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

Percentage Accurate: 97.0% → 97.0%

Time: 8.3s

Alternatives: 17

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: 97.0% 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: 97.0% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 96.8%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-/.f64 N/A

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

   4. clear-num N/A

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

   5. un-div-inv N/A

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

   6. lower-/.f64 N/A

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

   7. frac-2neg N/A

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

   8. lower-/.f64 N/A

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

   9. neg-sub0 N/A

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

   10. lift--.f64 N/A

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

   11. sub-neg N/A

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

   12. +-commutative N/A

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

   13. associate--r+ N/A

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

   14. neg-sub0 N/A

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

   15. remove-double-neg N/A

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

   16. lower--.f64 N/A

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

   17. neg-sub0 N/A

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

   18. lift--.f64 N/A

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

   19. sub-neg N/A

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

   20. +-commutative N/A

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

   21. associate--r+ N/A

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

   22. neg-sub0 N/A

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

   23. remove-double-neg N/A

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

   24. lower--.f64 96.9

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

4. Applied rewrites 96.9%

   \[\leadsto \color{blue}{\frac{t}{\frac{y - z}{y - x}}} \]
5. Add Preprocessing

Alternative 2: 70.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x - y}{z - y}\\ t_2 := \frac{-x}{y} \cdot t\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+131}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq -1 \cdot 10^{-145}:\\ \;\;\;\;\frac{x}{z} \cdot t\\ \mathbf{elif}\;t\_1 \leq 0.0002:\\ \;\;\;\;\frac{\left(-y\right) \cdot t}{z}\\ \mathbf{elif}\;t\_1 \leq 2:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{y}, t, t\right)\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+248}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot t}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (- x y) (- z y))) (t_2 (* (/ (- x) y) t)))
   (if (<= t_1 -5e+131)
     t_2
     (if (<= t_1 -1e-145)
       (* (/ x z) t)
       (if (<= t_1 0.0002)
         (/ (* (- y) t) z)
         (if (<= t_1 2.0)
           (fma (/ z y) t t)
           (if (<= t_1 2e+248) t_2 (/ (* x t) z))))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x - y) / (z - y);
	double t_2 = (-x / y) * t;
	double tmp;
	if (t_1 <= -5e+131) {
		tmp = t_2;
	} else if (t_1 <= -1e-145) {
		tmp = (x / z) * t;
	} else if (t_1 <= 0.0002) {
		tmp = (-y * t) / z;
	} else if (t_1 <= 2.0) {
		tmp = fma((z / y), t, t);
	} else if (t_1 <= 2e+248) {
		tmp = t_2;
	} else {
		tmp = (x * t) / z;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x - y) / Float64(z - y))
	t_2 = Float64(Float64(Float64(-x) / y) * t)
	tmp = 0.0
	if (t_1 <= -5e+131)
		tmp = t_2;
	elseif (t_1 <= -1e-145)
		tmp = Float64(Float64(x / z) * t);
	elseif (t_1 <= 0.0002)
		tmp = Float64(Float64(Float64(-y) * t) / z);
	elseif (t_1 <= 2.0)
		tmp = fma(Float64(z / y), t, t);
	elseif (t_1 <= 2e+248)
		tmp = t_2;
	else
		tmp = Float64(Float64(x * t) / z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x - y), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[((-x) / y), $MachinePrecision] * t), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+131], t$95$2, If[LessEqual[t$95$1, -1e-145], N[(N[(x / z), $MachinePrecision] * t), $MachinePrecision], If[LessEqual[t$95$1, 0.0002], N[(N[((-y) * t), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[t$95$1, 2.0], N[(N[(z / y), $MachinePrecision] * t + t), $MachinePrecision], If[LessEqual[t$95$1, 2e+248], t$95$2, N[(N[(x * t), $MachinePrecision] / z), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 (-.f64 x y) (-.f64 z y)) < -4.99999999999999995e131 or 2 < (/.f64 (-.f64 x y) (-.f64 z y)) < 2.00000000000000009e248

   1. Initial program 99.7%

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

   3. Taylor expanded in z around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. sub-neg N/A

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

      5. +-commutative N/A

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

      6. distribute-neg-in N/A

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

      7. unsub-neg N/A

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

      8. remove-double-neg N/A

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

      9. lower--.f64 68.6

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

   5. Applied rewrites 68.6%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 68.6%

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

   if -4.99999999999999995e131 < (/.f64 (-.f64 x y) (-.f64 z y)) < -9.99999999999999915e-146

   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 56.7

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

   5. Applied rewrites 56.7%

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

   if -9.99999999999999915e-146 < (/.f64 (-.f64 x y) (-.f64 z y)) < 2.0000000000000001e-4

   1. Initial program 93.3%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 94.6

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

   5. Applied rewrites 94.6%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 65.2%

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

   if 2.0000000000000001e-4 < (/.f64 (-.f64 x y) (-.f64 z y)) < 2

   1. Initial program 99.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. frac-2neg N/A

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

      5. lower-/.f64 N/A

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

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

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

      7. lower-*.f64 N/A

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

      8. neg-sub0 N/A

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

      9. lift--.f64 N/A

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

      10. sub-neg N/A

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

      11. +-commutative N/A

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

      12. associate--r+ N/A

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

      13. neg-sub0 N/A

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

      14. remove-double-neg N/A

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

      15. lower--.f64 N/A

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

      16. neg-sub0 N/A

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

      17. lift--.f64 N/A

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

      18. sub-neg N/A

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

      19. +-commutative N/A

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

      20. associate--r+ N/A

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

      21. neg-sub0 N/A

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

      22. remove-double-neg N/A

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

      23. lower--.f64 69.5

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

   4. Applied rewrites 69.5%

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

   5. Taylor expanded in x around 0

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 97.2

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

   7. Applied rewrites 97.2%

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

   8. Taylor expanded in z around 0

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

   10. Applied rewrites 96.5%

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

   if 2.00000000000000009e248 < (/.f64 (-.f64 x y) (-.f64 z y))

   1. Initial program 72.7%

      \[\frac{x - y}{z - y} \cdot t \]
   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 85.7

