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

Percentage Accurate: 94.6% → 95.9%

Time: 11.1s

Alternatives: 10

Speedup: 0.9×

• 21.1% 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: 94.6% 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: 95.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_1 := \frac{y \cdot \left(1 - z\right) - z \cdot t}{1 - z} \cdot \frac{x\_m}{z}\\ t_2 := x\_m \cdot \left(\frac{y}{z} + \frac{t}{z + -1}\right)\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq -4 \cdot 10^{+268}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+224}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \end{array} \]

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 (- 1.0 z)) (* z t)) (- 1.0 z)) (/ x_m z)))
        (t_2 (* x_m (+ (/ y z) (/ t (+ z -1.0))))))
   (* x_s (if (<= t_2 -4e+268) t_1 (if (<= t_2 2e+224) t_2 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 * (1.0 - z)) - (z * t)) / (1.0 - z)) * (x_m / z);
	double t_2 = x_m * ((y / z) + (t / (z + -1.0)));
	double tmp;
	if (t_2 <= -4e+268) {
		tmp = t_1;
	} else if (t_2 <= 2e+224) {
		tmp = t_2;
	} 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) :: t_2
    real(8) :: tmp
    t_1 = (((y * (1.0d0 - z)) - (z * t)) / (1.0d0 - z)) * (x_m / z)
    t_2 = x_m * ((y / z) + (t / (z + (-1.0d0))))
    if (t_2 <= (-4d+268)) then
        tmp = t_1
    else if (t_2 <= 2d+224) then
        tmp = t_2
    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 * (1.0 - z)) - (z * t)) / (1.0 - z)) * (x_m / z);
	double t_2 = x_m * ((y / z) + (t / (z + -1.0)));
	double tmp;
	if (t_2 <= -4e+268) {
		tmp = t_1;
	} else if (t_2 <= 2e+224) {
		tmp = t_2;
	} 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 * (1.0 - z)) - (z * t)) / (1.0 - z)) * (x_m / z)
	t_2 = x_m * ((y / z) + (t / (z + -1.0)))
	tmp = 0
	if t_2 <= -4e+268:
		tmp = t_1
	elif t_2 <= 2e+224:
		tmp = t_2
	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(Float64(Float64(Float64(y * Float64(1.0 - z)) - Float64(z * t)) / Float64(1.0 - z)) * Float64(x_m / z))
	t_2 = Float64(x_m * Float64(Float64(y / z) + Float64(t / Float64(z + -1.0))))
	tmp = 0.0
	if (t_2 <= -4e+268)
		tmp = t_1;
	elseif (t_2 <= 2e+224)
		tmp = t_2;
	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 * (1.0 - z)) - (z * t)) / (1.0 - z)) * (x_m / z);
	t_2 = x_m * ((y / z) + (t / (z + -1.0)));
	tmp = 0.0;
	if (t_2 <= -4e+268)
		tmp = t_1;
	elseif (t_2 <= 2e+224)
		tmp = t_2;
	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[(N[(N[(N[(y * N[(1.0 - z), $MachinePrecision]), $MachinePrecision] - N[(z * t), $MachinePrecision]), $MachinePrecision] / N[(1.0 - z), $MachinePrecision]), $MachinePrecision] * N[(x$95$m / z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(x$95$m * N[(N[(y / z), $MachinePrecision] + N[(t / N[(z + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * If[LessEqual[t$95$2, -4e+268], t$95$1, If[LessEqual[t$95$2, 2e+224], t$95$2, 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)))) < -3.9999999999999999e268 or 1.99999999999999994e224 < (*.f64 x (-.f64 (/.f64 y z) (/.f64 t (-.f64 #s(literal 1 binary64) z))))

   1. Initial program 84.7%

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

      1. *-commutative N/A

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

      2. frac-sub N/A

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

      3. associate-*l/ N/A

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

      4. *-commutative N/A

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

      5. times-frac N/A

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

      6. *-lowering-*.f64 N/A

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

      7. /-lowering-/.f64 N/A

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

      8. --lowering--.f64 N/A

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

      9. *-lowering-*.f64 N/A

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

      10. --lowering--.f64 N/A

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

      11. *-lowering-*.f64 N/A

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

      12. --lowering--.f64 N/A

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

      13. /-lowering-/.f64 97.4

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

   4. Applied egg-rr 97.4%

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

   if -3.9999999999999999e268 < (*.f64 x (-.f64 (/.f64 y z) (/.f64 t (-.f64 #s(literal 1 binary64) z)))) < 1.99999999999999994e224

