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

Percentage Accurate: 96.9% → 96.8%

Time: 9.7s

Alternatives: 15

Speedup: N/A×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 96.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 96.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 6 \cdot 10^{+23}:\\ \;\;\;\;\frac{t\_m \cdot \left(x - y\right)}{z - y}\\ \mathbf{else}:\\ \;\;\;\;\left(x - y\right) \cdot \frac{t\_m}{z - y}\\ \end{array} \end{array} \]

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
 (*
  t_s
  (if (<= t_m 6e+23) (/ (* t_m (- x y)) (- z y)) (* (- x y) (/ t_m (- z y))))))

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 tmp;
	if (t_m <= 6e+23) {
		tmp = (t_m * (x - y)) / (z - y);
	} else {
		tmp = (x - y) * (t_m / (z - y));
	}
	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) :: tmp
    if (t_m <= 6d+23) then
        tmp = (t_m * (x - y)) / (z - y)
    else
        tmp = (x - y) * (t_m / (z - y))
    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 tmp;
	if (t_m <= 6e+23) {
		tmp = (t_m * (x - y)) / (z - y);
	} else {
		tmp = (x - y) * (t_m / (z - y));
	}
	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):
	tmp = 0
	if t_m <= 6e+23:
		tmp = (t_m * (x - y)) / (z - y)
	else:
		tmp = (x - y) * (t_m / (z - y))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, y, z, t_m)
	tmp = 0.0
	if (t_m <= 6e+23)
		tmp = Float64(Float64(t_m * Float64(x - y)) / Float64(z - y));
	else
		tmp = Float64(Float64(x - y) * Float64(t_m / Float64(z - y)));
	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)
	tmp = 0.0;
	if (t_m <= 6e+23)
		tmp = (t_m * (x - y)) / (z - y);
	else
		tmp = (x - y) * (t_m / (z - y));
	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_] := N[(t$95$s * If[LessEqual[t$95$m, 6e+23], N[(N[(t$95$m * N[(x - y), $MachinePrecision]), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision], N[(N[(x - y), $MachinePrecision] * N[(t$95$m / N[(z - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 6.0000000000000002e23

   1. Initial program 97.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 86.8

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

   4. Applied rewrites 86.8%

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

   if 6.0000000000000002e23 < t

   1. Initial program 97.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. 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 99.8

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

   4. Applied rewrites 99.8%

      \[\leadsto \color{blue}{\frac{t}{z - y} \cdot \left(x - y\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 90.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 6 \cdot 10^{+23}:\\ \;\;\;\;\frac{t \cdot \left(x - y\right)}{z - y}\\ \mathbf{else}:\\ \;\;\;\;\left(x - y\right) \cdot \frac{t}{z - y}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 79.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := t\_m \cdot \frac{x}{-y}\\ t_3 := \frac{x - y}{z - y}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_3 \leq -10000000:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_3 \leq 10^{-21}:\\ \;\;\;\;\left(x - y\right) \cdot \frac{t\_m}{z}\\ \mathbf{elif}\;t\_3 \leq 20000000:\\ \;\;\;\;t\_m \cdot \frac{y}{y - z}\\ \mathbf{elif}\;t\_3 \leq 5 \cdot 10^{+173}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_m \cdot x}{z}\\ \end{array} \end{array} \end{array} \]

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)))) (t_3 (/ (- x y) (- z y))))
   (*
    t_s
    (if (<= t_3 -10000000.0)
      t_2
      (if (<= t_3 1e-21)
        (* (- x y) (/ t_m z))
        (if (<= t_3 20000000.0)
          (* t_m (/ y (- y z)))
          (if (<= t_3 5e+173) t_2 (/ (* t_m x) z))))))))

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);
	double t_3 = (x - y) / (z - y);
	double tmp;
	if (t_3 <= -10000000.0) {
		tmp = t_2;
	} else if (t_3 <= 1e-21) {
		tmp = (x - y) * (t_m / z);
	} else if (t_3 <= 20000000.0) {
		tmp = t_m * (y / (y - z));
	} else if (t_3 <= 5e+173) {
		tmp = t_2;
	} else {
		tmp = (t_m * x) / z;
	}
	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 * (x / -y)
    t_3 = (x - y) / (z - y)
    if (t_3 <= (-10000000.0d0)) then
        tmp = t_2
    else if (t_3 <= 1d-21) then
        tmp = (x - y) * (t_m / z)
    else if (t_3 <= 20000000.0d0) then
        tmp = t_m * (y / (y - z))
    else if (t_3 <= 5d+173) then
        tmp = t_2
    else
        tmp = (t_m * x) / z
    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);
	double t_3 = (x - y) / (z - y);
	double tmp;
	if (t_3 <= -10000000.0) {
		tmp = t_2;
	} else if (t_3 <= 1e-21) {
		tmp = (x - y) * (t_m / z);
	} else if (t_3 <= 20000000.0) {
		tmp = t_m * (y / (y - z));
	} else if (t_3 <= 5e+173) {
		tmp = t_2;
	} else {
		tmp = (t_m * x) / z;
	}
	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)
	t_3 = (x - y) / (z - y)
	tmp = 0
	if t_3 <= -10000000.0:
		tmp = t_2
	elif t_3 <= 1e-21:
		tmp = (x - y) * (t_m / z)
	elif t_3 <= 20000000.0:
		tmp = t_m * (y / (y - z))
	elif t_3 <= 5e+173:
		tmp = t_2
	else:
		tmp = (t_m * x) / z
	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(x / Float64(-y)))
	t_3 = Float64(Float64(x - y) / Float64(z - y))
	tmp = 0.0
	if (t_3 <= -10000000.0)
		tmp = t_2;
	elseif (t_3 <= 1e-21)
		tmp = Float64(Float64(x - y) * Float64(t_m / z));
	elseif (t_3 <= 20000000.0)
		tmp = Float64(t_m * Float64(y / Float64(y - z)));
	elseif (t_3 <= 5e+173)
		tmp = t_2;
	else
		tmp = Float64(Float64(t_m * x) / z);
	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);
	t_3 = (x - y) / (z - y);
	tmp = 0.0;
	if (t_3 <= -10000000.0)
		tmp = t_2;
	elseif (t_3 <= 1e-21)
		tmp = (x - y) * (t_m / z);
	elseif (t_3 <= 20000000.0)
		tmp = t_m * (y / (y - z));
	elseif (t_3 <= 5e+173)
		tmp = t_2;
	else
		tmp = (t_m * x) / z;
	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[(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, -10000000.0], t$95$2, If[LessEqual[t$95$3, 1e-21], N[(N[(x - y), $MachinePrecision] * N[(t$95$m / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 20000000.0], N[(t$95$m * N[(y / N[(y - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 5e+173], t$95$2, N[(N[(t$95$m * x), $MachinePrecision] / z), $MachinePrecision]]]]]), $MachinePrecision]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (-.f64 x y) (-.f64 z y)) < -1e7 or 2e7 < (/.f64 (-.f64 x y) (-.f64 z y)) < 5.00000000000000034e173