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

   5. Applied rewrites 85.7%

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

4. Final simplification 76.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{z - y} \leq -5 \cdot 10^{+131}:\\ \;\;\;\;\frac{-x}{y} \cdot t\\ \mathbf{elif}\;\frac{x - y}{z - y} \leq -1 \cdot 10^{-145}:\\ \;\;\;\;\frac{x}{z} \cdot t\\ \mathbf{elif}\;\frac{x - y}{z - y} \leq 0.0002:\\ \;\;\;\;\frac{\left(-y\right) \cdot t}{z}\\ \mathbf{elif}\;\frac{x - y}{z - y} \leq 2:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{y}, t, t\right)\\ \mathbf{elif}\;\frac{x - y}{z - y} \leq 2 \cdot 10^{+248}:\\ \;\;\;\;\frac{-x}{y} \cdot t\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot t}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 80.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x - y}{z - y}\\ t_2 := \frac{-x}{y} \cdot t\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+21}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 0.0002:\\ \;\;\;\;\frac{t}{z} \cdot \left(x - y\right)\\ \mathbf{elif}\;t\_1 \leq 2:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{y}, t, t\right)\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+248}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot t}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (- x y) (- z y))) (t_2 (* (/ (- x) y) t)))
   (if (<= t_1 -1e+21)
     t_2
     (if (<= t_1 0.0002)
       (* (/ t z) (- x y))
       (if (<= t_1 2.0)
         (fma (/ z y) t t)
         (if (<= t_1 2e+248) t_2 (/ (* x t) z)))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x - y) / (z - y);
	double t_2 = (-x / y) * t;
	double tmp;
	if (t_1 <= -1e+21) {
		tmp = t_2;
	} else if (t_1 <= 0.0002) {
		tmp = (t / z) * (x - y);
	} else if (t_1 <= 2.0) {
		tmp = fma((z / y), t, t);
	} else if (t_1 <= 2e+248) {
		tmp = t_2;
	} else {
		tmp = (x * t) / z;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x - y) / Float64(z - y))
	t_2 = Float64(Float64(Float64(-x) / y) * t)
	tmp = 0.0
	if (t_1 <= -1e+21)
		tmp = t_2;
	elseif (t_1 <= 0.0002)
		tmp = Float64(Float64(t / z) * Float64(x - y));
	elseif (t_1 <= 2.0)
		tmp = fma(Float64(z / y), t, t);
	elseif (t_1 <= 2e+248)
		tmp = t_2;
	else
		tmp = Float64(Float64(x * t) / z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x - y), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[((-x) / y), $MachinePrecision] * t), $MachinePrecision]}, If[LessEqual[t$95$1, -1e+21], t$95$2, If[LessEqual[t$95$1, 0.0002], N[(N[(t / z), $MachinePrecision] * N[(x - y), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2.0], N[(N[(z / y), $MachinePrecision] * t + t), $MachinePrecision], If[LessEqual[t$95$1, 2e+248], t$95$2, N[(N[(x * t), $MachinePrecision] / z), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (-.f64 x y) (-.f64 z y)) < -1e21 or 2 < (/.f64 (-.f64 x y) (-.f64 z y)) < 2.00000000000000009e248

   1. Initial program 99.7%

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

   3. Taylor expanded in z around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. sub-neg N/A

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

      5. +-commutative N/A

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

      6. distribute-neg-in N/A

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

      7. unsub-neg N/A

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

      8. remove-double-neg N/A

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

      9. lower--.f64 63.6

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

   5. Applied rewrites 63.6%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 63.6%

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

   if -1e21 < (/.f64 (-.f64 x y) (-.f64 z y)) < 2.0000000000000001e-4

   1. Initial program 95.3%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 89.2

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

   5. Applied rewrites 89.2%

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

   7. Applied rewrites 87.0%

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

   if 2.0000000000000001e-4 < (/.f64 (-.f64 x y) (-.f64 z y)) < 2

   1. Initial program 99.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. frac-2neg N/A

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

      5. lower-/.f64 N/A

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

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

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

      7. lower-*.f64 N/A

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

      8. neg-sub0 N/A

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

      9. lift--.f64 N/A

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

      10. sub-neg N/A

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

      11. +-commutative N/A

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

      12. associate--r+ N/A

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

      13. neg-sub0 N/A

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

      14. remove-double-neg N/A

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

      15. lower--.f64 N/A

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

      16. neg-sub0 N/A

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

      17. lift--.f64 N/A

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

      18. sub-neg N/A

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

      19. +-commutative N/A

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

      20. associate--r+ N/A

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

      21. neg-sub0 N/A

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

      22. remove-double-neg N/A

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

      23. lower--.f64 69.5

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

   4. Applied rewrites 69.5%

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

   5. Taylor expanded in x around 0

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 97.2

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

   7. Applied rewrites 97.2%

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

   8. Taylor expanded in z around 0

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

   10. Applied rewrites 96.5%

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

   if 2.00000000000000009e248 < (/.f64 (-.f64 x y) (-.f64 z y))

   1. Initial program 72.7%

      \[\frac{x - y}{z - y} \cdot t \]
   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 85.7

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

   5. Applied rewrites 85.7%

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

4. Final simplification 83.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{z - y} \leq -1 \cdot 10^{+21}:\\ \;\;\;\;\frac{-x}{y} \cdot t\\ \mathbf{elif}\;\frac{x - y}{z - y} \leq 0.0002:\\ \;\;\;\;\frac{t}{z} \cdot \left(x - y\right)\\ \mathbf{elif}\;\frac{x - y}{z - y} \leq 2:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{y}, t, t\right)\\ \mathbf{elif}\;\frac{x - y}{z - y} \leq 2 \cdot 10^{+248}:\\ \;\;\;\;\frac{-x}{y} \cdot t\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot t}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 93.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x - y}{z - y}\\ t_2 := \frac{t}{z - y} \cdot x\\ \mathbf{if}\;t\_1 \leq -5000:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 0.0002:\\ \;\;\;\;\frac{x - y}{z} \cdot t\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+51}:\\ \;\;\;\;\mathsf{fma}\left(t, \frac{z - x}{y}, t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (- x y) (- z y))) (t_2 (* (/ t (- z y)) x)))
   (if (<= t_1 -5000.0)
     t_2
     (if (<= t_1 0.0002)
       (* (/ (- x y) z) t)
       (if (<= t_1 2e+51) (fma t (/ (- z x) y) t) t_2)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x - y) / (z - y);
	double t_2 = (t / (z - y)) * x;
	double tmp;
	if (t_1 <= -5000.0) {
		tmp = t_2;
	} else if (t_1 <= 0.0002) {
		tmp = ((x - y) / z) * t;
	} else if (t_1 <= 2e+51) {
		tmp = fma(t, ((z - x) / y), t);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x - y) / Float64(z - y))
	t_2 = Float64(Float64(t / Float64(z - y)) * x)
	tmp = 0.0
	if (t_1 <= -5000.0)
		tmp = t_2;
	elseif (t_1 <= 0.0002)
		tmp = Float64(Float64(Float64(x - y) / z) * t);
	elseif (t_1 <= 2e+51)
		tmp = fma(t, Float64(Float64(z - x) / y), t);
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x - y), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t / N[(z - y), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[t$95$1, -5000.0], t$95$2, If[LessEqual[t$95$1, 0.0002], N[(N[(N[(x - y), $MachinePrecision] / z), $MachinePrecision] * t), $MachinePrecision], If[LessEqual[t$95$1, 2e+51], N[(t * N[(N[(z - x), $MachinePrecision] / y), $MachinePrecision] + t), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 x y) (-.f64 z y)) < -5e3 or 2e51 < (/.f64 (-.f64 x y) (-.f64 z y))

