Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, C

Percentage Accurate: 94.7% → 96.1%

Time: 8.8s

Alternatives: 12

Speedup: 0.5×

• 21.0% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \end{array} \]

FPCore

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

C

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

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = x * ((y / z) - (t / (1.0d0 - z)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_] := N[(x * N[(N[(y / z), $MachinePrecision] - N[(t / N[(1.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 94.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \end{array} \]

FPCore

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

C

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

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = x * ((y / z) - (t / (1.0d0 - z)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_] := N[(x * N[(N[(y / z), $MachinePrecision] - N[(t / N[(1.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 96.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_1 := x\_m \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right)\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(-t, z, \left(1 - z\right) \cdot y\right) \cdot \frac{x\_m}{\left(1 - z\right) \cdot z}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (let* ((t_1 (* x_m (- (/ y z) (/ t (- 1.0 z))))))
   (*
    x_s
    (if (<= t_1 (- INFINITY))
      (* (fma (- t) z (* (- 1.0 z) y)) (/ x_m (* (- 1.0 z) z)))
      t_1))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t) {
	double t_1 = x_m * ((y / z) - (t / (1.0 - z)));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = fma(-t, z, ((1.0 - z) * y)) * (x_m / ((1.0 - z) * z));
	} else {
		tmp = t_1;
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t)
	t_1 = Float64(x_m * Float64(Float64(y / z) - Float64(t / Float64(1.0 - z))))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(fma(Float64(-t), z, Float64(Float64(1.0 - z) * y)) * Float64(x_m / Float64(Float64(1.0 - z) * z)));
	else
		tmp = t_1;
	end
	return Float64(x_s * tmp)
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (*.f64 x (-.f64 (/.f64 y z) (/.f64 t (-.f64 #s(literal 1 binary64) z)))) < -inf.0

   1. Initial program 80.6%

      \[x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift--.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. frac-sub N/A

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

      7. associate-*l/ N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

      10. fp-cancel-sub-sign-inv N/A

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

      11. +-commutative N/A

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

      12. distribute-lft-neg-out N/A

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

      13. distribute-rgt-neg-out N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. lower-neg.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 N/A

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

      19. lower-/.f64 N/A

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

      20. *-commutative N/A

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

      21. lower-*.f64 93.1

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

   4. Applied rewrites 93.1%

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

   if -inf.0 < (*.f64 x (-.f64 (/.f64 y z) (/.f64 t (-.f64 #s(literal 1 binary64) z))))

   1. Initial program 95.8%

      \[x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \]
   2. Add Preprocessing
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 96.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_1 := \frac{y}{z} - \frac{t}{1 - z}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;y \cdot \frac{x\_m}{z}\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot t\_1\\ \end{array} \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (let* ((t_1 (- (/ y z) (/ t (- 1.0 z)))))
   (* x_s (if (<= t_1 (- INFINITY)) (* y (/ x_m z)) (* x_m t_1)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t) {
	double t_1 = (y / z) - (t / (1.0 - z));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = y * (x_m / z);
	} else {
		tmp = x_m * t_1;
	}
	return x_s * tmp;
}

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z, double t) {
	double t_1 = (y / z) - (t / (1.0 - z));
	double tmp;
	if (t_1 <= -Double.POSITIVE_INFINITY) {
		tmp = y * (x_m / z);
	} else {
		tmp = x_m * t_1;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z, t):
	t_1 = (y / z) - (t / (1.0 - z))
	tmp = 0
	if t_1 <= -math.inf:
		tmp = y * (x_m / z)
	else:
		tmp = x_m * t_1
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t)
	t_1 = Float64(Float64(y / z) - Float64(t / Float64(1.0 - z)))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(y * Float64(x_m / z));
	else
		tmp = Float64(x_m * t_1);
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z, t)
	t_1 = (y / z) - (t / (1.0 - z));
	tmp = 0.0;
	if (t_1 <= -Inf)
		tmp = y * (x_m / z);
	else
		tmp = x_m * t_1;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (-.f64 (/.f64 y z) (/.f64 t (-.f64 #s(literal 1 binary64) z))) < -inf.0

   1. Initial program 61.2%

      \[x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 61.2

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

   5. Applied rewrites 61.2%

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

   7. Applied rewrites 100.0%

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

   if -inf.0 < (-.f64 (/.f64 y z) (/.f64 t (-.f64 #s(literal 1 binary64) z)))