   1. Initial program 96.8%

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

4. Final simplification 97.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot \left(\frac{y}{z} + \frac{t}{z + -1}\right) \leq -4 \cdot 10^{+268}:\\ \;\;\;\;\frac{y \cdot \left(1 - z\right) - z \cdot t}{1 - z} \cdot \frac{x}{z}\\ \mathbf{elif}\;x \cdot \left(\frac{y}{z} + \frac{t}{z + -1}\right) \leq 2 \cdot 10^{+224}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} + \frac{t}{z + -1}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot \left(1 - z\right) - z \cdot t}{1 - z} \cdot \frac{x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_1 := y \cdot \frac{x\_m}{z}\\ t_2 := \frac{y}{z} + \frac{t}{z + -1}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+294}:\\ \;\;\;\;x\_m \cdot t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \end{array} \]

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))) (t_2 (+ (/ y z) (/ t (+ z -1.0)))))
   (*
    x_s
    (if (<= t_2 (- INFINITY)) t_1 (if (<= t_2 5e+294) (* x_m t_2) 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 t_2 = (y / z) + (t / (z + -1.0));
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = t_1;
	} else if (t_2 <= 5e+294) {
		tmp = x_m * t_2;
	} else {
		tmp = 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 * (x_m / z);
	double t_2 = (y / z) + (t / (z + -1.0));
	double tmp;
	if (t_2 <= -Double.POSITIVE_INFINITY) {
		tmp = t_1;
	} else if (t_2 <= 5e+294) {
		tmp = x_m * t_2;
	} 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)
	t_2 = (y / z) + (t / (z + -1.0))
	tmp = 0
	if t_2 <= -math.inf:
		tmp = t_1
	elif t_2 <= 5e+294:
		tmp = x_m * t_2
	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))
	t_2 = Float64(Float64(y / z) + Float64(t / Float64(z + -1.0)))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = t_1;
	elseif (t_2 <= 5e+294)
		tmp = Float64(x_m * t_2);
	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);
	t_2 = (y / z) + (t / (z + -1.0));
	tmp = 0.0;
	if (t_2 <= -Inf)
		tmp = t_1;
	elseif (t_2 <= 5e+294)
		tmp = x_m * t_2;
	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]}, Block[{t$95$2 = N[(N[(y / z), $MachinePrecision] + N[(t / N[(z + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * If[LessEqual[t$95$2, (-Infinity)], t$95$1, If[LessEqual[t$95$2, 5e+294], N[(x$95$m * t$95$2), $MachinePrecision], t$95$1]]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (/.f64 y z) (/.f64 t (-.f64 #s(literal 1 binary64) z))) < -inf.0 or 4.9999999999999999e294 < (-.f64 (/.f64 y z) (/.f64 t (-.f64 #s(literal 1 binary64) z)))

   1. Initial program 63.8%

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

   3. Taylor expanded in y around inf

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

      1. /-lowering-/.f64 63.8

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

   5. Simplified 63.8%

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

      1. associate-*r/ N/A

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

      2. /-lowering-/.f64 N/A

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

      3. *-commutative N/A

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

      4. *-lowering-*.f64 99.7

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

   7. Applied egg-rr 99.7%

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 N/A

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

      4. /-lowering-/.f64 99.9

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

   9. Applied egg-rr 99.9%

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

   if -inf.0 < (-.f64 (/.f64 y z) (/.f64 t (-.f64 #s(literal 1 binary64) z))) < 4.9999999999999999e294

   1. Initial program 97.4%

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

4. Final simplification 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y}{z} + \frac{t}{z + -1} \leq -\infty:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{elif}\;\frac{y}{z} + \frac{t}{z + -1} \leq 5 \cdot 10^{+294}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} + \frac{t}{z + -1}\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 95.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_1 := x\_m \cdot \frac{y + t}{z}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -1:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 2.6 \cdot 10^{-6}:\\ \;\;\;\;\frac{x\_m}{z} \cdot \mathsf{fma}\left(t, -z, y\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \end{array} \]

