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

Percentage Accurate: 96.9% → 96.9%

Time: 9.5s

Alternatives: 14

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: 96.9% 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: 96.9% 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 96.9%

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

Alternative 2: 69.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := t \cdot \frac{x}{z}\\ t_2 := \frac{x - y}{z - y}\\ t_3 := t \cdot \frac{x}{-y}\\ \mathbf{if}\;t\_2 \leq -5 \cdot 10^{+23}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_2 \leq -5 \cdot 10^{-111}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{-43}:\\ \;\;\;\;-\frac{y \cdot t}{z}\\ \mathbf{elif}\;t\_2 \leq 10^{-7}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 2:\\ \;\;\;\;\mathsf{fma}\left(t, \frac{z}{y}, t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* t (/ x z))) (t_2 (/ (- x y) (- z y))) (t_3 (* t (/ x (- y)))))
   (if (<= t_2 -5e+23)
     t_3
     (if (<= t_2 -5e-111)
       t_1
       (if (<= t_2 5e-43)
         (- (/ (* y t) z))
         (if (<= t_2 1e-7) t_1 (if (<= t_2 2.0) (fma t (/ z y) t) t_3)))))))

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 t_3 = t * (x / -y);
	double tmp;
	if (t_2 <= -5e+23) {
		tmp = t_3;
	} else if (t_2 <= -5e-111) {
		tmp = t_1;
	} else if (t_2 <= 5e-43) {
		tmp = -((y * t) / z);
	} else if (t_2 <= 1e-7) {
		tmp = t_1;
	} else if (t_2 <= 2.0) {
		tmp = fma(t, (z / y), t);
	} else {
		tmp = t_3;
	}
	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))
	t_3 = Float64(t * Float64(x / Float64(-y)))
	tmp = 0.0
	if (t_2 <= -5e+23)
		tmp = t_3;
	elseif (t_2 <= -5e-111)
		tmp = t_1;
	elseif (t_2 <= 5e-43)
		tmp = Float64(-Float64(Float64(y * t) / z));
	elseif (t_2 <= 1e-7)
		tmp = t_1;
	elseif (t_2 <= 2.0)
		tmp = fma(t, Float64(z / y), t);
	else
		tmp = t_3;
	end
	return 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]}, Block[{t$95$3 = N[(t * N[(x / (-y)), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -5e+23], t$95$3, If[LessEqual[t$95$2, -5e-111], t$95$1, If[LessEqual[t$95$2, 5e-43], (-N[(N[(y * t), $MachinePrecision] / z), $MachinePrecision]), If[LessEqual[t$95$2, 1e-7], t$95$1, If[LessEqual[t$95$2, 2.0], N[(t * N[(z / y), $MachinePrecision] + t), $MachinePrecision], t$95$3]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (-.f64 x y) (-.f64 z y)) < -4.9999999999999999e23 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 x around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 96.2

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

   5. Applied rewrites 96.2%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 58.0%

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

   if -4.9999999999999999e23 < (/.f64 (-.f64 x y) (-.f64 z y)) < -5.0000000000000003e-111 or 5.00000000000000019e-43 < (/.f64 (-.f64 x y) (-.f64 z y)) < 9.9999999999999995e-8

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 60.8

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

   5. Applied rewrites 60.8%

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

   if -5.0000000000000003e-111 < (/.f64 (-.f64 x y) (-.f64 z y)) < 5.00000000000000019e-43

   1. Initial program 91.3%

      \[\frac{x - y}{z - y} \cdot t \]
   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. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 92.2

