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

Percentage Accurate: 94.5% → 98.0%

Time: 9.2s

Alternatives: 12

Speedup: 0.9×

• 20.4% 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: 98.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{y}{z} - \frac{t}{1 - z}\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\left(\frac{1}{z} \cdot x\right) \cdot \mathsf{fma}\left(-t, z, y\right)\\ \mathbf{elif}\;t\_1 \leq 10^{+299}:\\ \;\;\;\;x \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (/ y z) (/ t (- 1.0 z)))))
   (if (<= t_1 (- INFINITY))
     (* (* (/ 1.0 z) x) (fma (- t) z y))
     (if (<= t_1 1e+299) (* x t_1) (/ (* x y) z)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (y / z) - (t / (1.0 - z));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = ((1.0 / z) * x) * fma(-t, z, y);
	} else if (t_1 <= 1e+299) {
		tmp = x * t_1;
	} else {
		tmp = (x * y) / z;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(y / z) - Float64(t / Float64(1.0 - z)))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(Float64(Float64(1.0 / z) * x) * fma(Float64(-t), z, y));
	elseif (t_1 <= 1e+299)
		tmp = Float64(x * t_1);
	else
		tmp = Float64(Float64(x * y) / z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(y / z), $MachinePrecision] - N[(t / N[(1.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(N[(1.0 / z), $MachinePrecision] * x), $MachinePrecision] * N[((-t) * z + y), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e+299], N[(x * t$95$1), $MachinePrecision], N[(N[(x * y), $MachinePrecision] / z), $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 68.6%

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

   3. Taylor expanded in z around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. mul-1-neg N/A

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

      4. unsub-neg N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

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

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

      9. unsub-neg N/A

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

      10. mul-1-neg N/A

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

      11. lower-*.f64 N/A

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

      12. +-commutative N/A

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

      13. associate-*r* N/A

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

      14. lower-fma.f64 N/A

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

      15. mul-1-neg N/A

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

      16. lower-neg.f64 99.8

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

   5. Applied rewrites 99.8%

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

   7. Applied rewrites 100.0%

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

   if -inf.0 < (-.f64 (/.f64 y z) (/.f64 t (-.f64 #s(literal 1 binary64) z))) < 1.0000000000000001e299

   1. Initial program 98.8%

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

   if 1.0000000000000001e299 < (-.f64 (/.f64 y z) (/.f64 t (-.f64 #s(literal 1 binary64) z)))

   1. Initial program 77.0%

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

   3. Taylor expanded in z around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. distribute-lft-out N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. distribute-rgt-out N/A

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

      11. lower-*.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. mul-1-neg N/A

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

      15. lower-neg.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 99.9

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 99.9%

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

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y}{z} - \frac{t}{1 - z} \leq -\infty:\\ \;\;\;\;\left(\frac{1}{z} \cdot x\right) \cdot \mathsf{fma}\left(-t, z, y\right)\\ \mathbf{elif}\;\frac{y}{z} - \frac{t}{1 - z} \leq 10^{+299}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 74.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{y}{z} \cdot x\\ \mathbf{if}\;z \leq -1.12 \cdot 10^{+82}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq -1 \cdot 10^{+18}:\\ \;\;\;\;\frac{x}{z - 1} \cdot t\\ \mathbf{elif}\;z \leq 8500000000000:\\ \;\;\;\;\left(\frac{y}{z} - t\right) \cdot x\\ \mathbf{elif}\;z \leq 1.3 \cdot 10^{+137}:\\ \;\;\;\;\frac{x \cdot t}{z - 1}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (/ y z) x)))
   (if (<= z -1.12e+82)
     t_1
     (if (<= z -1e+18)
       (* (/ x (- z 1.0)) t)
       (if (<= z 8500000000000.0)
         (* (- (/ y z) t) x)
         (if (<= z 1.3e+137) (/ (* x t) (- z 1.0)) t_1))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (y / z) * x;
	double tmp;
	if (z <= -1.12e+82) {
		tmp = t_1;
	} else if (z <= -1e+18) {
		tmp = (x / (z - 1.0)) * t;
	} else if (z <= 8500000000000.0) {
		tmp = ((y / z) - t) * x;
	} else if (z <= 1.3e+137) {
		tmp = (x * t) / (z - 1.0);
	} 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) :: tmp
    t_1 = (y / z) * x
    if (z <= (-1.12d+82)) then
        tmp = t_1
    else if (z <= (-1d+18)) then
        tmp = (x / (z - 1.0d0)) * t
    else if (z <= 8500000000000.0d0) then
        tmp = ((y / z) - t) * x
    else if (z <= 1.3d+137) then
        tmp = (x * t) / (z - 1.0d0)
    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 = (y / z) * x;
	double tmp;
	if (z <= -1.12e+82) {
		tmp = t_1;
	} else if (z <= -1e+18) {
		tmp = (x / (z - 1.0)) * t;
	} else if (z <= 8500000000000.0) {
		tmp = ((y / z) - t) * x;
	} else if (z <= 1.3e+137) {
		tmp = (x * t) / (z - 1.0);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (y / z) * x
	tmp = 0
	if z <= -1.12e+82:
		tmp = t_1
	elif z <= -1e+18:
		tmp = (x / (z - 1.0)) * t
	elif z <= 8500000000000.0:
		tmp = ((y / z) - t) * x
	elif z <= 1.3e+137:
		tmp = (x * t) / (z - 1.0)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(y / z) * x)
	tmp = 0.0
	if (z <= -1.12e+82)
		tmp = t_1;
	elseif (z <= -1e+18)
		tmp = Float64(Float64(x / Float64(z - 1.0)) * t);
	elseif (z <= 8500000000000.0)
		tmp = Float64(Float64(Float64(y / z) - t) * x);
	elseif (z <= 1.3e+137)
		tmp = Float64(Float64(x * t) / Float64(z - 1.0));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (y / z) * x;
	tmp = 0.0;
	if (z <= -1.12e+82)
		tmp = t_1;
	elseif (z <= -1e+18)
		tmp = (x / (z - 1.0)) * t;
	elseif (z <= 8500000000000.0)
		tmp = ((y / z) - t) * x;
	elseif (z <= 1.3e+137)
		tmp = (x * t) / (z - 1.0);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(y / z), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[z, -1.12e+82], t$95$1, If[LessEqual[z, -1e+18], N[(N[(x / N[(z - 1.0), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision], If[LessEqual[z, 8500000000000.0], N[(N[(N[(y / z), $MachinePrecision] - t), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[z, 1.3e+137], N[(N[(x * t), $MachinePrecision] / N[(z - 1.0), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 4 regimes