   1. Initial program 96.4%

      \[x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \]
   2. Add Preprocessing
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 87.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -2.05 \cdot 10^{+282}:\\ \;\;\;\;y \cdot \frac{x\_m}{z}\\ \mathbf{elif}\;z \leq -1150000000000 \lor \neg \left(z \leq 6 \cdot 10^{+17}\right):\\ \;\;\;\;\frac{\left(t + y\right) \cdot x\_m}{z}\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot \left(\frac{y}{z} - t\right)\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (*
  x_s
  (if (<= z -2.05e+282)
    (* y (/ x_m z))
    (if (or (<= z -1150000000000.0) (not (<= z 6e+17)))
      (/ (* (+ t y) x_m) z)
      (* x_m (- (/ y z) t))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (z <= -2.05e+282) {
		tmp = y * (x_m / z);
	} else if ((z <= -1150000000000.0) || !(z <= 6e+17)) {
		tmp = ((t + y) * x_m) / z;
	} else {
		tmp = x_m * ((y / z) - t);
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z, t)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (z <= (-2.05d+282)) then
        tmp = y * (x_m / z)
    else if ((z <= (-1150000000000.0d0)) .or. (.not. (z <= 6d+17))) then
        tmp = ((t + y) * x_m) / z
    else
        tmp = x_m * ((y / z) - t)
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (z <= -2.05e+282) {
		tmp = y * (x_m / z);
	} else if ((z <= -1150000000000.0) || !(z <= 6e+17)) {
		tmp = ((t + y) * x_m) / z;
	} else {
		tmp = x_m * ((y / z) - t);
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z, t):
	tmp = 0
	if z <= -2.05e+282:
		tmp = y * (x_m / z)
	elif (z <= -1150000000000.0) or not (z <= 6e+17):
		tmp = ((t + y) * x_m) / z
	else:
		tmp = x_m * ((y / z) - t)
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t)
	tmp = 0.0
	if (z <= -2.05e+282)
		tmp = Float64(y * Float64(x_m / z));
	elseif ((z <= -1150000000000.0) || !(z <= 6e+17))
		tmp = Float64(Float64(Float64(t + y) * x_m) / z);
	else
		tmp = Float64(x_m * Float64(Float64(y / z) - t));
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z, t)
	tmp = 0.0;
	if (z <= -2.05e+282)
		tmp = y * (x_m / z);
	elseif ((z <= -1150000000000.0) || ~((z <= 6e+17)))
		tmp = ((t + y) * x_m) / z;
	else
		tmp = x_m * ((y / z) - t);
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_] := N[(x$95$s * If[LessEqual[z, -2.05e+282], N[(y * N[(x$95$m / z), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[z, -1150000000000.0], N[Not[LessEqual[z, 6e+17]], $MachinePrecision]], N[(N[(N[(t + y), $MachinePrecision] * x$95$m), $MachinePrecision] / z), $MachinePrecision], N[(x$95$m * N[(N[(y / z), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if z < -2.04999999999999997e282

   1. Initial program 99.8%

      \[x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 91.8

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

   5. Applied rewrites 91.8%

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

   7. Applied rewrites 92.0%

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

   if -2.04999999999999997e282 < z < -1.15e12 or 6e17 < z

   1. Initial program 94.8%

      \[x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. fp-cancel-sub-sign-inv N/A