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

   5. Applied rewrites 92.2%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 74.0%

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

   if 9.9999999999999995e-8 < (/.f64 (-.f64 x y) (-.f64 z y)) < 2

   1. Initial program 100.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 73.6

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

   4. Applied rewrites 73.6%

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

   5. Taylor expanded in y around inf

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

      1. associate--l+ N/A

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

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

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

      3. div-sub N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

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

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

      7. associate-/l* N/A

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

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

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

      9. mul-1-neg N/A

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

      10. lower-fma.f64 N/A

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

   7. Applied rewrites 99.3%

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

   8. Taylor expanded in z around inf

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

   10. Applied rewrites 97.7%

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

4. Final simplification 75.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{z - y} \leq -5 \cdot 10^{+23}:\\ \;\;\;\;t \cdot \frac{x}{-y}\\ \mathbf{elif}\;\frac{x - y}{z - y} \leq -5 \cdot 10^{-111}:\\ \;\;\;\;t \cdot \frac{x}{z}\\ \mathbf{elif}\;\frac{x - y}{z - y} \leq 5 \cdot 10^{-43}:\\ \;\;\;\;-\frac{y \cdot t}{z}\\ \mathbf{elif}\;\frac{x - y}{z - y} \leq 10^{-7}:\\ \;\;\;\;t \cdot \frac{x}{z}\\ \mathbf{elif}\;\frac{x - y}{z - y} \leq 2:\\ \;\;\;\;\mathsf{fma}\left(t, \frac{z}{y}, t\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{x}{-y}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 69.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x - y}{z - y}\\ t_2 := t \cdot \frac{x}{-y}\\ \mathbf{if}\;t\_1 \leq -400:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-43}:\\ \;\;\;\;t \cdot \frac{-y}{z}\\ \mathbf{elif}\;t\_1 \leq 10^{-7}:\\ \;\;\;\;t \cdot \frac{x}{z}\\ \mathbf{elif}\;t\_1 \leq 2:\\ \;\;\;\;\mathsf{fma}\left(t, \frac{z}{y}, t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (- x y) (- z y))) (t_2 (* t (/ x (- y)))))
   (if (<= t_1 -400.0)
     t_2
     (if (<= t_1 5e-43)
       (* t (/ (- y) z))
       (if (<= t_1 1e-7)
         (* t (/ x 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 / -y);
	double tmp;
	if (t_1 <= -400.0) {
		tmp = t_2;
	} else if (t_1 <= 5e-43) {
		tmp = t * (-y / z);
	} else if (t_1 <= 1e-7) {
		tmp = t * (x / 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(-y)))
	tmp = 0.0
	if (t_1 <= -400.0)
		tmp = t_2;
	elseif (t_1 <= 5e-43)
		tmp = Float64(t * Float64(Float64(-y) / z));
	elseif (t_1 <= 1e-7)
		tmp = Float64(t * Float64(x / 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 / (-y)), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -400.0], t$95$2, If[LessEqual[t$95$1, 5e-43], N[(t * N[((-y) / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e-7], N[(t * N[(x / 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 4 regimes

2. if (/.f64 (-.f64 x y) (-.f64 z y)) < -400 or 2 < (/.f64 (-.f64 x y) (-.f64 z y))

   1. Initial program 97.3%

      \[\frac{x - y}{z - y} \cdot t \]
   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. lower-/.f64 N/A

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

      2. lower--.f64 95.2

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

   5. Applied rewrites 95.2%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 55.7%

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

   if -400 < (/.f64 (-.f64 x y) (-.f64 z y)) < 5.00000000000000019e-43

   1. Initial program 93.0%

      \[\frac{x - y}{z - y} \cdot t \]
   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. lower-/.f64 N/A

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

      2. lower--.f64 91.9

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

   5. Applied rewrites 91.9%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 67.7%

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

   if 5.00000000000000019e-43 < (/.f64 (-.f64 x y) (-.f64 z y)) < 9.9999999999999995e-8

   1. Initial program 99.4%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 75.5

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

   5. Applied rewrites 75.5%

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

   if 9.9999999999999995e-8 < (/.f64 (-.f64 x y) (-.f64 z y)) < 2

   1. Initial program 100.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 73.6

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

   4. Applied rewrites 73.6%

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

   5. Taylor expanded in y around inf

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

      1. associate--l+ N/A

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

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

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

      3. div-sub N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

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

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

      7. associate-/l* N/A

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

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

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

      9. mul-1-neg N/A

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

      10. lower-fma.f64 N/A

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

   7. Applied rewrites 99.3%

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

   8. Taylor expanded in z around inf

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

   10. Applied rewrites 97.7%

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

4. Final simplification 74.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{z - y} \leq -400:\\ \;\;\;\;t \cdot \frac{x}{-y}\\ \mathbf{elif}\;\frac{x - y}{z - y} \leq 5 \cdot 10^{-43}:\\ \;\;\;\;t \cdot \frac{-y}{z}\\ \mathbf{elif}\;\frac{x - y}{z - y} \leq 10^{-7}:\\ \;\;\;\;t \cdot \frac{x}{z}\\ \mathbf{elif}\;\frac{x - y}{z - y} \leq 2:\\ \;\;\;\;\mathsf{fma}\left(t, \frac{z}{y}, t\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{x}{-y}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 69.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x - y}{z - y}\\ t_2 := t \cdot \frac{x}{z}\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{-111}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-43}:\\ \;\;\;\;-\frac{y \cdot t}{z}\\ \mathbf{elif}\;t\_1 \leq 10^{-7}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 2000:\\ \;\;\;\;\mathsf{fma}\left(t, \frac{z}{y}, t\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot t}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (- x y) (- z y))) (t_2 (* t (/ x z))))
   (if (<= t_1 -5e-111)
     t_2
     (if (<= t_1 5e-43)
       (- (/ (* y t) z))
       (if (<= t_1 1e-7)
         t_2
         (if (<= t_1 2000.0) (fma t (/ z y) t) (/ (* x t) z)))))))