2. if z < -1.11999999999999998e82 or 1.3e137 < z

   1. Initial program 97.3%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 65.3

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

   5. Applied rewrites 65.3%

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

   if -1.11999999999999998e82 < z < -1e18

   1. Initial program 99.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift--.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. frac-sub N/A

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

      7. associate-*l/ N/A

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

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

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-neg.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 71.3

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

   4. Applied rewrites 71.3%

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

   5. Taylor expanded in y around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{t \cdot x}{1 - z}} \]
   6. 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. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. sub-neg N/A

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

      7. mul-1-neg N/A

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

      8. distribute-lft-in N/A

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

      9. metadata-eval N/A

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

      10. neg-mul-1 N/A

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

      11. mul-1-neg N/A

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

      12. remove-double-neg N/A

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

      13. lower-+.f64 63.7

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

   7. Applied rewrites 63.7%

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

   9. Applied rewrites 77.5%

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

   if -1e18 < z < 8.5e12

   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 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. mul-1-neg N/A

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

      4. unsub-neg N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

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

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

      9. unsub-neg N/A

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

      10. mul-1-neg N/A

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

      11. lower-*.f64 N/A

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

      12. +-commutative N/A

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

      13. associate-*r* N/A

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

      14. lower-fma.f64 N/A

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

      15. mul-1-neg N/A

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

      16. lower-neg.f64 94.2

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

   5. Applied rewrites 94.2%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 90.1%

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

   if 8.5e12 < z < 1.3e137

   1. Initial program 99.7%

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

   3. Taylor expanded in y around 0

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

      1. 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. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. sub-neg N/A