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

      5. metadata-eval N/A

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

      6. *-lft-identity N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 81.9

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

   5. Applied rewrites 81.9%

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

   if -1.15e12 < z < 6e17

   1. Initial program 93.0%

      \[x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. fp-cancel-sub-sign N/A

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

      4. div-sub N/A

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

      5. associate-/l* N/A

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

      6. *-inverses N/A

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

      7. *-rgt-identity N/A

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

      8. lower--.f64 N/A

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

      9. lower-/.f64 91.3

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

   5. Applied rewrites 91.3%

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

4. Final simplification 86.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.05 \cdot 10^{+282}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{elif}\;z \leq -1150000000000 \lor \neg \left(z \leq 6 \cdot 10^{+17}\right):\\ \;\;\;\;\frac{\left(t + y\right) \cdot x}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 74.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_1 := y \cdot \frac{x\_m}{z}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -2.2 \cdot 10^{-22}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 4.8 \cdot 10^{-44}:\\ \;\;\;\;x\_m \cdot \frac{t}{z - 1}\\ \mathbf{elif}\;y \leq 7 \cdot 10^{+159}:\\ \;\;\;\;\frac{\left(t + y\right) \cdot x\_m}{z}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (let* ((t_1 (* y (/ x_m z))))
   (*
    x_s
    (if (<= y -2.2e-22)
      t_1
      (if (<= y 4.8e-44)
        (* x_m (/ t (- z 1.0)))
        (if (<= y 7e+159) (/ (* (+ t y) x_m) z) t_1))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t) {
	double t_1 = y * (x_m / z);
	double tmp;
	if (y <= -2.2e-22) {
		tmp = t_1;
	} else if (y <= 4.8e-44) {
		tmp = x_m * (t / (z - 1.0));
	} else if (y <= 7e+159) {
		tmp = ((t + y) * x_m) / z;
	} else {
		tmp = t_1;
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z, t)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = y * (x_m / z)
    if (y <= (-2.2d-22)) then
        tmp = t_1
    else if (y <= 4.8d-44) then
        tmp = x_m * (t / (z - 1.0d0))
    else if (y <= 7d+159) then
        tmp = ((t + y) * x_m) / z
    else
        tmp = t_1
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z, double t) {
	double t_1 = y * (x_m / z);
	double tmp;
	if (y <= -2.2e-22) {
		tmp = t_1;
	} else if (y <= 4.8e-44) {
		tmp = x_m * (t / (z - 1.0));
	} else if (y <= 7e+159) {
		tmp = ((t + y) * x_m) / z;
	} else {
		tmp = t_1;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z, t):
	t_1 = y * (x_m / z)
	tmp = 0
	if y <= -2.2e-22:
		tmp = t_1
	elif y <= 4.8e-44:
		tmp = x_m * (t / (z - 1.0))
	elif y <= 7e+159:
		tmp = ((t + y) * x_m) / z
	else:
		tmp = t_1
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t)
	t_1 = Float64(y * Float64(x_m / z))
	tmp = 0.0
	if (y <= -2.2e-22)
		tmp = t_1;
	elseif (y <= 4.8e-44)
		tmp = Float64(x_m * Float64(t / Float64(z - 1.0)));
	elseif (y <= 7e+159)
		tmp = Float64(Float64(Float64(t + y) * x_m) / z);
	else
		tmp = t_1;
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z, t)
	t_1 = y * (x_m / z);
	tmp = 0.0;
	if (y <= -2.2e-22)
		tmp = t_1;
	elseif (y <= 4.8e-44)
		tmp = x_m * (t / (z - 1.0));
	elseif (y <= 7e+159)
		tmp = ((t + y) * x_m) / z;
	else
		tmp = t_1;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_] := Block[{t$95$1 = N[(y * N[(x$95$m / z), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * If[LessEqual[y, -2.2e-22], t$95$1, If[LessEqual[y, 4.8e-44], N[(x$95$m * N[(t / N[(z - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 7e+159], N[(N[(N[(t + y), $MachinePrecision] * x$95$m), $MachinePrecision] / z), $MachinePrecision], t$95$1]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if y < -2.2000000000000001e-22 or 6.9999999999999999e159 < y

   1. Initial program 91.9%

      \[x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 82.4

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

   5. Applied rewrites 82.4%

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

   7. Applied rewrites 87.3%

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

   if -2.2000000000000001e-22 < y < 4.80000000000000017e-44

   1. Initial program 95.0%

      \[x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. lower-/.f64 N/A

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

      4. *-lft-identity N/A

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

      5. metadata-eval N/A

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

      6. fp-cancel-sign-sub-inv N/A

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

      7. mul-1-neg N/A

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

      8. distribute-neg-in N/A

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

      9. metadata-eval N/A

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

      10. remove-double-neg N/A

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

      11. lower-+.f64 75.5

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

   5. Applied rewrites 75.5%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 40.9%

      \[\leadsto x \cdot \left(-t\right) \]

   9. Taylor expanded in t around 0

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

   11. Applied rewrites 75.5%

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

   if 4.80000000000000017e-44 < y < 6.9999999999999999e159

   1. Initial program 96.9%

      \[x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. fp-cancel-sub-sign-inv N/A

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

      5. metadata-eval N/A

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

      6. *-lft-identity N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 79.9

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

   5. Applied rewrites 79.9%

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

Alternative 5: 73.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_1 := y \cdot \frac{x\_m}{z}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -2.2 \cdot 10^{-22}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 7.4 \cdot 10^{-47}:\\ \;\;\;\;\frac{x\_m}{z - 1} \cdot t\\ \mathbf{elif}\;y \leq 7 \cdot 10^{+159}:\\ \;\;\;\;\frac{\left(t + y\right) \cdot x\_m}{z}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (let* ((t_1 (* y (/ x_m z))))
   (*
    x_s
    (if (<= y -2.2e-22)
      t_1
      (if (<= y 7.4e-47)
        (* (/ x_m (- z 1.0)) t)
        (if (<= y 7e+159) (/ (* (+ t y) x_m) z) t_1))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t) {
	double t_1 = y * (x_m / z);
	double tmp;
	if (y <= -2.2e-22) {
		tmp = t_1;
	} else if (y <= 7.4e-47) {
		tmp = (x_m / (z - 1.0)) * t;
	} else if (y <= 7e+159) {
		tmp = ((t + y) * x_m) / z;
	} else {
		tmp = t_1;
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z, t)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = y * (x_m / z)
    if (y <= (-2.2d-22)) then
        tmp = t_1
    else if (y <= 7.4d-47) then
        tmp = (x_m / (z - 1.0d0)) * t
    else if (y <= 7d+159) then
        tmp = ((t + y) * x_m) / z
    else
        tmp = t_1
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z, double t) {
	double t_1 = y * (x_m / z);
	double tmp;
	if (y <= -2.2e-22) {
		tmp = t_1;
	} else if (y <= 7.4e-47) {
		tmp = (x_m / (z - 1.0)) * t;
	} else if (y <= 7e+159) {
		tmp = ((t + y) * x_m) / z;
	} else {
		tmp = t_1;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z, t):
	t_1 = y * (x_m / z)
	tmp = 0
	if y <= -2.2e-22:
		tmp = t_1
	elif y <= 7.4e-47:
		tmp = (x_m / (z - 1.0)) * t
	elif y <= 7e+159:
		tmp = ((t + y) * x_m) / z
	else:
		tmp = t_1
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t)
	t_1 = Float64(y * Float64(x_m / z))
	tmp = 0.0
	if (y <= -2.2e-22)
		tmp = t_1;
	elseif (y <= 7.4e-47)
		tmp = Float64(Float64(x_m / Float64(z - 1.0)) * t);
	elseif (y <= 7e+159)
		tmp = Float64(Float64(Float64(t + y) * x_m) / z);
	else
		tmp = t_1;
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z, t)
	t_1 = y * (x_m / z);
	tmp = 0.0;
	if (y <= -2.2e-22)
		tmp = t_1;
	elseif (y <= 7.4e-47)
		tmp = (x_m / (z - 1.0)) * t;
	elseif (y <= 7e+159)
		tmp = ((t + y) * x_m) / z;
	else
		tmp = t_1;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_] := Block[{t$95$1 = N[(y * N[(x$95$m / z), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * If[LessEqual[y, -2.2e-22], t$95$1, If[LessEqual[y, 7.4e-47], N[(N[(x$95$m / N[(z - 1.0), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision], If[LessEqual[y, 7e+159], N[(N[(N[(t + y), $MachinePrecision] * x$95$m), $MachinePrecision] / z), $MachinePrecision], t$95$1]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if y < -2.2000000000000001e-22 or 6.9999999999999999e159 < y

