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

Percentage Accurate: 97.0% → 97.4%

Time: 7.6s

Alternatives: 12

Speedup: 0.5×

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: 97.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := t\_m \cdot \frac{x - y}{z - y}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq -2 \cdot 10^{+161}:\\ \;\;\;\;\frac{t\_m}{z - y} \cdot \left(x - y\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x y z t_m)
 :precision binary64
 (let* ((t_2 (* t_m (/ (- x y) (- z y)))))
   (* t_s (if (<= t_2 -2e+161) (* (/ t_m (- z y)) (- x y)) t_2))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double y, double z, double t_m) {
	double t_2 = t_m * ((x - y) / (z - y));
	double tmp;
	if (t_2 <= -2e+161) {
		tmp = (t_m / (z - y)) * (x - y);
	} else {
		tmp = t_2;
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, x, y, z, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t_m
    real(8) :: t_2
    real(8) :: tmp
    t_2 = t_m * ((x - y) / (z - y))
    if (t_2 <= (-2d+161)) then
        tmp = (t_m / (z - y)) * (x - y)
    else
        tmp = t_2
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double y, double z, double t_m) {
	double t_2 = t_m * ((x - y) / (z - y));
	double tmp;
	if (t_2 <= -2e+161) {
		tmp = (t_m / (z - y)) * (x - y);
	} else {
		tmp = t_2;
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, y, z, t_m):
	t_2 = t_m * ((x - y) / (z - y))
	tmp = 0
	if t_2 <= -2e+161:
		tmp = (t_m / (z - y)) * (x - y)
	else:
		tmp = t_2
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, y, z, t_m)
	t_2 = Float64(t_m * Float64(Float64(x - y) / Float64(z - y)))
	tmp = 0.0
	if (t_2 <= -2e+161)
		tmp = Float64(Float64(t_m / Float64(z - y)) * Float64(x - y));
	else
		tmp = t_2;
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, y, z, t_m)
	t_2 = t_m * ((x - y) / (z - y));
	tmp = 0.0;
	if (t_2 <= -2e+161)
		tmp = (t_m / (z - y)) * (x - y);
	else
		tmp = t_2;
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, y_, z_, t$95$m_] := Block[{t$95$2 = N[(t$95$m * N[(N[(x - y), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$2, -2e+161], N[(N[(t$95$m / N[(z - y), $MachinePrecision]), $MachinePrecision] * N[(x - y), $MachinePrecision]), $MachinePrecision], t$95$2]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (/.f64 (-.f64 x y) (-.f64 z y)) t) < -2.0000000000000001e161

   1. Initial program 87.6%

      \[\frac{x - y}{z - y} \cdot t \]
   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 96.7

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

   4. Applied rewrites 96.7%

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

   if -2.0000000000000001e161 < (*.f64 (/.f64 (-.f64 x y) (-.f64 z y)) t)

   1. Initial program 96.8%

      \[\frac{x - y}{z - y} \cdot t \]
   2. Add Preprocessing
3. Recombined 2 regimes into one program.

4. Final simplification 96.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \cdot \frac{x - y}{z - y} \leq -2 \cdot 10^{+161}:\\ \;\;\;\;\frac{t}{z - y} \cdot \left(x - y\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{x - y}{z - y}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 92.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{t\_m}{z - y} \cdot \left(x - y\right)\\ t_3 := \frac{x - y}{z - y}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_3 \leq -2 \cdot 10^{-97}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_3 \leq 5 \cdot 10^{-34}:\\ \;\;\;\;\frac{t\_m \cdot \left(x - y\right)}{z}\\ \mathbf{elif}\;t\_3 \leq 1:\\ \;\;\;\;\frac{y}{y - z} \cdot t\_m\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x y z t_m)
 :precision binary64
 (let* ((t_2 (* (/ t_m (- z y)) (- x y))) (t_3 (/ (- x y) (- z y))))
   (*
    t_s
    (if (<= t_3 -2e-97)
      t_2
      (if (<= t_3 5e-34)
        (/ (* t_m (- x y)) z)
        (if (<= t_3 1.0) (* (/ y (- y z)) t_m) t_2))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double y, double z, double t_m) {
	double t_2 = (t_m / (z - y)) * (x - y);
	double t_3 = (x - y) / (z - y);
	double tmp;
	if (t_3 <= -2e-97) {
		tmp = t_2;
	} else if (t_3 <= 5e-34) {
		tmp = (t_m * (x - y)) / z;
	} else if (t_3 <= 1.0) {
		tmp = (y / (y - z)) * t_m;
	} else {
		tmp = t_2;
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, x, y, z, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t_m
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_2 = (t_m / (z - y)) * (x - y)
    t_3 = (x - y) / (z - y)
    if (t_3 <= (-2d-97)) then
        tmp = t_2
    else if (t_3 <= 5d-34) then
        tmp = (t_m * (x - y)) / z
    else if (t_3 <= 1.0d0) then
        tmp = (y / (y - z)) * t_m
    else
        tmp = t_2
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double y, double z, double t_m) {
	double t_2 = (t_m / (z - y)) * (x - y);
	double t_3 = (x - y) / (z - y);
	double tmp;
	if (t_3 <= -2e-97) {
		tmp = t_2;
	} else if (t_3 <= 5e-34) {
		tmp = (t_m * (x - y)) / z;
	} else if (t_3 <= 1.0) {
		tmp = (y / (y - z)) * t_m;
	} else {
		tmp = t_2;
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, y, z, t_m):
	t_2 = (t_m / (z - y)) * (x - y)
	t_3 = (x - y) / (z - y)
	tmp = 0
	if t_3 <= -2e-97:
		tmp = t_2
	elif t_3 <= 5e-34:
		tmp = (t_m * (x - y)) / z
	elif t_3 <= 1.0:
		tmp = (y / (y - z)) * t_m
	else:
		tmp = t_2
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, y, z, t_m)
	t_2 = Float64(Float64(t_m / Float64(z - y)) * Float64(x - y))
	t_3 = Float64(Float64(x - y) / Float64(z - y))
	tmp = 0.0
	if (t_3 <= -2e-97)
		tmp = t_2;
	elseif (t_3 <= 5e-34)
		tmp = Float64(Float64(t_m * Float64(x - y)) / z);
	elseif (t_3 <= 1.0)
		tmp = Float64(Float64(y / Float64(y - z)) * t_m);
	else
		tmp = t_2;
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, y, z, t_m)
	t_2 = (t_m / (z - y)) * (x - y);
	t_3 = (x - y) / (z - y);
	tmp = 0.0;
	if (t_3 <= -2e-97)
		tmp = t_2;
	elseif (t_3 <= 5e-34)
		tmp = (t_m * (x - y)) / z;
	elseif (t_3 <= 1.0)
		tmp = (y / (y - z)) * t_m;
	else
		tmp = t_2;
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, y_, z_, t$95$m_] := Block[{t$95$2 = N[(N[(t$95$m / N[(z - y), $MachinePrecision]), $MachinePrecision] * N[(x - y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(x - y), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$3, -2e-97], t$95$2, If[LessEqual[t$95$3, 5e-34], N[(N[(t$95$m * N[(x - y), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[t$95$3, 1.0], N[(N[(y / N[(y - z), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision], t$95$2]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 x y) (-.f64 z y)) < -2.00000000000000007e-97 or 1 < (/.f64 (-.f64 x y) (-.f64 z y))