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

      6. mul-1-neg N/A

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

      7. +-commutative N/A

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

      8. distribute-neg-in N/A

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

      9. mul-1-neg N/A

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

      10. remove-double-neg N/A

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

      11. sub-neg N/A

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

      12. lower--.f64 81.0

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

   5. Applied rewrites 81.0%

      \[\leadsto \color{blue}{\frac{t \cdot x}{z - 1}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 80.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.12 \cdot 10^{+82}:\\ \;\;\;\;\frac{y}{z} \cdot x\\ \mathbf{elif}\;z \leq -1 \cdot 10^{+18}:\\ \;\;\;\;\frac{x}{z - 1} \cdot t\\ \mathbf{elif}\;z \leq 8500000000000:\\ \;\;\;\;\left(\frac{y}{z} - t\right) \cdot x\\ \mathbf{elif}\;z \leq 1.3 \cdot 10^{+137}:\\ \;\;\;\;\frac{x \cdot t}{z - 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z} \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 61.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{y}{z} \cdot x\\ \mathbf{if}\;z \leq -6.4 \cdot 10^{-6}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq -4 \cdot 10^{-53}:\\ \;\;\;\;\left(\left(-1 - z\right) \cdot x\right) \cdot t\\ \mathbf{elif}\;z \leq 8500000000000:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{elif}\;z \leq 1.3 \cdot 10^{+137}:\\ \;\;\;\;\frac{x \cdot t}{z}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (/ y z) x)))
   (if (<= z -6.4e-6)
     t_1
     (if (<= z -4e-53)
       (* (* (- -1.0 z) x) t)
       (if (<= z 8500000000000.0)
         (/ (* x y) z)
         (if (<= z 1.3e+137) (/ (* x t) z) t_1))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (y / z) * x;
	double tmp;
	if (z <= -6.4e-6) {
		tmp = t_1;
	} else if (z <= -4e-53) {
		tmp = ((-1.0 - z) * x) * t;
	} else if (z <= 8500000000000.0) {
		tmp = (x * y) / z;
	} else if (z <= 1.3e+137) {
		tmp = (x * t) / 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) :: tmp
    t_1 = (y / z) * x
    if (z <= (-6.4d-6)) then
        tmp = t_1
    else if (z <= (-4d-53)) then
        tmp = (((-1.0d0) - z) * x) * t
    else if (z <= 8500000000000.0d0) then
        tmp = (x * y) / z
    else if (z <= 1.3d+137) then
        tmp = (x * t) / 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 = (y / z) * x;
	double tmp;
	if (z <= -6.4e-6) {
		tmp = t_1;
	} else if (z <= -4e-53) {
		tmp = ((-1.0 - z) * x) * t;
	} else if (z <= 8500000000000.0) {
		tmp = (x * y) / z;
	} else if (z <= 1.3e+137) {
		tmp = (x * t) / z;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (y / z) * x
	tmp = 0
	if z <= -6.4e-6:
		tmp = t_1
	elif z <= -4e-53:
		tmp = ((-1.0 - z) * x) * t
	elif z <= 8500000000000.0:
		tmp = (x * y) / z
	elif z <= 1.3e+137:
		tmp = (x * t) / z
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(y / z) * x)
	tmp = 0.0
	if (z <= -6.4e-6)
		tmp = t_1;
	elseif (z <= -4e-53)
		tmp = Float64(Float64(Float64(-1.0 - z) * x) * t);
	elseif (z <= 8500000000000.0)
		tmp = Float64(Float64(x * y) / z);
	elseif (z <= 1.3e+137)
		tmp = Float64(Float64(x * t) / z);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (y / z) * x;
	tmp = 0.0;
	if (z <= -6.4e-6)
		tmp = t_1;
	elseif (z <= -4e-53)
		tmp = ((-1.0 - z) * x) * t;
	elseif (z <= 8500000000000.0)
		tmp = (x * y) / z;
	elseif (z <= 1.3e+137)
		tmp = (x * t) / z;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(y / z), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[z, -6.4e-6], t$95$1, If[LessEqual[z, -4e-53], N[(N[(N[(-1.0 - z), $MachinePrecision] * x), $MachinePrecision] * t), $MachinePrecision], If[LessEqual[z, 8500000000000.0], N[(N[(x * y), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[z, 1.3e+137], N[(N[(x * t), $MachinePrecision] / z), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 4 regimes

2. if z < -6.3999999999999997e-6 or 1.3e137 < z

   1. Initial program 97.7%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 61.8

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

   5. Applied rewrites 61.8%

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

   if -6.3999999999999997e-6 < z < -4.00000000000000012e-53

   1. Initial program 89.1%

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

   3. Taylor expanded in z around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. distribute-lft-out N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. distribute-rgt-out N/A

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

      11. lower-*.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. mul-1-neg N/A

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

      15. lower-neg.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 98.9

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

   5. Applied rewrites 98.9%

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

   6. Taylor expanded in t around inf

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

   8. Applied rewrites 84.4%

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

   if -4.00000000000000012e-53 < z < 8.5e12

   1. Initial program 91.7%

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

   3. Taylor expanded in z around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. distribute-lft-out N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. distribute-rgt-out N/A

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

      11. lower-*.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. mul-1-neg N/A

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

      15. lower-neg.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 93.0

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

   5. Applied rewrites 93.0%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 74.8%

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

   if 8.5e12 < z < 1.3e137

   1. Initial program 99.7%

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

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-neg.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 78.7

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

   4. Applied rewrites 78.7%

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

   5. Taylor expanded in y around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{t \cdot x}{1 - z}} \]
   6. 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. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. sub-neg N/A