   1. Initial program 91.9%

      \[x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 82.4

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

   5. Applied rewrites 82.4%

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

   7. Applied rewrites 87.3%

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

   if -2.2000000000000001e-22 < y < 7.4000000000000001e-47

   1. Initial program 95.0%

      \[x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

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

      1. distribute-rgt-in N/A

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

      2. fp-cancel-sign-sub-inv N/A

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

      3. mul-1-neg N/A

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

      4. distribute-rgt-out-- N/A

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

      5. *-commutative N/A

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

      6. fp-cancel-sub-sign-inv N/A

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

      7. *-commutative N/A

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

      8. mul-1-neg N/A

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

      9. distribute-rgt-in N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

   5. Applied rewrites 78.4%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 75.5%

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

   if 7.4000000000000001e-47 < y < 6.9999999999999999e159

   1. Initial program 96.9%

      \[x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. fp-cancel-sub-sign-inv N/A

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

      5. metadata-eval N/A

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

      6. *-lft-identity N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 79.9

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

   5. Applied rewrites 79.9%

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

Alternative 6: 94.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -0.135 \lor \neg \left(z \leq 5.9 \cdot 10^{+28}\right):\\ \;\;\;\;x\_m \cdot \frac{t + y}{z}\\ \mathbf{else}:\\ \;\;\;\;\left(y - z \cdot t\right) \cdot \frac{x\_m}{z}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (*
  x_s
  (if (or (<= z -0.135) (not (<= z 5.9e+28)))
    (* x_m (/ (+ t y) z))
    (* (- y (* z t)) (/ x_m z)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if ((z <= -0.135) || !(z <= 5.9e+28)) {
		tmp = x_m * ((t + y) / z);
	} else {
		tmp = (y - (z * t)) * (x_m / z);
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z, t)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((z <= (-0.135d0)) .or. (.not. (z <= 5.9d+28))) then
        tmp = x_m * ((t + y) / z)
    else
        tmp = (y - (z * t)) * (x_m / z)
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if ((z <= -0.135) || !(z <= 5.9e+28)) {
		tmp = x_m * ((t + y) / z);
	} else {
		tmp = (y - (z * t)) * (x_m / z);
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z, t):
	tmp = 0
	if (z <= -0.135) or not (z <= 5.9e+28):
		tmp = x_m * ((t + y) / z)
	else:
		tmp = (y - (z * t)) * (x_m / z)
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t)
	tmp = 0.0
	if ((z <= -0.135) || !(z <= 5.9e+28))
		tmp = Float64(x_m * Float64(Float64(t + y) / z));
	else
		tmp = Float64(Float64(y - Float64(z * t)) * Float64(x_m / z));
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z, t)
	tmp = 0.0;
	if ((z <= -0.135) || ~((z <= 5.9e+28)))
		tmp = x_m * ((t + y) / z);
	else
		tmp = (y - (z * t)) * (x_m / z);
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_] := N[(x$95$s * If[Or[LessEqual[z, -0.135], N[Not[LessEqual[z, 5.9e+28]], $MachinePrecision]], N[(x$95$m * N[(N[(t + y), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], N[(N[(y - N[(z * t), $MachinePrecision]), $MachinePrecision] * N[(x$95$m / z), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < -0.13500000000000001 or 5.9000000000000002e28 < z

   1. Initial program 95.1%

      \[x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. fp-cancel-sub-sign-inv N/A