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 <= -5e-111) {
		tmp = t_2;
	} else if (t_1 <= 5e-43) {
		tmp = -((y * t) / z);
	} else if (t_1 <= 1e-7) {
		tmp = t_2;
	} else if (t_1 <= 2000.0) {
		tmp = fma(t, (z / y), t);
	} else {
		tmp = (x * t) / z;
	}
	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 <= -5e-111)
		tmp = t_2;
	elseif (t_1 <= 5e-43)
		tmp = Float64(-Float64(Float64(y * t) / z));
	elseif (t_1 <= 1e-7)
		tmp = t_2;
	elseif (t_1 <= 2000.0)
		tmp = fma(t, Float64(z / y), t);
	else
		tmp = Float64(Float64(x * t) / 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]}, Block[{t$95$2 = N[(t * N[(x / z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5e-111], t$95$2, If[LessEqual[t$95$1, 5e-43], (-N[(N[(y * t), $MachinePrecision] / z), $MachinePrecision]), If[LessEqual[t$95$1, 1e-7], t$95$2, If[LessEqual[t$95$1, 2000.0], N[(t * N[(z / y), $MachinePrecision] + t), $MachinePrecision], N[(N[(x * t), $MachinePrecision] / z), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (-.f64 x y) (-.f64 z y)) < -5.0000000000000003e-111 or 5.00000000000000019e-43 < (/.f64 (-.f64 x y) (-.f64 z y)) < 9.9999999999999995e-8

   1. Initial program 96.9%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 47.4

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

   5. Applied rewrites 47.4%

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

   if -5.0000000000000003e-111 < (/.f64 (-.f64 x y) (-.f64 z y)) < 5.00000000000000019e-43

   1. Initial program 91.3%

      \[\frac{x - y}{z - y} \cdot t \]
   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. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 92.2

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

   5. Applied rewrites 92.2%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 74.0%

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

   if 9.9999999999999995e-8 < (/.f64 (-.f64 x y) (-.f64 z y)) < 2e3

   1. Initial program 100.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 73.9

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

   4. Applied rewrites 73.9%

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

   5. Taylor expanded in y around inf

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

      1. associate--l+ N/A

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

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

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

      3. div-sub N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

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

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

      7. associate-/l* N/A

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

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

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

      9. mul-1-neg N/A

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

      10. lower-fma.f64 N/A

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

   7. Applied rewrites 99.3%

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

   8. Taylor expanded in z around inf

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

   10. Applied rewrites 96.7%

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

   if 2e3 < (/.f64 (-.f64 x y) (-.f64 z y))

   1. Initial program 99.5%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 51.1

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

   5. Applied rewrites 51.1%

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

4. Final simplification 71.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{z - y} \leq -5 \cdot 10^{-111}:\\ \;\;\;\;t \cdot \frac{x}{z}\\ \mathbf{elif}\;\frac{x - y}{z - y} \leq 5 \cdot 10^{-43}:\\ \;\;\;\;-\frac{y \cdot t}{z}\\ \mathbf{elif}\;\frac{x - y}{z - y} \leq 10^{-7}:\\ \;\;\;\;t \cdot \frac{x}{z}\\ \mathbf{elif}\;\frac{x - y}{z - y} \leq 2000:\\ \;\;\;\;\mathsf{fma}\left(t, \frac{z}{y}, t\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot t}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 95.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x - y}{z - y}\\ t_2 := t \cdot \frac{x}{z - y}\\ \mathbf{if}\;t\_1 \leq -400:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 10^{-7}:\\ \;\;\;\;t \cdot \frac{x - y}{z}\\ \mathbf{elif}\;t\_1 \leq 2000:\\ \;\;\;\;\mathsf{fma}\left(t, \frac{z - x}{y}, t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (- x y) (- z y))) (t_2 (* t (/ x (- z y)))))
   (if (<= t_1 -400.0)
     t_2
     (if (<= t_1 1e-7)
       (* t (/ (- x y) z))
       (if (<= t_1 2000.0) (fma t (/ (- z x) 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 <= -400.0) {
		tmp = t_2;
	} else if (t_1 <= 1e-7) {
		tmp = t * ((x - y) / z);
	} else if (t_1 <= 2000.0) {
		tmp = fma(t, ((z - x) / 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 <= -400.0)
		tmp = t_2;
	elseif (t_1 <= 1e-7)
		tmp = Float64(t * Float64(Float64(x - y) / z));
	elseif (t_1 <= 2000.0)
		tmp = fma(t, Float64(Float64(z - x) / 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, -400.0], t$95$2, If[LessEqual[t$95$1, 1e-7], N[(t * N[(N[(x - y), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2000.0], N[(t * N[(N[(z - x), $MachinePrecision] / y), $MachinePrecision] + t), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 x y) (-.f64 z y)) < -400 or 2e3 < (/.f64 (-.f64 x y) (-.f64 z y))