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

      7. mul-1-neg N/A

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

      8. distribute-lft-in N/A

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

      9. metadata-eval N/A

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

      10. neg-mul-1 N/A

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

      11. mul-1-neg N/A

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

      12. remove-double-neg N/A

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

      13. lower-+.f64 81.0

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

   7. Applied rewrites 81.0%

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

   8. Taylor expanded in z around inf

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

   10. Applied rewrites 80.6%

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

4. Final simplification 70.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -6.4 \cdot 10^{-6}:\\ \;\;\;\;\frac{y}{z} \cdot x\\ \mathbf{elif}\;z \leq -4 \cdot 10^{-53}:\\ \;\;\;\;\left(\left(-1 - z\right) \cdot x\right) \cdot t\\ \mathbf{elif}\;z \leq 8500000000000:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{elif}\;z \leq 1.3 \cdot 10^{+137}:\\ \;\;\;\;\frac{x \cdot t}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z} \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 74.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{y}{z} \cdot x\\ \mathbf{if}\;z \leq -1.15 \cdot 10^{+83}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 8500000000000:\\ \;\;\;\;\left(\frac{y}{z} - t\right) \cdot x\\ \mathbf{elif}\;z \leq 1.3 \cdot 10^{+137}:\\ \;\;\;\;\frac{x \cdot t}{z - 1}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (/ y z) x)))
   (if (<= z -1.15e+83)
     t_1
     (if (<= z 8500000000000.0)
       (* (- (/ y z) t) x)
       (if (<= z 1.3e+137) (/ (* x t) (- z 1.0)) t_1)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (y / z) * x;
	double tmp;
	if (z <= -1.15e+83) {
		tmp = t_1;
	} else if (z <= 8500000000000.0) {
		tmp = ((y / z) - t) * x;
	} else if (z <= 1.3e+137) {
		tmp = (x * t) / (z - 1.0);
	} 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) :: tmp
    t_1 = (y / z) * x
    if (z <= (-1.15d+83)) then
        tmp = t_1
    else if (z <= 8500000000000.0d0) then
        tmp = ((y / z) - t) * x
    else if (z <= 1.3d+137) then
        tmp = (x * t) / (z - 1.0d0)
    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 = (y / z) * x;
	double tmp;
	if (z <= -1.15e+83) {
		tmp = t_1;
	} else if (z <= 8500000000000.0) {
		tmp = ((y / z) - t) * x;
	} else if (z <= 1.3e+137) {
		tmp = (x * t) / (z - 1.0);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (y / z) * x
	tmp = 0
	if z <= -1.15e+83:
		tmp = t_1
	elif z <= 8500000000000.0:
		tmp = ((y / z) - t) * x
	elif z <= 1.3e+137:
		tmp = (x * t) / (z - 1.0)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(y / z) * x)
	tmp = 0.0
	if (z <= -1.15e+83)
		tmp = t_1;
	elseif (z <= 8500000000000.0)
		tmp = Float64(Float64(Float64(y / z) - t) * x);
	elseif (z <= 1.3e+137)
		tmp = Float64(Float64(x * t) / Float64(z - 1.0));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (y / z) * x;
	tmp = 0.0;
	if (z <= -1.15e+83)
		tmp = t_1;
	elseif (z <= 8500000000000.0)
		tmp = ((y / z) - t) * x;
	elseif (z <= 1.3e+137)
		tmp = (x * t) / (z - 1.0);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(y / z), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[z, -1.15e+83], t$95$1, If[LessEqual[z, 8500000000000.0], N[(N[(N[(y / z), $MachinePrecision] - t), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[z, 1.3e+137], N[(N[(x * t), $MachinePrecision] / N[(z - 1.0), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.14999999999999997e83 or 1.3e137 < z

   1. Initial program 97.2%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 64.8

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

   5. Applied rewrites 64.8%

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

   if -1.14999999999999997e83 < z < 8.5e12

   1. Initial program 92.6%

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

   3. Taylor expanded in z around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. mul-1-neg N/A

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

      4. unsub-neg N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

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

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

      9. unsub-neg N/A

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

      10. mul-1-neg N/A

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

      11. lower-*.f64 N/A

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

      12. +-commutative N/A

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

      13. associate-*r* N/A

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

      14. lower-fma.f64 N/A

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

      15. mul-1-neg N/A

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

      16. lower-neg.f64 88.3

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

   5. Applied rewrites 88.3%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 85.3%

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

   if 8.5e12 < z < 1.3e137

   1. Initial program 99.7%

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

   3. Taylor expanded in y around 0

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

      1. 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. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. sub-neg N/A

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

      6. mul-1-neg N/A

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

      7. +-commutative N/A

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

      8. distribute-neg-in N/A

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

      9. mul-1-neg N/A

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

      10. remove-double-neg N/A

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

      11. sub-neg N/A

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

      12. lower--.f64 81.0

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

   5. Applied rewrites 81.0%

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

4. Final simplification 78.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.15 \cdot 10^{+83}:\\ \;\;\;\;\frac{y}{z} \cdot x\\ \mathbf{elif}\;z \leq 8500000000000:\\ \;\;\;\;\left(\frac{y}{z} - t\right) \cdot x\\ \mathbf{elif}\;z \leq 1.3 \cdot 10^{+137}:\\ \;\;\;\;\frac{x \cdot t}{z - 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z} \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 94.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t + y}{z} \cdot x\\ \mathbf{if}\;z \leq -63000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1:\\ \;\;\;\;\frac{\mathsf{fma}\left(-t, z, y\right) \cdot x}{z}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (/ (+ t y) z) x)))
   (if (<= z -63000.0) t_1 (if (<= z 1.0) (/ (* (fma (- t) z y) x) z) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = ((t + y) / z) * x;
	double tmp;
	if (z <= -63000.0) {
		tmp = t_1;
	} else if (z <= 1.0) {
		tmp = (fma(-t, z, y) * x) / z;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(Float64(t + y) / z) * x)
	tmp = 0.0
	if (z <= -63000.0)
		tmp = t_1;
	elseif (z <= 1.0)
		tmp = Float64(Float64(fma(Float64(-t), z, y) * x) / z);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(N[(t + y), $MachinePrecision] / z), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[z, -63000.0], t$95$1, If[LessEqual[z, 1.0], N[(N[(N[((-t) * z + y), $MachinePrecision] * x), $MachinePrecision] / z), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -63000 or 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. +-commutative N/A

