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

Percentage Accurate: 94.5% → 97.5%

Time: 11.0s

Alternatives: 12

Speedup: 0.3×

• 21.7% 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: 94.5% 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: 97.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_1 := y \cdot \left(1 - z\right) - z \cdot t\\ t_2 := x\_m \cdot \left(\frac{y}{z} + \frac{t}{z + -1}\right)\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;\frac{t\_1}{1 - z} \cdot \frac{x\_m}{z}\\ \mathbf{elif}\;t\_2 \leq 10^{+274}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\left(x\_m \cdot t\_1\right) \cdot \frac{1}{z \cdot \left(1 - z\right)}\\ \end{array} \end{array} \end{array} \]

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)))
        (t_2 (* x_m (+ (/ y z) (/ t (+ z -1.0))))))
   (*
    x_s
    (if (<= t_2 (- INFINITY))
      (* (/ t_1 (- 1.0 z)) (/ x_m z))
      (if (<= t_2 1e+274) t_2 (* (* x_m t_1) (/ 1.0 (* z (- 1.0 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 t_1 = (y * (1.0 - z)) - (z * t);
	double t_2 = x_m * ((y / z) + (t / (z + -1.0)));
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = (t_1 / (1.0 - z)) * (x_m / z);
	} else if (t_2 <= 1e+274) {
		tmp = t_2;
	} else {
		tmp = (x_m * t_1) * (1.0 / (z * (1.0 - z)));
	}
	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 * (1.0 - z)) - (z * t);
	double t_2 = x_m * ((y / z) + (t / (z + -1.0)));
	double tmp;
	if (t_2 <= -Double.POSITIVE_INFINITY) {
		tmp = (t_1 / (1.0 - z)) * (x_m / z);
	} else if (t_2 <= 1e+274) {
		tmp = t_2;
	} else {
		tmp = (x_m * t_1) * (1.0 / (z * (1.0 - 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):
	t_1 = (y * (1.0 - z)) - (z * t)
	t_2 = x_m * ((y / z) + (t / (z + -1.0)))
	tmp = 0
	if t_2 <= -math.inf:
		tmp = (t_1 / (1.0 - z)) * (x_m / z)
	elif t_2 <= 1e+274:
		tmp = t_2
	else:
		tmp = (x_m * t_1) * (1.0 / (z * (1.0 - 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)
	t_1 = Float64(Float64(y * Float64(1.0 - z)) - Float64(z * t))
	t_2 = Float64(x_m * Float64(Float64(y / z) + Float64(t / Float64(z + -1.0))))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = Float64(Float64(t_1 / Float64(1.0 - z)) * Float64(x_m / z));
	elseif (t_2 <= 1e+274)
		tmp = t_2;
	else
		tmp = Float64(Float64(x_m * t_1) * Float64(1.0 / Float64(z * Float64(1.0 - 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)
	t_1 = (y * (1.0 - z)) - (z * t);
	t_2 = x_m * ((y / z) + (t / (z + -1.0)));
	tmp = 0.0;
	if (t_2 <= -Inf)
		tmp = (t_1 / (1.0 - z)) * (x_m / z);
	elseif (t_2 <= 1e+274)
		tmp = t_2;
	else
		tmp = (x_m * t_1) * (1.0 / (z * (1.0 - 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_] := Block[{t$95$1 = N[(N[(y * N[(1.0 - z), $MachinePrecision]), $MachinePrecision] - N[(z * t), $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, (-Infinity)], N[(N[(t$95$1 / N[(1.0 - z), $MachinePrecision]), $MachinePrecision] * N[(x$95$m / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 1e+274], t$95$2, N[(N[(x$95$m * t$95$1), $MachinePrecision] * N[(1.0 / N[(z * N[(1.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 74.8%

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

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

      9. times-frac N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 99.9

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

   4. Applied rewrites 99.9%

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

   if -inf.0 < (*.f64 x (-.f64 (/.f64 y z) (/.f64 t (-.f64 #s(literal 1 binary64) z)))) < 9.99999999999999921e273

   1. Initial program 97.4%

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

   if 9.99999999999999921e273 < (*.f64 x (-.f64 (/.f64 y z) (/.f64 t (-.f64 #s(literal 1 binary64) z))))

   1. Initial program 87.2%

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

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower--.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-*.f64 97.5