   1. Initial program 93.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 91.2

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

   4. Applied rewrites 91.2%

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

   if -2.00000000000000007e-97 < (/.f64 (-.f64 x y) (-.f64 z y)) < 5.0000000000000003e-34

   1. Initial program 94.0%

      \[\frac{x - y}{z - y} \cdot t \]
   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 99.8

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

   5. Applied rewrites 99.8%

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

   if 5.0000000000000003e-34 < (/.f64 (-.f64 x y) (-.f64 z y)) < 1

   1. Initial program 100.0%

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

      1. lift-*.f64 N/A

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

      2. *-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 99.9

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

   4. Applied rewrites 99.9%

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

   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. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower--.f64 98.8

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

   7. Applied rewrites 98.8%

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

4. Final simplification 95.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{z - y} \leq -2 \cdot 10^{-97}:\\ \;\;\;\;\frac{t}{z - y} \cdot \left(x - y\right)\\ \mathbf{elif}\;\frac{x - y}{z - y} \leq 5 \cdot 10^{-34}:\\ \;\;\;\;\frac{t \cdot \left(x - y\right)}{z}\\ \mathbf{elif}\;\frac{x - y}{z - y} \leq 1:\\ \;\;\;\;\frac{y}{y - z} \cdot t\\ \mathbf{else}:\\ \;\;\;\;\frac{t}{z - y} \cdot \left(x - y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 93.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{t\_m}{z - y} \cdot x\\ t_3 := \frac{x - y}{z - y}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_3 \leq -10000000:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_3 \leq 5 \cdot 10^{-9}:\\ \;\;\;\;\frac{x - y}{z} \cdot t\_m\\ \mathbf{elif}\;t\_3 \leq 400000000:\\ \;\;\;\;\mathsf{fma}\left(t\_m, \frac{z - x}{y}, t\_m\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x y z t_m)
 :precision binary64
 (let* ((t_2 (* (/ t_m (- z y)) x)) (t_3 (/ (- x y) (- z y))))
   (*
    t_s
    (if (<= t_3 -10000000.0)
      t_2
      (if (<= t_3 5e-9)
        (* (/ (- x y) z) t_m)
        (if (<= t_3 400000000.0) (fma t_m (/ (- z x) y) t_m) t_2))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double y, double z, double t_m) {
	double t_2 = (t_m / (z - y)) * x;
	double t_3 = (x - y) / (z - y);
	double tmp;
	if (t_3 <= -10000000.0) {
		tmp = t_2;
	} else if (t_3 <= 5e-9) {
		tmp = ((x - y) / z) * t_m;
	} else if (t_3 <= 400000000.0) {
		tmp = fma(t_m, ((z - x) / y), t_m);
	} else {
		tmp = t_2;
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, y, z, t_m)
	t_2 = Float64(Float64(t_m / Float64(z - y)) * x)
	t_3 = Float64(Float64(x - y) / Float64(z - y))
	tmp = 0.0
	if (t_3 <= -10000000.0)
		tmp = t_2;
	elseif (t_3 <= 5e-9)
		tmp = Float64(Float64(Float64(x - y) / z) * t_m);
	elseif (t_3 <= 400000000.0)
		tmp = fma(t_m, Float64(Float64(z - x) / y), t_m);
	else
		tmp = t_2;
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, y_, z_, t$95$m_] := Block[{t$95$2 = N[(N[(t$95$m / N[(z - y), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]}, Block[{t$95$3 = N[(N[(x - y), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$3, -10000000.0], t$95$2, If[LessEqual[t$95$3, 5e-9], N[(N[(N[(x - y), $MachinePrecision] / z), $MachinePrecision] * t$95$m), $MachinePrecision], If[LessEqual[t$95$3, 400000000.0], N[(t$95$m * N[(N[(z - x), $MachinePrecision] / y), $MachinePrecision] + t$95$m), $MachinePrecision], t$95$2]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 x y) (-.f64 z y)) < -1e7 or 4e8 < (/.f64 (-.f64 x y) (-.f64 z y))

