmp = t;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(t * N[(x / z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x - y), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -4e+41], (-N[(x * N[(t / y), $MachinePrecision]), $MachinePrecision]), If[LessEqual[t$95$2, 3e-13], t$95$1, If[LessEqual[t$95$2, 2.0], t, t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 x y) (-.f64 z y)) < -4.00000000000000002e41

   1. Initial program 88.9%

      \[\frac{x - y}{z - y} \cdot t \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. div-sub N/A

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

      4. sub-neg N/A

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

      5. *-inverses N/A

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

      6. metadata-eval N/A

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

      7. distribute-lft-in N/A

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

      8. neg-mul-1 N/A

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

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

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

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

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

      11. mul-1-neg N/A

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

      12. neg-mul-1 N/A

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

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

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

      14. *-commutative N/A

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

      15. neg-mul-1 N/A

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

      16. remove-double-neg N/A

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

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

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

      18. mul-1-neg N/A

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

      19. distribute-neg-frac2 N/A

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

      20. /-lowering-/.f64 N/A

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

      21. neg-lowering-neg.f64 69.6

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

   5. Simplified 69.6%

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

   6. Taylor expanded in x around inf

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

      1. associate-*r/ N/A

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

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

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

      3. mul-1-neg N/A

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

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

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

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

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

      6. neg-lowering-neg.f64 72.5

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

   8. Simplified 72.5%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

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

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

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

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

      6. neg-lowering-neg.f64 74.9

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

   10. Applied egg-rr 74.9%

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

   if -4.00000000000000002e41 < (/.f64 (-.f64 x y) (-.f64 z y)) < 2.99999999999999984e-13 or 2 < (/.f64 (-.f64 x y) (-.f64 z y))

   1. Initial program 97.1%

      \[\frac{x - y}{z - y} \cdot t \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. /-lowering-/.f64 62.5

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

   5. Simplified 62.5%

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

   if 2.99999999999999984e-13 < (/.f64 (-.f64 x y) (-.f64 z y)) < 2

   1. Initial program 99.9%

      \[\frac{x - y}{z - y} \cdot t \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

   5. Simplified 97.3%

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

4. Final simplification 76.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{z - y} \leq -4 \cdot 10^{+41}:\\ \;\;\;\;-x \cdot \frac{t}{y}\\ \mathbf{elif}\;\frac{x - y}{z - y} \leq 3 \cdot 10^{-13}:\\ \;\;\;\;t \cdot \frac{x}{z}\\ \mathbf{elif}\;\frac{x - y}{z - y} \leq 2:\\ \;\;\;\;t\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 69.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x - y}{z - y}\\ t_2 := t \cdot \frac{x}{z}\\ \mathbf{if}\;t\_1 \leq 3 \cdot 10^{-13}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 2:\\ \;\;\;\;t\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (- x y) (- z y))) (t_2 (* t (/ x z))))
   (if (<= t_1 3e-13) t_2 (if (<= t_1 2.0) t t_2))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x - y) / (z - y);
	double t_2 = t * (x / z);
	double tmp;
	if (t_1 <= 3e-13) {
		tmp = t_2;
	} else if (t_1 <= 2.0) {
		tmp = t;
	} else {
		tmp = t_2;
	}
	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) :: t_2
    real(8) :: tmp
    t_1 = (x - y) / (z - y)
    t_2 = t * (x / z)
    if (t_1 <= 3d-13) then
        tmp = t_2
    else if (t_1 <= 2.0d0) then
        tmp = t
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (x - y) / (z - y);
	double t_2 = t * (x / z);
	double tmp;
	if (t_1 <= 3e-13) {
		tmp = t_2;
	} else if (t_1 <= 2.0) {
		tmp = t;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (x - y) / (z - y)
	t_2 = t * (x / z)
	tmp = 0
	if t_1 <= 3e-13:
		tmp = t_2
	elif t_1 <= 2.0:
		tmp = t
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x - y) / Float64(z - y))
	t_2 = Float64(t * Float64(x / z))
	tmp = 0.0
	if (t_1 <= 3e-13)
		tmp = t_2;
	elseif (t_1 <= 2.0)
		tmp = t;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (x - y) / (z - y);
	t_2 = t * (x / z);
	tmp = 0.0;
	if (t_1 <= 3e-13)
		tmp = t_2;
	elseif (t_1 <= 2.0)
		tmp = t;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x - y), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t * N[(x / z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 3e-13], t$95$2, If[LessEqual[t$95$1, 2.0], t, t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 x y) (-.f64 z y)) < 2.99999999999999984e-13 or 2 < (/.f64 (-.f64 x y) (-.f64 z y))