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

   4. Applied rewrites 97.5%

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

4. Final simplification 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot \left(\frac{y}{z} + \frac{t}{z + -1}\right) \leq -\infty:\\ \;\;\;\;\frac{y \cdot \left(1 - z\right) - z \cdot t}{1 - z} \cdot \frac{x}{z}\\ \mathbf{elif}\;x \cdot \left(\frac{y}{z} + \frac{t}{z + -1}\right) \leq 10^{+274}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} + \frac{t}{z + -1}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \left(y \cdot \left(1 - z\right) - z \cdot t\right)\right) \cdot \frac{1}{z \cdot \left(1 - z\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 97.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_1 := \frac{y}{z} + \frac{t}{z + -1}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;y \cdot \frac{x\_m}{z}\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+291}:\\ \;\;\;\;x\_m \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m \cdot y}{z}\\ \end{array} \end{array} \end{array} \]

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 (+ z -1.0)))))
   (*
    x_s
    (if (<= t_1 (- INFINITY))
      (* y (/ x_m z))
      (if (<= t_1 2e+291) (* x_m t_1) (/ (* x_m y) 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 t_1 = (y / z) + (t / (z + -1.0));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = y * (x_m / z);
	} else if (t_1 <= 2e+291) {
		tmp = x_m * t_1;
	} else {
		tmp = (x_m * y) / z;
	}
	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 / (z + -1.0));
	double tmp;
	if (t_1 <= -Double.POSITIVE_INFINITY) {
		tmp = y * (x_m / z);
	} else if (t_1 <= 2e+291) {
		tmp = x_m * t_1;
	} else {
		tmp = (x_m * y) / 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):
	t_1 = (y / z) + (t / (z + -1.0))
	tmp = 0
	if t_1 <= -math.inf:
		tmp = y * (x_m / z)
	elif t_1 <= 2e+291:
		tmp = x_m * t_1
	else:
		tmp = (x_m * y) / 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)
	t_1 = Float64(Float64(y / z) + Float64(t / Float64(z + -1.0)))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(y * Float64(x_m / z));
	elseif (t_1 <= 2e+291)
		tmp = Float64(x_m * t_1);
	else
		tmp = Float64(Float64(x_m * y) / 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)
	t_1 = (y / z) + (t / (z + -1.0));
	tmp = 0.0;
	if (t_1 <= -Inf)
		tmp = y * (x_m / z);
	elseif (t_1 <= 2e+291)
		tmp = x_m * t_1;
	else
		tmp = (x_m * y) / 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_] := Block[{t$95$1 = N[(N[(y / z), $MachinePrecision] + N[(t / N[(z + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * If[LessEqual[t$95$1, (-Infinity)], N[(y * N[(x$95$m / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e+291], N[(x$95$m * t$95$1), $MachinePrecision], N[(N[(x$95$m * y), $MachinePrecision] / z), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 70.7%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 99.9

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

   5. Applied rewrites 99.9%

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

   7. Applied rewrites 99.9%

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

   if -inf.0 < (-.f64 (/.f64 y z) (/.f64 t (-.f64 #s(literal 1 binary64) z))) < 1.9999999999999999e291

   1. Initial program 97.8%

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

   if 1.9999999999999999e291 < (-.f64 (/.f64 y z) (/.f64 t (-.f64 #s(literal 1 binary64) z)))

   1. Initial program 56.9%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 99.8

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

   5. Applied rewrites 99.8%

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

4. Final simplification 98.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y}{z} + \frac{t}{z + -1} \leq -\infty:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{elif}\;\frac{y}{z} + \frac{t}{z + -1} \leq 2 \cdot 10^{+291}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} + \frac{t}{z + -1}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 77.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -6.5 \cdot 10^{-106}:\\ \;\;\;\;\frac{x\_m \cdot \left(y + t\right)}{z}\\ \mathbf{elif}\;z \leq -7.6 \cdot 10^{-127}:\\ \;\;\;\;x\_m \cdot \left(-\mathsf{fma}\left(t, z, t\right)\right)\\ \mathbf{elif}\;z \leq 1:\\ \;\;\;\;y \cdot \frac{x\_m}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{z} \cdot \left(y + t\right)\\ \end{array} \end{array} \]

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 -6.5e-106)
    (/ (* x_m (+ y t)) z)
    (if (<= z -7.6e-127)
      (* x_m (- (fma t z t)))
      (if (<= z 1.0) (* y (/ x_m z)) (* (/ x_m z) (+ y 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 <= -6.5e-106) {
		tmp = (x_m * (y + t)) / z;
	} else if (z <= -7.6e-127) {
		tmp = x_m * -fma(t, z, t);
	} else if (z <= 1.0) {
		tmp = y * (x_m / z);
	} else {
		tmp = (x_m / z) * (y + 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 <= -6.5e-106)
		tmp = Float64(Float64(x_m * Float64(y + t)) / z);
	elseif (z <= -7.6e-127)
		tmp = Float64(x_m * Float64(-fma(t, z, t)));
	elseif (z <= 1.0)
		tmp = Float64(y * Float64(x_m / z));
	else
		tmp = Float64(Float64(x_m / z) * Float64(y + t));
	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_] := N[(x$95$s * If[LessEqual[z, -6.5e-106], N[(N[(x$95$m * N[(y + t), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[z, -7.6e-127], N[(x$95$m * (-N[(t * z + t), $MachinePrecision])), $MachinePrecision], If[LessEqual[z, 1.0], N[(y * N[(x$95$m / z), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m / z), $MachinePrecision] * N[(y + t), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]