   1. Initial program 91.8%

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

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

   5. Applied rewrites 93.3%

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

   if -1e7 < (/.f64 (-.f64 x y) (-.f64 z y)) < 5.0000000000000001e-9

   1. Initial program 95.8%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 92.5

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

   5. Applied rewrites 92.5%

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

   if 5.0000000000000001e-9 < (/.f64 (-.f64 x y) (-.f64 z y)) < 4e8

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf

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

      1. associate--l+ N/A

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

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

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

      3. div-sub N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

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

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

      7. associate-/l* N/A

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

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

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

      9. mul-1-neg N/A

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

      10. lower-fma.f64 N/A

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

   5. Applied rewrites 96.6%

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

Alternative 4: 93.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{t\_m}{z - y} \cdot x\\ t_3 := \frac{x - y}{z - y}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_3 \leq -10000000:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_3 \leq 5 \cdot 10^{-9}:\\ \;\;\;\;\frac{x - y}{z} \cdot t\_m\\ \mathbf{elif}\;t\_3 \leq 400000000:\\ \;\;\;\;\frac{y - x}{y} \cdot t\_m\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x y z t_m)
 :precision binary64
 (let* ((t_2 (* (/ t_m (- z y)) x)) (t_3 (/ (- x y) (- z y))))
   (*
    t_s
    (if (<= t_3 -10000000.0)
      t_2
      (if (<= t_3 5e-9)
        (* (/ (- x y) z) t_m)
        (if (<= t_3 400000000.0) (* (/ (- y x) y) t_m) t_2))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double y, double z, double t_m) {
	double t_2 = (t_m / (z - y)) * x;
	double t_3 = (x - y) / (z - y);
	double tmp;
	if (t_3 <= -10000000.0) {
		tmp = t_2;
	} else if (t_3 <= 5e-9) {
		tmp = ((x - y) / z) * t_m;
	} else if (t_3 <= 400000000.0) {
		tmp = ((y - x) / y) * t_m;
	} else {
		tmp = t_2;
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, x, y, z, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t_m
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_2 = (t_m / (z - y)) * x
    t_3 = (x - y) / (z - y)
    if (t_3 <= (-10000000.0d0)) then
        tmp = t_2
    else if (t_3 <= 5d-9) then
        tmp = ((x - y) / z) * t_m
    else if (t_3 <= 400000000.0d0) then
        tmp = ((y - x) / y) * t_m
    else
        tmp = t_2
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double y, double z, double t_m) {
	double t_2 = (t_m / (z - y)) * x;
	double t_3 = (x - y) / (z - y);
	double tmp;
	if (t_3 <= -10000000.0) {
		tmp = t_2;
	} else if (t_3 <= 5e-9) {
		tmp = ((x - y) / z) * t_m;
	} else if (t_3 <= 400000000.0) {
		tmp = ((y - x) / y) * t_m;
	} else {
		tmp = t_2;
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, y, z, t_m):
	t_2 = (t_m / (z - y)) * x
	t_3 = (x - y) / (z - y)
	tmp = 0
	if t_3 <= -10000000.0:
		tmp = t_2
	elif t_3 <= 5e-9:
		tmp = ((x - y) / z) * t_m
	elif t_3 <= 400000000.0:
		tmp = ((y - x) / y) * t_m
	else:
		tmp = t_2
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, y, z, t_m)
	t_2 = Float64(Float64(t_m / Float64(z - y)) * x)
	t_3 = Float64(Float64(x - y) / Float64(z - y))
	tmp = 0.0
	if (t_3 <= -10000000.0)
		tmp = t_2;
	elseif (t_3 <= 5e-9)
		tmp = Float64(Float64(Float64(x - y) / z) * t_m);
	elseif (t_3 <= 400000000.0)
		tmp = Float64(Float64(Float64(y - x) / y) * t_m);
	else
		tmp = t_2;
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, y, z, t_m)
	t_2 = (t_m / (z - y)) * x;
	t_3 = (x - y) / (z - y);
	tmp = 0.0;
	if (t_3 <= -10000000.0)
		tmp = t_2;
	elseif (t_3 <= 5e-9)
		tmp = ((x - y) / z) * t_m;
	elseif (t_3 <= 400000000.0)
		tmp = ((y - x) / y) * t_m;
	else
		tmp = t_2;
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, y_, z_, t$95$m_] := Block[{t$95$2 = N[(N[(t$95$m / N[(z - y), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]}, Block[{t$95$3 = N[(N[(x - y), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$3, -10000000.0], t$95$2, If[LessEqual[t$95$3, 5e-9], N[(N[(N[(x - y), $MachinePrecision] / z), $MachinePrecision] * t$95$m), $MachinePrecision], If[LessEqual[t$95$3, 400000000.0], N[(N[(N[(y - x), $MachinePrecision] / y), $MachinePrecision] * t$95$m), $MachinePrecision], t$95$2]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 x y) (-.f64 z y)) < -1e7 or 4e8 < (/.f64 (-.f64 x y) (-.f64 z y))

   1. Initial program 91.8%

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

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

   5. Applied rewrites 93.3%

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

   if -1e7 < (/.f64 (-.f64 x y) (-.f64 z y)) < 5.0000000000000001e-9

   1. Initial program 95.8%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 92.5

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

   5. Applied rewrites 92.5%

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

   if 5.0000000000000001e-9 < (/.f64 (-.f64 x y) (-.f64 z y)) < 4e8

   1. Initial program 100.0%

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

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{\left(-1 \cdot \frac{x - y}{y}\right)} \cdot t \]
   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 95.5