   1. Initial program 97.3%

      \[\frac{x - y}{z - y} \cdot t \]
   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. lower-/.f64 N/A

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

      2. lower--.f64 95.9

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

   5. Applied rewrites 95.9%

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

   if -400 < (/.f64 (-.f64 x y) (-.f64 z y)) < 9.9999999999999995e-8

   1. Initial program 93.5%

      \[\frac{x - y}{z - y} \cdot t \]
   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. lower-/.f64 N/A

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

      2. lower--.f64 92.1

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

   5. Applied rewrites 92.1%

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

   if 9.9999999999999995e-8 < (/.f64 (-.f64 x y) (-.f64 z y)) < 2e3

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf

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

      1. associate--l+ N/A

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

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

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

      3. div-sub N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

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

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

      7. associate-/l* N/A

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

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

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

      9. mul-1-neg N/A

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

      10. lower-fma.f64 N/A

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

   5. Applied rewrites 99.3%

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

4. Final simplification 95.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{z - y} \leq -400:\\ \;\;\;\;t \cdot \frac{x}{z - y}\\ \mathbf{elif}\;\frac{x - y}{z - y} \leq 10^{-7}:\\ \;\;\;\;t \cdot \frac{x - y}{z}\\ \mathbf{elif}\;\frac{x - y}{z - y} \leq 2000:\\ \;\;\;\;\mathsf{fma}\left(t, \frac{z - x}{y}, t\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{x}{z - y}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 95.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x - y}{z - y}\\ t_2 := t \cdot \frac{x}{z - y}\\ \mathbf{if}\;t\_1 \leq -400:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 10^{-7}:\\ \;\;\;\;t \cdot \frac{x - y}{z}\\ \mathbf{elif}\;t\_1 \leq 2000:\\ \;\;\;\;\mathsf{fma}\left(t, \frac{x}{-y}, t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (- x y) (- z y))) (t_2 (* t (/ x (- z y)))))
   (if (<= t_1 -400.0)
     t_2
     (if (<= t_1 1e-7)
       (* t (/ (- x y) z))
       (if (<= t_1 2000.0) (fma t (/ x (- 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 <= -400.0) {
		tmp = t_2;
	} else if (t_1 <= 1e-7) {
		tmp = t * ((x - y) / z);
	} else if (t_1 <= 2000.0) {
		tmp = fma(t, (x / -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 <= -400.0)
		tmp = t_2;
	elseif (t_1 <= 1e-7)
		tmp = Float64(t * Float64(Float64(x - y) / z));
	elseif (t_1 <= 2000.0)
		tmp = fma(t, Float64(x / Float64(-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, -400.0], t$95$2, If[LessEqual[t$95$1, 1e-7], N[(t * N[(N[(x - y), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2000.0], N[(t * N[(x / (-y)), $MachinePrecision] + t), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 x y) (-.f64 z y)) < -400 or 2e3 < (/.f64 (-.f64 x y) (-.f64 z y))

   1. Initial program 97.3%

      \[\frac{x - y}{z - y} \cdot t \]
   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. lower-/.f64 N/A

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

      2. lower--.f64 95.9

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

   5. Applied rewrites 95.9%

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

   if -400 < (/.f64 (-.f64 x y) (-.f64 z y)) < 9.9999999999999995e-8

   1. Initial program 93.5%

      \[\frac{x - y}{z - y} \cdot t \]
   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. lower-/.f64 N/A