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

      6. lower-+.f64 97.7

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

   5. Applied rewrites 97.7%

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

   if -63000 < z < 1

   1. Initial program 91.5%

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

   3. Taylor expanded in z around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. mul-1-neg N/A

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

      4. unsub-neg N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

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

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

      9. unsub-neg N/A

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

      10. mul-1-neg N/A

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

      11. lower-*.f64 N/A

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

      12. +-commutative N/A

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

      13. associate-*r* N/A

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

      14. lower-fma.f64 N/A

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

      15. mul-1-neg N/A

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

      16. lower-neg.f64 94.7

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

   5. Applied rewrites 94.7%

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

4. Final simplification 96.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -63000:\\ \;\;\;\;\frac{t + y}{z} \cdot x\\ \mathbf{elif}\;z \leq 1:\\ \;\;\;\;\frac{\mathsf{fma}\left(-t, z, y\right) \cdot x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{t + y}{z} \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 94.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t + y}{z} \cdot x\\ \mathbf{if}\;z \leq -63000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1:\\ \;\;\;\;\frac{x}{z} \cdot \left(y - t \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (/ (+ t y) z) x)))
   (if (<= z -63000.0) t_1 (if (<= z 1.0) (* (/ x z) (- y (* t z))) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = ((t + y) / z) * x;
	double tmp;
	if (z <= -63000.0) {
		tmp = t_1;
	} else if (z <= 1.0) {
		tmp = (x / z) * (y - (t * 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) :: tmp
    t_1 = ((t + y) / z) * x
    if (z <= (-63000.0d0)) then
        tmp = t_1
    else if (z <= 1.0d0) then
        tmp = (x / z) * (y - (t * 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 = ((t + y) / z) * x;
	double tmp;
	if (z <= -63000.0) {
		tmp = t_1;
	} else if (z <= 1.0) {
		tmp = (x / z) * (y - (t * z));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = ((t + y) / z) * x
	tmp = 0
	if z <= -63000.0:
		tmp = t_1
	elif z <= 1.0:
		tmp = (x / z) * (y - (t * z))
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(Float64(t + y) / z) * x)
	tmp = 0.0
	if (z <= -63000.0)
		tmp = t_1;
	elseif (z <= 1.0)
		tmp = Float64(Float64(x / z) * Float64(y - Float64(t * z)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = ((t + y) / z) * x;
	tmp = 0.0;
	if (z <= -63000.0)
		tmp = t_1;
	elseif (z <= 1.0)
		tmp = (x / z) * (y - (t * z));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(N[(t + y), $MachinePrecision] / z), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[z, -63000.0], t$95$1, If[LessEqual[z, 1.0], N[(N[(x / z), $MachinePrecision] * N[(y - N[(t * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -63000 or 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. +-commutative N/A

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

      6. lower-+.f64 97.7

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

   5. Applied rewrites 97.7%

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

   if -63000 < z < 1

   1. Initial program 91.5%

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

   3. Taylor expanded in z around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. mul-1-neg N/A

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

      4. unsub-neg N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

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

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

      9. unsub-neg N/A

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

      10. mul-1-neg N/A

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

      11. lower-*.f64 N/A

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

      12. +-commutative N/A

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

      13. associate-*r* N/A

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

      14. lower-fma.f64 N/A

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

      15. mul-1-neg N/A

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

      16. lower-neg.f64 94.7

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

   5. Applied rewrites 94.7%

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

   7. Applied rewrites 93.5%

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

   9. Applied rewrites 93.5%

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

4. Final simplification 95.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -63000:\\ \;\;\;\;\frac{t + y}{z} \cdot x\\ \mathbf{elif}\;z \leq 1:\\ \;\;\;\;\frac{x}{z} \cdot \left(y - t \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t + y}{z} \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 74.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{y}{z} \cdot x\\ \mathbf{if}\;z \leq -1.15 \cdot 10^{+83}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 8500000000000:\\ \;\;\;\;\left(\frac{y}{z} - t\right) \cdot x\\ \mathbf{elif}\;z \leq 1.3 \cdot 10^{+137}:\\ \;\;\;\;\frac{x \cdot t}{z}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (/ y z) x)))
   (if (<= z -1.15e+83)
     t_1
     (if (<= z 8500000000000.0)
       (* (- (/ y z) t) x)
       (if (<= z 1.3e+137) (/ (* x t) z) t_1)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (y / z) * x;
	double tmp;
	if (z <= -1.15e+83) {
		tmp = t_1;
	} else if (z <= 8500000000000.0) {
		tmp = ((y / z) - t) * x;
	} else if (z <= 1.3e+137) {
		tmp = (x * t) / 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) :: tmp
    t_1 = (y / z) * x
    if (z <= (-1.15d+83)) then
        tmp = t_1
    else if (z <= 8500000000000.0d0) then
        tmp = ((y / z) - t) * x
    else if (z <= 1.3d+137) then
        tmp = (x * t) / 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 = (y / z) * x;
	double tmp;
	if (z <= -1.15e+83) {
		tmp = t_1;
	} else if (z <= 8500000000000.0) {
		tmp = ((y / z) - t) * x;
	} else if (z <= 1.3e+137) {
		tmp = (x * t) / z;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (y / z) * x
	tmp = 0
	if z <= -1.15e+83:
		tmp = t_1
	elif z <= 8500000000000.0:
		tmp = ((y / z) - t) * x
	elif z <= 1.3e+137:
		tmp = (x * t) / z
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(y / z) * x)
	tmp = 0.0
	if (z <= -1.15e+83)
		tmp = t_1;
	elseif (z <= 8500000000000.0)
		tmp = Float64(Float64(Float64(y / z) - t) * x);
	elseif (z <= 1.3e+137)
		tmp = Float64(Float64(x * t) / z);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (y / z) * x;
	tmp = 0.0;
	if (z <= -1.15e+83)
		tmp = t_1;
	elseif (z <= 8500000000000.0)
		tmp = ((y / z) - t) * x;
	elseif (z <= 1.3e+137)
		tmp = (x * t) / z;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(y / z), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[z, -1.15e+83], t$95$1, If[LessEqual[z, 8500000000000.0], N[(N[(N[(y / z), $MachinePrecision] - t), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[z, 1.3e+137], N[(N[(x * t), $MachinePrecision] / z), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.14999999999999997e83 or 1.3e137 < z