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

   5. Applied rewrites 95.5%

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

Alternative 5: 92.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{t\_m}{z - y} \cdot x\\ t_3 := \frac{x - y}{z - y}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_3 \leq -10000000:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_3 \leq 5 \cdot 10^{-34}:\\ \;\;\;\;\frac{x - y}{z} \cdot t\_m\\ \mathbf{elif}\;t\_3 \leq 5:\\ \;\;\;\;\frac{y}{y - z} \cdot t\_m\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x y z t_m)
 :precision binary64
 (let* ((t_2 (* (/ t_m (- z y)) x)) (t_3 (/ (- x y) (- z y))))
   (*
    t_s
    (if (<= t_3 -10000000.0)
      t_2
      (if (<= t_3 5e-34)
        (* (/ (- x y) z) t_m)
        (if (<= t_3 5.0) (* (/ y (- y z)) t_m) t_2))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double y, double z, double t_m) {
	double t_2 = (t_m / (z - y)) * x;
	double t_3 = (x - y) / (z - y);
	double tmp;
	if (t_3 <= -10000000.0) {
		tmp = t_2;
	} else if (t_3 <= 5e-34) {
		tmp = ((x - y) / z) * t_m;
	} else if (t_3 <= 5.0) {
		tmp = (y / (y - z)) * t_m;
	} else {
		tmp = t_2;
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, x, y, z, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t_m
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_2 = (t_m / (z - y)) * x
    t_3 = (x - y) / (z - y)
    if (t_3 <= (-10000000.0d0)) then
        tmp = t_2
    else if (t_3 <= 5d-34) then
        tmp = ((x - y) / z) * t_m
    else if (t_3 <= 5.0d0) then
        tmp = (y / (y - z)) * t_m
    else
        tmp = t_2
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double y, double z, double t_m) {
	double t_2 = (t_m / (z - y)) * x;
	double t_3 = (x - y) / (z - y);
	double tmp;
	if (t_3 <= -10000000.0) {
		tmp = t_2;
	} else if (t_3 <= 5e-34) {
		tmp = ((x - y) / z) * t_m;
	} else if (t_3 <= 5.0) {
		tmp = (y / (y - z)) * t_m;
	} else {
		tmp = t_2;
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, y, z, t_m):
	t_2 = (t_m / (z - y)) * x
	t_3 = (x - y) / (z - y)
	tmp = 0
	if t_3 <= -10000000.0:
		tmp = t_2
	elif t_3 <= 5e-34:
		tmp = ((x - y) / z) * t_m
	elif t_3 <= 5.0:
		tmp = (y / (y - z)) * t_m
	else:
		tmp = t_2
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, y, z, t_m)
	t_2 = Float64(Float64(t_m / Float64(z - y)) * x)
	t_3 = Float64(Float64(x - y) / Float64(z - y))
	tmp = 0.0
	if (t_3 <= -10000000.0)
		tmp = t_2;
	elseif (t_3 <= 5e-34)
		tmp = Float64(Float64(Float64(x - y) / z) * t_m);
	elseif (t_3 <= 5.0)
		tmp = Float64(Float64(y / Float64(y - z)) * t_m);
	else
		tmp = t_2;
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, y, z, t_m)
	t_2 = (t_m / (z - y)) * x;
	t_3 = (x - y) / (z - y);
	tmp = 0.0;
	if (t_3 <= -10000000.0)
		tmp = t_2;
	elseif (t_3 <= 5e-34)
		tmp = ((x - y) / z) * t_m;
	elseif (t_3 <= 5.0)
		tmp = (y / (y - z)) * t_m;
	else
		tmp = t_2;
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, y_, z_, t$95$m_] := Block[{t$95$2 = N[(N[(t$95$m / N[(z - y), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]}, Block[{t$95$3 = N[(N[(x - y), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$3, -10000000.0], t$95$2, If[LessEqual[t$95$3, 5e-34], N[(N[(N[(x - y), $MachinePrecision] / z), $MachinePrecision] * t$95$m), $MachinePrecision], If[LessEqual[t$95$3, 5.0], N[(N[(y / N[(y - z), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision], t$95$2]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 x y) (-.f64 z y)) < -1e7 or 5 < (/.f64 (-.f64 x y) (-.f64 z y))

   1. Initial program 91.8%

      \[\frac{x - y}{z - y} \cdot t \]
   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 92.9

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

   5. Applied rewrites 92.9%

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

   if -1e7 < (/.f64 (-.f64 x y) (-.f64 z y)) < 5.0000000000000003e-34

   1. Initial program 95.5%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 92.6

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

   5. Applied rewrites 92.6%

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

   if 5.0000000000000003e-34 < (/.f64 (-.f64 x y) (-.f64 z y)) < 5

   1. Initial program 100.0%

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

      1. lift-*.f64 N/A

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

      2. *-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 99.9

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

   4. Applied rewrites 99.9%

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

   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. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower--.f64 95.4

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

   7. Applied rewrites 95.4%

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

4. Final simplification 93.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{z - y} \leq -10000000:\\ \;\;\;\;\frac{t}{z - y} \cdot x\\ \mathbf{elif}\;\frac{x - y}{z - y} \leq 5 \cdot 10^{-34}:\\ \;\;\;\;\frac{x - y}{z} \cdot t\\ \mathbf{elif}\;\frac{x - y}{z - y} \leq 5:\\ \;\;\;\;\frac{y}{y - z} \cdot t\\ \mathbf{else}:\\ \;\;\;\;\frac{t}{z - y} \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 91.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{t\_m}{z - y} \cdot x\\ t_3 := \frac{x - y}{z - y}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_3 \leq -10000000:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_3 \leq 5 \cdot 10^{-34}:\\ \;\;\;\;\frac{t\_m \cdot \left(x - y\right)}{z}\\ \mathbf{elif}\;t\_3 \leq 5:\\ \;\;\;\;\frac{y}{y - z} \cdot t\_m\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x y z t_m)
 :precision binary64
 (let* ((t_2 (* (/ t_m (- z y)) x)) (t_3 (/ (- x y) (- z y))))
   (*
    t_s
    (if (<= t_3 -10000000.0)
      t_2
      (if (<= t_3 5e-34)
        (/ (* t_m (- x y)) z)
        (if (<= t_3 5.0) (* (/ y (- y z)) t_m) t_2))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double y, double z, double t_m) {
	double t_2 = (t_m / (z - y)) * x;
	double t_3 = (x - y) / (z - y);
	double tmp;
	if (t_3 <= -10000000.0) {
		tmp = t_2;
	} else if (t_3 <= 5e-34) {
		tmp = (t_m * (x - y)) / z;
	} else if (t_3 <= 5.0) {
		tmp = (y / (y - z)) * t_m;
	} else {
		tmp = t_2;
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, x, y, z, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t_m
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_2 = (t_m / (z - y)) * x
    t_3 = (x - y) / (z - y)
    if (t_3 <= (-10000000.0d0)) then
        tmp = t_2
    else if (t_3 <= 5d-34) then
        tmp = (t_m * (x - y)) / z
    else if (t_3 <= 5.0d0) then
        tmp = (y / (y - z)) * t_m
    else
        tmp = t_2
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double y, double z, double t_m) {
	double t_2 = (t_m / (z - y)) * x;
	double t_3 = (x - y) / (z - y);
	double tmp;
	if (t_3 <= -10000000.0) {
		tmp = t_2;
	} else if (t_3 <= 5e-34) {
		tmp = (t_m * (x - y)) / z;
	} else if (t_3 <= 5.0) {
		tmp = (y / (y - z)) * t_m;
	} else {
		tmp = t_2;
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, y, z, t_m):
	t_2 = (t_m / (z - y)) * x
	t_3 = (x - y) / (z - y)
	tmp = 0
	if t_3 <= -10000000.0:
		tmp = t_2
	elif t_3 <= 5e-34:
		tmp = (t_m * (x - y)) / z
	elif t_3 <= 5.0:
		tmp = (y / (y - z)) * t_m
	else:
		tmp = t_2
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, y, z, t_m)
	t_2 = Float64(Float64(t_m / Float64(z - y)) * x)
	t_3 = Float64(Float64(x - y) / Float64(z - y))
	tmp = 0.0
	if (t_3 <= -10000000.0)
		tmp = t_2;
	elseif (t_3 <= 5e-34)
		tmp = Float64(Float64(t_m * Float64(x - y)) / z);
	elseif (t_3 <= 5.0)
		tmp = Float64(Float64(y / Float64(y - z)) * t_m);
	else
		tmp = t_2;
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, y, z, t_m)
	t_2 = (t_m / (z - y)) * x;
	t_3 = (x - y) / (z - y);
	tmp = 0.0;
	if (t_3 <= -10000000.0)
		tmp = t_2;
	elseif (t_3 <= 5e-34)
		tmp = (t_m * (x - y)) / z;
	elseif (t_3 <= 5.0)
		tmp = (y / (y - z)) * t_m;
	else
		tmp = t_2;
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, y_, z_, t$95$m_] := Block[{t$95$2 = N[(N[(t$95$m / N[(z - y), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]}, Block[{t$95$3 = N[(N[(x - y), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$3, -10000000.0], t$95$2, If[LessEqual[t$95$3, 5e-34], N[(N[(t$95$m * N[(x - y), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[t$95$3, 5.0], N[(N[(y / N[(y - z), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision], t$95$2]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 x y) (-.f64 z y)) < -1e7 or 5 < (/.f64 (-.f64 x y) (-.f64 z y))