   1. Initial program 95.2%

      \[\frac{x - y}{z - y} \cdot t \]
   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 89.2

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

   5. Applied rewrites 89.2%

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

   if -5e3 < (/.f64 (-.f64 x y) (-.f64 z y)) < 2.0000000000000001e-4

   1. Initial program 95.2%

      \[\frac{x - y}{z - y} \cdot t \]
   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 93.3

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

   5. Applied rewrites 93.3%

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

   if 2.0000000000000001e-4 < (/.f64 (-.f64 x y) (-.f64 z y)) < 2e51

   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-rgt-neg-in N/A

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

      9. mul-1-neg N/A

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

      10. lower-fma.f64 N/A

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

   5. Applied rewrites 98.2%

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

Alternative 5: 93.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x - y}{z - y}\\ t_2 := \frac{t}{z - y} \cdot x\\ \mathbf{if}\;t\_1 \leq -5000:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 0.0002:\\ \;\;\;\;\frac{x - y}{z} \cdot t\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+51}:\\ \;\;\;\;\left(1 - \frac{x}{y}\right) \cdot t\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (- x y) (- z y))) (t_2 (* (/ t (- z y)) x)))
   (if (<= t_1 -5000.0)
     t_2
     (if (<= t_1 0.0002)
       (* (/ (- x y) z) t)
       (if (<= t_1 2e+51) (* (- 1.0 (/ x y)) t) t_2)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x - y) / (z - y);
	double t_2 = (t / (z - y)) * x;
	double tmp;
	if (t_1 <= -5000.0) {
		tmp = t_2;
	} else if (t_1 <= 0.0002) {
		tmp = ((x - y) / z) * t;
	} else if (t_1 <= 2e+51) {
		tmp = (1.0 - (x / y)) * t;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = (x - y) / (z - y)
    t_2 = (t / (z - y)) * x
    if (t_1 <= (-5000.0d0)) then
        tmp = t_2
    else if (t_1 <= 0.0002d0) then
        tmp = ((x - y) / z) * t
    else if (t_1 <= 2d+51) then
        tmp = (1.0d0 - (x / y)) * t
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (x - y) / (z - y);
	double t_2 = (t / (z - y)) * x;
	double tmp;
	if (t_1 <= -5000.0) {
		tmp = t_2;
	} else if (t_1 <= 0.0002) {
		tmp = ((x - y) / z) * t;
	} else if (t_1 <= 2e+51) {
		tmp = (1.0 - (x / y)) * t;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (x - y) / (z - y)
	t_2 = (t / (z - y)) * x
	tmp = 0
	if t_1 <= -5000.0:
		tmp = t_2
	elif t_1 <= 0.0002:
		tmp = ((x - y) / z) * t
	elif t_1 <= 2e+51:
		tmp = (1.0 - (x / y)) * t
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x - y) / Float64(z - y))
	t_2 = Float64(Float64(t / Float64(z - y)) * x)
	tmp = 0.0
	if (t_1 <= -5000.0)
		tmp = t_2;
	elseif (t_1 <= 0.0002)
		tmp = Float64(Float64(Float64(x - y) / z) * t);
	elseif (t_1 <= 2e+51)
		tmp = Float64(Float64(1.0 - Float64(x / y)) * t);
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (x - y) / (z - y);
	t_2 = (t / (z - y)) * x;
	tmp = 0.0;
	if (t_1 <= -5000.0)
		tmp = t_2;
	elseif (t_1 <= 0.0002)
		tmp = ((x - y) / z) * t;
	elseif (t_1 <= 2e+51)
		tmp = (1.0 - (x / y)) * t;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x - y), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t / N[(z - y), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[t$95$1, -5000.0], t$95$2, If[LessEqual[t$95$1, 0.0002], N[(N[(N[(x - y), $MachinePrecision] / z), $MachinePrecision] * t), $MachinePrecision], If[LessEqual[t$95$1, 2e+51], N[(N[(1.0 - N[(x / y), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 x y) (-.f64 z y)) < -5e3 or 2e51 < (/.f64 (-.f64 x y) (-.f64 z y))

   1. Initial program 95.2%

      \[\frac{x - y}{z - y} \cdot t \]
   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 89.2

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

   5. Applied rewrites 89.2%

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

   if -5e3 < (/.f64 (-.f64 x y) (-.f64 z y)) < 2.0000000000000001e-4

   1. Initial program 95.2%

      \[\frac{x - y}{z - y} \cdot t \]
   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 93.3