   1. Initial program 97.2%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 64.8

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

   5. Applied rewrites 64.8%

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

   if -1.14999999999999997e83 < z < 8.5e12

   1. Initial program 92.6%

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

   3. Taylor expanded in z around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. mul-1-neg N/A

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

      4. unsub-neg N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

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

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

      9. unsub-neg N/A

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

      10. mul-1-neg N/A

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

      11. lower-*.f64 N/A

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

      12. +-commutative N/A

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

      13. associate-*r* N/A

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

      14. lower-fma.f64 N/A

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

      15. mul-1-neg N/A

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

      16. lower-neg.f64 88.3

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

   5. Applied rewrites 88.3%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 85.3%

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

   if 8.5e12 < z < 1.3e137

   1. Initial program 99.7%

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

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-neg.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 78.7

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

   4. Applied rewrites 78.7%

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

   5. Taylor expanded in y around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{t \cdot x}{1 - z}} \]
   6. 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. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. sub-neg N/A

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

      7. mul-1-neg N/A

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

      8. distribute-lft-in N/A

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

      9. metadata-eval N/A

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

      10. neg-mul-1 N/A

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

      11. mul-1-neg N/A

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

      12. remove-double-neg N/A

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

      13. lower-+.f64 81.0

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

   7. Applied rewrites 81.0%

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

   8. Taylor expanded in z around inf

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

   10. Applied rewrites 80.6%

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

4. Final simplification 78.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.15 \cdot 10^{+83}:\\ \;\;\;\;\frac{y}{z} \cdot x\\ \mathbf{elif}\;z \leq 8500000000000:\\ \;\;\;\;\left(\frac{y}{z} - t\right) \cdot x\\ \mathbf{elif}\;z \leq 1.3 \cdot 10^{+137}:\\ \;\;\;\;\frac{x \cdot t}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z} \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 63.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x \cdot t}{z}\\ \mathbf{if}\;t \leq -9.5 \cdot 10^{+62}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 5 \cdot 10^{-253}:\\ \;\;\;\;\frac{x}{z} \cdot y\\ \mathbf{elif}\;t \leq 2.05 \cdot 10^{+182}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (* x t) z)))
   (if (<= t -9.5e+62)
     t_1
     (if (<= t 5e-253)
       (* (/ x z) y)
       (if (<= t 2.05e+182) (/ (* x y) z) t_1)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x * t) / z;
	double tmp;
	if (t <= -9.5e+62) {
		tmp = t_1;
	} else if (t <= 5e-253) {
		tmp = (x / z) * y;
	} else if (t <= 2.05e+182) {
		tmp = (x * y) / 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) :: tmp
    t_1 = (x * t) / z
    if (t <= (-9.5d+62)) then
        tmp = t_1
    else if (t <= 5d-253) then
        tmp = (x / z) * y
    else if (t <= 2.05d+182) then
        tmp = (x * y) / 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 * t) / z;
	double tmp;
	if (t <= -9.5e+62) {
		tmp = t_1;
	} else if (t <= 5e-253) {
		tmp = (x / z) * y;
	} else if (t <= 2.05e+182) {
		tmp = (x * y) / z;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (x * t) / z
	tmp = 0
	if t <= -9.5e+62:
		tmp = t_1
	elif t <= 5e-253:
		tmp = (x / z) * y
	elif t <= 2.05e+182:
		tmp = (x * y) / z
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x * t) / z)
	tmp = 0.0
	if (t <= -9.5e+62)
		tmp = t_1;
	elseif (t <= 5e-253)
		tmp = Float64(Float64(x / z) * y);
	elseif (t <= 2.05e+182)
		tmp = Float64(Float64(x * y) / z);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (x * t) / z;
	tmp = 0.0;
	if (t <= -9.5e+62)
		tmp = t_1;
	elseif (t <= 5e-253)
		tmp = (x / z) * y;
	elseif (t <= 2.05e+182)
		tmp = (x * y) / z;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x * t), $MachinePrecision] / z), $MachinePrecision]}, If[LessEqual[t, -9.5e+62], t$95$1, If[LessEqual[t, 5e-253], N[(N[(x / z), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[t, 2.05e+182], N[(N[(x * y), $MachinePrecision] / z), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if t < -9.5000000000000003e62 or 2.05000000000000001e182 < t