   1. Initial program 91.8%

      \[\frac{x - y}{z - y} \cdot t \]
   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 92.9

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

   5. Applied rewrites 92.9%

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

   if -1e7 < (/.f64 (-.f64 x y) (-.f64 z y)) < 5.0000000000000003e-34

   1. Initial program 95.5%

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

   3. Taylor expanded in z around inf

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

      1. 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 92.0

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

   5. Applied rewrites 92.0%

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

   if 5.0000000000000003e-34 < (/.f64 (-.f64 x y) (-.f64 z y)) < 5

   1. Initial program 100.0%

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

      1. lift-*.f64 N/A

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

      2. *-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 99.9

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

   4. Applied rewrites 99.9%

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

   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. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower--.f64 95.4

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

   7. Applied rewrites 95.4%

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

4. Final simplification 93.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{z - y} \leq -10000000:\\ \;\;\;\;\frac{t}{z - y} \cdot x\\ \mathbf{elif}\;\frac{x - y}{z - y} \leq 5 \cdot 10^{-34}:\\ \;\;\;\;\frac{t \cdot \left(x - y\right)}{z}\\ \mathbf{elif}\;\frac{x - y}{z - y} \leq 5:\\ \;\;\;\;\frac{y}{y - z} \cdot t\\ \mathbf{else}:\\ \;\;\;\;\frac{t}{z - y} \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 90.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{t\_m}{z - y} \cdot x\\ t_3 := \frac{x - y}{z - y}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_3 \leq -10000000:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_3 \leq 2 \cdot 10^{-19}:\\ \;\;\;\;\frac{t\_m \cdot \left(x - y\right)}{z}\\ \mathbf{elif}\;t\_3 \leq 5:\\ \;\;\;\;1 \cdot t\_m\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x y z t_m)
 :precision binary64
 (let* ((t_2 (* (/ t_m (- z y)) x)) (t_3 (/ (- x y) (- z y))))
   (*
    t_s
    (if (<= t_3 -10000000.0)
      t_2
      (if (<= t_3 2e-19)
        (/ (* t_m (- x y)) z)
        (if (<= t_3 5.0) (* 1.0 t_m) t_2))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double y, double z, double t_m) {
	double t_2 = (t_m / (z - y)) * x;
	double t_3 = (x - y) / (z - y);
	double tmp;
	if (t_3 <= -10000000.0) {
		tmp = t_2;
	} else if (t_3 <= 2e-19) {
		tmp = (t_m * (x - y)) / z;
	} else if (t_3 <= 5.0) {
		tmp = 1.0 * t_m;
	} else {
		tmp = t_2;
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, x, y, z, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t_m
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_2 = (t_m / (z - y)) * x
    t_3 = (x - y) / (z - y)
    if (t_3 <= (-10000000.0d0)) then
        tmp = t_2
    else if (t_3 <= 2d-19) then
        tmp = (t_m * (x - y)) / z
    else if (t_3 <= 5.0d0) then
        tmp = 1.0d0 * t_m
    else
        tmp = t_2
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double y, double z, double t_m) {
	double t_2 = (t_m / (z - y)) * x;
	double t_3 = (x - y) / (z - y);
	double tmp;
	if (t_3 <= -10000000.0) {
		tmp = t_2;
	} else if (t_3 <= 2e-19) {
		tmp = (t_m * (x - y)) / z;
	} else if (t_3 <= 5.0) {
		tmp = 1.0 * t_m;
	} else {
		tmp = t_2;
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, y, z, t_m):
	t_2 = (t_m / (z - y)) * x
	t_3 = (x - y) / (z - y)
	tmp = 0
	if t_3 <= -10000000.0:
		tmp = t_2
	elif t_3 <= 2e-19:
		tmp = (t_m * (x - y)) / z
	elif t_3 <= 5.0:
		tmp = 1.0 * t_m
	else:
		tmp = t_2
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, y, z, t_m)
	t_2 = Float64(Float64(t_m / Float64(z - y)) * x)
	t_3 = Float64(Float64(x - y) / Float64(z - y))
	tmp = 0.0
	if (t_3 <= -10000000.0)
		tmp = t_2;
	elseif (t_3 <= 2e-19)
		tmp = Float64(Float64(t_m * Float64(x - y)) / z);
	elseif (t_3 <= 5.0)
		tmp = Float64(1.0 * t_m);
	else
		tmp = t_2;
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, y, z, t_m)
	t_2 = (t_m / (z - y)) * x;
	t_3 = (x - y) / (z - y);
	tmp = 0.0;
	if (t_3 <= -10000000.0)
		tmp = t_2;
	elseif (t_3 <= 2e-19)
		tmp = (t_m * (x - y)) / z;
	elseif (t_3 <= 5.0)
		tmp = 1.0 * t_m;
	else
		tmp = t_2;
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, y_, z_, t$95$m_] := Block[{t$95$2 = N[(N[(t$95$m / N[(z - y), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]}, Block[{t$95$3 = N[(N[(x - y), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$3, -10000000.0], t$95$2, If[LessEqual[t$95$3, 2e-19], N[(N[(t$95$m * N[(x - y), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[t$95$3, 5.0], N[(1.0 * t$95$m), $MachinePrecision], t$95$2]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 x y) (-.f64 z y)) < -1e7 or 5 < (/.f64 (-.f64 x y) (-.f64 z y))