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

   5. Applied rewrites 93.3%

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

   if 2.0000000000000001e-4 < (/.f64 (-.f64 x y) (-.f64 z y)) < 2e51

   1. Initial program 99.9%

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

   3. Taylor expanded in z around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. sub-neg N/A

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

      5. +-commutative N/A

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

      6. distribute-neg-in N/A

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

      7. unsub-neg N/A

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

      8. remove-double-neg N/A

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

      9. lower--.f64 97.1

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

   5. Applied rewrites 97.1%

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

   7. Applied rewrites 97.2%

      \[\leadsto \left(1 - \color{blue}{\frac{x}{y}}\right) \cdot t \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 6: 90.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x - y}{z - y}\\ t_2 := \frac{t}{z - y} \cdot x\\ \mathbf{if}\;t\_1 \leq -5000:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 0.0002:\\ \;\;\;\;\frac{\left(x - y\right) \cdot t}{z}\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+51}:\\ \;\;\;\;\left(1 - \frac{x}{y}\right) \cdot t\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (- x y) (- z y))) (t_2 (* (/ t (- z y)) x)))
   (if (<= t_1 -5000.0)
     t_2
     (if (<= t_1 0.0002)
       (/ (* (- x y) t) z)
       (if (<= t_1 2e+51) (* (- 1.0 (/ x y)) t) t_2)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x - y) / (z - y);
	double t_2 = (t / (z - y)) * x;
	double tmp;
	if (t_1 <= -5000.0) {
		tmp = t_2;
	} else if (t_1 <= 0.0002) {
		tmp = ((x - y) * t) / z;
	} else if (t_1 <= 2e+51) {
		tmp = (1.0 - (x / y)) * t;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = (x - y) / (z - y)
    t_2 = (t / (z - y)) * x
    if (t_1 <= (-5000.0d0)) then
        tmp = t_2
    else if (t_1 <= 0.0002d0) then
        tmp = ((x - y) * t) / z
    else if (t_1 <= 2d+51) then
        tmp = (1.0d0 - (x / y)) * t
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (x - y) / (z - y);
	double t_2 = (t / (z - y)) * x;
	double tmp;
	if (t_1 <= -5000.0) {
		tmp = t_2;
	} else if (t_1 <= 0.0002) {
		tmp = ((x - y) * t) / z;
	} else if (t_1 <= 2e+51) {
		tmp = (1.0 - (x / y)) * t;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (x - y) / (z - y)
	t_2 = (t / (z - y)) * x
	tmp = 0
	if t_1 <= -5000.0:
		tmp = t_2
	elif t_1 <= 0.0002:
		tmp = ((x - y) * t) / z
	elif t_1 <= 2e+51:
		tmp = (1.0 - (x / y)) * t
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x - y) / Float64(z - y))
	t_2 = Float64(Float64(t / Float64(z - y)) * x)
	tmp = 0.0
	if (t_1 <= -5000.0)
		tmp = t_2;
	elseif (t_1 <= 0.0002)
		tmp = Float64(Float64(Float64(x - y) * t) / z);
	elseif (t_1 <= 2e+51)
		tmp = Float64(Float64(1.0 - Float64(x / y)) * t);
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (x - y) / (z - y);
	t_2 = (t / (z - y)) * x;
	tmp = 0.0;
	if (t_1 <= -5000.0)
		tmp = t_2;
	elseif (t_1 <= 0.0002)
		tmp = ((x - y) * t) / z;
	elseif (t_1 <= 2e+51)
		tmp = (1.0 - (x / y)) * t;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x - y), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t / N[(z - y), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[t$95$1, -5000.0], t$95$2, If[LessEqual[t$95$1, 0.0002], N[(N[(N[(x - y), $MachinePrecision] * t), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[t$95$1, 2e+51], N[(N[(1.0 - N[(x / y), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 x y) (-.f64 z y)) < -5e3 or 2e51 < (/.f64 (-.f64 x y) (-.f64 z y))

   1. Initial program 95.2%

      \[\frac{x - y}{z - y} \cdot t \]
   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 89.2

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

   5. Applied rewrites 89.2%

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

   if -5e3 < (/.f64 (-.f64 x y) (-.f64 z y)) < 2.0000000000000001e-4

   1. Initial program 95.2%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 91.0

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

   5. Applied rewrites 91.0%

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

   if 2.0000000000000001e-4 < (/.f64 (-.f64 x y) (-.f64 z y)) < 2e51

   1. Initial program 99.9%

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

   3. Taylor expanded in z around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. sub-neg N/A

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

      5. +-commutative N/A

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

      6. distribute-neg-in N/A

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

      7. unsub-neg N/A

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

      8. remove-double-neg N/A

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

      9. lower--.f64 97.1

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

   5. Applied rewrites 97.1%

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

   7. Applied rewrites 97.2%

      \[\leadsto \left(1 - \color{blue}{\frac{x}{y}}\right) \cdot t \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 7: 91.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x - y}{z - y}\\ t_2 := \frac{t}{z - y} \cdot x\\ \mathbf{if}\;t\_1 \leq -5000:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-15}:\\ \;\;\;\;\frac{\left(x - y\right) \cdot t}{z}\\ \mathbf{elif}\;t\_1 \leq 2:\\ \;\;\;\;\frac{y}{y - z} \cdot t\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (- x y) (- z y))) (t_2 (* (/ t (- z y)) x)))
   (if (<= t_1 -5000.0)
     t_2
     (if (<= t_1 2e-15)
       (/ (* (- x y) t) z)
       (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 = (x - y) / (z - y);
	double t_2 = (t / (z - y)) * x;
	double tmp;
	if (t_1 <= -5000.0) {
		tmp = t_2;
	} else if (t_1 <= 2e-15) {
		tmp = ((x - y) * t) / z;
	} 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 = (x - y) / (z - y)
    t_2 = (t / (z - y)) * x
    if (t_1 <= (-5000.0d0)) then
        tmp = t_2
    else if (t_1 <= 2d-15) then
        tmp = ((x - y) * t) / z
    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 = (x - y) / (z - y);
	double t_2 = (t / (z - y)) * x;
	double tmp;
	if (t_1 <= -5000.0) {
		tmp = t_2;
	} else if (t_1 <= 2e-15) {
		tmp = ((x - y) * t) / z;
	} 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 = (x - y) / (z - y)
	t_2 = (t / (z - y)) * x
	tmp = 0
	if t_1 <= -5000.0:
		tmp = t_2
	elif t_1 <= 2e-15:
		tmp = ((x - y) * t) / z
	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(x - y) / Float64(z - y))
	t_2 = Float64(Float64(t / Float64(z - y)) * x)
	tmp = 0.0
	if (t_1 <= -5000.0)
		tmp = t_2;
	elseif (t_1 <= 2e-15)
		tmp = Float64(Float64(Float64(x - y) * t) / z);
	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 = (x - y) / (z - y);
	t_2 = (t / (z - y)) * x;
	tmp = 0.0;
	if (t_1 <= -5000.0)
		tmp = t_2;
	elseif (t_1 <= 2e-15)
		tmp = ((x - y) * t) / z;
	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[(x - y), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t / N[(z - y), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[t$95$1, -5000.0], t$95$2, If[LessEqual[t$95$1, 2e-15], N[(N[(N[(x - y), $MachinePrecision] * t), $MachinePrecision] / z), $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)) < -5e3 or 2 < (/.f64 (-.f64 x y) (-.f64 z y))

   1. Initial program 95.5%

      \[\frac{x - y}{z - y} \cdot t \]
   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 87.2