   1. Initial program 97.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift--.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. frac-sub N/A

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

      7. associate-*l/ N/A

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

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

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-neg.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 64.7

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

   4. Applied rewrites 64.7%

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

   5. Taylor expanded in y around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{t \cdot x}{1 - z}} \]
   6. 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. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. sub-neg N/A

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

      7. mul-1-neg N/A

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

      8. distribute-lft-in N/A

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

      9. metadata-eval N/A

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

      10. neg-mul-1 N/A

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

      11. mul-1-neg N/A

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

      12. remove-double-neg N/A

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

      13. lower-+.f64 72.9

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

   7. Applied rewrites 72.9%

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

   8. Taylor expanded in z around inf

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

   10. Applied rewrites 50.3%

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

   if -9.5000000000000003e62 < t < 4.99999999999999971e-253

   1. Initial program 93.9%

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

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-neg.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 66.3

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

   4. Applied rewrites 66.3%

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

   5. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   7. Applied rewrites 87.4%

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

   8. Taylor expanded in z around 0

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

   10. Applied rewrites 84.2%

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

   if 4.99999999999999971e-253 < t < 2.05000000000000001e182

   1. Initial program 93.2%

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

   3. Taylor expanded in z around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. distribute-lft-out N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. distribute-rgt-out N/A

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

      11. lower-*.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. mul-1-neg N/A

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

      15. lower-neg.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 64.2

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

   5. Applied rewrites 64.2%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 68.2%

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

4. Final simplification 67.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -9.5 \cdot 10^{+62}:\\ \;\;\;\;\frac{x \cdot t}{z}\\ \mathbf{elif}\;t \leq 5 \cdot 10^{-253}:\\ \;\;\;\;\frac{x}{z} \cdot y\\ \mathbf{elif}\;t \leq 2.05 \cdot 10^{+182}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot t}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 93.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t + y}{z} \cdot x\\ \mathbf{if}\;z \leq -63000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1:\\ \;\;\;\;\left(\frac{y}{z} - t\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (/ (+ t y) z) x)))
   (if (<= z -63000.0) t_1 (if (<= z 1.0) (* (- (/ y z) t) x) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = ((t + y) / z) * x;
	double tmp;
	if (z <= -63000.0) {
		tmp = t_1;
	} else if (z <= 1.0) {
		tmp = ((y / z) - t) * x;
	} 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) :: tmp
    t_1 = ((t + y) / z) * x
    if (z <= (-63000.0d0)) then
        tmp = t_1
    else if (z <= 1.0d0) then
        tmp = ((y / z) - t) * x
    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 = ((t + y) / z) * x;
	double tmp;
	if (z <= -63000.0) {
		tmp = t_1;
	} else if (z <= 1.0) {
		tmp = ((y / z) - t) * x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = ((t + y) / z) * x
	tmp = 0
	if z <= -63000.0:
		tmp = t_1
	elif z <= 1.0:
		tmp = ((y / z) - t) * x
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(Float64(t + y) / z) * x)
	tmp = 0.0
	if (z <= -63000.0)
		tmp = t_1;
	elseif (z <= 1.0)
		tmp = Float64(Float64(Float64(y / z) - t) * x);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = ((t + y) / z) * x;
	tmp = 0.0;
	if (z <= -63000.0)
		tmp = t_1;
	elseif (z <= 1.0)
		tmp = ((y / z) - t) * x;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(N[(t + y), $MachinePrecision] / z), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[z, -63000.0], t$95$1, If[LessEqual[z, 1.0], N[(N[(N[(y / z), $MachinePrecision] - t), $MachinePrecision] * x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -63000 or 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. +-commutative N/A

