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

Percentage Accurate: 97.2% → 97.2%

Time: 7.1s

Alternatives: 15

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: 97.2% 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: 97.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

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 * ((y - x) / (y - z))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.0%

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

3. Final simplification 98.0%

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

Alternative 2: 70.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{y - x}{y - z}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{-7}:\\ \;\;\;\;\frac{x}{z} \cdot t\\ \mathbf{elif}\;t\_1 \leq -2 \cdot 10^{-237}:\\ \;\;\;\;\frac{y}{-z} \cdot t\\ \mathbf{elif}\;t\_1 \leq 10^{-7}:\\ \;\;\;\;\frac{t}{z} \cdot x\\ \mathbf{elif}\;t\_1 \leq 2:\\ \;\;\;\;\mathsf{fma}\left(t, \frac{z}{y}, t\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t \cdot x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (- y x) (- y z))))
   (if (<= t_1 -1e-7)
     (* (/ x z) t)
     (if (<= t_1 -2e-237)
       (* (/ y (- z)) t)
       (if (<= t_1 1e-7)
         (* (/ t z) x)
         (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 = (y - x) / (y - z);
	double tmp;
	if (t_1 <= -1e-7) {
		tmp = (x / z) * t;
	} else if (t_1 <= -2e-237) {
		tmp = (y / -z) * t;
	} else if (t_1 <= 1e-7) {
		tmp = (t / z) * x;
	} 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(y - x) / Float64(y - z))
	tmp = 0.0
	if (t_1 <= -1e-7)
		tmp = Float64(Float64(x / z) * t);
	elseif (t_1 <= -2e-237)
		tmp = Float64(Float64(y / Float64(-z)) * t);
	elseif (t_1 <= 1e-7)
		tmp = Float64(Float64(t / z) * x);
	elseif (t_1 <= 2.0)
		tmp = fma(t, Float64(z / y), t);
	else
		tmp = Float64(Float64(t * x) / z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(y - x), $MachinePrecision] / N[(y - z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e-7], N[(N[(x / z), $MachinePrecision] * t), $MachinePrecision], If[LessEqual[t$95$1, -2e-237], N[(N[(y / (-z)), $MachinePrecision] * t), $MachinePrecision], If[LessEqual[t$95$1, 1e-7], N[(N[(t / z), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[t$95$1, 2.0], N[(t * N[(z / y), $MachinePrecision] + t), $MachinePrecision], N[(N[(t * x), $MachinePrecision] / z), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

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

   1. Initial program 99.6%

      \[\frac{x - y}{z - y} \cdot t \]
   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 64.2

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

   5. Applied rewrites 64.2%

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

   if -9.9999999999999995e-8 < (/.f64 (-.f64 x y) (-.f64 z y)) < -2e-237

   1. Initial program 99.7%

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

   4. Applied rewrites 99.8%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r/ N/A

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

      4. *-commutative N/A

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

      5. clear-num N/A

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

      6. un-div-inv N/A

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

      7. lower-/.f64 N/A

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

      8. frac-2neg N/A

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

      9. lift-neg.f64 N/A

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

      10. remove-double-neg N/A

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

      11. lower-/.f64 N/A

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

      12. neg-sub0 N/A

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

      13. lift--.f64 N/A

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

      14. sub-neg N/A

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

      15. +-commutative N/A

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

      16. associate--r+ N/A

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

      17. neg-sub0 N/A

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

      18. remove-double-neg N/A

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

      19. lower--.f64 87.3

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

   6. Applied rewrites 87.3%

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

   7. Taylor expanded in x around 0

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower--.f64 84.7

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

   9. Applied rewrites 84.7%

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

   10. Taylor expanded in y around 0

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

   12. Applied rewrites 84.7%

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

   if -2e-237 < (/.f64 (-.f64 x y) (-.f64 z y)) < 9.9999999999999995e-8

   1. Initial program 92.4%

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

   3. Taylor expanded in x around inf

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

      1. associate-*l/ N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower--.f64 71.4