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

      2. lower--.f64 92.1

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

   5. Applied rewrites 92.1%

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

   if 9.9999999999999995e-8 < (/.f64 (-.f64 x y) (-.f64 z y)) < 2e3

   1. Initial program 100.0%

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

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

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

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

      10. *-commutative N/A

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

      11. neg-mul-1 N/A

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

      12. remove-double-neg N/A

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

      13. neg-mul-1 N/A

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

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

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

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

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

      16. mul-1-neg N/A

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

      17. lower-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. lower-/.f64 N/A

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

      21. lower-neg.f64 99.2

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

   5. Applied rewrites 99.2%

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

4. Final simplification 95.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{z - y} \leq -400:\\ \;\;\;\;t \cdot \frac{x}{z - y}\\ \mathbf{elif}\;\frac{x - y}{z - y} \leq 10^{-7}:\\ \;\;\;\;t \cdot \frac{x - y}{z}\\ \mathbf{elif}\;\frac{x - y}{z - y} \leq 2000:\\ \;\;\;\;\mathsf{fma}\left(t, \frac{x}{-y}, t\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{x}{z - y}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 93.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x - y}{z - y}\\ t_2 := t \cdot \frac{x}{z - y}\\ \mathbf{if}\;t\_1 \leq -400:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 10^{-7}:\\ \;\;\;\;\left(x - y\right) \cdot \frac{t}{z}\\ \mathbf{elif}\;t\_1 \leq 2000:\\ \;\;\;\;\mathsf{fma}\left(t, \frac{x}{-y}, t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (- x y) (- z y))) (t_2 (* t (/ x (- z y)))))
   (if (<= t_1 -400.0)
     t_2
     (if (<= t_1 1e-7)
       (* (- x y) (/ t z))
       (if (<= t_1 2000.0) (fma t (/ x (- 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 <= -400.0) {
		tmp = t_2;
	} else if (t_1 <= 1e-7) {
		tmp = (x - y) * (t / z);
	} else if (t_1 <= 2000.0) {
		tmp = fma(t, (x / -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 <= -400.0)
		tmp = t_2;
	elseif (t_1 <= 1e-7)
		tmp = Float64(Float64(x - y) * Float64(t / z));
	elseif (t_1 <= 2000.0)
		tmp = fma(t, Float64(x / Float64(-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, -400.0], t$95$2, If[LessEqual[t$95$1, 1e-7], N[(N[(x - y), $MachinePrecision] * N[(t / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2000.0], N[(t * N[(x / (-y)), $MachinePrecision] + t), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 x y) (-.f64 z y)) < -400 or 2e3 < (/.f64 (-.f64 x y) (-.f64 z y))

   1. Initial program 97.3%

      \[\frac{x - y}{z - y} \cdot t \]
   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. lower-/.f64 N/A

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

      2. lower--.f64 95.9

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

   5. Applied rewrites 95.9%

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

   if -400 < (/.f64 (-.f64 x y) (-.f64 z y)) < 9.9999999999999995e-8

   1. Initial program 93.5%

      \[\frac{x - y}{z - y} \cdot t \]
   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. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 89.8

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

   5. Applied rewrites 89.8%

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

   if 9.9999999999999995e-8 < (/.f64 (-.f64 x y) (-.f64 z y)) < 2e3

   1. Initial program 100.0%

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

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

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

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

      10. *-commutative N/A

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

      11. neg-mul-1 N/A

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

      12. remove-double-neg N/A

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

      13. neg-mul-1 N/A

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

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

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

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

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

      16. mul-1-neg N/A

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

      17. lower-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. lower-/.f64 N/A

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

      21. lower-neg.f64 99.2

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

   5. Applied rewrites 99.2%

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

4. Final simplification 94.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{z - y} \leq -400:\\ \;\;\;\;t \cdot \frac{x}{z - y}\\ \mathbf{elif}\;\frac{x - y}{z - y} \leq 10^{-7}:\\ \;\;\;\;\left(x - y\right) \cdot \frac{t}{z}\\ \mathbf{elif}\;\frac{x - y}{z - y} \leq 2000:\\ \;\;\;\;\mathsf{fma}\left(t, \frac{x}{-y}, t\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{x}{z - y}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 79.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x - y}{z - y}\\ t_2 := t \cdot \frac{x}{-y}\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+23}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 10^{-7}:\\ \;\;\;\;\left(x - y\right) \cdot \frac{t}{z}\\ \mathbf{elif}\;t\_1 \leq 2:\\ \;\;\;\;\mathsf{fma}\left(t, \frac{z}{y}, t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (- x y) (- z y))) (t_2 (* t (/ x (- y)))))
   (if (<= t_1 -5e+23)
     t_2
     (if (<= t_1 1e-7)
       (* (- 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 / -y);
	double tmp;
	if (t_1 <= -5e+23) {
		tmp = t_2;
	} else if (t_1 <= 1e-7) {
		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(-y)))
	tmp = 0.0
	if (t_1 <= -5e+23)
		tmp = t_2;
	elseif (t_1 <= 1e-7)
		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 / (-y)), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+23], t$95$2, If[LessEqual[t$95$1, 1e-7], 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)) < -4.9999999999999999e23 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 x around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 96.2

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

   5. Applied rewrites 96.2%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 58.0%

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

   if -4.9999999999999999e23 < (/.f64 (-.f64 x y) (-.f64 z y)) < 9.9999999999999995e-8

   1. Initial program 93.9%

      \[\frac{x - y}{z - y} \cdot t \]
   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. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 88.3