   1. Initial program 91.8%

      \[\frac{x - y}{z - y} \cdot t \]
   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 92.9

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

   5. Applied rewrites 92.9%

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

   if -1e7 < (/.f64 (-.f64 x y) (-.f64 z y)) < 2e-19

   1. Initial program 95.8%

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

   3. Taylor expanded in z around inf

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

      1. 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 92.5

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

   5. Applied rewrites 92.5%

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

   if 2e-19 < (/.f64 (-.f64 x y) (-.f64 z y)) < 5

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf

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

   5. Applied rewrites 91.4%

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

4. Final simplification 92.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{z - y} \leq -10000000:\\ \;\;\;\;\frac{t}{z - y} \cdot x\\ \mathbf{elif}\;\frac{x - y}{z - y} \leq 2 \cdot 10^{-19}:\\ \;\;\;\;\frac{t \cdot \left(x - y\right)}{z}\\ \mathbf{elif}\;\frac{x - y}{z - y} \leq 5:\\ \;\;\;\;1 \cdot t\\ \mathbf{else}:\\ \;\;\;\;\frac{t}{z - y} \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 90.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{t\_m}{z - y} \cdot x\\ t_3 := \frac{x - y}{z - y}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_3 \leq -500000:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_3 \leq 5 \cdot 10^{-9}:\\ \;\;\;\;\frac{t\_m}{z} \cdot \left(x - y\right)\\ \mathbf{elif}\;t\_3 \leq 5:\\ \;\;\;\;1 \cdot t\_m\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x y z t_m)
 :precision binary64
 (let* ((t_2 (* (/ t_m (- z y)) x)) (t_3 (/ (- x y) (- z y))))
   (*
    t_s
    (if (<= t_3 -500000.0)
      t_2
      (if (<= t_3 5e-9)
        (* (/ t_m z) (- x y))
        (if (<= t_3 5.0) (* 1.0 t_m) t_2))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double y, double z, double t_m) {
	double t_2 = (t_m / (z - y)) * x;
	double t_3 = (x - y) / (z - y);
	double tmp;
	if (t_3 <= -500000.0) {
		tmp = t_2;
	} else if (t_3 <= 5e-9) {
		tmp = (t_m / z) * (x - y);
	} else if (t_3 <= 5.0) {
		tmp = 1.0 * t_m;
	} else {
		tmp = t_2;
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, x, y, z, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t_m
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_2 = (t_m / (z - y)) * x
    t_3 = (x - y) / (z - y)
    if (t_3 <= (-500000.0d0)) then
        tmp = t_2
    else if (t_3 <= 5d-9) then
        tmp = (t_m / z) * (x - y)
    else if (t_3 <= 5.0d0) then
        tmp = 1.0d0 * t_m
    else
        tmp = t_2
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double y, double z, double t_m) {
	double t_2 = (t_m / (z - y)) * x;
	double t_3 = (x - y) / (z - y);
	double tmp;
	if (t_3 <= -500000.0) {
		tmp = t_2;
	} else if (t_3 <= 5e-9) {
		tmp = (t_m / z) * (x - y);
	} else if (t_3 <= 5.0) {
		tmp = 1.0 * t_m;
	} else {
		tmp = t_2;
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, y, z, t_m):
	t_2 = (t_m / (z - y)) * x
	t_3 = (x - y) / (z - y)
	tmp = 0
	if t_3 <= -500000.0:
		tmp = t_2
	elif t_3 <= 5e-9:
		tmp = (t_m / z) * (x - y)
	elif t_3 <= 5.0:
		tmp = 1.0 * t_m
	else:
		tmp = t_2
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, y, z, t_m)
	t_2 = Float64(Float64(t_m / Float64(z - y)) * x)
	t_3 = Float64(Float64(x - y) / Float64(z - y))
	tmp = 0.0
	if (t_3 <= -500000.0)
		tmp = t_2;
	elseif (t_3 <= 5e-9)
		tmp = Float64(Float64(t_m / z) * Float64(x - y));
	elseif (t_3 <= 5.0)
		tmp = Float64(1.0 * t_m);
	else
		tmp = t_2;
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, y, z, t_m)
	t_2 = (t_m / (z - y)) * x;
	t_3 = (x - y) / (z - y);
	tmp = 0.0;
	if (t_3 <= -500000.0)
		tmp = t_2;
	elseif (t_3 <= 5e-9)
		tmp = (t_m / z) * (x - y);
	elseif (t_3 <= 5.0)
		tmp = 1.0 * t_m;
	else
		tmp = t_2;
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, y_, z_, t$95$m_] := Block[{t$95$2 = N[(N[(t$95$m / N[(z - y), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]}, Block[{t$95$3 = N[(N[(x - y), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$3, -500000.0], t$95$2, If[LessEqual[t$95$3, 5e-9], N[(N[(t$95$m / z), $MachinePrecision] * N[(x - y), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 5.0], N[(1.0 * t$95$m), $MachinePrecision], t$95$2]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 x y) (-.f64 z y)) < -5e5 or 5 < (/.f64 (-.f64 x y) (-.f64 z y))

   1. Initial program 92.0%

      \[\frac{x - y}{z - y} \cdot t \]
   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 91.6

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

   5. Applied rewrites 91.6%

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

   if -5e5 < (/.f64 (-.f64 x y) (-.f64 z y)) < 5.0000000000000001e-9

   1. Initial program 95.7%

      \[\frac{x - y}{z - y} \cdot t \]
   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 92.4