   1. Initial program 95.8%

      \[\frac{x - y}{z - y} \cdot t \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. /-lowering-/.f64 58.5

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

   5. Simplified 58.5%

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

   if 2.99999999999999984e-13 < (/.f64 (-.f64 x y) (-.f64 z y)) < 2

   1. Initial program 99.9%

      \[\frac{x - y}{z - y} \cdot t \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

   5. Simplified 97.3%

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

4. Final simplification 72.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{z - y} \leq 3 \cdot 10^{-13}:\\ \;\;\;\;t \cdot \frac{x}{z}\\ \mathbf{elif}\;\frac{x - y}{z - y} \leq 2:\\ \;\;\;\;t\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 68.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x - y}{z - y}\\ t_2 := x \cdot \frac{t}{z}\\ \mathbf{if}\;t\_1 \leq 3 \cdot 10^{-13}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 2:\\ \;\;\;\;t\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (- x y) (- z y))) (t_2 (* x (/ t z))))
   (if (<= t_1 3e-13) t_2 (if (<= t_1 2.0) t t_2))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x - y) / (z - y);
	double t_2 = x * (t / z);
	double tmp;
	if (t_1 <= 3e-13) {
		tmp = t_2;
	} else if (t_1 <= 2.0) {
		tmp = t;
	} else {
		tmp = t_2;
	}
	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) :: t_2
    real(8) :: tmp
    t_1 = (x - y) / (z - y)
    t_2 = x * (t / z)
    if (t_1 <= 3d-13) then
        tmp = t_2
    else if (t_1 <= 2.0d0) then
        tmp = t
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (x - y) / (z - y);
	double t_2 = x * (t / z);
	double tmp;
	if (t_1 <= 3e-13) {
		tmp = t_2;
	} else if (t_1 <= 2.0) {
		tmp = t;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (x - y) / (z - y)
	t_2 = x * (t / z)
	tmp = 0
	if t_1 <= 3e-13:
		tmp = t_2
	elif t_1 <= 2.0:
		tmp = t
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x - y) / Float64(z - y))
	t_2 = Float64(x * Float64(t / z))
	tmp = 0.0
	if (t_1 <= 3e-13)
		tmp = t_2;
	elseif (t_1 <= 2.0)
		tmp = t;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (x - y) / (z - y);
	t_2 = x * (t / z);
	tmp = 0.0;
	if (t_1 <= 3e-13)
		tmp = t_2;
	elseif (t_1 <= 2.0)
		tmp = t;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x - y), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(x * N[(t / z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 3e-13], t$95$2, If[LessEqual[t$95$1, 2.0], t, t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 x y) (-.f64 z y)) < 2.99999999999999984e-13 or 2 < (/.f64 (-.f64 x y) (-.f64 z y))

   1. Initial program 95.8%

      \[\frac{x - y}{z - y} \cdot t \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

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

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

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

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

      5. /-lowering-/.f64 71.4

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

   5. Simplified 71.4%

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

   6. Taylor expanded in x around inf

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

   8. Simplified 52.8%

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

   if 2.99999999999999984e-13 < (/.f64 (-.f64 x y) (-.f64 z y)) < 2

   1. Initial program 99.9%

      \[\frac{x - y}{z - y} \cdot t \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

   5. Simplified 97.3%

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

Alternative 10: 96.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 97.3%

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

Alternative 11: 35.1% accurate, 23.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_] := t

Derivation

1. Initial program 97.3%

   \[\frac{x - y}{z - y} \cdot t \]
2. Add Preprocessing

3. Taylor expanded in y around inf

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

5. Simplified 37.1%

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

Developer Target 1: 96.8% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024199 
(FPCore (x y z t)
  :name "Numeric.Signal.Multichannel:$cput from hsignal-0.2.7.1"
  :precision binary64

  :alt
  (! :herbie-platform default (/ t (/ (- z y) (- x y))))

  (* (/ (- x y) (- z y)) t))