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

   5. Applied rewrites 88.3%

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

   if 9.9999999999999995e-8 < (/.f64 (-.f64 x y) (-.f64 z y)) < 2

   1. Initial program 100.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 73.6

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

   4. Applied rewrites 73.6%

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

   5. Taylor expanded in y around inf

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

      1. associate--l+ N/A

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

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

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

      3. div-sub N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

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

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

      7. associate-/l* N/A

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

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

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

      9. mul-1-neg N/A

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

      10. lower-fma.f64 N/A

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

   7. Applied rewrites 99.3%

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

   8. Taylor expanded in z around inf

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

   10. Applied rewrites 97.7%

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

4. Final simplification 82.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{z - y} \leq -5 \cdot 10^{+23}:\\ \;\;\;\;t \cdot \frac{x}{-y}\\ \mathbf{elif}\;\frac{x - y}{z - y} \leq 10^{-7}:\\ \;\;\;\;\left(x - y\right) \cdot \frac{t}{z}\\ \mathbf{elif}\;\frac{x - y}{z - y} \leq 2:\\ \;\;\;\;\mathsf{fma}\left(t, \frac{z}{y}, t\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{x}{-y}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 93.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x - y}{z - y}\\ \mathbf{if}\;t\_1 \leq 10^{-7}:\\ \;\;\;\;\left(x - y\right) \cdot \frac{t}{z - y}\\ \mathbf{elif}\;t\_1 \leq 2000:\\ \;\;\;\;\mathsf{fma}\left(t, \frac{z - x}{y}, t\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{x}{z - y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (- x y) (- z y))))
   (if (<= t_1 1e-7)
     (* (- x y) (/ t (- z y)))
     (if (<= t_1 2000.0) (fma t (/ (- z x) y) t) (* 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-7) {
		tmp = (x - y) * (t / (z - y));
	} else if (t_1 <= 2000.0) {
		tmp = fma(t, ((z - x) / y), t);
	} 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-7)
		tmp = Float64(Float64(x - y) * Float64(t / Float64(z - y)));
	elseif (t_1 <= 2000.0)
		tmp = fma(t, Float64(Float64(z - x) / y), t);
	else
		tmp = Float64(t * Float64(x / Float64(z - y)));
	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, 1e-7], N[(N[(x - y), $MachinePrecision] * N[(t / N[(z - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2000.0], N[(t * N[(N[(z - x), $MachinePrecision] / y), $MachinePrecision] + t), $MachinePrecision], N[(t * N[(x / N[(z - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 x y) (-.f64 z y)) < 9.9999999999999995e-8

   1. Initial program 94.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 91.7

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

   4. Applied rewrites 91.7%

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

   if 9.9999999999999995e-8 < (/.f64 (-.f64 x y) (-.f64 z y)) < 2e3

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf

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

      1. associate--l+ N/A

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

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

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

      3. div-sub N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

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

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

      7. associate-/l* N/A

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

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

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

      9. mul-1-neg N/A

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

      10. lower-fma.f64 N/A

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

   5. Applied rewrites 99.3%

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

   if 2e3 < (/.f64 (-.f64 x y) (-.f64 z y))

   1. Initial program 99.5%

      \[\frac{x - y}{z - y} \cdot t \]
   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. lower-/.f64 N/A

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

      2. lower--.f64 99.5

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

   5. Applied rewrites 99.5%

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

4. Final simplification 95.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{z - y} \leq 10^{-7}:\\ \;\;\;\;\left(x - y\right) \cdot \frac{t}{z - y}\\ \mathbf{elif}\;\frac{x - y}{z - y} \leq 2000:\\ \;\;\;\;\mathsf{fma}\left(t, \frac{z - x}{y}, t\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{x}{z - y}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 79.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x - y}{z - y}\\ t_2 := \frac{x}{-y}\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+23}:\\ \;\;\;\;t \cdot t\_2\\ \mathbf{elif}\;t\_1 \leq 10^{-7}:\\ \;\;\;\;\left(x - y\right) \cdot \frac{t}{z}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t, t\_2, t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (- x y) (- z y))) (t_2 (/ x (- y))))
   (if (<= t_1 -5e+23)
     (* t t_2)
     (if (<= t_1 1e-7) (* (- x y) (/ t z)) (fma t t_2 t)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x - y) / (z - y);
	double t_2 = x / -y;
	double tmp;
	if (t_1 <= -5e+23) {
		tmp = t * t_2;
	} else if (t_1 <= 1e-7) {
		tmp = (x - y) * (t / z);
	} else {
		tmp = fma(t, t_2, t);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x - y) / Float64(z - y))
	t_2 = Float64(x / Float64(-y))
	tmp = 0.0
	if (t_1 <= -5e+23)
		tmp = Float64(t * t_2);
	elseif (t_1 <= 1e-7)
		tmp = Float64(Float64(x - y) * Float64(t / z));
	else
		tmp = fma(t, t_2, t);
	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[(x / (-y)), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+23], N[(t * t$95$2), $MachinePrecision], If[LessEqual[t$95$1, 1e-7], N[(N[(x - y), $MachinePrecision] * N[(t / z), $MachinePrecision]), $MachinePrecision], N[(t * t$95$2 + t), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 x y) (-.f64 z y)) < -4.9999999999999999e23

   1. Initial program 94.6%

      \[\frac{x - y}{z - y} \cdot t \]
   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. lower-/.f64 N/A

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

      2. lower--.f64 94.6

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

   5. Applied rewrites 94.6%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 66.0%

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

   if -4.9999999999999999e23 < (/.f64 (-.f64 x y) (-.f64 z y)) < 9.9999999999999995e-8