Derivation

1. Split input into 4 regimes

2. if z < -6.4999999999999997e-106

   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

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

      1. *-commutative N/A

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

      2. remove-double-neg N/A

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

      3. neg-mul-1 N/A

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

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

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

      5. neg-mul-1 N/A

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

      6. mul-1-neg N/A

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

      7. remove-double-neg N/A

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

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

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

      9. *-commutative N/A

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

      10. mul-1-neg N/A

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

      11. lower-/.f64 N/A

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

   5. Applied rewrites 85.6%

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

   if -6.4999999999999997e-106 < z < -7.60000000000000005e-127

   1. Initial program 100.0%

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

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

      5. mul-1-neg N/A

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

      6. +-commutative N/A

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

      7. distribute-neg-in N/A

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

      8. mul-1-neg N/A

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

      9. remove-double-neg N/A

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

      10. metadata-eval N/A

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

      11. lower-+.f64 85.7

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

   5. Applied rewrites 85.7%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 85.7%

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

   if -7.60000000000000005e-127 < z < 1

   1. Initial program 87.8%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 73.4

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

   5. Applied rewrites 73.4%

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

   7. Applied rewrites 77.5%

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

   if 1 < z

   1. Initial program 98.2%

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

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower--.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-*.f64 45.5

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

   4. Applied rewrites 45.5%

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

   5. Taylor expanded in z around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

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

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

      5. sub-neg N/A

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

      6. mul-1-neg N/A

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

      7. distribute-neg-out N/A

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

      8. remove-double-neg N/A

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

      9. mul-1-neg N/A

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

      10. sub-neg N/A

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

      11. remove-double-neg N/A

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

      12. lower-*.f64 N/A

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

      13. cancel-sign-sub-inv N/A

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

      14. metadata-eval N/A

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

      15. *-lft-identity N/A

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

      16. lower-+.f64 90.4

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

   7. Applied rewrites 90.4%

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

   9. Applied rewrites 90.7%

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

4. Final simplification 83.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -6.5 \cdot 10^{-106}:\\ \;\;\;\;\frac{x \cdot \left(y + t\right)}{z}\\ \mathbf{elif}\;z \leq -7.6 \cdot 10^{-127}:\\ \;\;\;\;x \cdot \left(-\mathsf{fma}\left(t, z, t\right)\right)\\ \mathbf{elif}\;z \leq 1:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \left(y + t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 77.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_1 := \frac{x\_m \cdot \left(y + t\right)}{z}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -6.5 \cdot 10^{-106}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq -7.6 \cdot 10^{-127}:\\ \;\;\;\;x\_m \cdot \left(-\mathsf{fma}\left(t, z, t\right)\right)\\ \mathbf{elif}\;z \leq 5.2 \cdot 10^{+16}:\\ \;\;\;\;y \cdot \frac{x\_m}{z}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \end{array} \]

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 (/ (* x_m (+ y t)) z)))
   (*
    x_s
    (if (<= z -6.5e-106)
      t_1
      (if (<= z -7.6e-127)
        (* x_m (- (fma t z t)))
        (if (<= z 5.2e+16) (* 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 = (x_m * (y + t)) / z;
	double tmp;
	if (z <= -6.5e-106) {
		tmp = t_1;
	} else if (z <= -7.6e-127) {
		tmp = x_m * -fma(t, z, t);
	} else if (z <= 5.2e+16) {
		tmp = 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(Float64(x_m * Float64(y + t)) / z)
	tmp = 0.0
	if (z <= -6.5e-106)
		tmp = t_1;
	elseif (z <= -7.6e-127)
		tmp = Float64(x_m * Float64(-fma(t, z, t)));
	elseif (z <= 5.2e+16)
		tmp = Float64(y * Float64(x_m / 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[(N[(x$95$m * N[(y + t), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]}, N[(x$95$s * If[LessEqual[z, -6.5e-106], t$95$1, If[LessEqual[z, -7.6e-127], N[(x$95$m * (-N[(t * z + t), $MachinePrecision])), $MachinePrecision], If[LessEqual[z, 5.2e+16], N[(y * N[(x$95$m / z), $MachinePrecision]), $MachinePrecision], t$95$1]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if z < -6.4999999999999997e-106 or 5.2e16 < z