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

      3. metadata-eval N/A

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

      4. *-lft-identity N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 95.1

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

   5. Applied rewrites 95.1%

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

   if -0.13500000000000001 < z < 5.9000000000000002e28

   1. Initial program 93.0%

      \[x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. associate-*r* N/A

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

      8. distribute-rgt-out N/A

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

      9. lower-*.f64 N/A

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

      10. associate-*r* N/A

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

      11. mul-1-neg N/A

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

      12. fp-cancel-sub-sign N/A

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

      13. lower--.f64 N/A

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

      14. lower-*.f64 91.9

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

   5. Applied rewrites 91.9%

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

   7. Applied rewrites 93.6%

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

4. Final simplification 94.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -0.135 \lor \neg \left(z \leq 5.9 \cdot 10^{+28}\right):\\ \;\;\;\;x \cdot \frac{t + y}{z}\\ \mathbf{else}:\\ \;\;\;\;\left(y - z \cdot t\right) \cdot \frac{x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 93.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -1150000000000 \lor \neg \left(z \leq 1\right):\\ \;\;\;\;x\_m \cdot \frac{t + y}{z}\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot \left(\frac{y}{z} - t\right)\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (*
  x_s
  (if (or (<= z -1150000000000.0) (not (<= z 1.0)))
    (* x_m (/ (+ t y) z))
    (* x_m (- (/ y z) t)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if ((z <= -1150000000000.0) || !(z <= 1.0)) {
		tmp = x_m * ((t + y) / z);
	} else {
		tmp = x_m * ((y / z) - t);
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z, t)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((z <= (-1150000000000.0d0)) .or. (.not. (z <= 1.0d0))) then
        tmp = x_m * ((t + y) / z)
    else
        tmp = x_m * ((y / z) - t)
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if ((z <= -1150000000000.0) || !(z <= 1.0)) {
		tmp = x_m * ((t + y) / z);
	} else {
		tmp = x_m * ((y / z) - t);
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z, t):
	tmp = 0
	if (z <= -1150000000000.0) or not (z <= 1.0):
		tmp = x_m * ((t + y) / z)
	else:
		tmp = x_m * ((y / z) - t)
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t)
	tmp = 0.0
	if ((z <= -1150000000000.0) || !(z <= 1.0))
		tmp = Float64(x_m * Float64(Float64(t + y) / z));
	else
		tmp = Float64(x_m * Float64(Float64(y / z) - t));
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z, t)
	tmp = 0.0;
	if ((z <= -1150000000000.0) || ~((z <= 1.0)))
		tmp = x_m * ((t + y) / z);
	else
		tmp = x_m * ((y / z) - t);
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_] := N[(x$95$s * If[Or[LessEqual[z, -1150000000000.0], N[Not[LessEqual[z, 1.0]], $MachinePrecision]], N[(x$95$m * N[(N[(t + y), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], N[(x$95$m * N[(N[(y / z), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < -1.15e12 or 1 < z

   1. Initial program 95.2%

      \[x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. fp-cancel-sub-sign-inv N/A

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

      3. metadata-eval N/A

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

      4. *-lft-identity N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 95.2

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

   5. Applied rewrites 95.2%

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

   if -1.15e12 < z < 1

   1. Initial program 92.8%

      \[x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. fp-cancel-sub-sign N/A

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

      4. div-sub N/A

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

      5. associate-/l* N/A

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

      6. *-inverses N/A

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

      7. *-rgt-identity N/A

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

      8. lower--.f64 N/A

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

      9. lower-/.f64 91.1

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

   5. Applied rewrites 91.1%

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

4. Final simplification 93.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1150000000000 \lor \neg \left(z \leq 1\right):\\ \;\;\;\;x \cdot \frac{t + y}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 72.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -2.2 \cdot 10^{-22} \lor \neg \left(y \leq 2.4 \cdot 10^{-6}\right):\\ \;\;\;\;y \cdot \frac{x\_m}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{z - 1} \cdot t\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (*
  x_s
  (if (or (<= y -2.2e-22) (not (<= y 2.4e-6)))
    (* y (/ x_m z))
    (* (/ x_m (- z 1.0)) t))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if ((y <= -2.2e-22) || !(y <= 2.4e-6)) {
		tmp = y * (x_m / z);
	} else {
		tmp = (x_m / (z - 1.0)) * t;
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z, t)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((y <= (-2.2d-22)) .or. (.not. (y <= 2.4d-6))) then
        tmp = y * (x_m / z)
    else
        tmp = (x_m / (z - 1.0d0)) * t
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if ((y <= -2.2e-22) || !(y <= 2.4e-6)) {
		tmp = y * (x_m / z);
	} else {
		tmp = (x_m / (z - 1.0)) * t;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z, t):
	tmp = 0
	if (y <= -2.2e-22) or not (y <= 2.4e-6):
		tmp = y * (x_m / z)
	else:
		tmp = (x_m / (z - 1.0)) * t
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t)
	tmp = 0.0
	if ((y <= -2.2e-22) || !(y <= 2.4e-6))
		tmp = Float64(y * Float64(x_m / z));
	else
		tmp = Float64(Float64(x_m / Float64(z - 1.0)) * t);
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z, t)
	tmp = 0.0;
	if ((y <= -2.2e-22) || ~((y <= 2.4e-6)))
		tmp = y * (x_m / z);
	else
		tmp = (x_m / (z - 1.0)) * t;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_] := N[(x$95$s * If[Or[LessEqual[y, -2.2e-22], N[Not[LessEqual[y, 2.4e-6]], $MachinePrecision]], N[(y * N[(x$95$m / z), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m / N[(z - 1.0), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < -2.2000000000000001e-22 or 2.3999999999999999e-6 < y