   1. Initial program 93.9%

      \[\frac{x - y}{z - y} \cdot t \]
   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. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 88.3

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

   5. Applied rewrites 88.3%

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

   if 9.9999999999999995e-8 < (/.f64 (-.f64 x y) (-.f64 z y))

   1. Initial program 99.8%

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

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

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

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

      10. *-commutative N/A

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

      11. neg-mul-1 N/A

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

      12. remove-double-neg N/A

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

      13. neg-mul-1 N/A

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

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

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

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

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

      16. mul-1-neg N/A

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

      17. lower-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. lower-/.f64 N/A

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

      21. lower-neg.f64 84.7

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

   5. Applied rewrites 84.7%

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

4. Final simplification 83.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{z - y} \leq -5 \cdot 10^{+23}:\\ \;\;\;\;t \cdot \frac{x}{-y}\\ \mathbf{elif}\;\frac{x - y}{z - y} \leq 10^{-7}:\\ \;\;\;\;\left(x - y\right) \cdot \frac{t}{z}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t, \frac{x}{-y}, t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 69.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x - y}{z - y}\\ \mathbf{if}\;t\_1 \leq 10^{-7}:\\ \;\;\;\;t \cdot \frac{x}{z}\\ \mathbf{elif}\;t\_1 \leq 2000:\\ \;\;\;\;\mathsf{fma}\left(t, \frac{z}{y}, t\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot t}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (- x y) (- z y))))
   (if (<= t_1 1e-7)
     (* t (/ x z))
     (if (<= t_1 2000.0) (fma t (/ z y) t) (/ (* x t) z)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x - y) / (z - y);
	double tmp;
	if (t_1 <= 1e-7) {
		tmp = t * (x / z);
	} else if (t_1 <= 2000.0) {
		tmp = fma(t, (z / y), t);
	} else {
		tmp = (x * t) / 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 <= 1e-7)
		tmp = Float64(t * Float64(x / z));
	elseif (t_1 <= 2000.0)
		tmp = fma(t, Float64(z / y), t);
	else
		tmp = Float64(Float64(x * t) / 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, 1e-7], N[(t * N[(x / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2000.0], N[(t * N[(z / y), $MachinePrecision] + t), $MachinePrecision], N[(N[(x * t), $MachinePrecision] / z), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 x y) (-.f64 z y)) < 9.9999999999999995e-8

   1. Initial program 94.1%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 53.7

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

   5. Applied rewrites 53.7%

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

   if 9.9999999999999995e-8 < (/.f64 (-.f64 x y) (-.f64 z y)) < 2e3

   1. Initial program 100.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 73.9

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

   4. Applied rewrites 73.9%

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

   5. Taylor expanded in y around inf

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

      1. associate--l+ N/A

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

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

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

      3. div-sub N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

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

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

      7. associate-/l* N/A

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

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

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

      9. mul-1-neg N/A

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

      10. lower-fma.f64 N/A

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

   7. Applied rewrites 99.3%

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

   8. Taylor expanded in z around inf

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

   10. Applied rewrites 96.7%

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

   if 2e3 < (/.f64 (-.f64 x y) (-.f64 z y))

   1. Initial program 99.5%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 51.1

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

   5. Applied rewrites 51.1%

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

4. Final simplification 68.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{z - y} \leq 10^{-7}:\\ \;\;\;\;t \cdot \frac{x}{z}\\ \mathbf{elif}\;\frac{x - y}{z - y} \leq 2000:\\ \;\;\;\;\mathsf{fma}\left(t, \frac{z}{y}, t\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot t}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 69.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x - y}{z - y}\\ \mathbf{if}\;t\_1 \leq 10^{-7}:\\ \;\;\;\;t \cdot \frac{x}{z}\\ \mathbf{elif}\;t\_1 \leq 2000:\\ \;\;\;\;t \cdot 1\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot t}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (- x y) (- z y))))
   (if (<= t_1 1e-7)
     (* t (/ x z))
     (if (<= t_1 2000.0) (* t 1.0) (/ (* x t) z)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x - y) / (z - y);
	double tmp;
	if (t_1 <= 1e-7) {
		tmp = t * (x / z);
	} else if (t_1 <= 2000.0) {
		tmp = t * 1.0;
	} else {
		tmp = (x * t) / 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) :: t_1
    real(8) :: tmp
    t_1 = (x - y) / (z - y)
    if (t_1 <= 1d-7) then
        tmp = t * (x / z)
    else if (t_1 <= 2000.0d0) then
        tmp = t * 1.0d0
    else
        tmp = (x * t) / z
    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-7) {
		tmp = t * (x / z);
	} else if (t_1 <= 2000.0) {
		tmp = t * 1.0;
	} else {
		tmp = (x * t) / z;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (x - y) / (z - y)
	tmp = 0
	if t_1 <= 1e-7:
		tmp = t * (x / z)
	elif t_1 <= 2000.0:
		tmp = t * 1.0
	else:
		tmp = (x * t) / 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 <= 1e-7)
		tmp = Float64(t * Float64(x / z));
	elseif (t_1 <= 2000.0)
		tmp = Float64(t * 1.0);
	else
		tmp = Float64(Float64(x * t) / z);
	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-7)
		tmp = t * (x / z);
	elseif (t_1 <= 2000.0)
		tmp = t * 1.0;
	else
		tmp = (x * t) / z;
	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-7], N[(t * N[(x / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2000.0], N[(t * 1.0), $MachinePrecision], N[(N[(x * t), $MachinePrecision] / z), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 x y) (-.f64 z y)) < 9.9999999999999995e-8