   1. Initial program 96.7%

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

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

      2. remove-double-neg N/A

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

      3. neg-mul-1 N/A

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

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

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

      5. neg-mul-1 N/A

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

      6. mul-1-neg N/A

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

      7. remove-double-neg N/A

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

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

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

      9. *-commutative N/A

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

      10. mul-1-neg N/A

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

      11. lower-/.f64 N/A

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

   5. Applied rewrites 87.2%

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

   if -6.4999999999999997e-106 < z < -7.60000000000000005e-127

   1. Initial program 100.0%

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

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

      5. mul-1-neg N/A

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

      6. +-commutative N/A

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

      7. distribute-neg-in N/A

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

      8. mul-1-neg N/A

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

      9. remove-double-neg N/A

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

      10. metadata-eval N/A

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

      11. lower-+.f64 85.7

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

   5. Applied rewrites 85.7%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 85.7%

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

   if -7.60000000000000005e-127 < z < 5.2e16

   1. Initial program 88.3%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 74.4

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

   5. Applied rewrites 74.4%

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

   7. Applied rewrites 78.3%

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

Alternative 5: 63.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_1 := x\_m \cdot \left(-t\right)\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;t \leq -9 \cdot 10^{+175}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 3.1 \cdot 10^{-212}:\\ \;\;\;\;\frac{x\_m \cdot y}{z}\\ \mathbf{elif}\;t \leq 7.5 \cdot 10^{+176}:\\ \;\;\;\;y \cdot \frac{x\_m}{z}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \end{array} \]

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 (* x_m (- t))))
   (*
    x_s
    (if (<= t -9e+175)
      t_1
      (if (<= t 3.1e-212)
        (/ (* x_m y) z)
        (if (<= t 7.5e+176) (* 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 = x_m * -t;
	double tmp;
	if (t <= -9e+175) {
		tmp = t_1;
	} else if (t <= 3.1e-212) {
		tmp = (x_m * y) / z;
	} else if (t <= 7.5e+176) {
		tmp = 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 = x_m * -t
    if (t <= (-9d+175)) then
        tmp = t_1
    else if (t <= 3.1d-212) then
        tmp = (x_m * y) / z
    else if (t <= 7.5d+176) then
        tmp = 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 = x_m * -t;
	double tmp;
	if (t <= -9e+175) {
		tmp = t_1;
	} else if (t <= 3.1e-212) {
		tmp = (x_m * y) / z;
	} else if (t <= 7.5e+176) {
		tmp = 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 = x_m * -t
	tmp = 0
	if t <= -9e+175:
		tmp = t_1
	elif t <= 3.1e-212:
		tmp = (x_m * y) / z
	elif t <= 7.5e+176:
		tmp = 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(x_m * Float64(-t))
	tmp = 0.0
	if (t <= -9e+175)
		tmp = t_1;
	elseif (t <= 3.1e-212)
		tmp = Float64(Float64(x_m * y) / z);
	elseif (t <= 7.5e+176)
		tmp = Float64(y * Float64(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 = x_m * -t;
	tmp = 0.0;
	if (t <= -9e+175)
		tmp = t_1;
	elseif (t <= 3.1e-212)
		tmp = (x_m * y) / z;
	elseif (t <= 7.5e+176)
		tmp = 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[(x$95$m * (-t)), $MachinePrecision]}, N[(x$95$s * If[LessEqual[t, -9e+175], t$95$1, If[LessEqual[t, 3.1e-212], N[(N[(x$95$m * y), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[t, 7.5e+176], N[(y * N[(x$95$m / z), $MachinePrecision]), $MachinePrecision], t$95$1]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if t < -8.99999999999999979e175 or 7.499999999999999e176 < t

   1. Initial program 96.6%

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

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

      5. mul-1-neg N/A

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

      6. +-commutative N/A

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

      7. distribute-neg-in N/A

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

      8. mul-1-neg N/A

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

      9. remove-double-neg N/A

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

      10. metadata-eval N/A

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

      11. lower-+.f64 80.8

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

   5. Applied rewrites 80.8%

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

   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 43.8%

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

   if -8.99999999999999979e175 < t < 3.10000000000000006e-212

   1. Initial program 91.7%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 76.4

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

   5. Applied rewrites 76.4%

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

   if 3.10000000000000006e-212 < t < 7.499999999999999e176

   1. Initial program 93.5%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 68.4

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

   5. Applied rewrites 68.4%

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

   7. Applied rewrites 74.4%

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

Alternative 6: 91.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -3.8 \cdot 10^{-15}:\\ \;\;\;\;\frac{x\_m \cdot \left(y + t\right)}{z}\\ \mathbf{elif}\;z \leq 1:\\ \;\;\;\;x\_m \cdot \left(\frac{y}{z} - t\right)\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot \frac{y + t}{z}\\ \end{array} \end{array} \]