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

   5. Applied rewrites 71.4%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 71.4%

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

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

   1. Initial program 99.9%

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

   4. Applied rewrites 100.0%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r/ N/A

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

      4. *-commutative N/A

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

      5. clear-num N/A

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

      6. un-div-inv N/A

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

      7. lower-/.f64 N/A

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

      8. frac-2neg N/A

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

      9. lift-neg.f64 N/A

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

      10. remove-double-neg N/A

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

      11. lower-/.f64 N/A

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

      12. neg-sub0 N/A

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

      13. lift--.f64 N/A

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

      14. sub-neg N/A

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

      15. +-commutative N/A

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

      16. associate--r+ N/A

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

      17. neg-sub0 N/A

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

      18. remove-double-neg N/A

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

      19. lower--.f64 76.0

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

   6. Applied rewrites 76.0%

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

   7. Taylor expanded in x around 0

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower--.f64 98.6

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

   9. Applied rewrites 98.6%

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

   10. Taylor expanded in y around inf

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

   12. Applied rewrites 98.7%

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

   if 2 < (/.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 59.7

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

   5. Applied rewrites 59.7%

      \[\leadsto \color{blue}{\frac{t \cdot x}{z}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 78.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y - x}{y - z} \leq -1 \cdot 10^{-7}:\\ \;\;\;\;\frac{x}{z} \cdot t\\ \mathbf{elif}\;\frac{y - x}{y - z} \leq -2 \cdot 10^{-237}:\\ \;\;\;\;\frac{y}{-z} \cdot t\\ \mathbf{elif}\;\frac{y - x}{y - z} \leq 10^{-7}:\\ \;\;\;\;\frac{t}{z} \cdot x\\ \mathbf{elif}\;\frac{y - x}{y - z} \leq 2:\\ \;\;\;\;\mathsf{fma}\left(t, \frac{z}{y}, t\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t \cdot x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 69.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{y - x}{y - z}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{-7}:\\ \;\;\;\;\frac{x}{z} \cdot t\\ \mathbf{elif}\;t\_1 \leq -2 \cdot 10^{-237}:\\ \;\;\;\;\left(-y\right) \cdot \frac{t}{z}\\ \mathbf{elif}\;t\_1 \leq 10^{-7}:\\ \;\;\;\;\frac{t}{z} \cdot x\\ \mathbf{elif}\;t\_1 \leq 2:\\ \;\;\;\;\mathsf{fma}\left(t, \frac{z}{y}, t\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t \cdot x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (- y x) (- y z))))
   (if (<= t_1 -1e-7)
     (* (/ x z) t)
     (if (<= t_1 -2e-237)
       (* (- y) (/ t z))
       (if (<= t_1 1e-7)
         (* (/ t z) x)
         (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 = (y - x) / (y - z);
	double tmp;
	if (t_1 <= -1e-7) {
		tmp = (x / z) * t;
	} else if (t_1 <= -2e-237) {
		tmp = -y * (t / z);
	} else if (t_1 <= 1e-7) {
		tmp = (t / z) * x;
	} 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(y - x) / Float64(y - z))
	tmp = 0.0
	if (t_1 <= -1e-7)
		tmp = Float64(Float64(x / z) * t);
	elseif (t_1 <= -2e-237)
		tmp = Float64(Float64(-y) * Float64(t / z));
	elseif (t_1 <= 1e-7)
		tmp = Float64(Float64(t / z) * x);
	elseif (t_1 <= 2.0)
		tmp = fma(t, Float64(z / y), t);
	else
		tmp = Float64(Float64(t * x) / z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(y - x), $MachinePrecision] / N[(y - z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e-7], N[(N[(x / z), $MachinePrecision] * t), $MachinePrecision], If[LessEqual[t$95$1, -2e-237], N[((-y) * N[(t / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e-7], N[(N[(t / z), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[t$95$1, 2.0], N[(t * N[(z / y), $MachinePrecision] + t), $MachinePrecision], N[(N[(t * x), $MachinePrecision] / z), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

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

   1. Initial program 99.6%

      \[\frac{x - y}{z - y} \cdot t \]
   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 64.2

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

   5. Applied rewrites 64.2%

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

   if -9.9999999999999995e-8 < (/.f64 (-.f64 x y) (-.f64 z y)) < -2e-237

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0

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

      1. associate-*r/ N/A

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

      2. div-add-rev N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. mul-1-neg N/A

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

      6. *-commutative N/A

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

      7. distribute-rgt-in N/A

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

      8. +-commutative N/A

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

      9. sub-neg N/A

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

      10. *-commutative N/A

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

      11. associate-/l* N/A

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

      12. lower-*.f64 N/A

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

      13. lower--.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower--.f64 87.3

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

   5. Applied rewrites 87.3%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 87.3%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 75.6%

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

   if -2e-237 < (/.f64 (-.f64 x y) (-.f64 z y)) < 9.9999999999999995e-8

   1. Initial program 92.4%

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

   3. Taylor expanded in x around inf

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

      1. associate-*l/ N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower--.f64 71.4

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

   5. Applied rewrites 71.4%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 71.4%

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

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

   1. Initial program 99.9%

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

   4. Applied rewrites 100.0%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r/ N/A

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

      4. *-commutative N/A

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

      5. clear-num N/A

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

      6. un-div-inv N/A

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

      7. lower-/.f64 N/A

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

      8. frac-2neg N/A

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

      9. lift-neg.f64 N/A

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

      10. remove-double-neg N/A

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

      11. lower-/.f64 N/A

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

      12. neg-sub0 N/A

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

      13. lift--.f64 N/A

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

      14. sub-neg N/A

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

      15. +-commutative N/A

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

      16. associate--r+ N/A

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

      17. neg-sub0 N/A

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

      18. remove-double-neg N/A

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

      19. lower--.f64 76.0

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

   6. Applied rewrites 76.0%

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

   7. Taylor expanded in x around 0

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower--.f64 98.6

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

   9. Applied rewrites 98.6%

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

   10. Taylor expanded in y around inf

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

   12. Applied rewrites 98.7%

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

   if 2 < (/.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 59.7

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

   5. Applied rewrites 59.7%

      \[\leadsto \color{blue}{\frac{t \cdot x}{z}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 77.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y - x}{y - z} \leq -1 \cdot 10^{-7}:\\ \;\;\;\;\frac{x}{z} \cdot t\\ \mathbf{elif}\;\frac{y - x}{y - z} \leq -2 \cdot 10^{-237}:\\ \;\;\;\;\left(-y\right) \cdot \frac{t}{z}\\ \mathbf{elif}\;\frac{y - x}{y - z} \leq 10^{-7}:\\ \;\;\;\;\frac{t}{z} \cdot x\\ \mathbf{elif}\;\frac{y - x}{y - z} \leq 2:\\ \;\;\;\;\mathsf{fma}\left(t, \frac{z}{y}, t\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t \cdot x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 95.7% accurate, 0.2× speedup

Math

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

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (- y x) (- y z))) (t_2 (* (/ x (- z y)) t)))
   (if (<= t_1 -10.0)
     t_2
     (if (<= t_1 1e-7)
       (* (/ (- x y) z) t)
       (if (<= t_1 2.0) (fma (- t) (/ (- x z) y) t) t_2)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (y - x) / (y - z);
	double t_2 = (x / (z - y)) * t;
	double tmp;
	if (t_1 <= -10.0) {
		tmp = t_2;
	} else if (t_1 <= 1e-7) {
		tmp = ((x - y) / z) * t;
	} else if (t_1 <= 2.0) {
		tmp = fma(-t, ((x - z) / y), t);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(y - x) / Float64(y - z))
	t_2 = Float64(Float64(x / Float64(z - y)) * t)
	tmp = 0.0
	if (t_1 <= -10.0)
		tmp = t_2;
	elseif (t_1 <= 1e-7)
		tmp = Float64(Float64(Float64(x - y) / z) * t);
	elseif (t_1 <= 2.0)
		tmp = fma(Float64(-t), Float64(Float64(x - z) / y), t);
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(y - x), $MachinePrecision] / N[(y - z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x / N[(z - y), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]}, If[LessEqual[t$95$1, -10.0], t$95$2, If[LessEqual[t$95$1, 1e-7], N[(N[(N[(x - y), $MachinePrecision] / z), $MachinePrecision] * t), $MachinePrecision], If[LessEqual[t$95$1, 2.0], N[((-t) * N[(N[(x - z), $MachinePrecision] / y), $MachinePrecision] + t), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 x y) (-.f64 z y)) < -10 or 2 < (/.f64 (-.f64 x y) (-.f64 z y))

   1. Initial program 99.6%

      \[\frac{x - y}{z - y} \cdot t \]
   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 97.8