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 t) z))))
   (*
    x_s
    (if (<= z -1.0)
      t_1
      (if (<= z 2.6e-6) (* (/ x_m z) (fma t (- z) y)) 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 + t) / z);
	double tmp;
	if (z <= -1.0) {
		tmp = t_1;
	} else if (z <= 2.6e-6) {
		tmp = (x_m / z) * fma(t, -z, y);
	} 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 + t) / z))
	tmp = 0.0
	if (z <= -1.0)
		tmp = t_1;
	elseif (z <= 2.6e-6)
		tmp = Float64(Float64(x_m / z) * fma(t, Float64(-z), y));
	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 + t), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * If[LessEqual[z, -1.0], t$95$1, If[LessEqual[z, 2.6e-6], N[(N[(x$95$m / z), $MachinePrecision] * N[(t * (-z) + y), $MachinePrecision]), $MachinePrecision], t$95$1]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1 or 2.60000000000000009e-6 < z

   1. Initial program 95.9%

      \[x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \]
   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. /-lowering-/.f64 N/A

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

      2. cancel-sign-sub-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. +-lowering-+.f64 95.1

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

   5. Simplified 95.1%

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

   if -1 < z < 2.60000000000000009e-6

   1. Initial program 90.4%

      \[x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \]
   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. /-lowering-/.f64 N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. --lowering--.f64 N/A

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

      5. *-commutative N/A

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

      6. *-lowering-*.f64 88.5

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

   5. Simplified 88.5%

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

      1. clear-num N/A

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

      2. associate-*r/ N/A

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

      3. div-inv N/A

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

      4. times-frac N/A

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

      5. flip-- N/A

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

      6. clear-num N/A

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

      7. clear-num N/A

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

      8. flip-- N/A

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

      9. *-lowering-*.f64 N/A

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

      10. /-lowering-/.f64 N/A

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

      11. sub-neg N/A

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

      12. +-commutative N/A

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

      13. *-commutative N/A

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

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

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

      15. accelerator-lowering-fma.f64 N/A

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

      16. neg-lowering-neg.f64 92.3

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

   7. Applied egg-rr 92.3%

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

Alternative 4: 93.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_1 := x\_m \cdot \frac{y + t}{z}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -1:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 2.6 \cdot 10^{-6}:\\ \;\;\;\;x\_m \cdot \left(\frac{y}{z} - t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \end{array} \]

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 t) z))))
   (* x_s (if (<= z -1.0) t_1 (if (<= z 2.6e-6) (* x_m (- (/ y z) t)) 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 + t) / z);
	double tmp;
	if (z <= -1.0) {
		tmp = t_1;
	} else if (z <= 2.6e-6) {
		tmp = x_m * ((y / z) - t);
	} 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 = x_m * ((y + t) / z)
    if (z <= (-1.0d0)) then
        tmp = t_1
    else if (z <= 2.6d-6) then
        tmp = x_m * ((y / z) - t)
    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 = x_m * ((y + t) / z);
	double tmp;
	if (z <= -1.0) {
		tmp = t_1;
	} else if (z <= 2.6e-6) {
		tmp = x_m * ((y / z) - t);
	} 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 = x_m * ((y + t) / z)
	tmp = 0
	if z <= -1.0:
		tmp = t_1
	elif z <= 2.6e-6:
		tmp = x_m * ((y / z) - t)
	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 + t) / z))
	tmp = 0.0
	if (z <= -1.0)
		tmp = t_1;
	elseif (z <= 2.6e-6)
		tmp = Float64(x_m * Float64(Float64(y / z) - t));
	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 = x_m * ((y + t) / z);
	tmp = 0.0;
	if (z <= -1.0)
		tmp = t_1;
	elseif (z <= 2.6e-6)
		tmp = x_m * ((y / z) - t);
	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[(x$95$m * N[(N[(y + t), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * If[LessEqual[z, -1.0], t$95$1, If[LessEqual[z, 2.6e-6], N[(x$95$m * N[(N[(y / z), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision], t$95$1]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1 or 2.60000000000000009e-6 < z