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 (<= z -3.8e-15)
    (/ (* x_m (+ y t)) z)
    (if (<= z 1.0) (* x_m (- (/ y z) t)) (* x_m (/ (+ y 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 (z <= -3.8e-15) {
		tmp = (x_m * (y + t)) / z;
	} else if (z <= 1.0) {
		tmp = x_m * ((y / z) - t);
	} else {
		tmp = x_m * ((y + 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 (z <= (-3.8d-15)) then
        tmp = (x_m * (y + t)) / z
    else if (z <= 1.0d0) then
        tmp = x_m * ((y / z) - t)
    else
        tmp = x_m * ((y + 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 (z <= -3.8e-15) {
		tmp = (x_m * (y + t)) / z;
	} else if (z <= 1.0) {
		tmp = x_m * ((y / z) - t);
	} else {
		tmp = x_m * ((y + 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 z <= -3.8e-15:
		tmp = (x_m * (y + t)) / z
	elif z <= 1.0:
		tmp = x_m * ((y / z) - t)
	else:
		tmp = x_m * ((y + 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 (z <= -3.8e-15)
		tmp = Float64(Float64(x_m * Float64(y + t)) / z);
	elseif (z <= 1.0)
		tmp = Float64(x_m * Float64(Float64(y / z) - t));
	else
		tmp = Float64(x_m * Float64(Float64(y + 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 (z <= -3.8e-15)
		tmp = (x_m * (y + t)) / z;
	elseif (z <= 1.0)
		tmp = x_m * ((y / z) - t);
	else
		tmp = x_m * ((y + 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[LessEqual[z, -3.8e-15], N[(N[(x$95$m * N[(y + t), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[z, 1.0], N[(x$95$m * N[(N[(y / z), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision], N[(x$95$m * N[(N[(y + t), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if z < -3.8000000000000002e-15

   1. Initial program 94.4%

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

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

      2. remove-double-neg N/A

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

      3. neg-mul-1 N/A

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

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

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

      5. neg-mul-1 N/A

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

      6. mul-1-neg N/A

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

      7. remove-double-neg N/A

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

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

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

      9. *-commutative N/A

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

      10. mul-1-neg N/A

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

      11. lower-/.f64 N/A

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

   5. Applied rewrites 94.8%

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

   if -3.8000000000000002e-15 < z < 1

   1. Initial program 90.6%

      \[x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \]
   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. lower-/.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. lower--.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. lower-*.f64 89.3

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

   5. Applied rewrites 89.3%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 89.3%

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

   if 1 < z

   1. Initial program 98.2%

      \[x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \]
   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. 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. lower-+.f64 98.2

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

   5. Applied rewrites 98.2%

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

Alternative 7: 88.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -3.8 \cdot 10^{-15}:\\ \;\;\;\;\frac{x\_m \cdot \left(y + t\right)}{z}\\ \mathbf{elif}\;z \leq 1:\\ \;\;\;\;x\_m \cdot \left(\frac{y}{z} - t\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{z} \cdot \left(y + t\right)\\ \end{array} \end{array} \]

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 -3.8e-15)
    (/ (* x_m (+ y t)) z)
    (if (<= z 1.0) (* x_m (- (/ y z) t)) (* (/ x_m z) (+ y 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 <= -3.8e-15) {
		tmp = (x_m * (y + t)) / z;
	} else if (z <= 1.0) {
		tmp = x_m * ((y / z) - t);
	} else {
		tmp = (x_m / z) * (y + 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 <= (-3.8d-15)) then
        tmp = (x_m * (y + t)) / z
    else if (z <= 1.0d0) then
        tmp = x_m * ((y / z) - t)
    else
        tmp = (x_m / z) * (y + 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 <= -3.8e-15) {
		tmp = (x_m * (y + t)) / z;
	} else if (z <= 1.0) {
		tmp = x_m * ((y / z) - t);
	} else {
		tmp = (x_m / z) * (y + 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 <= -3.8e-15:
		tmp = (x_m * (y + t)) / z
	elif z <= 1.0:
		tmp = x_m * ((y / z) - t)
	else:
		tmp = (x_m / z) * (y + 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 <= -3.8e-15)
		tmp = Float64(Float64(x_m * Float64(y + t)) / z);
	elseif (z <= 1.0)
		tmp = Float64(x_m * Float64(Float64(y / z) - t));
	else
		tmp = Float64(Float64(x_m / z) * Float64(y + 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 <= -3.8e-15)
		tmp = (x_m * (y + t)) / z;
	elseif (z <= 1.0)
		tmp = x_m * ((y / z) - t);
	else
		tmp = (x_m / z) * (y + 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, -3.8e-15], N[(N[(x$95$m * N[(y + t), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[z, 1.0], N[(x$95$m * N[(N[(y / z), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m / z), $MachinePrecision] * N[(y + t), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if z < -3.8000000000000002e-15