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

   5. Applied rewrites 97.8%

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

   if -10 < (/.f64 (-.f64 x y) (-.f64 z y)) < 9.9999999999999995e-8

   1. Initial program 94.7%

      \[\frac{x - y}{z - y} \cdot t \]
   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 93.9

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

   5. Applied rewrites 93.9%

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

   if 9.9999999999999995e-8 < (/.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

      \[\leadsto \color{blue}{\left(t + -1 \cdot \frac{t \cdot x}{y}\right) - -1 \cdot \frac{t \cdot z}{y}} \]
   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-lft-neg-in N/A

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

      9. neg-mul-1 N/A

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

      10. lower-fma.f64 N/A

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

      11. neg-mul-1 N/A

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

      12. lower-neg.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower--.f64 100.0

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

   5. Applied rewrites 100.0%

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

4. Final simplification 97.1%

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

Alternative 5: 95.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{y - x}{y - z}\\ t_2 := \frac{x}{z - y} \cdot t\\ \mathbf{if}\;t\_1 \leq -10:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-17}:\\ \;\;\;\;\frac{x - y}{z} \cdot t\\ \mathbf{elif}\;t\_1 \leq 2:\\ \;\;\;\;\frac{y}{y - z} \cdot t\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (- y x) (- y z))) (t_2 (* (/ x (- z y)) t)))
   (if (<= t_1 -10.0)
     t_2
     (if (<= t_1 5e-17)
       (* (/ (- x y) z) t)
       (if (<= t_1 2.0) (* (/ y (- y z)) t) t_2)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (y - x) / (y - z);
	double t_2 = (x / (z - y)) * t;
	double tmp;
	if (t_1 <= -10.0) {
		tmp = t_2;
	} else if (t_1 <= 5e-17) {
		tmp = ((x - y) / z) * t;
	} else if (t_1 <= 2.0) {
		tmp = (y / (y - z)) * 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 = (y - x) / (y - z)
    t_2 = (x / (z - y)) * t
    if (t_1 <= (-10.0d0)) then
        tmp = t_2
    else if (t_1 <= 5d-17) then
        tmp = ((x - y) / z) * t
    else if (t_1 <= 2.0d0) then
        tmp = (y / (y - z)) * 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 = (y - x) / (y - z);
	double t_2 = (x / (z - y)) * t;
	double tmp;
	if (t_1 <= -10.0) {
		tmp = t_2;
	} else if (t_1 <= 5e-17) {
		tmp = ((x - y) / z) * t;
	} else if (t_1 <= 2.0) {
		tmp = (y / (y - z)) * t;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (y - x) / (y - z)
	t_2 = (x / (z - y)) * t
	tmp = 0
	if t_1 <= -10.0:
		tmp = t_2
	elif t_1 <= 5e-17:
		tmp = ((x - y) / z) * t
	elif t_1 <= 2.0:
		tmp = (y / (y - z)) * t
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(y - x) / Float64(y - z))
	t_2 = Float64(Float64(x / Float64(z - y)) * t)
	tmp = 0.0
	if (t_1 <= -10.0)
		tmp = t_2;
	elseif (t_1 <= 5e-17)
		tmp = Float64(Float64(Float64(x - y) / z) * t);
	elseif (t_1 <= 2.0)
		tmp = Float64(Float64(y / Float64(y - z)) * t);
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (y - x) / (y - z);
	t_2 = (x / (z - y)) * t;
	tmp = 0.0;
	if (t_1 <= -10.0)
		tmp = t_2;
	elseif (t_1 <= 5e-17)
		tmp = ((x - y) / z) * t;
	elseif (t_1 <= 2.0)
		tmp = (y / (y - z)) * t;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(y - x), $MachinePrecision] / N[(y - z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x / N[(z - y), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]}, If[LessEqual[t$95$1, -10.0], t$95$2, If[LessEqual[t$95$1, 5e-17], N[(N[(N[(x - y), $MachinePrecision] / z), $MachinePrecision] * t), $MachinePrecision], If[LessEqual[t$95$1, 2.0], N[(N[(y / N[(y - z), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 x y) (-.f64 z y)) < -10 or 2 < (/.f64 (-.f64 x y) (-.f64 z y))

   1. Initial program 99.6%

      \[\frac{x - y}{z - y} \cdot t \]
   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 97.8

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

   5. Applied rewrites 97.8%

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

   if -10 < (/.f64 (-.f64 x y) (-.f64 z y)) < 4.9999999999999999e-17

   1. Initial program 94.6%

      \[\frac{x - y}{z - y} \cdot t \]
   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 94.6