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

   5. Applied rewrites 87.2%

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

   if -5e3 < (/.f64 (-.f64 x y) (-.f64 z y)) < 2.0000000000000002e-15

   1. Initial program 95.0%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 92.1

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

   5. Applied rewrites 92.1%

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

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

   1. Initial program 99.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. frac-2neg N/A

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

      5. lower-/.f64 N/A

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

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

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

      7. lower-*.f64 N/A

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

      8. neg-sub0 N/A

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

      9. lift--.f64 N/A

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

      10. sub-neg N/A

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

      11. +-commutative N/A

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

      12. associate--r+ N/A

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

      13. neg-sub0 N/A

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

      14. remove-double-neg N/A

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

      15. lower--.f64 N/A

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

      16. neg-sub0 N/A

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

      17. lift--.f64 N/A

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

      18. sub-neg N/A

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

      19. +-commutative N/A

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

      20. associate--r+ N/A

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

      21. neg-sub0 N/A

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

      22. remove-double-neg N/A

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

      23. lower--.f64 70.2

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

   4. Applied rewrites 70.2%

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

   5. Taylor expanded in x around 0

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 97.2

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

   7. Applied rewrites 97.2%

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

Alternative 8: 91.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x - y}{z - y}\\ t_2 := \frac{t}{z - y} \cdot x\\ \mathbf{if}\;t\_1 \leq -5000:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 0.0002:\\ \;\;\;\;\frac{\left(x - y\right) \cdot t}{z}\\ \mathbf{elif}\;t\_1 \leq 2:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{y}, t, t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (- x y) (- z y))) (t_2 (* (/ t (- z y)) x)))
   (if (<= t_1 -5000.0)
     t_2
     (if (<= t_1 0.0002)
       (/ (* (- x y) t) z)
       (if (<= t_1 2.0) (fma (/ z y) t t) t_2)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x - y) / (z - y);
	double t_2 = (t / (z - y)) * x;
	double tmp;
	if (t_1 <= -5000.0) {
		tmp = t_2;
	} else if (t_1 <= 0.0002) {
		tmp = ((x - y) * t) / z;
	} else if (t_1 <= 2.0) {
		tmp = fma((z / y), t, t);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x - y) / Float64(z - y))
	t_2 = Float64(Float64(t / Float64(z - y)) * x)
	tmp = 0.0
	if (t_1 <= -5000.0)
		tmp = t_2;
	elseif (t_1 <= 0.0002)
		tmp = Float64(Float64(Float64(x - y) * t) / z);
	elseif (t_1 <= 2.0)
		tmp = fma(Float64(z / y), t, t);
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x - y), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t / N[(z - y), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[t$95$1, -5000.0], t$95$2, If[LessEqual[t$95$1, 0.0002], N[(N[(N[(x - y), $MachinePrecision] * t), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[t$95$1, 2.0], N[(N[(z / y), $MachinePrecision] * t + t), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 x y) (-.f64 z y)) < -5e3 or 2 < (/.f64 (-.f64 x y) (-.f64 z y))

   1. Initial program 95.5%

      \[\frac{x - y}{z - y} \cdot t \]
   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 87.2

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

   5. Applied rewrites 87.2%

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

   if -5e3 < (/.f64 (-.f64 x y) (-.f64 z y)) < 2.0000000000000001e-4

   1. Initial program 95.2%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 91.0

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

   5. Applied rewrites 91.0%

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

   if 2.0000000000000001e-4 < (/.f64 (-.f64 x y) (-.f64 z y)) < 2

   1. Initial program 99.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. frac-2neg N/A

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

      5. lower-/.f64 N/A

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

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

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

      7. lower-*.f64 N/A

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

      8. neg-sub0 N/A

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

      9. lift--.f64 N/A

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

      10. sub-neg N/A

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

      11. +-commutative N/A

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

      12. associate--r+ N/A

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

      13. neg-sub0 N/A

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

      14. remove-double-neg N/A

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

      15. lower--.f64 N/A

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

      16. neg-sub0 N/A

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

      17. lift--.f64 N/A

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

      18. sub-neg N/A

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

      19. +-commutative N/A

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

      20. associate--r+ N/A

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

      21. neg-sub0 N/A

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

      22. remove-double-neg N/A

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

      23. lower--.f64 69.5

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

   4. Applied rewrites 69.5%

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

   5. Taylor expanded in x around 0

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 97.2

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

   7. Applied rewrites 97.2%

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

   8. Taylor expanded in z around 0

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

   10. Applied rewrites 96.5%

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

Alternative 9: 91.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x - y}{z - y}\\ t_2 := \frac{t}{z - y} \cdot x\\ \mathbf{if}\;t\_1 \leq -5000:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 0.0002:\\ \;\;\;\;\frac{t}{z} \cdot \left(x - y\right)\\ \mathbf{elif}\;t\_1 \leq 2:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{y}, t, t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (- x y) (- z y))) (t_2 (* (/ t (- z y)) x)))
   (if (<= t_1 -5000.0)
     t_2
     (if (<= t_1 0.0002)
       (* (/ t z) (- x y))
       (if (<= t_1 2.0) (fma (/ z y) t t) t_2)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x - y) / (z - y);
	double t_2 = (t / (z - y)) * x;
	double tmp;
	if (t_1 <= -5000.0) {
		tmp = t_2;
	} else if (t_1 <= 0.0002) {
		tmp = (t / z) * (x - y);
	} else if (t_1 <= 2.0) {
		tmp = fma((z / y), t, t);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x - y) / Float64(z - y))
	t_2 = Float64(Float64(t / Float64(z - y)) * x)
	tmp = 0.0
	if (t_1 <= -5000.0)
		tmp = t_2;
	elseif (t_1 <= 0.0002)
		tmp = Float64(Float64(t / z) * Float64(x - y));
	elseif (t_1 <= 2.0)
		tmp = fma(Float64(z / y), t, t);
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x - y), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t / N[(z - y), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[t$95$1, -5000.0], t$95$2, If[LessEqual[t$95$1, 0.0002], N[(N[(t / z), $MachinePrecision] * N[(x - y), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2.0], N[(N[(z / y), $MachinePrecision] * t + t), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 x y) (-.f64 z y)) < -5e3 or 2 < (/.f64 (-.f64 x y) (-.f64 z y))

   1. Initial program 95.5%

      \[\frac{x - y}{z - y} \cdot t \]
   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 87.2