   1. Initial program 94.4%

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

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

      2. remove-double-neg N/A

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

      3. neg-mul-1 N/A

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

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

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

      5. neg-mul-1 N/A

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

      6. mul-1-neg N/A

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

      7. remove-double-neg N/A

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

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

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

      9. *-commutative N/A

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

      10. mul-1-neg N/A

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

      11. lower-/.f64 N/A

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

   5. Applied rewrites 94.8%

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

   if -3.8000000000000002e-15 < z < 1

   1. Initial program 90.6%

      \[x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \]
   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. lower-/.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. lower--.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. lower-*.f64 89.3

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

   5. Applied rewrites 89.3%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 89.3%

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

   if 1 < z

   1. Initial program 98.2%

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

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower--.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-*.f64 45.5

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

   4. Applied rewrites 45.5%

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

   5. Taylor expanded in z around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

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

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

      5. sub-neg N/A

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

      6. mul-1-neg N/A

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

      7. distribute-neg-out N/A

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

      8. remove-double-neg N/A

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

      9. mul-1-neg N/A

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

      10. sub-neg N/A

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

      11. remove-double-neg N/A

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

      12. lower-*.f64 N/A

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

      13. cancel-sign-sub-inv N/A

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

      14. metadata-eval N/A

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

      15. *-lft-identity N/A

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

      16. lower-+.f64 90.4

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

   7. Applied rewrites 90.4%

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

   9. Applied rewrites 90.7%

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

4. Final simplification 91.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.8 \cdot 10^{-15}:\\ \;\;\;\;\frac{x \cdot \left(y + t\right)}{z}\\ \mathbf{elif}\;z \leq 1:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - t\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \left(y + t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 75.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_1 := \frac{x\_m \cdot \left(y + t\right)}{z}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -2.3 \cdot 10^{-160}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 2.5 \cdot 10^{-31}:\\ \;\;\;\;\frac{x\_m \cdot t}{z + -1}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \end{array} \]

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 (<= y -2.3e-160)
      t_1
      (if (<= y 2.5e-31) (/ (* x_m t) (+ z -1.0)) 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 (y <= -2.3e-160) {
		tmp = t_1;
	} else if (y <= 2.5e-31) {
		tmp = (x_m * t) / (z + -1.0);
	} 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 (y <= (-2.3d-160)) then
        tmp = t_1
    else if (y <= 2.5d-31) then
        tmp = (x_m * t) / (z + (-1.0d0))
    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 (y <= -2.3e-160) {
		tmp = t_1;
	} else if (y <= 2.5e-31) {
		tmp = (x_m * t) / (z + -1.0);
	} 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 y <= -2.3e-160:
		tmp = t_1
	elif y <= 2.5e-31:
		tmp = (x_m * t) / (z + -1.0)
	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(x_m * Float64(y + t)) / z)
	tmp = 0.0
	if (y <= -2.3e-160)
		tmp = t_1;
	elseif (y <= 2.5e-31)
		tmp = Float64(Float64(x_m * t) / Float64(z + -1.0));
	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 (y <= -2.3e-160)
		tmp = t_1;
	elseif (y <= 2.5e-31)
		tmp = (x_m * t) / (z + -1.0);
	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[(x$95$m * N[(y + t), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]}, N[(x$95$s * If[LessEqual[y, -2.3e-160], t$95$1, If[LessEqual[y, 2.5e-31], N[(N[(x$95$m * t), $MachinePrecision] / N[(z + -1.0), $MachinePrecision]), $MachinePrecision], t$95$1]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -2.29999999999999985e-160 or 2.5e-31 < y