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

   5. Applied rewrites 94.6%

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

   if 4.9999999999999999e-17 < (/.f64 (-.f64 x y) (-.f64 z y)) < 2

   1. Initial program 99.9%

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

   4. Applied rewrites 100.0%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r/ N/A

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

      4. *-commutative N/A

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

      5. clear-num N/A

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

      6. un-div-inv N/A

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

      7. lower-/.f64 N/A

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

      8. frac-2neg N/A

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

      9. lift-neg.f64 N/A

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

      10. remove-double-neg N/A

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

      11. lower-/.f64 N/A

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

      12. neg-sub0 N/A

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

      13. lift--.f64 N/A

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

      14. sub-neg N/A

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

      15. +-commutative N/A

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

      16. associate--r+ N/A

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

      17. neg-sub0 N/A

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

      18. remove-double-neg N/A

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

      19. lower--.f64 76.6

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

   6. Applied rewrites 76.6%

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

   7. Taylor expanded in x around 0

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower--.f64 98.7

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

   9. Applied rewrites 98.7%

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

4. Final simplification 97.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y - x}{y - z} \leq -10:\\ \;\;\;\;\frac{x}{z - y} \cdot t\\ \mathbf{elif}\;\frac{y - x}{y - z} \leq 5 \cdot 10^{-17}:\\ \;\;\;\;\frac{x - y}{z} \cdot t\\ \mathbf{elif}\;\frac{y - x}{y - z} \leq 2:\\ \;\;\;\;\frac{y}{y - z} \cdot t\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z - y} \cdot t\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 93.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{y - x}{y - z}\\ t_2 := \frac{x}{z - y} \cdot t\\ \mathbf{if}\;t\_1 \leq -10:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-17}:\\ \;\;\;\;\frac{t}{z} \cdot \left(x - y\right)\\ \mathbf{elif}\;t\_1 \leq 2:\\ \;\;\;\;\frac{y}{y - z} \cdot t\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (- y x) (- y z))) (t_2 (* (/ x (- z y)) t)))
   (if (<= t_1 -10.0)
     t_2
     (if (<= t_1 5e-17)
       (* (/ t z) (- x y))
       (if (<= t_1 2.0) (* (/ y (- y z)) t) t_2)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (y - x) / (y - z);
	double t_2 = (x / (z - y)) * t;
	double tmp;
	if (t_1 <= -10.0) {
		tmp = t_2;
	} else if (t_1 <= 5e-17) {
		tmp = (t / z) * (x - y);
	} else if (t_1 <= 2.0) {
		tmp = (y / (y - z)) * 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 = (y - x) / (y - z)
    t_2 = (x / (z - y)) * t
    if (t_1 <= (-10.0d0)) then
        tmp = t_2
    else if (t_1 <= 5d-17) then
        tmp = (t / z) * (x - y)
    else if (t_1 <= 2.0d0) then
        tmp = (y / (y - z)) * 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 = (y - x) / (y - z);
	double t_2 = (x / (z - y)) * t;
	double tmp;
	if (t_1 <= -10.0) {
		tmp = t_2;
	} else if (t_1 <= 5e-17) {
		tmp = (t / z) * (x - y);
	} else if (t_1 <= 2.0) {
		tmp = (y / (y - z)) * t;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (y - x) / (y - z)
	t_2 = (x / (z - y)) * t
	tmp = 0
	if t_1 <= -10.0:
		tmp = t_2
	elif t_1 <= 5e-17:
		tmp = (t / z) * (x - y)
	elif t_1 <= 2.0:
		tmp = (y / (y - z)) * t
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(y - x) / Float64(y - z))
	t_2 = Float64(Float64(x / Float64(z - y)) * t)
	tmp = 0.0
	if (t_1 <= -10.0)
		tmp = t_2;
	elseif (t_1 <= 5e-17)
		tmp = Float64(Float64(t / z) * Float64(x - y));
	elseif (t_1 <= 2.0)
		tmp = Float64(Float64(y / Float64(y - z)) * t);
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (y - x) / (y - z);
	t_2 = (x / (z - y)) * t;
	tmp = 0.0;
	if (t_1 <= -10.0)
		tmp = t_2;
	elseif (t_1 <= 5e-17)
		tmp = (t / z) * (x - y);
	elseif (t_1 <= 2.0)
		tmp = (y / (y - z)) * t;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(y - x), $MachinePrecision] / N[(y - z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x / N[(z - y), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]}, If[LessEqual[t$95$1, -10.0], t$95$2, If[LessEqual[t$95$1, 5e-17], N[(N[(t / z), $MachinePrecision] * N[(x - y), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2.0], N[(N[(y / N[(y - z), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 x y) (-.f64 z y)) < -10 or 2 < (/.f64 (-.f64 x y) (-.f64 z y))

   1. Initial program 99.6%

      \[\frac{x - y}{z - y} \cdot t \]
   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 97.8

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

   5. Applied rewrites 97.8%

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

   if -10 < (/.f64 (-.f64 x y) (-.f64 z y)) < 4.9999999999999999e-17

   1. Initial program 94.6%

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

   3. Taylor expanded in x around 0

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

      1. associate-*r/ N/A

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

      2. div-add-rev N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. mul-1-neg N/A