   1. Initial program 97.1%

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

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

      2. sub-neg N/A

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

      3. *-inverses N/A

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

      4. metadata-eval N/A

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

      5. distribute-lft-in N/A

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

      6. metadata-eval N/A

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

      7. +-commutative N/A

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

      8. mul-1-neg N/A

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

      9. unsub-neg N/A

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

      10. lower--.f64 N/A

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

      11. lower-/.f64 71.5

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

   5. Applied rewrites 71.5%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 69.6%

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

   if -1e7 < (/.f64 (-.f64 x y) (-.f64 z y)) < 9.99999999999999908e-22

   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. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 89.4

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

   5. Applied rewrites 89.4%

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

   if 9.99999999999999908e-22 < (/.f64 (-.f64 x y) (-.f64 z y)) < 2e7

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

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

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

      5. mul-1-neg N/A

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

      6. lower-*.f64 N/A

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

      7. mul-1-neg N/A

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

      8. distribute-neg-frac2 N/A

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

      9. lower-/.f64 N/A

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

      10. sub-neg N/A

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

      11. +-commutative N/A

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

      12. distribute-neg-in N/A

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

      13. remove-double-neg N/A

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

      14. lower-+.f64 N/A

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

      15. lower-neg.f64 75.5

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

   5. Applied rewrites 75.5%

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

   7. Applied rewrites 98.0%

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

   if 5.00000000000000034e173 < (/.f64 (-.f64 x y) (-.f64 z y))

   1. Initial program 91.9%

      \[\frac{x - y}{z - y} \cdot t \]
   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 80.9

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

   5. Applied rewrites 80.9%

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

4. Final simplification 86.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{z - y} \leq -10000000:\\ \;\;\;\;t \cdot \frac{x}{-y}\\ \mathbf{elif}\;\frac{x - y}{z - y} \leq 10^{-21}:\\ \;\;\;\;\left(x - y\right) \cdot \frac{t}{z}\\ \mathbf{elif}\;\frac{x - y}{z - y} \leq 20000000:\\ \;\;\;\;t \cdot \frac{y}{y - z}\\ \mathbf{elif}\;\frac{x - y}{z - y} \leq 5 \cdot 10^{+173}:\\ \;\;\;\;t \cdot \frac{x}{-y}\\ \mathbf{else}:\\ \;\;\;\;\frac{t \cdot x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 78.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := t\_m \cdot \frac{x}{-y}\\ t_3 := \frac{x - y}{z - y}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_3 \leq -10000000:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_3 \leq 0.0004:\\ \;\;\;\;\left(x - y\right) \cdot \frac{t\_m}{z}\\ \mathbf{elif}\;t\_3 \leq 2:\\ \;\;\;\;\mathsf{fma}\left(t\_m, \frac{z}{y}, t\_m\right)\\ \mathbf{elif}\;t\_3 \leq 5 \cdot 10^{+173}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_m \cdot x}{z}\\ \end{array} \end{array} \end{array} \]

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)))) (t_3 (/ (- x y) (- z y))))
   (*
    t_s
    (if (<= t_3 -10000000.0)
      t_2
      (if (<= t_3 0.0004)
        (* (- x y) (/ t_m z))
        (if (<= t_3 2.0)
          (fma t_m (/ z y) t_m)
          (if (<= t_3 5e+173) t_2 (/ (* t_m x) z))))))))

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);
	double t_3 = (x - y) / (z - y);
	double tmp;
	if (t_3 <= -10000000.0) {
		tmp = t_2;
	} else if (t_3 <= 0.0004) {
		tmp = (x - y) * (t_m / z);
	} else if (t_3 <= 2.0) {
		tmp = fma(t_m, (z / y), t_m);
	} else if (t_3 <= 5e+173) {
		tmp = t_2;
	} else {
		tmp = (t_m * x) / z;
	}
	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(x / Float64(-y)))
	t_3 = Float64(Float64(x - y) / Float64(z - y))
	tmp = 0.0
	if (t_3 <= -10000000.0)
		tmp = t_2;
	elseif (t_3 <= 0.0004)
		tmp = Float64(Float64(x - y) * Float64(t_m / z));
	elseif (t_3 <= 2.0)
		tmp = fma(t_m, Float64(z / y), t_m);
	elseif (t_3 <= 5e+173)
		tmp = t_2;
	else
		tmp = Float64(Float64(t_m * x) / z);
	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[(t$95$m * 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, -10000000.0], t$95$2, If[LessEqual[t$95$3, 0.0004], N[(N[(x - y), $MachinePrecision] * N[(t$95$m / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 2.0], N[(t$95$m * N[(z / y), $MachinePrecision] + t$95$m), $MachinePrecision], If[LessEqual[t$95$3, 5e+173], t$95$2, N[(N[(t$95$m * x), $MachinePrecision] / z), $MachinePrecision]]]]]), $MachinePrecision]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (-.f64 x y) (-.f64 z y)) < -1e7 or 2 < (/.f64 (-.f64 x y) (-.f64 z y)) < 5.00000000000000034e173

   1. Initial program 97.1%

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

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

      2. sub-neg N/A

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

      3. *-inverses N/A

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

      4. metadata-eval N/A

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

      5. distribute-lft-in N/A

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

      6. metadata-eval N/A

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

      7. +-commutative N/A

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

      8. mul-1-neg N/A

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

      9. unsub-neg N/A

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

      10. lower--.f64 N/A

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

      11. lower-/.f64 70.5

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

   5. Applied rewrites 70.5%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 68.6%

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

   if -1e7 < (/.f64 (-.f64 x y) (-.f64 z y)) < 4.00000000000000019e-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. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 86.7

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

   5. Applied rewrites 86.7%

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

   if 4.00000000000000019e-4 < (/.f64 (-.f64 x y) (-.f64 z y)) < 2

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

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

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

      5. mul-1-neg N/A

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

      6. lower-*.f64 N/A

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

      7. mul-1-neg N/A

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

      8. distribute-neg-frac2 N/A

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

      9. lower-/.f64 N/A

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

      10. sub-neg N/A

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

      11. +-commutative N/A

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

      12. distribute-neg-in N/A

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

      13. remove-double-neg N/A

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

      14. lower-+.f64 N/A

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

      15. lower-neg.f64 76.3

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

   5. Applied rewrites 76.3%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 98.7%

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

   if 5.00000000000000034e173 < (/.f64 (-.f64 x y) (-.f64 z y))

   1. Initial program 91.9%

      \[\frac{x - y}{z - y} \cdot t \]
   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 80.9

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

   5. Applied rewrites 80.9%

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

4. Final simplification 85.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{z - y} \leq -10000000:\\ \;\;\;\;t \cdot \frac{x}{-y}\\ \mathbf{elif}\;\frac{x - y}{z - y} \leq 0.0004:\\ \;\;\;\;\left(x - y\right) \cdot \frac{t}{z}\\ \mathbf{elif}\;\frac{x - y}{z - y} \leq 2:\\ \;\;\;\;\mathsf{fma}\left(t, \frac{z}{y}, t\right)\\ \mathbf{elif}\;\frac{x - y}{z - y} \leq 5 \cdot 10^{+173}:\\ \;\;\;\;t \cdot \frac{x}{-y}\\ \mathbf{else}:\\ \;\;\;\;\frac{t \cdot x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 69.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := t\_m \cdot \frac{x}{-y}\\ t_3 := \frac{x - y}{z - y}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_3 \leq -10000000:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_3 \leq 0.0004:\\ \;\;\;\;t\_m \cdot \frac{x}{z}\\ \mathbf{elif}\;t\_3 \leq 2:\\ \;\;\;\;\mathsf{fma}\left(t\_m, \frac{z}{y}, t\_m\right)\\ \mathbf{elif}\;t\_3 \leq 5 \cdot 10^{+173}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_m \cdot x}{z}\\ \end{array} \end{array} \end{array} \]