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

   5. Applied rewrites 92.4%

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

   7. Applied rewrites 86.3%

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

   if 5.0000000000000001e-9 < (/.f64 (-.f64 x y) (-.f64 z y)) < 5

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf

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

   5. Applied rewrites 92.3%

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

Alternative 9: 68.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{t\_m}{z} \cdot x\\ t_3 := \frac{x - y}{z - y}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_3 \leq -0.005:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_3 \leq 5 \cdot 10^{-9}:\\ \;\;\;\;\frac{-y}{z} \cdot t\_m\\ \mathbf{elif}\;t\_3 \leq 400000000:\\ \;\;\;\;1 \cdot t\_m\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x y z t_m)
 :precision binary64
 (let* ((t_2 (* (/ t_m z) x)) (t_3 (/ (- x y) (- z y))))
   (*
    t_s
    (if (<= t_3 -0.005)
      t_2
      (if (<= t_3 5e-9)
        (* (/ (- y) z) t_m)
        (if (<= t_3 400000000.0) (* 1.0 t_m) t_2))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double y, double z, double t_m) {
	double t_2 = (t_m / z) * x;
	double t_3 = (x - y) / (z - y);
	double tmp;
	if (t_3 <= -0.005) {
		tmp = t_2;
	} else if (t_3 <= 5e-9) {
		tmp = (-y / z) * t_m;
	} else if (t_3 <= 400000000.0) {
		tmp = 1.0 * t_m;
	} else {
		tmp = t_2;
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, x, y, z, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t_m
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_2 = (t_m / z) * x
    t_3 = (x - y) / (z - y)
    if (t_3 <= (-0.005d0)) then
        tmp = t_2
    else if (t_3 <= 5d-9) then
        tmp = (-y / z) * t_m
    else if (t_3 <= 400000000.0d0) then
        tmp = 1.0d0 * t_m
    else
        tmp = t_2
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double y, double z, double t_m) {
	double t_2 = (t_m / z) * x;
	double t_3 = (x - y) / (z - y);
	double tmp;
	if (t_3 <= -0.005) {
		tmp = t_2;
	} else if (t_3 <= 5e-9) {
		tmp = (-y / z) * t_m;
	} else if (t_3 <= 400000000.0) {
		tmp = 1.0 * t_m;
	} else {
		tmp = t_2;
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, y, z, t_m):
	t_2 = (t_m / z) * x
	t_3 = (x - y) / (z - y)
	tmp = 0
	if t_3 <= -0.005:
		tmp = t_2
	elif t_3 <= 5e-9:
		tmp = (-y / z) * t_m
	elif t_3 <= 400000000.0:
		tmp = 1.0 * t_m
	else:
		tmp = t_2
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, y, z, t_m)
	t_2 = Float64(Float64(t_m / z) * x)
	t_3 = Float64(Float64(x - y) / Float64(z - y))
	tmp = 0.0
	if (t_3 <= -0.005)
		tmp = t_2;
	elseif (t_3 <= 5e-9)
		tmp = Float64(Float64(Float64(-y) / z) * t_m);
	elseif (t_3 <= 400000000.0)
		tmp = Float64(1.0 * t_m);
	else
		tmp = t_2;
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, y, z, t_m)
	t_2 = (t_m / z) * x;
	t_3 = (x - y) / (z - y);
	tmp = 0.0;
	if (t_3 <= -0.005)
		tmp = t_2;
	elseif (t_3 <= 5e-9)
		tmp = (-y / z) * t_m;
	elseif (t_3 <= 400000000.0)
		tmp = 1.0 * t_m;
	else
		tmp = t_2;
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, y_, z_, t$95$m_] := Block[{t$95$2 = N[(N[(t$95$m / z), $MachinePrecision] * x), $MachinePrecision]}, Block[{t$95$3 = N[(N[(x - y), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$3, -0.005], t$95$2, If[LessEqual[t$95$3, 5e-9], N[(N[((-y) / z), $MachinePrecision] * t$95$m), $MachinePrecision], If[LessEqual[t$95$3, 400000000.0], N[(1.0 * t$95$m), $MachinePrecision], t$95$2]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 x y) (-.f64 z y)) < -0.0050000000000000001 or 4e8 < (/.f64 (-.f64 x y) (-.f64 z y))

   1. Initial program 92.0%

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

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

      3. lower-*.f64 49.2

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

   5. Applied rewrites 49.2%

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

   7. Applied rewrites 55.4%

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

   if -0.0050000000000000001 < (/.f64 (-.f64 x y) (-.f64 z y)) < 5.0000000000000001e-9

   1. Initial program 95.6%

      \[\frac{x - y}{z - y} \cdot t \]
   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 92.3

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

   5. Applied rewrites 92.3%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 69.6%

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

   if 5.0000000000000001e-9 < (/.f64 (-.f64 x y) (-.f64 z y)) < 4e8

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf

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

   5. Applied rewrites 91.4%

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

4. Final simplification 71.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{z - y} \leq -0.005:\\ \;\;\;\;\frac{t}{z} \cdot x\\ \mathbf{elif}\;\frac{x - y}{z - y} \leq 5 \cdot 10^{-9}:\\ \;\;\;\;\frac{-y}{z} \cdot t\\ \mathbf{elif}\;\frac{x - y}{z - y} \leq 400000000:\\ \;\;\;\;1 \cdot t\\ \mathbf{else}:\\ \;\;\;\;\frac{t}{z} \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 78.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{x - y}{z - y}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq 5 \cdot 10^{-9}:\\ \;\;\;\;\frac{t\_m}{z} \cdot \left(x - y\right)\\ \mathbf{elif}\;t\_2 \leq 400000000:\\ \;\;\;\;1 \cdot t\_m\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_m}{z} \cdot x\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x y z t_m)
 :precision binary64
 (let* ((t_2 (/ (- x y) (- z y))))
   (*
    t_s
    (if (<= t_2 5e-9)
      (* (/ t_m z) (- x y))
      (if (<= t_2 400000000.0) (* 1.0 t_m) (* (/ t_m z) x))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double y, double z, double t_m) {
	double t_2 = (x - y) / (z - y);
	double tmp;
	if (t_2 <= 5e-9) {
		tmp = (t_m / z) * (x - y);
	} else if (t_2 <= 400000000.0) {
		tmp = 1.0 * t_m;
	} else {
		tmp = (t_m / z) * x;
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, x, y, z, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t_m
    real(8) :: t_2
    real(8) :: tmp
    t_2 = (x - y) / (z - y)
    if (t_2 <= 5d-9) then
        tmp = (t_m / z) * (x - y)
    else if (t_2 <= 400000000.0d0) then
        tmp = 1.0d0 * t_m
    else
        tmp = (t_m / z) * x
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double y, double z, double t_m) {
	double t_2 = (x - y) / (z - y);
	double tmp;
	if (t_2 <= 5e-9) {
		tmp = (t_m / z) * (x - y);
	} else if (t_2 <= 400000000.0) {
		tmp = 1.0 * t_m;
	} else {
		tmp = (t_m / z) * x;
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, y, z, t_m):
	t_2 = (x - y) / (z - y)
	tmp = 0
	if t_2 <= 5e-9:
		tmp = (t_m / z) * (x - y)
	elif t_2 <= 400000000.0:
		tmp = 1.0 * t_m
	else:
		tmp = (t_m / z) * x
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, y, z, t_m)
	t_2 = Float64(Float64(x - y) / Float64(z - y))
	tmp = 0.0
	if (t_2 <= 5e-9)
		tmp = Float64(Float64(t_m / z) * Float64(x - y));
	elseif (t_2 <= 400000000.0)
		tmp = Float64(1.0 * t_m);
	else
		tmp = Float64(Float64(t_m / z) * x);
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, y, z, t_m)
	t_2 = (x - y) / (z - y);
	tmp = 0.0;
	if (t_2 <= 5e-9)
		tmp = (t_m / z) * (x - y);
	elseif (t_2 <= 400000000.0)
		tmp = 1.0 * t_m;
	else
		tmp = (t_m / z) * x;
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, y_, z_, t$95$m_] := Block[{t$95$2 = N[(N[(x - y), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$2, 5e-9], N[(N[(t$95$m / z), $MachinePrecision] * N[(x - y), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 400000000.0], N[(1.0 * t$95$m), $MachinePrecision], N[(N[(t$95$m / z), $MachinePrecision] * x), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 x y) (-.f64 z y)) < 5.0000000000000001e-9