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

      6. *-commutative N/A

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

      7. distribute-rgt-in N/A

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

      8. +-commutative N/A

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

      9. sub-neg N/A

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

      10. *-commutative N/A

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

      11. associate-/l* N/A

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

      12. lower-*.f64 N/A

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

      13. lower--.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower--.f64 90.4

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

   5. Applied rewrites 90.4%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 90.4%

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

   if 4.9999999999999999e-17 < (/.f64 (-.f64 x y) (-.f64 z y)) < 2

   1. Initial program 99.9%

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

   4. Applied rewrites 100.0%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r/ N/A

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

      4. *-commutative N/A

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

      5. clear-num N/A

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

      6. un-div-inv N/A

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

      7. lower-/.f64 N/A

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

      8. frac-2neg N/A

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

      9. lift-neg.f64 N/A

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

      10. remove-double-neg N/A

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

      11. lower-/.f64 N/A

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

      12. neg-sub0 N/A

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

      13. lift--.f64 N/A

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

      14. sub-neg N/A

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

      15. +-commutative N/A

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

      16. associate--r+ N/A

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

      17. neg-sub0 N/A

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

      18. remove-double-neg N/A

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

      19. lower--.f64 76.6

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

   6. Applied rewrites 76.6%

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

   7. Taylor expanded in x around 0

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower--.f64 98.7

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

   9. Applied rewrites 98.7%

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

4. Final simplification 95.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y - x}{y - z} \leq -10:\\ \;\;\;\;\frac{x}{z - y} \cdot t\\ \mathbf{elif}\;\frac{y - x}{y - z} \leq 5 \cdot 10^{-17}:\\ \;\;\;\;\frac{t}{z} \cdot \left(x - y\right)\\ \mathbf{elif}\;\frac{y - x}{y - z} \leq 2:\\ \;\;\;\;\frac{y}{y - z} \cdot t\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z - y} \cdot t\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 92.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{y - x}{y - z}\\ \mathbf{if}\;t\_1 \leq -10:\\ \;\;\;\;\frac{t}{z - y} \cdot x\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-17}:\\ \;\;\;\;\frac{t}{z} \cdot \left(x - y\right)\\ \mathbf{elif}\;t\_1 \leq 2:\\ \;\;\;\;\frac{y}{y - z} \cdot t\\ \mathbf{else}:\\ \;\;\;\;\frac{t \cdot x}{z - y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (- y x) (- y z))))
   (if (<= t_1 -10.0)
     (* (/ t (- z y)) x)
     (if (<= t_1 5e-17)
       (* (/ t z) (- x y))
       (if (<= t_1 2.0) (* (/ y (- y z)) t) (/ (* t x) (- z y)))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (y - x) / (y - z);
	double tmp;
	if (t_1 <= -10.0) {
		tmp = (t / (z - y)) * x;
	} else if (t_1 <= 5e-17) {
		tmp = (t / z) * (x - y);
	} else if (t_1 <= 2.0) {
		tmp = (y / (y - z)) * t;
	} 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 = (y - x) / (y - z)
    if (t_1 <= (-10.0d0)) then
        tmp = (t / (z - y)) * x
    else if (t_1 <= 5d-17) then
        tmp = (t / z) * (x - y)
    else if (t_1 <= 2.0d0) then
        tmp = (y / (y - z)) * t
    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 = (y - x) / (y - z);
	double tmp;
	if (t_1 <= -10.0) {
		tmp = (t / (z - y)) * x;
	} else if (t_1 <= 5e-17) {
		tmp = (t / z) * (x - y);
	} else if (t_1 <= 2.0) {
		tmp = (y / (y - z)) * t;
	} else {
		tmp = (t * x) / (z - y);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (y - x) / (y - z)
	tmp = 0
	if t_1 <= -10.0:
		tmp = (t / (z - y)) * x
	elif t_1 <= 5e-17:
		tmp = (t / z) * (x - y)
	elif t_1 <= 2.0:
		tmp = (y / (y - z)) * t
	else:
		tmp = (t * x) / (z - y)
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(y - x) / Float64(y - z))
	tmp = 0.0
	if (t_1 <= -10.0)
		tmp = Float64(Float64(t / Float64(z - y)) * x);
	elseif (t_1 <= 5e-17)
		tmp = Float64(Float64(t / z) * Float64(x - y));
	elseif (t_1 <= 2.0)
		tmp = Float64(Float64(y / Float64(y - z)) * t);
	else
		tmp = Float64(Float64(t * x) / Float64(z - y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (y - x) / (y - z);
	tmp = 0.0;
	if (t_1 <= -10.0)
		tmp = (t / (z - y)) * x;
	elseif (t_1 <= 5e-17)
		tmp = (t / z) * (x - y);
	elseif (t_1 <= 2.0)
		tmp = (y / (y - z)) * t;
	else
		tmp = (t * x) / (z - y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(y - x), $MachinePrecision] / N[(y - z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -10.0], N[(N[(t / N[(z - y), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[t$95$1, 5e-17], N[(N[(t / z), $MachinePrecision] * N[(x - y), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2.0], N[(N[(y / N[(y - z), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision], N[(N[(t * x), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (-.f64 x y) (-.f64 z y)) < -10

   1. Initial program 99.6%

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

   3. Taylor expanded in x around inf

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

      1. associate-*l/ N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower--.f64 80.7

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

   5. Applied rewrites 80.7%

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

   if -10 < (/.f64 (-.f64 x y) (-.f64 z y)) < 4.9999999999999999e-17

   1. Initial program 94.6%

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

   3. Taylor expanded in x around 0

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

      1. associate-*r/ N/A

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

      2. div-add-rev N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. mul-1-neg N/A

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

      6. *-commutative N/A

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

      7. distribute-rgt-in N/A

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

      8. +-commutative N/A

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

      9. sub-neg N/A

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

      10. *-commutative N/A

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

      11. associate-/l* N/A

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

      12. lower-*.f64 N/A

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

      13. lower--.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower--.f64 90.4

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

   5. Applied rewrites 90.4%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 90.4%

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

   if 4.9999999999999999e-17 < (/.f64 (-.f64 x y) (-.f64 z y)) < 2

   1. Initial program 99.9%

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

   4. Applied rewrites 100.0%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r/ N/A

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

      4. *-commutative N/A

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

      5. clear-num N/A

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

      6. un-div-inv N/A

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

      7. lower-/.f64 N/A

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

      8. frac-2neg N/A

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

      9. lift-neg.f64 N/A

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

      10. remove-double-neg N/A

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

      11. lower-/.f64 N/A

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

      12. neg-sub0 N/A

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

      13. lift--.f64 N/A

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

      14. sub-neg N/A

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

      15. +-commutative N/A

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

      16. associate--r+ N/A

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

      17. neg-sub0 N/A

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

      18. remove-double-neg N/A

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

      19. lower--.f64 76.6

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

   6. Applied rewrites 76.6%

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

   7. Taylor expanded in x around 0

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower--.f64 98.7

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

   9. Applied rewrites 98.7%

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

   if 2 < (/.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

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

      1. associate-*l/ N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower--.f64 93.2

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

   5. Applied rewrites 93.2%

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

   7. Applied rewrites 94.5%

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

4. Final simplification 91.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y - x}{y - z} \leq -10:\\ \;\;\;\;\frac{t}{z - y} \cdot x\\ \mathbf{elif}\;\frac{y - x}{y - z} \leq 5 \cdot 10^{-17}:\\ \;\;\;\;\frac{t}{z} \cdot \left(x - y\right)\\ \mathbf{elif}\;\frac{y - x}{y - z} \leq 2:\\ \;\;\;\;\frac{y}{y - z} \cdot t\\ \mathbf{else}:\\ \;\;\;\;\frac{t \cdot x}{z - y}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 92.0% accurate, 0.3× speedup

Math

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

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (- y x) (- y z))))
   (if (<= t_1 -10.0)
     (* (/ t (- z y)) x)
     (if (<= t_1 1e-7)
       (* (/ t z) (- x y))
       (if (<= t_1 2.0) (fma t (/ z y) t) (/ (* t x) (- z y)))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (y - x) / (y - z);
	double tmp;
	if (t_1 <= -10.0) {
		tmp = (t / (z - y)) * x;
	} else if (t_1 <= 1e-7) {
		tmp = (t / z) * (x - y);
	} else if (t_1 <= 2.0) {
		tmp = fma(t, (z / y), t);
	} else {
		tmp = (t * x) / (z - y);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(y - x) / Float64(y - z))
	tmp = 0.0
	if (t_1 <= -10.0)
		tmp = Float64(Float64(t / Float64(z - y)) * x);
	elseif (t_1 <= 1e-7)
		tmp = Float64(Float64(t / z) * Float64(x - y));
	elseif (t_1 <= 2.0)
		tmp = fma(t, Float64(z / y), t);
	else
		tmp = Float64(Float64(t * x) / Float64(z - y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(y - x), $MachinePrecision] / N[(y - z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -10.0], N[(N[(t / N[(z - y), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[t$95$1, 1e-7], N[(N[(t / z), $MachinePrecision] * N[(x - y), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2.0], N[(t * N[(z / y), $MachinePrecision] + t), $MachinePrecision], N[(N[(t * x), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (-.f64 x y) (-.f64 z y)) < -10

   1. Initial program 99.6%

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

   3. Taylor expanded in x around inf

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

      1. associate-*l/ N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower--.f64 80.7