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)))) (t_3 (/ (- x y) (- z y))))
   (*
    t_s
    (if (<= t_3 -10000000.0)
      t_2
      (if (<= t_3 0.0004)
        (* t_m (/ x z))
        (if (<= t_3 2.0)
          (fma t_m (/ z y) t_m)
          (if (<= t_3 5e+173) t_2 (/ (* t_m x) z))))))))

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);
	double t_3 = (x - y) / (z - y);
	double tmp;
	if (t_3 <= -10000000.0) {
		tmp = t_2;
	} else if (t_3 <= 0.0004) {
		tmp = t_m * (x / z);
	} else if (t_3 <= 2.0) {
		tmp = fma(t_m, (z / y), t_m);
	} else if (t_3 <= 5e+173) {
		tmp = t_2;
	} else {
		tmp = (t_m * x) / z;
	}
	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(x / Float64(-y)))
	t_3 = Float64(Float64(x - y) / Float64(z - y))
	tmp = 0.0
	if (t_3 <= -10000000.0)
		tmp = t_2;
	elseif (t_3 <= 0.0004)
		tmp = Float64(t_m * Float64(x / z));
	elseif (t_3 <= 2.0)
		tmp = fma(t_m, Float64(z / y), t_m);
	elseif (t_3 <= 5e+173)
		tmp = t_2;
	else
		tmp = Float64(Float64(t_m * x) / z);
	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[(t$95$m * 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, -10000000.0], t$95$2, If[LessEqual[t$95$3, 0.0004], N[(t$95$m * N[(x / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 2.0], N[(t$95$m * N[(z / y), $MachinePrecision] + t$95$m), $MachinePrecision], If[LessEqual[t$95$3, 5e+173], t$95$2, N[(N[(t$95$m * x), $MachinePrecision] / z), $MachinePrecision]]]]]), $MachinePrecision]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (-.f64 x y) (-.f64 z y)) < -1e7 or 2 < (/.f64 (-.f64 x y) (-.f64 z y)) < 5.00000000000000034e173

   1. Initial program 97.1%

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

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

      2. sub-neg N/A

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

      3. *-inverses N/A

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

      4. metadata-eval N/A

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

      5. distribute-lft-in N/A

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

      6. metadata-eval N/A

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

      7. +-commutative N/A

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

      8. mul-1-neg N/A

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

      9. unsub-neg N/A

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

      10. lower--.f64 N/A

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

      11. lower-/.f64 70.5

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

   5. Applied rewrites 70.5%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 68.6%

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

   if -1e7 < (/.f64 (-.f64 x y) (-.f64 z y)) < 4.00000000000000019e-4

   1. Initial program 95.2%

      \[\frac{x - y}{z - y} \cdot t \]
   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 69.2

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

   5. Applied rewrites 69.2%

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

   if 4.00000000000000019e-4 < (/.f64 (-.f64 x y) (-.f64 z y)) < 2

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

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

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

      5. mul-1-neg N/A

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

      6. lower-*.f64 N/A

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

      7. mul-1-neg N/A

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

      8. distribute-neg-frac2 N/A

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

      9. lower-/.f64 N/A

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

      10. sub-neg N/A

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

      11. +-commutative N/A

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

      12. distribute-neg-in N/A

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

      13. remove-double-neg N/A

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

      14. lower-+.f64 N/A

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

      15. lower-neg.f64 76.3

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

   5. Applied rewrites 76.3%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 98.7%

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

   if 5.00000000000000034e173 < (/.f64 (-.f64 x y) (-.f64 z y))

   1. Initial program 91.9%

      \[\frac{x - y}{z - y} \cdot t \]
   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 80.9

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

   5. Applied rewrites 80.9%

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

4. Final simplification 79.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{z - y} \leq -10000000:\\ \;\;\;\;t \cdot \frac{x}{-y}\\ \mathbf{elif}\;\frac{x - y}{z - y} \leq 0.0004:\\ \;\;\;\;t \cdot \frac{x}{z}\\ \mathbf{elif}\;\frac{x - y}{z - y} \leq 2:\\ \;\;\;\;\mathsf{fma}\left(t, \frac{z}{y}, t\right)\\ \mathbf{elif}\;\frac{x - y}{z - y} \leq 5 \cdot 10^{+173}:\\ \;\;\;\;t \cdot \frac{x}{-y}\\ \mathbf{else}:\\ \;\;\;\;\frac{t \cdot x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 94.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{x - y}{z - y}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq -1 \cdot 10^{-5}:\\ \;\;\;\;\left(x - y\right) \cdot \frac{t\_m}{z - y}\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{-21}:\\ \;\;\;\;t\_m \cdot \frac{x - y}{z}\\ \mathbf{elif}\;t\_2 \leq 2:\\ \;\;\;\;t\_m \cdot \frac{y}{y - z}\\ \mathbf{else}:\\ \;\;\;\;t\_m \cdot \frac{x}{z - y}\\ \end{array} \end{array} \end{array} \]

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 -1e-5)
      (* (- x y) (/ t_m (- z y)))
      (if (<= t_2 2e-21)
        (* t_m (/ (- x y) z))
        (if (<= t_2 2.0) (* t_m (/ y (- y z))) (* t_m (/ x (- z y)))))))))

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 <= -1e-5) {
		tmp = (x - y) * (t_m / (z - y));
	} else if (t_2 <= 2e-21) {
		tmp = t_m * ((x - y) / z);
	} else if (t_2 <= 2.0) {
		tmp = t_m * (y / (y - z));
	} else {
		tmp = t_m * (x / (z - y));
	}
	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 <= (-1d-5)) then
        tmp = (x - y) * (t_m / (z - y))
    else if (t_2 <= 2d-21) then
        tmp = t_m * ((x - y) / z)
    else if (t_2 <= 2.0d0) then
        tmp = t_m * (y / (y - z))
    else
        tmp = t_m * (x / (z - y))
    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 <= -1e-5) {
		tmp = (x - y) * (t_m / (z - y));
	} else if (t_2 <= 2e-21) {
		tmp = t_m * ((x - y) / z);
	} else if (t_2 <= 2.0) {
		tmp = t_m * (y / (y - z));
	} else {
		tmp = t_m * (x / (z - y));
	}
	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 <= -1e-5:
		tmp = (x - y) * (t_m / (z - y))
	elif t_2 <= 2e-21:
		tmp = t_m * ((x - y) / z)
	elif t_2 <= 2.0:
		tmp = t_m * (y / (y - z))
	else:
		tmp = t_m * (x / (z - y))
	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 <= -1e-5)
		tmp = Float64(Float64(x - y) * Float64(t_m / Float64(z - y)));
	elseif (t_2 <= 2e-21)
		tmp = Float64(t_m * Float64(Float64(x - y) / z));
	elseif (t_2 <= 2.0)
		tmp = Float64(t_m * Float64(y / Float64(y - z)));
	else
		tmp = Float64(t_m * Float64(x / Float64(z - y)));
	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 <= -1e-5)
		tmp = (x - y) * (t_m / (z - y));
	elseif (t_2 <= 2e-21)
		tmp = t_m * ((x - y) / z);
	elseif (t_2 <= 2.0)
		tmp = t_m * (y / (y - z));
	else
		tmp = t_m * (x / (z - y));
	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, -1e-5], N[(N[(x - y), $MachinePrecision] * N[(t$95$m / N[(z - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 2e-21], N[(t$95$m * N[(N[(x - y), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 2.0], N[(t$95$m * N[(y / N[(y - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$m * N[(x / N[(z - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (-.f64 x y) (-.f64 z y)) < -1.00000000000000008e-5