   1. Initial program 95.9%

      \[x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \]
   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. /-lowering-/.f64 N/A

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

      2. cancel-sign-sub-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. +-lowering-+.f64 95.1

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

   5. Simplified 95.1%

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

   if -1 < z < 2.60000000000000009e-6

   1. Initial program 90.4%

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

   3. Taylor expanded in z around 0

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

   5. Simplified 88.5%

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

Alternative 5: 76.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_1 := x\_m \cdot \frac{t}{z + -1}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;t \leq -1.1 \cdot 10^{+97}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 3.4 \cdot 10^{+52}:\\ \;\;\;\;x\_m \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \end{array} \]

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 (/ t (+ z -1.0)))))
   (* x_s (if (<= t -1.1e+97) t_1 (if (<= t 3.4e+52) (* x_m (/ y 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 * (t / (z + -1.0));
	double tmp;
	if (t <= -1.1e+97) {
		tmp = t_1;
	} else if (t <= 3.4e+52) {
		tmp = x_m * (y / 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 = x_m * (t / (z + (-1.0d0)))
    if (t <= (-1.1d+97)) then
        tmp = t_1
    else if (t <= 3.4d+52) then
        tmp = x_m * (y / 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 = x_m * (t / (z + -1.0));
	double tmp;
	if (t <= -1.1e+97) {
		tmp = t_1;
	} else if (t <= 3.4e+52) {
		tmp = x_m * (y / 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 = x_m * (t / (z + -1.0))
	tmp = 0
	if t <= -1.1e+97:
		tmp = t_1
	elif t <= 3.4e+52:
		tmp = x_m * (y / 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(t / Float64(z + -1.0)))
	tmp = 0.0
	if (t <= -1.1e+97)
		tmp = t_1;
	elseif (t <= 3.4e+52)
		tmp = Float64(x_m * Float64(y / 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 = x_m * (t / (z + -1.0));
	tmp = 0.0;
	if (t <= -1.1e+97)
		tmp = t_1;
	elseif (t <= 3.4e+52)
		tmp = x_m * (y / 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[(x$95$m * N[(t / N[(z + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * If[LessEqual[t, -1.1e+97], t$95$1, If[LessEqual[t, 3.4e+52], N[(x$95$m * N[(y / z), $MachinePrecision]), $MachinePrecision], t$95$1]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -1.1e97 or 3.4e52 < t

   1. Initial program 94.2%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. *-lowering-*.f64 N/A

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

      7. mul-1-neg N/A

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

      8. distribute-neg-frac2 N/A

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

      9. /-lowering-/.f64 N/A

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

      10. sub-neg N/A

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

      11. mul-1-neg N/A

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

      12. distribute-neg-in N/A

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

      13. metadata-eval N/A

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

      14. mul-1-neg N/A

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

      15. remove-double-neg N/A

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

      16. +-commutative N/A

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

      17. +-lowering-+.f64 77.1

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

   5. Simplified 77.1%

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

   if -1.1e97 < t < 3.4e52

   1. Initial program 92.8%

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

   3. Taylor expanded in y around inf

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

      1. /-lowering-/.f64 79.3

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

   5. Simplified 79.3%

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

Alternative 6: 74.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_1 := x\_m \cdot \frac{t}{z}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -5.5 \cdot 10^{+55}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 2.15 \cdot 10^{+33}:\\ \;\;\;\;x\_m \cdot \left(\frac{y}{z} - t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \end{array} \]