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

   5. Applied rewrites 80.7%

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

   if -10 < (/.f64 (-.f64 x y) (-.f64 z y)) < 9.9999999999999995e-8

   1. Initial program 94.7%

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

   3. Taylor expanded in x around 0

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

      1. associate-*r/ N/A

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

      2. div-add-rev N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. mul-1-neg N/A

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

      6. *-commutative N/A

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

      7. distribute-rgt-in N/A

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

      8. +-commutative N/A

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

      9. sub-neg N/A

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

      10. *-commutative N/A

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

      11. associate-/l* N/A

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

      12. lower-*.f64 N/A

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

      13. lower--.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower--.f64 90.6

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

   5. Applied rewrites 90.6%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 89.9%

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

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

   1. Initial program 99.9%

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

   4. Applied rewrites 100.0%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r/ N/A

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

      4. *-commutative N/A

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

      5. clear-num N/A

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

      6. un-div-inv N/A

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

      7. lower-/.f64 N/A

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

      8. frac-2neg N/A

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

      9. lift-neg.f64 N/A

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

      10. remove-double-neg N/A

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

      11. lower-/.f64 N/A

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

      12. neg-sub0 N/A

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

      13. lift--.f64 N/A

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

      14. sub-neg N/A

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

      15. +-commutative N/A

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

      16. associate--r+ N/A

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

      17. neg-sub0 N/A

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

      18. remove-double-neg N/A

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

      19. lower--.f64 76.0

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

   6. Applied rewrites 76.0%

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

   7. Taylor expanded in x around 0

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower--.f64 98.6

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

   9. Applied rewrites 98.6%

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

   10. Taylor expanded in y around inf

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

   12. Applied rewrites 98.7%

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

   if 2 < (/.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

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

      1. associate-*l/ N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower--.f64 93.2

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

   5. Applied rewrites 93.2%

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

   7. Applied rewrites 94.5%

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

4. Final simplification 91.3%

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

Alternative 9: 92.0% accurate, 0.3× speedup

Math

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

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (- y x) (- y z))) (t_2 (* (/ t (- z y)) x)))
   (if (<= t_1 -10.0)
     t_2
     (if (<= t_1 1e-7)
       (* (/ t z) (- x y))
       (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 = (y - x) / (y - z);
	double t_2 = (t / (z - y)) * x;
	double tmp;
	if (t_1 <= -10.0) {
		tmp = t_2;
	} else if (t_1 <= 1e-7) {
		tmp = (t / z) * (x - y);
	} 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(y - x) / Float64(y - z))
	t_2 = Float64(Float64(t / Float64(z - y)) * x)
	tmp = 0.0
	if (t_1 <= -10.0)
		tmp = t_2;
	elseif (t_1 <= 1e-7)
		tmp = Float64(Float64(t / z) * Float64(x - y));
	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[(y - x), $MachinePrecision] / N[(y - z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t / N[(z - y), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[t$95$1, -10.0], t$95$2, If[LessEqual[t$95$1, 1e-7], N[(N[(t / z), $MachinePrecision] * N[(x - y), $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)) < -10 or 2 < (/.f64 (-.f64 x y) (-.f64 z y))

   1. Initial program 99.6%

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

   3. Taylor expanded in x around inf

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

      1. associate-*l/ N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower--.f64 85.7

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

   5. Applied rewrites 85.7%

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

   if -10 < (/.f64 (-.f64 x y) (-.f64 z y)) < 9.9999999999999995e-8

   1. Initial program 94.7%

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

   3. Taylor expanded in x around 0

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

      1. associate-*r/ N/A

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

      2. div-add-rev N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. mul-1-neg N/A

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

      6. *-commutative N/A

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

      7. distribute-rgt-in N/A

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

      8. +-commutative N/A

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

      9. sub-neg N/A

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

      10. *-commutative N/A

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

      11. associate-/l* N/A

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

      12. lower-*.f64 N/A

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

      13. lower--.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower--.f64 90.6

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

   5. Applied rewrites 90.6%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 89.9%

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

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

   1. Initial program 99.9%

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

   4. Applied rewrites 100.0%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r/ N/A

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

      4. *-commutative N/A

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

      5. clear-num N/A

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

      6. un-div-inv N/A

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

      7. lower-/.f64 N/A

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

      8. frac-2neg N/A

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

      9. lift-neg.f64 N/A

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

      10. remove-double-neg N/A

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

      11. lower-/.f64 N/A

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

      12. neg-sub0 N/A

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

      13. lift--.f64 N/A

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

      14. sub-neg N/A

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

      15. +-commutative N/A

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

      16. associate--r+ N/A

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

      17. neg-sub0 N/A

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

      18. remove-double-neg N/A

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

      19. lower--.f64 76.0

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

   6. Applied rewrites 76.0%

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

   7. Taylor expanded in x around 0

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower--.f64 98.6

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

   9. Applied rewrites 98.6%

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

   10. Taylor expanded in y around inf

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

   12. Applied rewrites 98.7%

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

4. Final simplification 91.1%

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

Alternative 10: 79.3% accurate, 0.4× speedup

Math

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

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (- y x) (- y z))))
   (if (<= t_1 1e-7)
     (* (/ t z) (- x y))
     (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 = (y - x) / (y - z);
	double tmp;
	if (t_1 <= 1e-7) {
		tmp = (t / z) * (x - y);
	} 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(y - x) / Float64(y - z))
	tmp = 0.0
	if (t_1 <= 1e-7)
		tmp = Float64(Float64(t / z) * Float64(x - y));
	elseif (t_1 <= 2.0)
		tmp = fma(t, Float64(z / y), t);
	else
		tmp = Float64(Float64(t * x) / z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(y - x), $MachinePrecision] / N[(y - z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 1e-7], N[(N[(t / z), $MachinePrecision] * N[(x - y), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2.0], N[(t * N[(z / y), $MachinePrecision] + t), $MachinePrecision], N[(N[(t * x), $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 96.5%

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

   3. Taylor expanded in x around 0

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

      1. associate-*r/ N/A

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

      2. div-add-rev N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. mul-1-neg N/A