   1. Initial program 93.0%

      \[x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 79.0

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

   5. Applied rewrites 79.0%

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

   7. Applied rewrites 82.8%

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

   if -2.2000000000000001e-22 < y < 2.3999999999999999e-6

   1. Initial program 95.1%

      \[x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

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

      1. distribute-rgt-in N/A

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

      2. fp-cancel-sign-sub-inv N/A

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

      3. mul-1-neg N/A

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

      4. distribute-rgt-out-- N/A

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

      5. *-commutative N/A

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

      6. fp-cancel-sub-sign-inv N/A

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

      7. *-commutative N/A

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

      8. mul-1-neg N/A

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

      9. distribute-rgt-in N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

   5. Applied rewrites 78.3%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 75.1%

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

4. Final simplification 78.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.2 \cdot 10^{-22} \lor \neg \left(y \leq 2.4 \cdot 10^{-6}\right):\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z - 1} \cdot t\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 65.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -6.8 \cdot 10^{-26} \lor \neg \left(y \leq 3.8 \cdot 10^{-145}\right):\\ \;\;\;\;y \cdot \frac{x\_m}{z}\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot \frac{t}{z}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (*
  x_s
  (if (or (<= y -6.8e-26) (not (<= y 3.8e-145)))
    (* y (/ x_m z))
    (* x_m (/ t z)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if ((y <= -6.8e-26) || !(y <= 3.8e-145)) {
		tmp = y * (x_m / z);
	} else {
		tmp = x_m * (t / z);
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z, t)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((y <= (-6.8d-26)) .or. (.not. (y <= 3.8d-145))) then
        tmp = y * (x_m / z)
    else
        tmp = x_m * (t / z)
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if ((y <= -6.8e-26) || !(y <= 3.8e-145)) {
		tmp = y * (x_m / z);
	} else {
		tmp = x_m * (t / z);
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z, t):
	tmp = 0
	if (y <= -6.8e-26) or not (y <= 3.8e-145):
		tmp = y * (x_m / z)
	else:
		tmp = x_m * (t / z)
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t)
	tmp = 0.0
	if ((y <= -6.8e-26) || !(y <= 3.8e-145))
		tmp = Float64(y * Float64(x_m / z));
	else
		tmp = Float64(x_m * Float64(t / z));
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z, t)
	tmp = 0.0;
	if ((y <= -6.8e-26) || ~((y <= 3.8e-145)))
		tmp = y * (x_m / z);
	else
		tmp = x_m * (t / z);
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_] := N[(x$95$s * If[Or[LessEqual[y, -6.8e-26], N[Not[LessEqual[y, 3.8e-145]], $MachinePrecision]], N[(y * N[(x$95$m / z), $MachinePrecision]), $MachinePrecision], N[(x$95$m * N[(t / z), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < -6.80000000000000026e-26 or 3.8000000000000002e-145 < y