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

      6. lower-+.f64 97.7

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

   5. Applied rewrites 97.7%

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

   if -63000 < z < 1

   1. Initial program 91.5%

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

   3. Taylor expanded in z around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. mul-1-neg N/A

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

      4. unsub-neg N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

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

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

      9. unsub-neg N/A

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

      10. mul-1-neg N/A

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

      11. lower-*.f64 N/A

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

      12. +-commutative N/A

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

      13. associate-*r* N/A

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

      14. lower-fma.f64 N/A

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

      15. mul-1-neg N/A

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

      16. lower-neg.f64 94.7

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

   5. Applied rewrites 94.7%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 90.5%

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

4. Final simplification 94.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -63000:\\ \;\;\;\;\frac{t + y}{z} \cdot x\\ \mathbf{elif}\;z \leq 1:\\ \;\;\;\;\left(\frac{y}{z} - t\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{t + y}{z} \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 59.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x \cdot y}{z}\\ \mathbf{if}\;y \leq -6.2 \cdot 10^{-153}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 3.9 \cdot 10^{-14}:\\ \;\;\;\;\left(-x\right) \cdot t\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (* x y) z)))
   (if (<= y -6.2e-153) t_1 (if (<= y 3.9e-14) (* (- x) t) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x * y) / z;
	double tmp;
	if (y <= -6.2e-153) {
		tmp = t_1;
	} else if (y <= 3.9e-14) {
		tmp = -x * t;
	} 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) :: tmp
    t_1 = (x * y) / z
    if (y <= (-6.2d-153)) then
        tmp = t_1
    else if (y <= 3.9d-14) then
        tmp = -x * t
    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;
	double tmp;
	if (y <= -6.2e-153) {
		tmp = t_1;
	} else if (y <= 3.9e-14) {
		tmp = -x * t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (x * y) / z
	tmp = 0
	if y <= -6.2e-153:
		tmp = t_1
	elif y <= 3.9e-14:
		tmp = -x * t
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x * y) / z)
	tmp = 0.0
	if (y <= -6.2e-153)
		tmp = t_1;
	elseif (y <= 3.9e-14)
		tmp = Float64(Float64(-x) * t);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (x * y) / z;
	tmp = 0.0;
	if (y <= -6.2e-153)
		tmp = t_1;
	elseif (y <= 3.9e-14)
		tmp = -x * t;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x * y), $MachinePrecision] / z), $MachinePrecision]}, If[LessEqual[y, -6.2e-153], t$95$1, If[LessEqual[y, 3.9e-14], N[((-x) * t), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -6.1999999999999999e-153 or 3.8999999999999998e-14 < y

   1. Initial program 93.5%

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

   3. Taylor expanded in z around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. distribute-lft-out N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. distribute-rgt-out N/A

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

      11. lower-*.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. mul-1-neg N/A

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

      15. lower-neg.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 57.4

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

   5. Applied rewrites 57.4%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 69.5%

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

   if -6.1999999999999999e-153 < y < 3.8999999999999998e-14

   1. Initial program 98.5%

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

   3. Taylor expanded in z around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. mul-1-neg N/A

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

      4. unsub-neg N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

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

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

      9. unsub-neg N/A

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

      10. mul-1-neg N/A

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

      11. lower-*.f64 N/A

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

      12. +-commutative N/A

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

      13. associate-*r* N/A

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

      14. lower-fma.f64 N/A

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

      15. mul-1-neg N/A

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

      16. lower-neg.f64 59.4

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

   5. Applied rewrites 59.4%

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

   6. Taylor expanded in t around inf

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

   8. Applied rewrites 50.4%

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

Alternative 11: 60.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 8500000000000:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot t}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z 8500000000000.0) (/ (* x y) z) (/ (* x t) z)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 8500000000000.0) {
		tmp = (x * y) / z;
	} else {
		tmp = (x * t) / z;
	}
	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) :: tmp
    if (z <= 8500000000000.0d0) then
        tmp = (x * y) / z
    else
        tmp = (x * t) / z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 8500000000000.0) {
		tmp = (x * y) / z;
	} else {
		tmp = (x * t) / z;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= 8500000000000.0:
		tmp = (x * y) / z
	else:
		tmp = (x * t) / z
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= 8500000000000.0)
		tmp = Float64(Float64(x * y) / z);
	else
		tmp = Float64(Float64(x * t) / z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= 8500000000000.0)
		tmp = (x * y) / z;
	else
		tmp = (x * t) / z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, 8500000000000.0], N[(N[(x * y), $MachinePrecision] / z), $MachinePrecision], N[(N[(x * t), $MachinePrecision] / z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 8.5e12

   1. Initial program 93.5%

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

   3. Taylor expanded in z around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. distribute-lft-out N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. distribute-rgt-out N/A

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

      11. lower-*.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. mul-1-neg N/A

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

      15. lower-neg.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 67.3

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

   5. Applied rewrites 67.3%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 65.4%

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

   if 8.5e12 < z

   1. Initial program 99.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift--.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. frac-sub N/A

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

      7. associate-*l/ N/A

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

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

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-neg.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 50.8

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

   4. Applied rewrites 50.8%

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

   5. Taylor expanded in y around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{t \cdot x}{1 - z}} \]
   6. 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. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. sub-neg N/A

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

      7. mul-1-neg N/A

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

      8. distribute-lft-in N/A

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

      9. metadata-eval N/A

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

      10. neg-mul-1 N/A

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

      11. mul-1-neg N/A

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

      12. remove-double-neg N/A

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

      13. lower-+.f64 61.1

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

   7. Applied rewrites 61.1%

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

   8. Taylor expanded in z around inf

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

   10. Applied rewrites 61.0%

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

4. Final simplification 64.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 8500000000000:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot t}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 23.5% accurate, 4.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.9%

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

3. Taylor expanded in z around 0

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

   1. lower-/.f64 N/A

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

   2. +-commutative N/A

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

   3. mul-1-neg N/A

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

   4. unsub-neg N/A

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

   5. associate-*r* N/A

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

   6. *-commutative N/A

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

   7. associate-*l* N/A

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

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

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

   9. unsub-neg N/A

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

   10. mul-1-neg N/A

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

   11. lower-*.f64 N/A

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

   12. +-commutative N/A

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

   13. associate-*r* N/A

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

   14. lower-fma.f64 N/A

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

   15. mul-1-neg N/A

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

   16. lower-neg.f64 63.1

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

5. Applied rewrites 63.1%

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

6. Taylor expanded in t around inf

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

8. Applied rewrites 25.7%

   \[\leadsto \left(-x\right) \cdot \color{blue}{t} \]
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 2024235 
(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)))))