   1. Initial program 94.1%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 53.7

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

   5. Applied rewrites 53.7%

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

   if 9.9999999999999995e-8 < (/.f64 (-.f64 x y) (-.f64 z y)) < 2e3

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf

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

   5. Applied rewrites 96.6%

      \[\leadsto \color{blue}{1} \cdot t \]

   if 2e3 < (/.f64 (-.f64 x y) (-.f64 z y))

   1. Initial program 99.5%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 51.1

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

   5. Applied rewrites 51.1%

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

4. Final simplification 67.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{z - y} \leq 10^{-7}:\\ \;\;\;\;t \cdot \frac{x}{z}\\ \mathbf{elif}\;\frac{x - y}{z - y} \leq 2000:\\ \;\;\;\;t \cdot 1\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot t}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 68.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x - y}{z - y}\\ t_2 := \frac{x \cdot t}{z}\\ \mathbf{if}\;t\_1 \leq 10^{-7}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 2000:\\ \;\;\;\;t \cdot 1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (- x y) (- z y))) (t_2 (/ (* x t) z)))
   (if (<= t_1 1e-7) t_2 (if (<= t_1 2000.0) (* t 1.0) 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 <= 1e-7) {
		tmp = t_2;
	} else if (t_1 <= 2000.0) {
		tmp = t * 1.0;
	} 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 <= 1d-7) then
        tmp = t_2
    else if (t_1 <= 2000.0d0) then
        tmp = t * 1.0d0
    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 <= 1e-7) {
		tmp = t_2;
	} else if (t_1 <= 2000.0) {
		tmp = t * 1.0;
	} 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 <= 1e-7:
		tmp = t_2
	elif t_1 <= 2000.0:
		tmp = t * 1.0
	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(Float64(x * t) / z)
	tmp = 0.0
	if (t_1 <= 1e-7)
		tmp = t_2;
	elseif (t_1 <= 2000.0)
		tmp = Float64(t * 1.0);
	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 <= 1e-7)
		tmp = t_2;
	elseif (t_1 <= 2000.0)
		tmp = t * 1.0;
	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[(N[(x * t), $MachinePrecision] / z), $MachinePrecision]}, If[LessEqual[t$95$1, 1e-7], t$95$2, If[LessEqual[t$95$1, 2000.0], N[(t * 1.0), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 x y) (-.f64 z y)) < 9.9999999999999995e-8 or 2e3 < (/.f64 (-.f64 x y) (-.f64 z y))

   1. Initial program 95.3%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 51.5

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

   5. Applied rewrites 51.5%

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

   if 9.9999999999999995e-8 < (/.f64 (-.f64 x y) (-.f64 z y)) < 2e3

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf

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

   5. Applied rewrites 96.6%

      \[\leadsto \color{blue}{1} \cdot t \]
3. Recombined 2 regimes into one program.

4. Final simplification 66.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{z - y} \leq 10^{-7}:\\ \;\;\;\;\frac{x \cdot t}{z}\\ \mathbf{elif}\;\frac{x - y}{z - y} \leq 2000:\\ \;\;\;\;t \cdot 1\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot t}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 35.0% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ t \cdot 1 \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (* t 1.0))

C

double code(double x, double y, double z, double t) {
	return t * 1.0;
}

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 = t * 1.0d0
end function

Java

public static double code(double x, double y, double z, double t) {
	return t * 1.0;
}

Python

def code(x, y, z, t):
	return t * 1.0

Julia

function code(x, y, z, t)
	return Float64(t * 1.0)
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = t * 1.0;
end

Wolfram

code[x_, y_, z_, t_] := N[(t * 1.0), $MachinePrecision]

Derivation

1. Initial program 96.9%

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

3. Taylor expanded in y around inf

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

5. Applied rewrites 35.5%

   \[\leadsto \color{blue}{1} \cdot t \]

6. Final simplification 35.5%

   \[\leadsto t \cdot 1 \]
7. 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 2024221 
(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))