   1. Initial program 92.9%

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

   3. Taylor expanded in z around inf

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

      1. 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 73.3

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

   5. Applied rewrites 73.3%

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

   7. Applied rewrites 71.2%

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

   if 5.0000000000000001e-9 < (/.f64 (-.f64 x y) (-.f64 z y)) < 4e8

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf

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

   5. Applied rewrites 91.4%

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

   if 4e8 < (/.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. *-commutative N/A

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

      3. lower-*.f64 47.9

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

   5. Applied rewrites 47.9%

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

   7. Applied rewrites 58.5%

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

Alternative 11: 68.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{t\_m}{z} \cdot x\\ t_3 := \frac{x - y}{z - y}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_3 \leq 5 \cdot 10^{-34}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_3 \leq 400000000:\\ \;\;\;\;1 \cdot t\_m\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x y z t_m)
 :precision binary64
 (let* ((t_2 (* (/ t_m z) x)) (t_3 (/ (- x y) (- z y))))
   (* t_s (if (<= t_3 5e-34) t_2 (if (<= t_3 400000000.0) (* 1.0 t_m) t_2)))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double y, double z, double t_m) {
	double t_2 = (t_m / z) * x;
	double t_3 = (x - y) / (z - y);
	double tmp;
	if (t_3 <= 5e-34) {
		tmp = t_2;
	} else if (t_3 <= 400000000.0) {
		tmp = 1.0 * t_m;
	} else {
		tmp = t_2;
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, x, y, z, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t_m
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_2 = (t_m / z) * x
    t_3 = (x - y) / (z - y)
    if (t_3 <= 5d-34) then
        tmp = t_2
    else if (t_3 <= 400000000.0d0) then
        tmp = 1.0d0 * t_m
    else
        tmp = t_2
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double y, double z, double t_m) {
	double t_2 = (t_m / z) * x;
	double t_3 = (x - y) / (z - y);
	double tmp;
	if (t_3 <= 5e-34) {
		tmp = t_2;
	} else if (t_3 <= 400000000.0) {
		tmp = 1.0 * t_m;
	} else {
		tmp = t_2;
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, y, z, t_m):
	t_2 = (t_m / z) * x
	t_3 = (x - y) / (z - y)
	tmp = 0
	if t_3 <= 5e-34:
		tmp = t_2
	elif t_3 <= 400000000.0:
		tmp = 1.0 * t_m
	else:
		tmp = t_2
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, y, z, t_m)
	t_2 = Float64(Float64(t_m / z) * x)
	t_3 = Float64(Float64(x - y) / Float64(z - y))
	tmp = 0.0
	if (t_3 <= 5e-34)
		tmp = t_2;
	elseif (t_3 <= 400000000.0)
		tmp = Float64(1.0 * t_m);
	else
		tmp = t_2;
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, y, z, t_m)
	t_2 = (t_m / z) * x;
	t_3 = (x - y) / (z - y);
	tmp = 0.0;
	if (t_3 <= 5e-34)
		tmp = t_2;
	elseif (t_3 <= 400000000.0)
		tmp = 1.0 * t_m;
	else
		tmp = t_2;
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, y_, z_, t$95$m_] := Block[{t$95$2 = N[(N[(t$95$m / z), $MachinePrecision] * x), $MachinePrecision]}, Block[{t$95$3 = N[(N[(x - y), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$3, 5e-34], t$95$2, If[LessEqual[t$95$3, 400000000.0], N[(1.0 * t$95$m), $MachinePrecision], t$95$2]]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 x y) (-.f64 z y)) < 5.0000000000000003e-34 or 4e8 < (/.f64 (-.f64 x y) (-.f64 z y))

   1. Initial program 93.3%

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

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

      3. lower-*.f64 52.9

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

   5. Applied rewrites 52.9%

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

   7. Applied rewrites 55.8%

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

   if 5.0000000000000003e-34 < (/.f64 (-.f64 x y) (-.f64 z y)) < 4e8

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf

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

   5. Applied rewrites 87.0%

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

Alternative 12: 35.6% accurate, 3.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \left(1 \cdot t\_m\right) \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x y z t_m) :precision binary64 (* t_s (* 1.0 t_m)))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double y, double z, double t_m) {
	return t_s * (1.0 * t_m);
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, x, y, z, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t_m
    code = t_s * (1.0d0 * t_m)
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double y, double z, double t_m) {
	return t_s * (1.0 * t_m);
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, y, z, t_m):
	return t_s * (1.0 * t_m)

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, y, z, t_m)
	return Float64(t_s * Float64(1.0 * t_m))
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp = code(t_s, x, y, z, t_m)
	tmp = t_s * (1.0 * t_m);
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, y_, z_, t$95$m_] := N[(t$95$s * N[(1.0 * t$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 95.7%

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

3. Taylor expanded in y around inf

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

5. Applied rewrites 34.1%

   \[\leadsto \color{blue}{1} \cdot t \]
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 2024277 
(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))