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 (/ t z))))
   (*
    x_s
    (if (<= z -5.5e+55) t_1 (if (<= z 2.15e+33) (* x_m (- (/ y z) t)) 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 * (t / z);
	double tmp;
	if (z <= -5.5e+55) {
		tmp = t_1;
	} else if (z <= 2.15e+33) {
		tmp = x_m * ((y / z) - t);
	} 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 = x_m * (t / z)
    if (z <= (-5.5d+55)) then
        tmp = t_1
    else if (z <= 2.15d+33) then
        tmp = x_m * ((y / z) - t)
    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 = x_m * (t / z);
	double tmp;
	if (z <= -5.5e+55) {
		tmp = t_1;
	} else if (z <= 2.15e+33) {
		tmp = x_m * ((y / z) - t);
	} 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 = x_m * (t / z)
	tmp = 0
	if z <= -5.5e+55:
		tmp = t_1
	elif z <= 2.15e+33:
		tmp = x_m * ((y / z) - t)
	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(t / z))
	tmp = 0.0
	if (z <= -5.5e+55)
		tmp = t_1;
	elseif (z <= 2.15e+33)
		tmp = Float64(x_m * Float64(Float64(y / z) - t));
	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 = x_m * (t / z);
	tmp = 0.0;
	if (z <= -5.5e+55)
		tmp = t_1;
	elseif (z <= 2.15e+33)
		tmp = x_m * ((y / z) - t);
	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[(x$95$m * N[(t / z), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * If[LessEqual[z, -5.5e+55], t$95$1, If[LessEqual[z, 2.15e+33], N[(x$95$m * N[(N[(y / z), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision], t$95$1]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -5.5000000000000004e55 or 2.15000000000000014e33 < z

   1. Initial program 95.2%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. *-lowering-*.f64 N/A

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

      7. mul-1-neg N/A

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

      8. distribute-neg-frac2 N/A

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

      9. /-lowering-/.f64 N/A

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

      10. sub-neg N/A

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

      11. mul-1-neg N/A

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

      12. distribute-neg-in N/A

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

      13. metadata-eval N/A

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

      14. mul-1-neg N/A

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

      15. remove-double-neg N/A

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

      16. +-commutative N/A

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

      17. +-lowering-+.f64 64.7

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

   5. Simplified 64.7%

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

   6. Taylor expanded in z around inf

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

      1. /-lowering-/.f64 64.7

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

   8. Simplified 64.7%

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

   if -5.5000000000000004e55 < z < 2.15000000000000014e33

   1. Initial program 91.9%

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

   3. Taylor expanded in z around 0

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

   5. Simplified 86.9%

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

Alternative 7: 73.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_1 := t \cdot \frac{x\_m}{z + -1}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;t \leq -1.3 \cdot 10^{+97}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 7 \cdot 10^{+51}:\\ \;\;\;\;x\_m \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \end{array} \]

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 (* t (/ x_m (+ z -1.0)))))
   (* x_s (if (<= t -1.3e+97) t_1 (if (<= t 7e+51) (* x_m (/ y 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 = t * (x_m / (z + -1.0));
	double tmp;
	if (t <= -1.3e+97) {
		tmp = t_1;
	} else if (t <= 7e+51) {
		tmp = x_m * (y / 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 = t * (x_m / (z + (-1.0d0)))
    if (t <= (-1.3d+97)) then
        tmp = t_1
    else if (t <= 7d+51) then
        tmp = x_m * (y / 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 = t * (x_m / (z + -1.0));
	double tmp;
	if (t <= -1.3e+97) {
		tmp = t_1;
	} else if (t <= 7e+51) {
		tmp = x_m * (y / 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 = t * (x_m / (z + -1.0))
	tmp = 0
	if t <= -1.3e+97:
		tmp = t_1
	elif t <= 7e+51:
		tmp = x_m * (y / 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(t * Float64(x_m / Float64(z + -1.0)))
	tmp = 0.0
	if (t <= -1.3e+97)
		tmp = t_1;
	elseif (t <= 7e+51)
		tmp = Float64(x_m * Float64(y / 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 = t * (x_m / (z + -1.0));
	tmp = 0.0;
	if (t <= -1.3e+97)
		tmp = t_1;
	elseif (t <= 7e+51)
		tmp = x_m * (y / 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[(t * N[(x$95$m / N[(z + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * If[LessEqual[t, -1.3e+97], t$95$1, If[LessEqual[t, 7e+51], N[(x$95$m * N[(y / z), $MachinePrecision]), $MachinePrecision], t$95$1]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -1.3e97 or 7e51 < t

   1. Initial program 94.2%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. distribute-neg-frac2 N/A

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

      4. div-inv N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. /-lowering-/.f64 N/A