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

      6. *-commutative N/A

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

      7. distribute-rgt-in N/A

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

      8. +-commutative N/A

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

      9. sub-neg N/A

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

      10. *-commutative N/A

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

      11. associate-/l* N/A

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

      12. lower-*.f64 N/A

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

      13. lower--.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower--.f64 88.0

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

   5. Applied rewrites 88.0%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 77.9%

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

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

   1. Initial program 99.9%

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

   4. Applied rewrites 100.0%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r/ N/A

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

      4. *-commutative N/A

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

      5. clear-num N/A

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

      6. un-div-inv N/A

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

      7. lower-/.f64 N/A

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

      8. frac-2neg N/A

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

      9. lift-neg.f64 N/A

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

      10. remove-double-neg N/A

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

      11. lower-/.f64 N/A

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

      12. neg-sub0 N/A

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

      13. lift--.f64 N/A

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

      14. sub-neg N/A

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

      15. +-commutative N/A

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

      16. associate--r+ N/A

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

      17. neg-sub0 N/A

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

      18. remove-double-neg N/A

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

      19. lower--.f64 76.0

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

   6. Applied rewrites 76.0%

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

   7. Taylor expanded in x around 0

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower--.f64 98.6

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

   9. Applied rewrites 98.6%

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

   10. Taylor expanded in y around inf

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

   12. Applied rewrites 98.7%

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

   if 2 < (/.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 59.7

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

   5. Applied rewrites 59.7%

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

4. Final simplification 81.7%

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

Alternative 11: 69.9% accurate, 0.4× speedup

Math

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

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (- y x) (- y z))))
   (if (<= t_1 1e-7)
     (* (/ x z) t)
     (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 = (y - x) / (y - z);
	double tmp;
	if (t_1 <= 1e-7) {
		tmp = (x / z) * t;
	} 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(y - x) / Float64(y - z))
	tmp = 0.0
	if (t_1 <= 1e-7)
		tmp = Float64(Float64(x / z) * t);
	elseif (t_1 <= 2.0)
		tmp = fma(t, Float64(z / y), t);
	else
		tmp = Float64(Float64(t * x) / z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(y - x), $MachinePrecision] / N[(y - z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 1e-7], N[(N[(x / z), $MachinePrecision] * t), $MachinePrecision], If[LessEqual[t$95$1, 2.0], N[(t * N[(z / y), $MachinePrecision] + t), $MachinePrecision], N[(N[(t * x), $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 96.5%

      \[\frac{x - y}{z - y} \cdot t \]
   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 61.2

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

   5. Applied rewrites 61.2%

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

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

   1. Initial program 99.9%

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

   4. Applied rewrites 100.0%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r/ N/A

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

      4. *-commutative N/A

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

      5. clear-num N/A

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

      6. un-div-inv N/A

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

      7. lower-/.f64 N/A

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

      8. frac-2neg N/A

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

      9. lift-neg.f64 N/A

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

      10. remove-double-neg N/A

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

      11. lower-/.f64 N/A

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

      12. neg-sub0 N/A

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

      13. lift--.f64 N/A

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

      14. sub-neg N/A

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

      15. +-commutative N/A

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

      16. associate--r+ N/A

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

      17. neg-sub0 N/A

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

      18. remove-double-neg N/A

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

      19. lower--.f64 76.0

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

   6. Applied rewrites 76.0%

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

   7. Taylor expanded in x around 0

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower--.f64 98.6

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

   9. Applied rewrites 98.6%

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

   10. Taylor expanded in y around inf

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

   12. Applied rewrites 98.7%

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

   if 2 < (/.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 59.7

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

   5. Applied rewrites 59.7%

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

4. Final simplification 72.4%

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

Alternative 12: 69.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{y - x}{y - z}\\ \mathbf{if}\;t\_1 \leq 5 \cdot 10^{-17}:\\ \;\;\;\;\frac{x}{z} \cdot t\\ \mathbf{elif}\;t\_1 \leq 2:\\ \;\;\;\;1 \cdot t\\ \mathbf{else}:\\ \;\;\;\;\frac{t \cdot x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (- y x) (- y z))))
   (if (<= t_1 5e-17)
     (* (/ x z) t)
     (if (<= t_1 2.0) (* 1.0 t) (/ (* t x) z)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (y - x) / (y - z);
	double tmp;
	if (t_1 <= 5e-17) {
		tmp = (x / z) * t;
	} else if (t_1 <= 2.0) {
		tmp = 1.0 * t;
	} else {
		tmp = (t * x) / 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 = (y - x) / (y - z)
    if (t_1 <= 5d-17) then
        tmp = (x / z) * t
    else if (t_1 <= 2.0d0) then
        tmp = 1.0d0 * t
    else
        tmp = (t * x) / z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (y - x) / (y - z);
	double tmp;
	if (t_1 <= 5e-17) {
		tmp = (x / z) * t;
	} else if (t_1 <= 2.0) {
		tmp = 1.0 * t;
	} else {
		tmp = (t * x) / z;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (y - x) / (y - z)
	tmp = 0
	if t_1 <= 5e-17:
		tmp = (x / z) * t
	elif t_1 <= 2.0:
		tmp = 1.0 * t
	else:
		tmp = (t * x) / z
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(y - x) / Float64(y - z))
	tmp = 0.0
	if (t_1 <= 5e-17)
		tmp = Float64(Float64(x / z) * t);
	elseif (t_1 <= 2.0)
		tmp = Float64(1.0 * t);
	else
		tmp = Float64(Float64(t * x) / z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (y - x) / (y - z);
	tmp = 0.0;
	if (t_1 <= 5e-17)
		tmp = (x / z) * t;
	elseif (t_1 <= 2.0)
		tmp = 1.0 * t;
	else
		tmp = (t * x) / z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(y - x), $MachinePrecision] / N[(y - z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 5e-17], N[(N[(x / z), $MachinePrecision] * t), $MachinePrecision], If[LessEqual[t$95$1, 2.0], N[(1.0 * t), $MachinePrecision], N[(N[(t * x), $MachinePrecision] / z), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 x y) (-.f64 z y)) < 4.9999999999999999e-17

   1. Initial program 96.5%

      \[\frac{x - y}{z - y} \cdot t \]
   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 62.0

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

   5. Applied rewrites 62.0%

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

   if 4.9999999999999999e-17 < (/.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