   1. Initial program 91.8%

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

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

      2. remove-double-neg N/A

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

      3. neg-mul-1 N/A

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

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

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

      5. neg-mul-1 N/A

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

      6. mul-1-neg N/A

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

      7. remove-double-neg N/A

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

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

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

      9. *-commutative N/A

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

      10. mul-1-neg N/A

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

      11. lower-/.f64 N/A

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

   5. Applied rewrites 84.6%

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

   if -2.29999999999999985e-160 < y < 2.5e-31

   1. Initial program 96.1%

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

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower--.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-*.f64 73.2

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

   4. Applied rewrites 73.2%

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

   5. Taylor expanded in y around inf

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

   7. Applied rewrites 73.3%

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

   8. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. +-commutative N/A

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

      6. lft-mult-inverse N/A

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

      7. distribute-rgt-in N/A

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

      8. +-commutative N/A

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

      9. metadata-eval N/A

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

      10. sub-neg N/A

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

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

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

      12. sub-neg N/A

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

      13. metadata-eval N/A

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

      14. +-commutative N/A

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

      15. distribute-neg-in N/A

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

      16. metadata-eval N/A

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

      17. distribute-neg-frac N/A

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

      18. metadata-eval N/A

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

      19. distribute-rgt-in N/A

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

      20. *-lft-identity N/A

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

      21. metadata-eval N/A

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

      22. distribute-neg-frac N/A

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

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

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

      24. lft-mult-inverse N/A

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

   10. Applied rewrites 77.9%

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

4. Final simplification 82.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.3 \cdot 10^{-160}:\\ \;\;\;\;\frac{x \cdot \left(y + t\right)}{z}\\ \mathbf{elif}\;y \leq 2.5 \cdot 10^{-31}:\\ \;\;\;\;\frac{x \cdot t}{z + -1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(y + t\right)}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 68.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_1 := x\_m \cdot \frac{t}{z}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;t \leq -2.1 \cdot 10^{+94}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 4.4 \cdot 10^{+75}:\\ \;\;\;\;\frac{x\_m \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \end{array} \]

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 -2.1e+94) t_1 (if (<= t 4.4e+75) (/ (* 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 <= -2.1e+94) {
		tmp = t_1;
	} else if (t <= 4.4e+75) {
		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 <= (-2.1d+94)) then
        tmp = t_1
    else if (t <= 4.4d+75) 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 <= -2.1e+94) {
		tmp = t_1;
	} else if (t <= 4.4e+75) {
		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 <= -2.1e+94:
		tmp = t_1
	elif t <= 4.4e+75:
		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 <= -2.1e+94)
		tmp = t_1;
	elseif (t <= 4.4e+75)
		tmp = Float64(Float64(x_m * 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 <= -2.1e+94)
		tmp = t_1;
	elseif (t <= 4.4e+75)
		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, -2.1e+94], t$95$1, If[LessEqual[t, 4.4e+75], N[(N[(x$95$m * y), $MachinePrecision] / z), $MachinePrecision], t$95$1]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -2.09999999999999989e94 or 4.40000000000000024e75 < t

   1. Initial program 94.1%

      \[x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \]
   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. 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. lower-+.f64 64.2

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

   5. Applied rewrites 64.2%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 54.3%

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

   if -2.09999999999999989e94 < t < 4.40000000000000024e75

   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. lower-/.f64 N/A

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

      2. lower-*.f64 79.8

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

   5. Applied rewrites 79.8%

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

Alternative 10: 61.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_1 := \frac{x\_m \cdot y}{z}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -5.4 \cdot 10^{-148}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 1.8 \cdot 10^{+52}:\\ \;\;\;\;\frac{x\_m \cdot t}{z}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \end{array} \]

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)))
   (* x_s (if (<= y -5.4e-148) t_1 (if (<= y 1.8e+52) (/ (* x_m t) 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;
	double tmp;
	if (y <= -5.4e-148) {
		tmp = t_1;
	} else if (y <= 1.8e+52) {
		tmp = (x_m * t) / 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 * y) / z
    if (y <= (-5.4d-148)) then
        tmp = t_1
    else if (y <= 1.8d+52) then
        tmp = (x_m * t) / 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 * y) / z;
	double tmp;
	if (y <= -5.4e-148) {
		tmp = t_1;
	} else if (y <= 1.8e+52) {
		tmp = (x_m * t) / 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 * y) / z
	tmp = 0
	if y <= -5.4e-148:
		tmp = t_1
	elif y <= 1.8e+52:
		tmp = (x_m * t) / 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(Float64(x_m * y) / z)
	tmp = 0.0
	if (y <= -5.4e-148)
		tmp = t_1;
	elseif (y <= 1.8e+52)
		tmp = Float64(Float64(x_m * t) / 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 * y) / z;
	tmp = 0.0;
	if (y <= -5.4e-148)
		tmp = t_1;
	elseif (y <= 1.8e+52)
		tmp = (x_m * t) / 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[(N[(x$95$m * y), $MachinePrecision] / z), $MachinePrecision]}, N[(x$95$s * If[LessEqual[y, -5.4e-148], t$95$1, If[LessEqual[y, 1.8e+52], N[(N[(x$95$m * t), $MachinePrecision] / z), $MachinePrecision], t$95$1]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -5.39999999999999976e-148 or 1.8e52 < y