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

      7. sub-neg N/A

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

      8. distribute-neg-in N/A

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

      9. metadata-eval N/A

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

      10. remove-double-neg N/A

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

      11. +-lowering-+.f64 N/A

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

      12. /-lowering-/.f64 94.2

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

   4. Applied egg-rr 94.2%

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-commutative N/A

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

      4. associate-*l/ N/A

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

      5. *-lft-identity N/A

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

      6. times-frac N/A

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

      7. clear-num N/A

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

      8. unpow-1 N/A

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

      9. accelerator-lowering-fma.f64 N/A

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

      10. unpow-1 N/A

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

      11. clear-num N/A

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

      12. /-lowering-/.f64 N/A

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

      13. /-lowering-/.f64 N/A

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

      14. associate-*r* N/A

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

      15. div-inv N/A

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

      16. /-lowering-/.f64 N/A

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

      17. *-commutative N/A

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

      18. *-lowering-*.f64 N/A

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

      19. +-lowering-+.f64 84.8

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

   6. Applied egg-rr 84.8%

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

   7. Taylor expanded in y around 0

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

      1. associate-/l* N/A

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

      2. *-lowering-*.f64 N/A

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

      3. /-lowering-/.f64 N/A

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

      4. sub-neg N/A

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

      5. metadata-eval N/A

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

      6. +-lowering-+.f64 66.9

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

   9. Simplified 66.9%

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

   if -1.3e97 < t < 7e51

   1. Initial program 92.8%

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

   3. Taylor expanded in y around inf

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

      1. /-lowering-/.f64 79.3

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

   5. Simplified 79.3%

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

Alternative 8: 68.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_1 := x\_m \cdot \frac{t}{z}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;t \leq -1.9 \cdot 10^{+97}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 6.5 \cdot 10^{+72}:\\ \;\;\;\;x\_m \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \end{array} \]

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 (/ t z))))
   (* x_s (if (<= t -1.9e+97) t_1 (if (<= t 6.5e+72) (* x_m (/ y 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 * (t / z);
	double tmp;
	if (t <= -1.9e+97) {
		tmp = t_1;
	} else if (t <= 6.5e+72) {
		tmp = x_m * (y / 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 = x_m * (t / z)
    if (t <= (-1.9d+97)) then
        tmp = t_1
    else if (t <= 6.5d+72) then
        tmp = x_m * (y / 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 = x_m * (t / z);
	double tmp;
	if (t <= -1.9e+97) {
		tmp = t_1;
	} else if (t <= 6.5e+72) {
		tmp = x_m * (y / 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 = x_m * (t / z)
	tmp = 0
	if t <= -1.9e+97:
		tmp = t_1
	elif t <= 6.5e+72:
		tmp = x_m * (y / 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(t / z))
	tmp = 0.0
	if (t <= -1.9e+97)
		tmp = t_1;
	elseif (t <= 6.5e+72)
		tmp = Float64(x_m * Float64(y / 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 = x_m * (t / z);
	tmp = 0.0;
	if (t <= -1.9e+97)
		tmp = t_1;
	elseif (t <= 6.5e+72)
		tmp = x_m * (y / 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[(x$95$m * N[(t / z), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * If[LessEqual[t, -1.9e+97], t$95$1, If[LessEqual[t, 6.5e+72], N[(x$95$m * N[(y / z), $MachinePrecision]), $MachinePrecision], t$95$1]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -1.90000000000000018e97 or 6.5000000000000001e72 < t

   1. Initial program 93.9%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. *-lowering-*.f64 N/A

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

      7. mul-1-neg N/A

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

      8. distribute-neg-frac2 N/A

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

      9. /-lowering-/.f64 N/A

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

      10. sub-neg N/A

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

      11. mul-1-neg N/A

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

      12. distribute-neg-in N/A

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

      13. metadata-eval N/A

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

      14. mul-1-neg N/A

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

      15. remove-double-neg N/A

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

      16. +-commutative N/A

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

      17. +-lowering-+.f64 77.8

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

   5. Simplified 77.8%

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

   6. Taylor expanded in z around inf

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

      1. /-lowering-/.f64 62.1

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

   8. Simplified 62.1%

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

   if -1.90000000000000018e97 < t < 6.5000000000000001e72

   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

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

      1. /-lowering-/.f64 78.8

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

   5. Simplified 78.8%

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

Alternative 9: 46.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_1 := x\_m \cdot \frac{t}{z}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -0.76:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 3.4 \cdot 10^{-6}:\\ \;\;\;\;-x\_m \cdot \mathsf{fma}\left(z, t, t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \end{array} \]