   1. Initial program 95.4%

      \[\frac{x - y}{z - y} \cdot t \]
   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 95.5

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

   4. Applied rewrites 95.5%

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

   if -1.00000000000000008e-5 < (/.f64 (-.f64 x y) (-.f64 z y)) < 1.99999999999999982e-21

   1. Initial program 94.9%

      \[\frac{x - y}{z - y} \cdot t \]
   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 93.8

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

   5. Applied rewrites 93.8%

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

   if 1.99999999999999982e-21 < (/.f64 (-.f64 x y) (-.f64 z y)) < 2

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

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

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

      5. mul-1-neg N/A

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

      6. lower-*.f64 N/A

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

      7. mul-1-neg N/A

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

      8. distribute-neg-frac2 N/A

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

      9. lower-/.f64 N/A

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

      10. sub-neg N/A

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

      11. +-commutative N/A

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

      12. distribute-neg-in N/A

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

      13. remove-double-neg N/A

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

      14. lower-+.f64 N/A

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

      15. lower-neg.f64 77.0

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

   5. Applied rewrites 77.0%

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

   7. Applied rewrites 100.0%

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

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

   1. Initial program 97.5%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 96.2

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

   5. Applied rewrites 96.2%

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

4. Final simplification 96.7%

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

Alternative 6: 94.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := t\_m \cdot \frac{x}{z - y}\\ t_3 := \frac{x - y}{z - y}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_3 \leq -10000000:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_3 \leq 2 \cdot 10^{-21}:\\ \;\;\;\;t\_m \cdot \frac{x - y}{z}\\ \mathbf{elif}\;t\_3 \leq 2:\\ \;\;\;\;t\_m \cdot \frac{y}{y - z}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \end{array} \]

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 (- z y)))) (t_3 (/ (- x y) (- z y))))
   (*
    t_s
    (if (<= t_3 -10000000.0)
      t_2
      (if (<= t_3 2e-21)
        (* t_m (/ (- x y) z))
        (if (<= t_3 2.0) (* t_m (/ y (- y z))) 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 / (z - y));
	double t_3 = (x - y) / (z - y);
	double tmp;
	if (t_3 <= -10000000.0) {
		tmp = t_2;
	} else if (t_3 <= 2e-21) {
		tmp = t_m * ((x - y) / z);
	} else if (t_3 <= 2.0) {
		tmp = t_m * (y / (y - z));
	} 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 * (x / (z - y))
    t_3 = (x - y) / (z - y)
    if (t_3 <= (-10000000.0d0)) then
        tmp = t_2
    else if (t_3 <= 2d-21) then
        tmp = t_m * ((x - y) / z)
    else if (t_3 <= 2.0d0) then
        tmp = t_m * (y / (y - z))
    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 / (z - y));
	double t_3 = (x - y) / (z - y);
	double tmp;
	if (t_3 <= -10000000.0) {
		tmp = t_2;
	} else if (t_3 <= 2e-21) {
		tmp = t_m * ((x - y) / z);
	} else if (t_3 <= 2.0) {
		tmp = t_m * (y / (y - z));
	} 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 / (z - y))
	t_3 = (x - y) / (z - y)
	tmp = 0
	if t_3 <= -10000000.0:
		tmp = t_2
	elif t_3 <= 2e-21:
		tmp = t_m * ((x - y) / z)
	elif t_3 <= 2.0:
		tmp = t_m * (y / (y - z))
	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(x / Float64(z - y)))
	t_3 = Float64(Float64(x - y) / Float64(z - y))
	tmp = 0.0
	if (t_3 <= -10000000.0)
		tmp = t_2;
	elseif (t_3 <= 2e-21)
		tmp = Float64(t_m * Float64(Float64(x - y) / z));
	elseif (t_3 <= 2.0)
		tmp = Float64(t_m * Float64(y / Float64(y - z)));
	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 / (z - y));
	t_3 = (x - y) / (z - y);
	tmp = 0.0;
	if (t_3 <= -10000000.0)
		tmp = t_2;
	elseif (t_3 <= 2e-21)
		tmp = t_m * ((x - y) / z);
	elseif (t_3 <= 2.0)
		tmp = t_m * (y / (y - z));
	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[(x / N[(z - y), $MachinePrecision]), $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, -10000000.0], t$95$2, If[LessEqual[t$95$3, 2e-21], N[(t$95$m * N[(N[(x - y), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 2.0], N[(t$95$m * N[(y / N[(y - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 96.3%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 94.7

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

   5. Applied rewrites 94.7%

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

   if -1e7 < (/.f64 (-.f64 x y) (-.f64 z y)) < 1.99999999999999982e-21

   1. Initial program 95.1%

      \[\frac{x - y}{z - y} \cdot t \]
   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 93.4

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

   5. Applied rewrites 93.4%

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

   if 1.99999999999999982e-21 < (/.f64 (-.f64 x y) (-.f64 z y)) < 2

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

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

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

      5. mul-1-neg N/A

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

      6. lower-*.f64 N/A

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

      7. mul-1-neg N/A

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

      8. distribute-neg-frac2 N/A

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

      9. lower-/.f64 N/A

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

      10. sub-neg N/A

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

      11. +-commutative N/A

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

      12. distribute-neg-in N/A

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

      13. remove-double-neg N/A

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

      14. lower-+.f64 N/A

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

      15. lower-neg.f64 77.0

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

   5. Applied rewrites 77.0%

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

   7. Applied rewrites 100.0%

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

4. Final simplification 96.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{z - y} \leq -10000000:\\ \;\;\;\;t \cdot \frac{x}{z - y}\\ \mathbf{elif}\;\frac{x - y}{z - y} \leq 2 \cdot 10^{-21}:\\ \;\;\;\;t \cdot \frac{x - y}{z}\\ \mathbf{elif}\;\frac{x - y}{z - y} \leq 2:\\ \;\;\;\;t \cdot \frac{y}{y - z}\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{x}{z - y}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 91.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := t\_m \cdot \frac{x}{z - y}\\ t_3 := \frac{x - y}{z - y}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_3 \leq -5 \cdot 10^{-62}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_3 \leq 10^{-21}:\\ \;\;\;\;\left(x - y\right) \cdot \frac{t\_m}{z}\\ \mathbf{elif}\;t\_3 \leq 2:\\ \;\;\;\;t\_m \cdot \frac{y}{y - z}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \end{array} \]