   1. Initial program 93.7%

      \[x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 71.7

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

   5. Applied rewrites 71.7%

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

   7. Applied rewrites 74.1%

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

   if -6.80000000000000026e-26 < y < 3.8000000000000002e-145

   1. Initial program 94.6%

      \[x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. lower-/.f64 N/A

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

      4. *-lft-identity N/A

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

      5. metadata-eval N/A

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

      6. fp-cancel-sign-sub-inv N/A

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

      7. mul-1-neg N/A

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

      8. distribute-neg-in N/A

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

      9. metadata-eval N/A

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

      10. remove-double-neg N/A

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

      11. lower-+.f64 79.1

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

   5. Applied rewrites 79.1%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 54.3%

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

4. Final simplification 66.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -6.8 \cdot 10^{-26} \lor \neg \left(y \leq 3.8 \cdot 10^{-145}\right):\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{t}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 64.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -6.8 \cdot 10^{-26} \lor \neg \left(y \leq 3.8 \cdot 10^{-145}\right):\\ \;\;\;\;y \cdot \frac{x\_m}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{z} \cdot t\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (*
  x_s
  (if (or (<= y -6.8e-26) (not (<= y 3.8e-145)))
    (* y (/ x_m z))
    (* (/ x_m z) t))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if ((y <= -6.8e-26) || !(y <= 3.8e-145)) {
		tmp = y * (x_m / z);
	} else {
		tmp = (x_m / z) * t;
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z, t)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((y <= (-6.8d-26)) .or. (.not. (y <= 3.8d-145))) then
        tmp = y * (x_m / z)
    else
        tmp = (x_m / z) * t
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if ((y <= -6.8e-26) || !(y <= 3.8e-145)) {
		tmp = y * (x_m / z);
	} else {
		tmp = (x_m / z) * t;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z, t):
	tmp = 0
	if (y <= -6.8e-26) or not (y <= 3.8e-145):
		tmp = y * (x_m / z)
	else:
		tmp = (x_m / z) * t
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t)
	tmp = 0.0
	if ((y <= -6.8e-26) || !(y <= 3.8e-145))
		tmp = Float64(y * Float64(x_m / z));
	else
		tmp = Float64(Float64(x_m / z) * t);
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z, t)
	tmp = 0.0;
	if ((y <= -6.8e-26) || ~((y <= 3.8e-145)))
		tmp = y * (x_m / z);
	else
		tmp = (x_m / z) * t;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_] := N[(x$95$s * If[Or[LessEqual[y, -6.8e-26], N[Not[LessEqual[y, 3.8e-145]], $MachinePrecision]], N[(y * N[(x$95$m / z), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m / z), $MachinePrecision] * t), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < -6.80000000000000026e-26 or 3.8000000000000002e-145 < y

   1. Initial program 93.7%

      \[x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 71.7

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

   5. Applied rewrites 71.7%

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

   7. Applied rewrites 74.1%

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

   if -6.80000000000000026e-26 < y < 3.8000000000000002e-145

   1. Initial program 94.6%

      \[x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

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

      1. distribute-rgt-in N/A

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

      2. fp-cancel-sign-sub-inv N/A

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

      3. mul-1-neg N/A

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

      4. distribute-rgt-out-- N/A

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

      5. *-commutative N/A

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

      6. fp-cancel-sub-sign-inv N/A

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

      7. *-commutative N/A

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

      8. mul-1-neg N/A

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

      9. distribute-rgt-in N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

   5. Applied rewrites 78.1%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 51.4%

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

   9. Taylor expanded in y around 0

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

   11. Applied rewrites 54.2%

       \[\leadsto \frac{x}{z} \cdot t \]
3. Recombined 2 regimes into one program.

4. Final simplification 66.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -6.8 \cdot 10^{-26} \lor \neg \left(y \leq 3.8 \cdot 10^{-145}\right):\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot t\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 62.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;t \leq 5 \cdot 10^{+93}:\\ \;\;\;\;y \cdot \frac{x\_m}{z}\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot \left(-t\right)\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (* x_s (if (<= t 5e+93) (* y (/ x_m z)) (* x_m (- t)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (t <= 5e+93) {
		tmp = y * (x_m / z);
	} else {
		tmp = x_m * -t;
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z, t)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= 5d+93) then
        tmp = y * (x_m / z)
    else
        tmp = x_m * -t
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (t <= 5e+93) {
		tmp = y * (x_m / z);
	} else {
		tmp = x_m * -t;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z, t):
	tmp = 0
	if t <= 5e+93:
		tmp = y * (x_m / z)
	else:
		tmp = x_m * -t
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t)
	tmp = 0.0
	if (t <= 5e+93)
		tmp = Float64(y * Float64(x_m / z));
	else
		tmp = Float64(x_m * Float64(-t));
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z, t)
	tmp = 0.0;
	if (t <= 5e+93)
		tmp = y * (x_m / z);
	else
		tmp = x_m * -t;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_] := N[(x$95$s * If[LessEqual[t, 5e+93], N[(y * N[(x$95$m / z), $MachinePrecision]), $MachinePrecision], N[(x$95$m * (-t)), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 5.0000000000000001e93

   1. Initial program 92.9%

      \[x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 64.4

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

   5. Applied rewrites 64.4%

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

   7. Applied rewrites 67.0%

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

   if 5.0000000000000001e93 < t

   1. Initial program 99.9%

      \[x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. lower-/.f64 N/A

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

      4. *-lft-identity N/A

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

      5. metadata-eval N/A

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

      6. fp-cancel-sign-sub-inv N/A

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

      7. mul-1-neg N/A

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

      8. distribute-neg-in N/A

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

      9. metadata-eval N/A

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

      10. remove-double-neg N/A

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

      11. lower-+.f64 82.4

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

   5. Applied rewrites 82.4%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 46.3%

      \[\leadsto x \cdot \left(-t\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 22.8% accurate, 4.3× speedup