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 (/ t z))))
   (*
    x_s
    (if (<= z -0.76) t_1 (if (<= z 3.4e-6) (- (* x_m (fma z t t))) 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 * (t / z);
	double tmp;
	if (z <= -0.76) {
		tmp = t_1;
	} else if (z <= 3.4e-6) {
		tmp = -(x_m * fma(z, t, t));
	} 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(t / z))
	tmp = 0.0
	if (z <= -0.76)
		tmp = t_1;
	elseif (z <= 3.4e-6)
		tmp = Float64(-Float64(x_m * fma(z, t, t)));
	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[(t / z), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * If[LessEqual[z, -0.76], t$95$1, If[LessEqual[z, 3.4e-6], (-N[(x$95$m * N[(z * t + t), $MachinePrecision]), $MachinePrecision]), t$95$1]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -0.76000000000000001 or 3.40000000000000006e-6 < z

   1. Initial program 95.9%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. *-lowering-*.f64 N/A

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

      7. mul-1-neg N/A

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

      8. distribute-neg-frac2 N/A

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

      9. /-lowering-/.f64 N/A

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

      10. sub-neg N/A

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

      11. mul-1-neg N/A

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

      12. distribute-neg-in N/A

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

      13. metadata-eval N/A

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

      14. mul-1-neg N/A

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

      15. remove-double-neg N/A

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

      16. +-commutative N/A

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

      17. +-lowering-+.f64 60.4

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

   5. Simplified 60.4%

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

   6. Taylor expanded in z around inf

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

      1. /-lowering-/.f64 59.9

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

   8. Simplified 59.9%

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

   if -0.76000000000000001 < z < 3.40000000000000006e-6

   1. Initial program 90.5%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. *-lowering-*.f64 N/A

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

      7. mul-1-neg N/A

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

      8. distribute-neg-frac2 N/A

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

      9. /-lowering-/.f64 N/A

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

      10. sub-neg N/A

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

      11. mul-1-neg N/A

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

      12. distribute-neg-in N/A

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

      13. metadata-eval N/A

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

      14. mul-1-neg N/A

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

      15. remove-double-neg N/A

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

      16. +-commutative N/A

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

      17. +-lowering-+.f64 29.7

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

   5. Simplified 29.7%

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

   6. Taylor expanded in z around 0

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

      1. distribute-lft-out N/A

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

      2. mul-1-neg N/A

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

      3. neg-lowering-neg.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. accelerator-lowering-fma.f64 28.5

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

   8. Simplified 28.5%

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

4. Final simplification 45.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -0.76:\\ \;\;\;\;x \cdot \frac{t}{z}\\ \mathbf{elif}\;z \leq 3.4 \cdot 10^{-6}:\\ \;\;\;\;-x \cdot \mathsf{fma}\left(z, t, t\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{t}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 22.9% 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 93.3%

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

3. Taylor expanded in y around 0

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

   1. *-commutative N/A

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

   2. associate-/l* N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. associate-*r* N/A

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

   6. *-lowering-*.f64 N/A

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

   7. mul-1-neg N/A

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

   8. distribute-neg-frac2 N/A

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

   9. /-lowering-/.f64 N/A

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

   10. sub-neg N/A

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

   11. mul-1-neg N/A

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

   12. distribute-neg-in N/A

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

   13. metadata-eval N/A

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

   14. mul-1-neg N/A

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

   15. remove-double-neg N/A

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

   16. +-commutative N/A

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

   17. +-lowering-+.f64 45.9

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

5. Simplified 45.9%

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

6. Taylor expanded in z around 0

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

   1. mul-1-neg N/A

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

   2. neg-lowering-neg.f64 19.9

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

8. Simplified 19.9%

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

Developer Target 1: 94.9% 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 2024204 
(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)))))