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 (- z y)))) (t_3 (/ (- x y) (- z y))))
   (*
    t_s
    (if (<= t_3 -5e-62)
      t_2
      (if (<= t_3 1e-21)
        (* (- x y) (/ t_m z))
        (if (<= t_3 2.0) (* t_m (/ y (- y z))) 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 / (z - y));
	double t_3 = (x - y) / (z - y);
	double tmp;
	if (t_3 <= -5e-62) {
		tmp = t_2;
	} else if (t_3 <= 1e-21) {
		tmp = (x - y) * (t_m / z);
	} else if (t_3 <= 2.0) {
		tmp = t_m * (y / (y - z));
	} 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 * (x / (z - y))
    t_3 = (x - y) / (z - y)
    if (t_3 <= (-5d-62)) then
        tmp = t_2
    else if (t_3 <= 1d-21) then
        tmp = (x - y) * (t_m / z)
    else if (t_3 <= 2.0d0) then
        tmp = t_m * (y / (y - z))
    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 / (z - y));
	double t_3 = (x - y) / (z - y);
	double tmp;
	if (t_3 <= -5e-62) {
		tmp = t_2;
	} else if (t_3 <= 1e-21) {
		tmp = (x - y) * (t_m / z);
	} else if (t_3 <= 2.0) {
		tmp = t_m * (y / (y - z));
	} 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 / (z - y))
	t_3 = (x - y) / (z - y)
	tmp = 0
	if t_3 <= -5e-62:
		tmp = t_2
	elif t_3 <= 1e-21:
		tmp = (x - y) * (t_m / z)
	elif t_3 <= 2.0:
		tmp = t_m * (y / (y - z))
	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(x / Float64(z - y)))
	t_3 = Float64(Float64(x - y) / Float64(z - y))
	tmp = 0.0
	if (t_3 <= -5e-62)
		tmp = t_2;
	elseif (t_3 <= 1e-21)
		tmp = Float64(Float64(x - y) * Float64(t_m / z));
	elseif (t_3 <= 2.0)
		tmp = Float64(t_m * Float64(y / Float64(y - z)));
	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 / (z - y));
	t_3 = (x - y) / (z - y);
	tmp = 0.0;
	if (t_3 <= -5e-62)
		tmp = t_2;
	elseif (t_3 <= 1e-21)
		tmp = (x - y) * (t_m / z);
	elseif (t_3 <= 2.0)
		tmp = t_m * (y / (y - z));
	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[(x / N[(z - y), $MachinePrecision]), $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, -5e-62], t$95$2, If[LessEqual[t$95$3, 1e-21], N[(N[(x - y), $MachinePrecision] * N[(t$95$m / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 2.0], N[(t$95$m * N[(y / N[(y - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 x y) (-.f64 z y)) < -5.0000000000000002e-62 or 2 < (/.f64 (-.f64 x y) (-.f64 z y))

   1. Initial program 96.7%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 90.5

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

   5. Applied rewrites 90.5%

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

   if -5.0000000000000002e-62 < (/.f64 (-.f64 x y) (-.f64 z y)) < 9.99999999999999908e-22

   1. Initial program 94.3%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 95.2

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

   5. Applied rewrites 95.2%

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

   if 9.99999999999999908e-22 < (/.f64 (-.f64 x y) (-.f64 z y)) < 2

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

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

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

      5. mul-1-neg N/A

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

      6. lower-*.f64 N/A

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

      7. mul-1-neg N/A

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

      8. distribute-neg-frac2 N/A

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

      9. lower-/.f64 N/A

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

      10. sub-neg N/A

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

      11. +-commutative N/A

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

      12. distribute-neg-in N/A

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

      13. remove-double-neg N/A

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

      14. lower-+.f64 N/A

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

      15. lower-neg.f64 76.3

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

   5. Applied rewrites 76.3%

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

   7. Applied rewrites 98.9%

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

4. Final simplification 94.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{z - y} \leq -5 \cdot 10^{-62}:\\ \;\;\;\;t \cdot \frac{x}{z - y}\\ \mathbf{elif}\;\frac{x - y}{z - y} \leq 10^{-21}:\\ \;\;\;\;\left(x - y\right) \cdot \frac{t}{z}\\ \mathbf{elif}\;\frac{x - y}{z - y} \leq 2:\\ \;\;\;\;t \cdot \frac{y}{y - z}\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{x}{z - y}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 91.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{t\_m \cdot x}{z - y}\\ t_3 := \frac{x - y}{z - y}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_3 \leq -10000000:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_3 \leq 10^{-21}:\\ \;\;\;\;\left(x - y\right) \cdot \frac{t\_m}{z}\\ \mathbf{elif}\;t\_3 \leq 2:\\ \;\;\;\;t\_m \cdot \frac{y}{y - z}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \end{array} \]

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) (- z y))) (t_3 (/ (- x y) (- z y))))
   (*
    t_s
    (if (<= t_3 -10000000.0)
      t_2
      (if (<= t_3 1e-21)
        (* (- x y) (/ t_m z))
        (if (<= t_3 2.0) (* t_m (/ y (- y z))) 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) / (z - y);
	double t_3 = (x - y) / (z - y);
	double tmp;
	if (t_3 <= -10000000.0) {
		tmp = t_2;
	} else if (t_3 <= 1e-21) {
		tmp = (x - y) * (t_m / z);
	} else if (t_3 <= 2.0) {
		tmp = t_m * (y / (y - z));
	} 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 * x) / (z - y)
    t_3 = (x - y) / (z - y)
    if (t_3 <= (-10000000.0d0)) then
        tmp = t_2
    else if (t_3 <= 1d-21) then
        tmp = (x - y) * (t_m / z)
    else if (t_3 <= 2.0d0) then
        tmp = t_m * (y / (y - z))
    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) / (z - y);
	double t_3 = (x - y) / (z - y);
	double tmp;
	if (t_3 <= -10000000.0) {
		tmp = t_2;
	} else if (t_3 <= 1e-21) {
		tmp = (x - y) * (t_m / z);
	} else if (t_3 <= 2.0) {
		tmp = t_m * (y / (y - z));
	} 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) / (z - y)
	t_3 = (x - y) / (z - y)
	tmp = 0
	if t_3 <= -10000000.0:
		tmp = t_2
	elif t_3 <= 1e-21:
		tmp = (x - y) * (t_m / z)
	elif t_3 <= 2.0:
		tmp = t_m * (y / (y - z))
	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 * x) / Float64(z - y))
	t_3 = Float64(Float64(x - y) / Float64(z - y))
	tmp = 0.0
	if (t_3 <= -10000000.0)
		tmp = t_2;
	elseif (t_3 <= 1e-21)
		tmp = Float64(Float64(x - y) * Float64(t_m / z));
	elseif (t_3 <= 2.0)
		tmp = Float64(t_m * Float64(y / Float64(y - z)));
	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) / (z - y);
	t_3 = (x - y) / (z - y);
	tmp = 0.0;
	if (t_3 <= -10000000.0)
		tmp = t_2;
	elseif (t_3 <= 1e-21)
		tmp = (x - y) * (t_m / z);
	elseif (t_3 <= 2.0)
		tmp = t_m * (y / (y - z));
	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 * x), $MachinePrecision] / N[(z - 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, -10000000.0], t$95$2, If[LessEqual[t$95$3, 1e-21], N[(N[(x - y), $MachinePrecision] * N[(t$95$m / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 2.0], N[(t$95$m * N[(y / N[(y - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 96.3%

      \[\frac{x - y}{z - y} \cdot t \]
   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. lower-/.f64 96.9