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

   5. Applied rewrites 87.2%

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

   if -5e3 < (/.f64 (-.f64 x y) (-.f64 z y)) < 2.0000000000000001e-4

   1. Initial program 95.2%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 91.0

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

   5. Applied rewrites 91.0%

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

   7. Applied rewrites 88.7%

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

   if 2.0000000000000001e-4 < (/.f64 (-.f64 x y) (-.f64 z y)) < 2

   1. Initial program 99.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. frac-2neg N/A

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

      5. lower-/.f64 N/A

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

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

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

      7. lower-*.f64 N/A

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

      8. neg-sub0 N/A

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

      9. lift--.f64 N/A

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

      10. sub-neg N/A

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

      11. +-commutative N/A

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

      12. associate--r+ N/A

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

      13. neg-sub0 N/A

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

      14. remove-double-neg N/A

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

      15. lower--.f64 N/A

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

      16. neg-sub0 N/A

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

      17. lift--.f64 N/A

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

      18. sub-neg N/A

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

      19. +-commutative N/A

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

      20. associate--r+ N/A

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

      21. neg-sub0 N/A

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

      22. remove-double-neg N/A

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

      23. lower--.f64 69.5

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

   4. Applied rewrites 69.5%

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

   5. Taylor expanded in x around 0

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 97.2

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

   7. Applied rewrites 97.2%

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

   8. Taylor expanded in z around 0

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

   10. Applied rewrites 96.5%

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

Alternative 10: 69.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x - y}{z - y}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{-145}:\\ \;\;\;\;\frac{x}{z} \cdot t\\ \mathbf{elif}\;t\_1 \leq 0.0002:\\ \;\;\;\;\frac{\left(-y\right) \cdot t}{z}\\ \mathbf{elif}\;t\_1 \leq 2:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{y}, t, t\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t}{z} \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (- x y) (- z y))))
   (if (<= t_1 -1e-145)
     (* (/ x z) t)
     (if (<= t_1 0.0002)
       (/ (* (- y) t) z)
       (if (<= t_1 2.0) (fma (/ z y) t t) (* (/ t z) x))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x - y) / (z - y);
	double tmp;
	if (t_1 <= -1e-145) {
		tmp = (x / z) * t;
	} else if (t_1 <= 0.0002) {
		tmp = (-y * t) / z;
	} else if (t_1 <= 2.0) {
		tmp = fma((z / y), t, t);
	} else {
		tmp = (t / z) * x;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x - y) / Float64(z - y))
	tmp = 0.0
	if (t_1 <= -1e-145)
		tmp = Float64(Float64(x / z) * t);
	elseif (t_1 <= 0.0002)
		tmp = Float64(Float64(Float64(-y) * t) / z);
	elseif (t_1 <= 2.0)
		tmp = fma(Float64(z / y), t, t);
	else
		tmp = Float64(Float64(t / z) * x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x - y), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e-145], N[(N[(x / z), $MachinePrecision] * t), $MachinePrecision], If[LessEqual[t$95$1, 0.0002], N[(N[((-y) * t), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[t$95$1, 2.0], N[(N[(z / y), $MachinePrecision] * t + t), $MachinePrecision], N[(N[(t / z), $MachinePrecision] * x), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (-.f64 x y) (-.f64 z y)) < -9.99999999999999915e-146

   1. Initial program 99.7%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 53.9

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

   5. Applied rewrites 53.9%

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

   if -9.99999999999999915e-146 < (/.f64 (-.f64 x y) (-.f64 z y)) < 2.0000000000000001e-4

   1. Initial program 93.3%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 94.6

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

   5. Applied rewrites 94.6%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 65.2%

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

   if 2.0000000000000001e-4 < (/.f64 (-.f64 x y) (-.f64 z y)) < 2

   1. Initial program 99.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. frac-2neg N/A

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

      5. lower-/.f64 N/A

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

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

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

      7. lower-*.f64 N/A

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

      8. neg-sub0 N/A

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

      9. lift--.f64 N/A

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

      10. sub-neg N/A

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

      11. +-commutative N/A

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

      12. associate--r+ N/A

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

      13. neg-sub0 N/A

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

      14. remove-double-neg N/A

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

      15. lower--.f64 N/A

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

      16. neg-sub0 N/A

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

      17. lift--.f64 N/A

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

      18. sub-neg N/A

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

      19. +-commutative N/A

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

      20. associate--r+ N/A

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

      21. neg-sub0 N/A

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

      22. remove-double-neg N/A

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

      23. lower--.f64 69.5

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

   4. Applied rewrites 69.5%

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

   5. Taylor expanded in x around 0

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 97.2

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

   7. Applied rewrites 97.2%

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

   8. Taylor expanded in z around 0

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

   10. Applied rewrites 96.5%

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

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

   1. Initial program 91.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. frac-2neg N/A

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

      8. lower-/.f64 N/A

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

      9. neg-sub0 N/A

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

      10. lift--.f64 N/A

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

      11. sub-neg N/A

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

      12. +-commutative N/A

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

      13. associate--r+ N/A

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

      14. neg-sub0 N/A

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

      15. remove-double-neg N/A

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

      16. lower--.f64 N/A

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

      17. neg-sub0 N/A

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

      18. lift--.f64 N/A

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

      19. sub-neg N/A

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

      20. +-commutative N/A

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

      21. associate--r+ N/A

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

      22. neg-sub0 N/A

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

      23. remove-double-neg N/A

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

      24. lower--.f64 92.4

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

   4. Applied rewrites 92.4%

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

   5. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 47.3

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

   7. Applied rewrites 47.3%

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

   9. Applied rewrites 47.4%

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

Alternative 11: 92.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x - y}{z - y}\\ t_2 := \frac{t}{z - y}\\ \mathbf{if}\;t\_1 \leq 0.998:\\ \;\;\;\;t\_2 \cdot \left(x - y\right)\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+51}:\\ \;\;\;\;\mathsf{fma}\left(t, \frac{z - x}{y}, t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2 \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (- x y) (- z y))) (t_2 (/ t (- z y))))
   (if (<= t_1 0.998)
     (* t_2 (- x y))
     (if (<= t_1 2e+51) (fma t (/ (- z x) y) t) (* t_2 x)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x - y) / (z - y);
	double t_2 = t / (z - y);
	double tmp;
	if (t_1 <= 0.998) {
		tmp = t_2 * (x - y);
	} else if (t_1 <= 2e+51) {
		tmp = fma(t, ((z - x) / y), t);
	} else {
		tmp = t_2 * x;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x - y) / Float64(z - y))
	t_2 = Float64(t / Float64(z - y))
	tmp = 0.0
	if (t_1 <= 0.998)
		tmp = Float64(t_2 * Float64(x - y));
	elseif (t_1 <= 2e+51)
		tmp = fma(t, Float64(Float64(z - x) / y), t);
	else
		tmp = Float64(t_2 * x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x - y), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t / N[(z - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 0.998], N[(t$95$2 * N[(x - y), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e+51], N[(t * N[(N[(z - x), $MachinePrecision] / y), $MachinePrecision] + t), $MachinePrecision], N[(t$95$2 * x), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 x y) (-.f64 z y)) < 0.998