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

   5. Applied rewrites 95.9%

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

   if 2 < (/.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 59.7

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

   5. Applied rewrites 59.7%

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

4. Final simplification 72.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y - x}{y - z} \leq 5 \cdot 10^{-17}:\\ \;\;\;\;\frac{x}{z} \cdot t\\ \mathbf{elif}\;\frac{y - x}{y - z} \leq 2:\\ \;\;\;\;1 \cdot t\\ \mathbf{else}:\\ \;\;\;\;\frac{t \cdot x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 68.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{y - x}{y - z}\\ \mathbf{if}\;t\_1 \leq 5 \cdot 10^{-17}:\\ \;\;\;\;\frac{t}{z} \cdot x\\ \mathbf{elif}\;t\_1 \leq 2:\\ \;\;\;\;1 \cdot t\\ \mathbf{else}:\\ \;\;\;\;\frac{t \cdot x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (- y x) (- y z))))
   (if (<= t_1 5e-17)
     (* (/ t z) x)
     (if (<= t_1 2.0) (* 1.0 t) (/ (* t x) z)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (y - x) / (y - z);
	double tmp;
	if (t_1 <= 5e-17) {
		tmp = (t / z) * x;
	} else if (t_1 <= 2.0) {
		tmp = 1.0 * t;
	} else {
		tmp = (t * x) / 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 = (y - x) / (y - z)
    if (t_1 <= 5d-17) then
        tmp = (t / z) * x
    else if (t_1 <= 2.0d0) then
        tmp = 1.0d0 * t
    else
        tmp = (t * x) / z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (y - x) / (y - z);
	double tmp;
	if (t_1 <= 5e-17) {
		tmp = (t / z) * x;
	} else if (t_1 <= 2.0) {
		tmp = 1.0 * t;
	} else {
		tmp = (t * x) / z;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (y - x) / (y - z)
	tmp = 0
	if t_1 <= 5e-17:
		tmp = (t / z) * x
	elif t_1 <= 2.0:
		tmp = 1.0 * t
	else:
		tmp = (t * x) / z
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(y - x) / Float64(y - z))
	tmp = 0.0
	if (t_1 <= 5e-17)
		tmp = Float64(Float64(t / z) * x);
	elseif (t_1 <= 2.0)
		tmp = Float64(1.0 * t);
	else
		tmp = Float64(Float64(t * x) / z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (y - x) / (y - z);
	tmp = 0.0;
	if (t_1 <= 5e-17)
		tmp = (t / z) * x;
	elseif (t_1 <= 2.0)
		tmp = 1.0 * t;
	else
		tmp = (t * x) / z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(y - x), $MachinePrecision] / N[(y - z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 5e-17], N[(N[(t / z), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[t$95$1, 2.0], N[(1.0 * t), $MachinePrecision], N[(N[(t * x), $MachinePrecision] / z), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 x y) (-.f64 z y)) < 4.9999999999999999e-17

   1. Initial program 96.5%

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

   3. Taylor expanded in x around inf

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

      1. associate-*l/ N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower--.f64 69.0

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

   5. Applied rewrites 69.0%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 60.4%

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

   if 4.9999999999999999e-17 < (/.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

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

   5. Applied rewrites 95.9%

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

   if 2 < (/.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 59.7

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

   5. Applied rewrites 59.7%

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

4. Final simplification 71.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y - x}{y - z} \leq 5 \cdot 10^{-17}:\\ \;\;\;\;\frac{t}{z} \cdot x\\ \mathbf{elif}\;\frac{y - x}{y - z} \leq 2:\\ \;\;\;\;1 \cdot t\\ \mathbf{else}:\\ \;\;\;\;\frac{t \cdot x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 68.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{y - x}{y - z}\\ t_2 := \frac{t}{z} \cdot x\\ \mathbf{if}\;t\_1 \leq 5 \cdot 10^{-17}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 2:\\ \;\;\;\;1 \cdot t\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (- y x) (- y z))) (t_2 (* (/ t z) x)))
   (if (<= t_1 5e-17) t_2 (if (<= t_1 2.0) (* 1.0 t) t_2))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (y - x) / (y - z);
	double t_2 = (t / z) * x;
	double tmp;
	if (t_1 <= 5e-17) {
		tmp = t_2;
	} else if (t_1 <= 2.0) {
		tmp = 1.0 * 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 = (y - x) / (y - z)
    t_2 = (t / z) * x
    if (t_1 <= 5d-17) then
        tmp = t_2
    else if (t_1 <= 2.0d0) then
        tmp = 1.0d0 * 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 = (y - x) / (y - z);
	double t_2 = (t / z) * x;
	double tmp;
	if (t_1 <= 5e-17) {
		tmp = t_2;
	} else if (t_1 <= 2.0) {
		tmp = 1.0 * t;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (y - x) / (y - z)
	t_2 = (t / z) * x
	tmp = 0
	if t_1 <= 5e-17:
		tmp = t_2
	elif t_1 <= 2.0:
		tmp = 1.0 * t
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(y - x) / Float64(y - z))
	t_2 = Float64(Float64(t / z) * x)
	tmp = 0.0
	if (t_1 <= 5e-17)
		tmp = t_2;
	elseif (t_1 <= 2.0)
		tmp = Float64(1.0 * t);
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (y - x) / (y - z);
	t_2 = (t / z) * x;
	tmp = 0.0;
	if (t_1 <= 5e-17)
		tmp = t_2;
	elseif (t_1 <= 2.0)
		tmp = 1.0 * t;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(y - x), $MachinePrecision] / N[(y - z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t / z), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[t$95$1, 5e-17], t$95$2, If[LessEqual[t$95$1, 2.0], N[(1.0 * t), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 x y) (-.f64 z y)) < 4.9999999999999999e-17 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

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

      1. associate-*l/ N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower--.f64 74.0

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

   5. Applied rewrites 74.0%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 59.5%

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

   if 4.9999999999999999e-17 < (/.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

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

   5. Applied rewrites 95.9%

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

4. Final simplification 70.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y - x}{y - z} \leq 5 \cdot 10^{-17}:\\ \;\;\;\;\frac{t}{z} \cdot x\\ \mathbf{elif}\;\frac{y - x}{y - z} \leq 2:\\ \;\;\;\;1 \cdot t\\ \mathbf{else}:\\ \;\;\;\;\frac{t}{z} \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 35.3% accurate, 3.8× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z, double t) {
	return 1.0 * 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 = 1.0d0 * t
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.0%

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

3. Taylor expanded in y around inf

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

5. Applied rewrites 32.7%

   \[\leadsto \color{blue}{1} \cdot t \]
6. Add Preprocessing

Developer Target 1: 97.1% 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 2024298 
(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))