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

   4. Applied rewrites 96.9%

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

   5. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 88.2

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

   7. Applied rewrites 88.2%

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

   if -1e7 < (/.f64 (-.f64 x y) (-.f64 z y)) < 9.99999999999999908e-22

   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. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 89.4

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

   5. Applied rewrites 89.4%

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

   if 9.99999999999999908e-22 < (/.f64 (-.f64 x y) (-.f64 z y)) < 2

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

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

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

      5. mul-1-neg N/A

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

      6. lower-*.f64 N/A

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

      7. mul-1-neg N/A

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

      8. distribute-neg-frac2 N/A

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

      9. lower-/.f64 N/A

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

      10. sub-neg N/A

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

      11. +-commutative N/A

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

      12. distribute-neg-in N/A

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

      13. remove-double-neg N/A

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

      14. lower-+.f64 N/A

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

      15. lower-neg.f64 76.3

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

   5. Applied rewrites 76.3%

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

   7. Applied rewrites 98.9%

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

4. Final simplification 92.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{z - y} \leq -10000000:\\ \;\;\;\;\frac{t \cdot x}{z - y}\\ \mathbf{elif}\;\frac{x - y}{z - y} \leq 10^{-21}:\\ \;\;\;\;\left(x - y\right) \cdot \frac{t}{z}\\ \mathbf{elif}\;\frac{x - y}{z - y} \leq 2:\\ \;\;\;\;t \cdot \frac{y}{y - z}\\ \mathbf{else}:\\ \;\;\;\;\frac{t \cdot x}{z - y}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 69.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{x - y}{z - y}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq 0.0004:\\ \;\;\;\;t\_m \cdot \frac{x}{z}\\ \mathbf{elif}\;t\_2 \leq 2:\\ \;\;\;\;\mathsf{fma}\left(t\_m, \frac{z}{y}, t\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_m \cdot x}{z}\\ \end{array} \end{array} \end{array} \]

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 0.0004)
      (* t_m (/ x z))
      (if (<= t_2 2.0) (fma t_m (/ z y) t_m) (/ (* t_m x) z))))))

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 <= 0.0004) {
		tmp = t_m * (x / z);
	} else if (t_2 <= 2.0) {
		tmp = fma(t_m, (z / y), t_m);
	} else {
		tmp = (t_m * x) / z;
	}
	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 <= 0.0004)
		tmp = Float64(t_m * Float64(x / z));
	elseif (t_2 <= 2.0)
		tmp = fma(t_m, Float64(z / y), t_m);
	else
		tmp = Float64(Float64(t_m * x) / z);
	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[(x - y), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$2, 0.0004], N[(t$95$m * N[(x / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 2.0], N[(t$95$m * N[(z / y), $MachinePrecision] + t$95$m), $MachinePrecision], N[(N[(t$95$m * x), $MachinePrecision] / z), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 x y) (-.f64 z y)) < 4.00000000000000019e-4

   1. Initial program 95.2%

      \[\frac{x - y}{z - y} \cdot t \]
   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 59.2

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

   5. Applied rewrites 59.2%

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

   if 4.00000000000000019e-4 < (/.f64 (-.f64 x y) (-.f64 z y)) < 2

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

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

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

      5. mul-1-neg N/A

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

      6. lower-*.f64 N/A

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

      7. mul-1-neg N/A

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

      8. distribute-neg-frac2 N/A

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

      9. lower-/.f64 N/A

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

      10. sub-neg N/A

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

      11. +-commutative N/A

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

      12. distribute-neg-in N/A

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

      13. remove-double-neg N/A

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

      14. lower-+.f64 N/A

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

      15. lower-neg.f64 76.3

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

   5. Applied rewrites 76.3%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 98.7%

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

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

   1. Initial program 97.5%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 45.0

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

   5. Applied rewrites 45.0%

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

4. Final simplification 70.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{z - y} \leq 0.0004:\\ \;\;\;\;t \cdot \frac{x}{z}\\ \mathbf{elif}\;\frac{x - y}{z - y} \leq 2:\\ \;\;\;\;\mathsf{fma}\left(t, \frac{z}{y}, t\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t \cdot x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 69.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{x - y}{z - y}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq 2 \cdot 10^{-21}:\\ \;\;\;\;t\_m \cdot \frac{x}{z}\\ \mathbf{elif}\;t\_2 \leq 2:\\ \;\;\;\;t\_m \cdot 1\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_m \cdot x}{z}\\ \end{array} \end{array} \end{array} \]

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 2e-21)
      (* t_m (/ x z))
      (if (<= t_2 2.0) (* t_m 1.0) (/ (* t_m x) z))))))

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 <= 2e-21) {
		tmp = t_m * (x / z);
	} else if (t_2 <= 2.0) {
		tmp = t_m * 1.0;
	} else {
		tmp = (t_m * x) / z;
	}
	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 <= 2d-21) then
        tmp = t_m * (x / z)
    else if (t_2 <= 2.0d0) then
        tmp = t_m * 1.0d0
    else
        tmp = (t_m * x) / z
    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 <= 2e-21) {
		tmp = t_m * (x / z);
	} else if (t_2 <= 2.0) {
		tmp = t_m * 1.0;
	} else {
		tmp = (t_m * x) / z;
	}
	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 <= 2e-21:
		tmp = t_m * (x / z)
	elif t_2 <= 2.0:
		tmp = t_m * 1.0
	else:
		tmp = (t_m * x) / z
	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 <= 2e-21)
		tmp = Float64(t_m * Float64(x / z));
	elseif (t_2 <= 2.0)
		tmp = Float64(t_m * 1.0);
	else
		tmp = Float64(Float64(t_m * x) / z);
	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 <= 2e-21)
		tmp = t_m * (x / z);
	elseif (t_2 <= 2.0)
		tmp = t_m * 1.0;
	else
		tmp = (t_m * x) / z;
	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, 2e-21], N[(t$95$m * N[(x / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 2.0], N[(t$95$m * 1.0), $MachinePrecision], N[(N[(t$95$m * x), $MachinePrecision] / z), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 x y) (-.f64 z y)) < 1.99999999999999982e-21

   1. Initial program 95.1%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 60.6

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

   5. Applied rewrites 60.6%

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

   if 1.99999999999999982e-21 < (/.f64 (-.f64 x y) (-.f64 z y)) < 2

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf

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

   5. Applied rewrites 95.2%

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

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

   1. Initial program 97.5%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 45.0

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

   5. Applied rewrites 45.0%

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

4. Final simplification 70.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{z - y} \leq 2 \cdot 10^{-21}:\\ \;\;\;\;t \cdot \frac{x}{z}\\ \mathbf{elif}\;\frac{x - y}{z - y} \leq 2:\\ \;\;\;\;t \cdot 1\\ \mathbf{else}:\\ \;\;\;\;\frac{t \cdot x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 67.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{x - y}{z - y}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq 10^{-21}:\\ \;\;\;\;x \cdot \frac{t\_m}{z}\\ \mathbf{elif}\;t\_2 \leq 2:\\ \;\;\;\;t\_m \cdot 1\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_m \cdot x}{z}\\ \end{array} \end{array} \end{array} \]