   1. Initial program 96.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 88.2

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

   4. Applied rewrites 88.2%

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

   if 0.998 < (/.f64 (-.f64 x y) (-.f64 z y)) < 2e51

   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-rgt-neg-in N/A

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

      9. mul-1-neg N/A

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

      10. lower-fma.f64 N/A

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

   5. Applied rewrites 98.8%

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

   if 2e51 < (/.f64 (-.f64 x y) (-.f64 z y))

   1. Initial program 90.8%

      \[\frac{x - y}{z - y} \cdot t \]
   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 95.3

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

   5. Applied rewrites 95.3%

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

Alternative 12: 69.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x - y}{z - y}\\ \mathbf{if}\;t\_1 \leq 0.0002:\\ \;\;\;\;\frac{x}{z} \cdot t\\ \mathbf{elif}\;t\_1 \leq 2:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{y}, t, t\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t}{z} \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (- x y) (- z y))))
   (if (<= t_1 0.0002)
     (* (/ x z) t)
     (if (<= t_1 2.0) (fma (/ z y) t t) (* (/ t z) x)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x - y) / (z - y);
	double tmp;
	if (t_1 <= 0.0002) {
		tmp = (x / z) * t;
	} else if (t_1 <= 2.0) {
		tmp = fma((z / y), t, t);
	} else {
		tmp = (t / z) * x;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x - y) / Float64(z - y))
	tmp = 0.0
	if (t_1 <= 0.0002)
		tmp = Float64(Float64(x / z) * t);
	elseif (t_1 <= 2.0)
		tmp = fma(Float64(z / y), t, t);
	else
		tmp = Float64(Float64(t / z) * x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x - y), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 0.0002], N[(N[(x / z), $MachinePrecision] * t), $MachinePrecision], If[LessEqual[t$95$1, 2.0], N[(N[(z / y), $MachinePrecision] * t + t), $MachinePrecision], N[(N[(t / z), $MachinePrecision] * x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 x y) (-.f64 z y)) < 2.0000000000000001e-4

   1. Initial program 96.7%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 55.9

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

   5. Applied rewrites 55.9%

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

   if 2.0000000000000001e-4 < (/.f64 (-.f64 x y) (-.f64 z y)) < 2

   1. Initial program 99.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. frac-2neg N/A

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

      5. lower-/.f64 N/A

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

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

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

      7. lower-*.f64 N/A

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

      8. neg-sub0 N/A

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

      9. lift--.f64 N/A

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

      10. sub-neg N/A

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

      11. +-commutative N/A

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

      12. associate--r+ N/A

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

      13. neg-sub0 N/A

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

      14. remove-double-neg N/A

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

      15. lower--.f64 N/A

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

      16. neg-sub0 N/A

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

      17. lift--.f64 N/A

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

      18. sub-neg N/A

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

      19. +-commutative N/A

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

      20. associate--r+ N/A

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

      21. neg-sub0 N/A

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

      22. remove-double-neg N/A

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

      23. lower--.f64 69.5

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

   4. Applied rewrites 69.5%

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

   5. Taylor expanded in x around 0

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 97.2

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

   7. Applied rewrites 97.2%

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

   8. Taylor expanded in z around 0

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

   10. Applied rewrites 96.5%

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

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

   1. Initial program 91.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. frac-2neg N/A

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

      8. lower-/.f64 N/A

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

      9. neg-sub0 N/A

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

      10. lift--.f64 N/A

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

      11. sub-neg N/A

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

      12. +-commutative N/A

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

      13. associate--r+ N/A

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

      14. neg-sub0 N/A

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

      15. remove-double-neg N/A

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

      16. lower--.f64 N/A

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

      17. neg-sub0 N/A

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

      18. lift--.f64 N/A

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

      19. sub-neg N/A

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

      20. +-commutative N/A

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

      21. associate--r+ N/A

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

      22. neg-sub0 N/A

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

      23. remove-double-neg N/A

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

      24. lower--.f64 92.4

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

   4. Applied rewrites 92.4%

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

   5. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 47.3

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

   7. Applied rewrites 47.3%

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

   9. Applied rewrites 47.4%

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

Alternative 13: 69.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x - y}{z - y}\\ \mathbf{if}\;t\_1 \leq 2 \cdot 10^{-15}:\\ \;\;\;\;\frac{x}{z} \cdot t\\ \mathbf{elif}\;t\_1 \leq 2:\\ \;\;\;\;1 \cdot t\\ \mathbf{else}:\\ \;\;\;\;\frac{t}{z} \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (- x y) (- z y))))
   (if (<= t_1 2e-15)
     (* (/ x z) t)
     (if (<= t_1 2.0) (* 1.0 t) (* (/ t z) x)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x - y) / (z - y);
	double tmp;
	if (t_1 <= 2e-15) {
		tmp = (x / z) * t;
	} else if (t_1 <= 2.0) {
		tmp = 1.0 * t;
	} else {
		tmp = (t / z) * x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (x - y) / (z - y)
    if (t_1 <= 2d-15) then
        tmp = (x / z) * t
    else if (t_1 <= 2.0d0) then
        tmp = 1.0d0 * t
    else
        tmp = (t / z) * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (x - y) / (z - y);
	double tmp;
	if (t_1 <= 2e-15) {
		tmp = (x / z) * t;
	} else if (t_1 <= 2.0) {
		tmp = 1.0 * t;
	} else {
		tmp = (t / z) * x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (x - y) / (z - y)
	tmp = 0
	if t_1 <= 2e-15:
		tmp = (x / z) * t
	elif t_1 <= 2.0:
		tmp = 1.0 * t
	else:
		tmp = (t / z) * x
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x - y) / Float64(z - y))
	tmp = 0.0
	if (t_1 <= 2e-15)
		tmp = Float64(Float64(x / z) * t);
	elseif (t_1 <= 2.0)
		tmp = Float64(1.0 * t);
	else
		tmp = Float64(Float64(t / z) * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (x - y) / (z - y);
	tmp = 0.0;
	if (t_1 <= 2e-15)
		tmp = (x / z) * t;
	elseif (t_1 <= 2.0)
		tmp = 1.0 * t;
	else
		tmp = (t / z) * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x - y), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 2e-15], N[(N[(x / z), $MachinePrecision] * t), $MachinePrecision], If[LessEqual[t$95$1, 2.0], N[(1.0 * t), $MachinePrecision], N[(N[(t / z), $MachinePrecision] * x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 96.7%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 56.7

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

   5. Applied rewrites 56.7%

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

   if 2.0000000000000002e-15 < (/.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 93.6%