   1. Initial program 90.7%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 82.8

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

   5. Applied rewrites 82.8%

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

   if -5.39999999999999976e-148 < y < 1.8e52

   1. Initial program 96.8%

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

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower--.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-*.f64 69.9

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

   4. Applied rewrites 69.9%

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

   5. Taylor expanded in z around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

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

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

      5. sub-neg N/A

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

      6. mul-1-neg N/A

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

      7. distribute-neg-out N/A

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

      8. remove-double-neg N/A

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

      9. mul-1-neg N/A

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

      10. sub-neg N/A

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

      11. remove-double-neg N/A

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

      12. lower-*.f64 N/A

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

      13. cancel-sign-sub-inv N/A

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

      14. metadata-eval N/A

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

      15. *-lft-identity N/A

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

      16. lower-+.f64 64.5

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

   7. Applied rewrites 64.5%

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

   8. Taylor expanded in y around 0

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

   10. Applied rewrites 50.8%

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

4. Final simplification 68.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5.4 \cdot 10^{-148}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{elif}\;y \leq 1.8 \cdot 10^{+52}:\\ \;\;\;\;\frac{x \cdot t}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 63.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_1 := x\_m \cdot \left(-t\right)\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;t \leq -2.6 \cdot 10^{+84}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 7.5 \cdot 10^{+176}:\\ \;\;\;\;y \cdot \frac{x\_m}{z}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \end{array} \]

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 (* x_m (- t))))
   (* x_s (if (<= t -2.6e+84) t_1 (if (<= t 7.5e+176) (* 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 = x_m * -t;
	double tmp;
	if (t <= -2.6e+84) {
		tmp = t_1;
	} else if (t <= 7.5e+176) {
		tmp = 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 = x_m * -t
    if (t <= (-2.6d+84)) then
        tmp = t_1
    else if (t <= 7.5d+176) then
        tmp = 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 = x_m * -t;
	double tmp;
	if (t <= -2.6e+84) {
		tmp = t_1;
	} else if (t <= 7.5e+176) {
		tmp = 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 = x_m * -t
	tmp = 0
	if t <= -2.6e+84:
		tmp = t_1
	elif t <= 7.5e+176:
		tmp = 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(x_m * Float64(-t))
	tmp = 0.0
	if (t <= -2.6e+84)
		tmp = t_1;
	elseif (t <= 7.5e+176)
		tmp = Float64(y * Float64(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 = x_m * -t;
	tmp = 0.0;
	if (t <= -2.6e+84)
		tmp = t_1;
	elseif (t <= 7.5e+176)
		tmp = 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[(x$95$m * (-t)), $MachinePrecision]}, N[(x$95$s * If[LessEqual[t, -2.6e+84], t$95$1, If[LessEqual[t, 7.5e+176], N[(y * N[(x$95$m / z), $MachinePrecision]), $MachinePrecision], t$95$1]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -2.6000000000000001e84 or 7.499999999999999e176 < t

   1. Initial program 96.4%

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

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

      5. mul-1-neg N/A

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

      6. +-commutative N/A

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

      7. distribute-neg-in N/A

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

      8. mul-1-neg N/A

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

      9. remove-double-neg N/A

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

      10. metadata-eval N/A

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

      11. lower-+.f64 79.3

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

   5. Applied rewrites 79.3%

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

   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 45.3%

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

   if -2.6000000000000001e84 < t < 7.499999999999999e176

   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. lower-/.f64 N/A

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

      2. lower-*.f64 76.6

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

   5. Applied rewrites 76.6%

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

   7. Applied rewrites 76.2%

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

Alternative 12: 22.7% 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 93.4%

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

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

   5. mul-1-neg N/A

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

   6. +-commutative N/A

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

   7. distribute-neg-in N/A

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

   8. mul-1-neg N/A

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

   9. remove-double-neg N/A

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

   10. metadata-eval N/A

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

   11. lower-+.f64 49.5

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

5. Applied rewrites 49.5%

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

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 27.1%

   \[\leadsto x \cdot \left(-t\right) \]
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 2024219 
(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)))))