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 1e-21)
      (* x (/ t_m z))
      (if (<= t_2 2.0) (* t_m 1.0) (/ (* t_m x) z))))))

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 <= 1e-21) {
		tmp = x * (t_m / z);
	} else if (t_2 <= 2.0) {
		tmp = t_m * 1.0;
	} else {
		tmp = (t_m * x) / z;
	}
	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 <= 1d-21) then
        tmp = x * (t_m / z)
    else if (t_2 <= 2.0d0) then
        tmp = t_m * 1.0d0
    else
        tmp = (t_m * x) / z
    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 <= 1e-21) {
		tmp = x * (t_m / z);
	} else if (t_2 <= 2.0) {
		tmp = t_m * 1.0;
	} else {
		tmp = (t_m * x) / z;
	}
	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 <= 1e-21:
		tmp = x * (t_m / z)
	elif t_2 <= 2.0:
		tmp = t_m * 1.0
	else:
		tmp = (t_m * x) / z
	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 <= 1e-21)
		tmp = Float64(x * Float64(t_m / z));
	elseif (t_2 <= 2.0)
		tmp = Float64(t_m * 1.0);
	else
		tmp = Float64(Float64(t_m * x) / z);
	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 <= 1e-21)
		tmp = x * (t_m / z);
	elseif (t_2 <= 2.0)
		tmp = t_m * 1.0;
	else
		tmp = (t_m * x) / z;
	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, 1e-21], N[(x * N[(t$95$m / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 2.0], N[(t$95$m * 1.0), $MachinePrecision], N[(N[(t$95$m * x), $MachinePrecision] / z), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 x y) (-.f64 z y)) < 9.99999999999999908e-22

   1. Initial program 95.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. frac-2neg N/A

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

      4. div-inv N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. neg-sub0 N/A

         \[\leadsto \color{blue}{\left(0 - \left(x - y\right)\right)} \cdot \left(\frac{1}{\mathsf{neg}\left(\left(z - y\right)\right)} \cdot t\right) \]

      8. lift--.f64 N/A

         \[\leadsto \left(0 - \color{blue}{\left(x - y\right)}\right) \cdot \left(\frac{1}{\mathsf{neg}\left(\left(z - y\right)\right)} \cdot t\right) \]

      9. sub-neg N/A

         \[\leadsto \left(0 - \color{blue}{\left(x + \left(\mathsf{neg}\left(y\right)\right)\right)}\right) \cdot \left(\frac{1}{\mathsf{neg}\left(\left(z - y\right)\right)} \cdot t\right) \]

      10. +-commutative N/A

          \[\leadsto \left(0 - \color{blue}{\left(\left(\mathsf{neg}\left(y\right)\right) + x\right)}\right) \cdot \left(\frac{1}{\mathsf{neg}\left(\left(z - y\right)\right)} \cdot t\right) \]

      11. associate--r+ N/A

          \[\leadsto \color{blue}{\left(\left(0 - \left(\mathsf{neg}\left(y\right)\right)\right) - x\right)} \cdot \left(\frac{1}{\mathsf{neg}\left(\left(z - y\right)\right)} \cdot t\right) \]

      12. neg-sub0 N/A

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

      13. remove-double-neg N/A

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

      14. lower--.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. neg-sub0 N/A

          \[\leadsto \left(y - x\right) \cdot \left(\frac{1}{\color{blue}{0 - \left(z - y\right)}} \cdot t\right) \]

      18. lift--.f64 N/A

          \[\leadsto \left(y - x\right) \cdot \left(\frac{1}{0 - \color{blue}{\left(z - y\right)}} \cdot t\right) \]

      19. sub-neg N/A

          \[\leadsto \left(y - x\right) \cdot \left(\frac{1}{0 - \color{blue}{\left(z + \left(\mathsf{neg}\left(y\right)\right)\right)}} \cdot t\right) \]

      20. +-commutative N/A

          \[\leadsto \left(y - x\right) \cdot \left(\frac{1}{0 - \color{blue}{\left(\left(\mathsf{neg}\left(y\right)\right) + z\right)}} \cdot t\right) \]

      21. associate--r+ N/A

          \[\leadsto \left(y - x\right) \cdot \left(\frac{1}{\color{blue}{\left(0 - \left(\mathsf{neg}\left(y\right)\right)\right) - z}} \cdot t\right) \]

      22. neg-sub0 N/A

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

      23. remove-double-neg N/A

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

      24. lower--.f64 91.6

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

   4. Applied rewrites 91.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-/r/ N/A

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

      5. lift--.f64 N/A

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

      6. sub-neg N/A

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

      7. lift-neg.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. un-div-inv N/A

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

      10. lower-/.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. lift-neg.f64 N/A

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

      13. sub-neg N/A

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

      14. lift--.f64 N/A

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

      15. lower-/.f64 91.3

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

   6. Applied rewrites 91.3%

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

   7. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 54.7

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

   9. Applied rewrites 54.7%

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

   11. Applied rewrites 57.5%

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

   if 9.99999999999999908e-22 < (/.f64 (-.f64 x y) (-.f64 z y)) < 2

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf

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

   5. Applied rewrites 94.3%

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

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

   1. Initial program 97.5%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 45.0

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

   5. Applied rewrites 45.0%

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

4. Final simplification 68.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{z - y} \leq 10^{-21}:\\ \;\;\;\;x \cdot \frac{t}{z}\\ \mathbf{elif}\;\frac{x - y}{z - y} \leq 2:\\ \;\;\;\;t \cdot 1\\ \mathbf{else}:\\ \;\;\;\;\frac{t \cdot x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 67.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{t\_m \cdot x}{z}\\ t_3 := \frac{x - y}{z - y}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_3 \leq 2 \cdot 10^{-21}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_3 \leq 2:\\ \;\;\;\;t\_m \cdot 1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \end{array} \]

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) z)) (t_3 (/ (- x y) (- z y))))
   (* t_s (if (<= t_3 2e-21) t_2 (if (<= t_3 2.0) (* t_m 1.0) 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) / z;
	double t_3 = (x - y) / (z - y);
	double tmp;
	if (t_3 <= 2e-21) {
		tmp = t_2;
	} else if (t_3 <= 2.0) {
		tmp = t_m * 1.0;
	} 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 * x) / z
    t_3 = (x - y) / (z - y)
    if (t_3 <= 2d-21) then
        tmp = t_2
    else if (t_3 <= 2.0d0) then
        tmp = t_m * 1.0d0
    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) / z;
	double t_3 = (x - y) / (z - y);
	double tmp;
	if (t_3 <= 2e-21) {
		tmp = t_2;
	} else if (t_3 <= 2.0) {
		tmp = t_m * 1.0;
	} 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) / z
	t_3 = (x - y) / (z - y)
	tmp = 0
	if t_3 <= 2e-21:
		tmp = t_2
	elif t_3 <= 2.0:
		tmp = t_m * 1.0
	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 * x) / z)
	t_3 = Float64(Float64(x - y) / Float64(z - y))
	tmp = 0.0
	if (t_3 <= 2e-21)
		tmp = t_2;
	elseif (t_3 <= 2.0)
		tmp = Float64(t_m * 1.0);
	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) / z;
	t_3 = (x - y) / (z - y);
	tmp = 0.0;
	if (t_3 <= 2e-21)
		tmp = t_2;
	elseif (t_3 <= 2.0)
		tmp = t_m * 1.0;
	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 * x), $MachinePrecision] / z), $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-21], t$95$2, If[LessEqual[t$95$3, 2.0], N[(t$95$m * 1.0), $MachinePrecision], t$95$2]]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 x y) (-.f64 z y)) < 1.99999999999999982e-21 or 2 < (/.f64 (-.f64 x y) (-.f64 z y))