Math

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

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t) :precision binary64 (* x_s (* x_m (- t))))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.1%

   \[x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \]
2. Add Preprocessing

3. Taylor expanded in y around 0

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

   1. mul-1-neg N/A

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

   2. distribute-neg-frac2 N/A

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

   3. lower-/.f64 N/A

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

   4. *-lft-identity N/A

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

   5. metadata-eval N/A

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

   6. fp-cancel-sign-sub-inv N/A

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

   7. mul-1-neg N/A

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

   8. distribute-neg-in N/A

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

   9. metadata-eval N/A

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

   10. remove-double-neg N/A

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

   11. lower-+.f64 48.6

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

5. Applied rewrites 48.6%

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

6. Taylor expanded in z around 0

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

8. Applied rewrites 25.2%

   \[\leadsto x \cdot \left(-t\right) \]
9. Add Preprocessing

Developer Target 1: 95.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \left(\frac{y}{z} - t \cdot \frac{1}{1 - z}\right)\\ t_2 := x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right)\\ \mathbf{if}\;t\_2 < -7.623226303312042 \cdot 10^{-196}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 < 1.4133944927702302 \cdot 10^{-211}:\\ \;\;\;\;\frac{y \cdot x}{z} + \left(-\frac{t \cdot x}{1 - z}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* x (- (/ y z) (* t (/ 1.0 (- 1.0 z))))))
        (t_2 (* x (- (/ y z) (/ t (- 1.0 z))))))
   (if (< t_2 -7.623226303312042e-196)
     t_1
     (if (< t_2 1.4133944927702302e-211)
       (+ (/ (* y x) z) (- (/ (* t x) (- 1.0 z))))
       t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x * ((y / z) - (t * (1.0 / (1.0 - z))));
	double t_2 = x * ((y / z) - (t / (1.0 - z)));
	double tmp;
	if (t_2 < -7.623226303312042e-196) {
		tmp = t_1;
	} else if (t_2 < 1.4133944927702302e-211) {
		tmp = ((y * x) / z) + -((t * x) / (1.0 - z));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = x * ((y / z) - (t * (1.0d0 / (1.0d0 - z))))
    t_2 = x * ((y / z) - (t / (1.0d0 - z)))
    if (t_2 < (-7.623226303312042d-196)) then
        tmp = t_1
    else if (t_2 < 1.4133944927702302d-211) then
        tmp = ((y * x) / z) + -((t * x) / (1.0d0 - z))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = x * ((y / z) - (t * (1.0 / (1.0 - z))));
	double t_2 = x * ((y / z) - (t / (1.0 - z)));
	double tmp;
	if (t_2 < -7.623226303312042e-196) {
		tmp = t_1;
	} else if (t_2 < 1.4133944927702302e-211) {
		tmp = ((y * x) / z) + -((t * x) / (1.0 - z));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = x * ((y / z) - (t * (1.0 / (1.0 - z))))
	t_2 = x * ((y / z) - (t / (1.0 - z)))
	tmp = 0
	if t_2 < -7.623226303312042e-196:
		tmp = t_1
	elif t_2 < 1.4133944927702302e-211:
		tmp = ((y * x) / z) + -((t * x) / (1.0 - z))
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(x * Float64(Float64(y / z) - Float64(t * Float64(1.0 / Float64(1.0 - z)))))
	t_2 = Float64(x * Float64(Float64(y / z) - Float64(t / Float64(1.0 - z))))
	tmp = 0.0
	if (t_2 < -7.623226303312042e-196)
		tmp = t_1;
	elseif (t_2 < 1.4133944927702302e-211)
		tmp = Float64(Float64(Float64(y * x) / z) + Float64(-Float64(Float64(t * x) / Float64(1.0 - z))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = x * ((y / z) - (t * (1.0 / (1.0 - z))));
	t_2 = x * ((y / z) - (t / (1.0 - z)));
	tmp = 0.0;
	if (t_2 < -7.623226303312042e-196)
		tmp = t_1;
	elseif (t_2 < 1.4133944927702302e-211)
		tmp = ((y * x) / z) + -((t * x) / (1.0 - z));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x * N[(N[(y / z), $MachinePrecision] - N[(t * N[(1.0 / N[(1.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(x * N[(N[(y / z), $MachinePrecision] - N[(t / N[(1.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Less[t$95$2, -7.623226303312042e-196], t$95$1, If[Less[t$95$2, 1.4133944927702302e-211], N[(N[(N[(y * x), $MachinePrecision] / z), $MachinePrecision] + (-N[(N[(t * x), $MachinePrecision] / N[(1.0 - z), $MachinePrecision]), $MachinePrecision])), $MachinePrecision], t$95$1]]]]

Reproduce

herbie shell --seed 2024337 
(FPCore (x y z t)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, C"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< (* x (- (/ y z) (/ t (- 1 z)))) -3811613151656021/5000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (* x (- (/ y z) (* t (/ 1 (- 1 z))))) (if (< (* x (- (/ y z) (/ t (- 1 z)))) 7066972463851151/50000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (+ (/ (* y x) z) (- (/ (* t x) (- 1 z)))) (* x (- (/ y z) (* t (/ 1 (- 1 z))))))))

  (* x (- (/ y z) (/ t (- 1.0 z)))))