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

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

   1. Initial program 91.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. frac-2neg N/A

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

      8. lower-/.f64 N/A

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

      9. neg-sub0 N/A

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

      10. lift--.f64 N/A

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

      11. sub-neg N/A

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

      12. +-commutative N/A

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

      13. associate--r+ N/A

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

      14. neg-sub0 N/A

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

      15. remove-double-neg N/A

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

      16. lower--.f64 N/A

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

      17. neg-sub0 N/A

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

      18. lift--.f64 N/A

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

      19. sub-neg N/A

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

      20. +-commutative N/A

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

      21. associate--r+ N/A

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

      22. neg-sub0 N/A

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

      23. remove-double-neg N/A

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

      24. lower--.f64 92.4

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

   4. Applied rewrites 92.4%

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

   5. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 47.3

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

   7. Applied rewrites 47.3%

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

   9. Applied rewrites 47.4%

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

Alternative 14: 67.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x - y}{z - y}\\ \mathbf{if}\;t\_1 \leq 2 \cdot 10^{-15}:\\ \;\;\;\;\frac{x \cdot t}{z}\\ \mathbf{elif}\;t\_1 \leq 2:\\ \;\;\;\;1 \cdot t\\ \mathbf{else}:\\ \;\;\;\;\frac{t}{z} \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (- x y) (- z y))))
   (if (<= t_1 2e-15)
     (/ (* x t) z)
     (if (<= t_1 2.0) (* 1.0 t) (* (/ t z) x)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x - y) / (z - y);
	double tmp;
	if (t_1 <= 2e-15) {
		tmp = (x * t) / z;
	} else if (t_1 <= 2.0) {
		tmp = 1.0 * t;
	} else {
		tmp = (t / z) * x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (x - y) / (z - y)
    if (t_1 <= 2d-15) then
        tmp = (x * t) / z
    else if (t_1 <= 2.0d0) then
        tmp = 1.0d0 * t
    else
        tmp = (t / z) * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (x - y) / (z - y);
	double tmp;
	if (t_1 <= 2e-15) {
		tmp = (x * t) / z;
	} else if (t_1 <= 2.0) {
		tmp = 1.0 * t;
	} else {
		tmp = (t / z) * x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (x - y) / (z - y)
	tmp = 0
	if t_1 <= 2e-15:
		tmp = (x * t) / z
	elif t_1 <= 2.0:
		tmp = 1.0 * t
	else:
		tmp = (t / z) * x
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x - y) / Float64(z - y))
	tmp = 0.0
	if (t_1 <= 2e-15)
		tmp = Float64(Float64(x * t) / z);
	elseif (t_1 <= 2.0)
		tmp = Float64(1.0 * t);
	else
		tmp = Float64(Float64(t / z) * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (x - y) / (z - y);
	tmp = 0.0;
	if (t_1 <= 2e-15)
		tmp = (x * t) / z;
	elseif (t_1 <= 2.0)
		tmp = 1.0 * t;
	else
		tmp = (t / z) * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x - y), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 2e-15], N[(N[(x * t), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[t$95$1, 2.0], N[(1.0 * t), $MachinePrecision], N[(N[(t / z), $MachinePrecision] * x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 96.7%

      \[\frac{x - y}{z - y} \cdot t \]
   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 55.9

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

   5. Applied rewrites 55.9%

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

   if 2.0000000000000002e-15 < (/.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 93.6%

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

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

   1. Initial program 91.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. frac-2neg N/A

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

      8. lower-/.f64 N/A

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

      9. neg-sub0 N/A

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

      10. lift--.f64 N/A

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

      11. sub-neg N/A

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

      12. +-commutative N/A

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

      13. associate--r+ N/A

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

      14. neg-sub0 N/A

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

      15. remove-double-neg N/A

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

      16. lower--.f64 N/A

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

      17. neg-sub0 N/A

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

      18. lift--.f64 N/A

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

      19. sub-neg N/A

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

      20. +-commutative N/A

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

      21. associate--r+ N/A

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

      22. neg-sub0 N/A

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

      23. remove-double-neg N/A

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

      24. lower--.f64 92.4

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

   4. Applied rewrites 92.4%

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

   5. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 47.3

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

   7. Applied rewrites 47.3%

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

   9. Applied rewrites 47.4%

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

4. Final simplification 66.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{z - y} \leq 2 \cdot 10^{-15}:\\ \;\;\;\;\frac{x \cdot t}{z}\\ \mathbf{elif}\;\frac{x - y}{z - y} \leq 2:\\ \;\;\;\;1 \cdot t\\ \mathbf{else}:\\ \;\;\;\;\frac{t}{z} \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 67.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x - y}{z - y}\\ t_2 := \frac{x \cdot t}{z}\\ \mathbf{if}\;t\_1 \leq 2 \cdot 10^{-15}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 2:\\ \;\;\;\;1 \cdot t\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (- x y) (- z y))) (t_2 (/ (* x t) z)))
   (if (<= t_1 2e-15) 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 = (x - y) / (z - y);
	double t_2 = (x * t) / z;
	double tmp;
	if (t_1 <= 2e-15) {
		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 = (x - y) / (z - y)
    t_2 = (x * t) / z
    if (t_1 <= 2d-15) 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 = (x - y) / (z - y);
	double t_2 = (x * t) / z;
	double tmp;
	if (t_1 <= 2e-15) {
		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 = (x - y) / (z - y)
	t_2 = (x * t) / z
	tmp = 0
	if t_1 <= 2e-15:
		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(x - y) / Float64(z - y))
	t_2 = Float64(Float64(x * t) / z)
	tmp = 0.0
	if (t_1 <= 2e-15)
		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 = (x - y) / (z - y);
	t_2 = (x * t) / z;
	tmp = 0.0;
	if (t_1 <= 2e-15)
		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[(x - y), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x * t), $MachinePrecision] / z), $MachinePrecision]}, If[LessEqual[t$95$1, 2e-15], 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)) < 2.0000000000000002e-15 or 2 < (/.f64 (-.f64 x y) (-.f64 z y))

   1. Initial program 95.3%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 53.5

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

   5. Applied rewrites 53.5%

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

   if 2.0000000000000002e-15 < (/.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 93.6%

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

4. Final simplification 66.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{z - y} \leq 2 \cdot 10^{-15}:\\ \;\;\;\;\frac{x \cdot t}{z}\\ \mathbf{elif}\;\frac{x - y}{z - y} \leq 2:\\ \;\;\;\;1 \cdot t\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot t}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 97.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 96.8%

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

Alternative 17: 35.5% 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 96.8%

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

3. Taylor expanded in y around inf

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

5. Applied rewrites 34.0%

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

Developer Target 1: 97.0% 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 2024276 
(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))