   1. Initial program 95.7%

      \[\frac{x - y}{z - y} \cdot t \]
   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 52.5

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

   5. Applied rewrites 52.5%

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

   if 1.99999999999999982e-21 < (/.f64 (-.f64 x y) (-.f64 z y)) < 2

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf

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

   5. Applied rewrites 95.2%

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

4. Final simplification 68.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{z - y} \leq 2 \cdot 10^{-21}:\\ \;\;\;\;\frac{t \cdot x}{z}\\ \mathbf{elif}\;\frac{x - y}{z - y} \leq 2:\\ \;\;\;\;t \cdot 1\\ \mathbf{else}:\\ \;\;\;\;\frac{t \cdot x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 35.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \cdot \frac{x - y}{z - y} \leq 5 \cdot 10^{+188}:\\ \;\;\;\;t\_m \cdot 1\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{t\_m}{y}\\ \end{array} \end{array} \]

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
 (*
  t_s
  (if (<= (* t_m (/ (- x y) (- z y))) 5e+188) (* t_m 1.0) (* y (/ t_m y)))))

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 tmp;
	if ((t_m * ((x - y) / (z - y))) <= 5e+188) {
		tmp = t_m * 1.0;
	} else {
		tmp = y * (t_m / y);
	}
	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) :: tmp
    if ((t_m * ((x - y) / (z - y))) <= 5d+188) then
        tmp = t_m * 1.0d0
    else
        tmp = y * (t_m / y)
    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 tmp;
	if ((t_m * ((x - y) / (z - y))) <= 5e+188) {
		tmp = t_m * 1.0;
	} else {
		tmp = y * (t_m / y);
	}
	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):
	tmp = 0
	if (t_m * ((x - y) / (z - y))) <= 5e+188:
		tmp = t_m * 1.0
	else:
		tmp = y * (t_m / y)
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, y, z, t_m)
	tmp = 0.0
	if (Float64(t_m * Float64(Float64(x - y) / Float64(z - y))) <= 5e+188)
		tmp = Float64(t_m * 1.0);
	else
		tmp = Float64(y * Float64(t_m / y));
	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)
	tmp = 0.0;
	if ((t_m * ((x - y) / (z - y))) <= 5e+188)
		tmp = t_m * 1.0;
	else
		tmp = y * (t_m / y);
	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_] := N[(t$95$s * If[LessEqual[N[(t$95$m * N[(N[(x - y), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e+188], N[(t$95$m * 1.0), $MachinePrecision], N[(y * N[(t$95$m / y), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (/.f64 (-.f64 x y) (-.f64 z y)) t) < 5.0000000000000001e188

   1. Initial program 97.8%

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

   3. Taylor expanded in y around inf

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

   5. Applied rewrites 37.1%

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

   if 5.0000000000000001e188 < (*.f64 (/.f64 (-.f64 x y) (-.f64 z y)) t)

   1. Initial program 93.4%

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

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

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

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

      5. mul-1-neg N/A

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

      6. lower-*.f64 N/A

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

      7. mul-1-neg N/A

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

      8. distribute-neg-frac2 N/A

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

      9. lower-/.f64 N/A

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

      10. sub-neg N/A

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

      11. +-commutative N/A

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

      12. distribute-neg-in N/A

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

      13. remove-double-neg N/A

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

      14. lower-+.f64 N/A

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

      15. lower-neg.f64 43.1

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

   5. Applied rewrites 43.1%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 49.5%

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

4. Final simplification 38.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \cdot \frac{x - y}{z - y} \leq 5 \cdot 10^{+188}:\\ \;\;\;\;t \cdot 1\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{t}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 96.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.2 \cdot 10^{-17}:\\ \;\;\;\;t\_m \cdot \frac{x - y}{z - y}\\ \mathbf{else}:\\ \;\;\;\;\left(x - y\right) \cdot \frac{t\_m}{z - y}\\ \end{array} \end{array} \]

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
 (*
  t_s
  (if (<= t_m 2.2e-17)
    (* t_m (/ (- x y) (- z y)))
    (* (- x y) (/ t_m (- z y))))))

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 tmp;
	if (t_m <= 2.2e-17) {
		tmp = t_m * ((x - y) / (z - y));
	} else {
		tmp = (x - y) * (t_m / (z - y));
	}
	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) :: tmp
    if (t_m <= 2.2d-17) then
        tmp = t_m * ((x - y) / (z - y))
    else
        tmp = (x - y) * (t_m / (z - y))
    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 tmp;
	if (t_m <= 2.2e-17) {
		tmp = t_m * ((x - y) / (z - y));
	} else {
		tmp = (x - y) * (t_m / (z - y));
	}
	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):
	tmp = 0
	if t_m <= 2.2e-17:
		tmp = t_m * ((x - y) / (z - y))
	else:
		tmp = (x - y) * (t_m / (z - y))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, y, z, t_m)
	tmp = 0.0
	if (t_m <= 2.2e-17)
		tmp = Float64(t_m * Float64(Float64(x - y) / Float64(z - y)));
	else
		tmp = Float64(Float64(x - y) * Float64(t_m / Float64(z - y)));
	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)
	tmp = 0.0;
	if (t_m <= 2.2e-17)
		tmp = t_m * ((x - y) / (z - y));
	else
		tmp = (x - y) * (t_m / (z - y));
	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_] := N[(t$95$s * If[LessEqual[t$95$m, 2.2e-17], N[(t$95$m * N[(N[(x - y), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x - y), $MachinePrecision] * N[(t$95$m / N[(z - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 2.2e-17

   1. Initial program 97.2%

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

   if 2.2e-17 < t

   1. Initial program 97.3%

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

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

   4. Applied rewrites 99.8%

      \[\leadsto \color{blue}{\frac{t}{z - y} \cdot \left(x - y\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.2 \cdot 10^{-17}:\\ \;\;\;\;t \cdot \frac{x - y}{z - y}\\ \mathbf{else}:\\ \;\;\;\;\left(x - y\right) \cdot \frac{t}{z - y}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 34.1% accurate, 3.8× speedup

Math

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

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 (* t_m 1.0)))

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 * (t_m * 1.0);
}

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 * (t_m * 1.0d0)
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 * (t_m * 1.0);
}

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 * (t_m * 1.0)

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(t_m * 1.0))
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 * (t_m * 1.0);
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[(t$95$m * 1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 97.3%

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

3. Taylor expanded in y around inf

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

5. Applied rewrites 37.0%

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

6. Final simplification 37.0%

   \[\leadsto t \cdot 1 \]
7. 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 2024234 
(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))