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

Percentage Accurate: 94.3% → 95.7%

Time: 8.6s

Alternatives: 11

Speedup: 0.9×

• 20.9% 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: 94.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 95.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{y}{z} - \frac{t}{1 - z}\\ \mathbf{if}\;t\_1 \leq 2 \cdot 10^{+287}:\\ \;\;\;\;x \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (/ y z) (/ t (- 1.0 z)))))
   (if (<= t_1 2e+287) (* x t_1) (* (/ x z) y))))

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 <= 2e+287) {
		tmp = x * t_1;
	} else {
		tmp = (x / z) * y;
	}
	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) - (t / (1.0d0 - z))
    if (t_1 <= 2d+287) then
        tmp = x * t_1
    else
        tmp = (x / z) * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (y / z) - (t / (1.0 - z));
	double tmp;
	if (t_1 <= 2e+287) {
		tmp = x * t_1;
	} else {
		tmp = (x / z) * y;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (y / z) - (t / (1.0 - z))
	tmp = 0
	if t_1 <= 2e+287:
		tmp = x * t_1
	else:
		tmp = (x / z) * y
	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 <= 2e+287)
		tmp = Float64(x * t_1);
	else
		tmp = Float64(Float64(x / z) * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (y / z) - (t / (1.0 - z));
	tmp = 0.0;
	if (t_1 <= 2e+287)
		tmp = x * t_1;
	else
		tmp = (x / z) * y;
	end
	tmp_2 = 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, 2e+287], N[(x * t$95$1), $MachinePrecision], N[(N[(x / z), $MachinePrecision] * y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (/.f64 y z) (/.f64 t (-.f64 #s(literal 1 binary64) z))) < 2.0000000000000002e287

   1. Initial program 95.8%

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

   if 2.0000000000000002e287 < (-.f64 (/.f64 y z) (/.f64 t (-.f64 #s(literal 1 binary64) z)))

   1. Initial program 71.1%

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

   3. Taylor expanded in y around inf

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

      1. associate-*l/ N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 100.0

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

   5. Applied rewrites 100.0%

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

Alternative 2: 73.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{y}{z} \cdot x\\ \mathbf{if}\;z \leq -3.7 \cdot 10^{+121}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq -800:\\ \;\;\;\;\frac{x \cdot t}{z - 1}\\ \mathbf{elif}\;z \leq 7 \cdot 10^{+34}:\\ \;\;\;\;\left(\frac{y}{z} - t\right) \cdot x\\ \mathbf{elif}\;z \leq 1.82 \cdot 10^{+245}:\\ \;\;\;\;\frac{t}{z} \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (/ y z) x)))
   (if (<= z -3.7e+121)
     t_1
     (if (<= z -800.0)
       (/ (* x t) (- z 1.0))
       (if (<= z 7e+34)
         (* (- (/ y z) t) x)
         (if (<= z 1.82e+245) (* (/ t z) x) t_1))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (y / z) * x;
	double tmp;
	if (z <= -3.7e+121) {
		tmp = t_1;
	} else if (z <= -800.0) {
		tmp = (x * t) / (z - 1.0);
	} else if (z <= 7e+34) {
		tmp = ((y / z) - t) * x;
	} else if (z <= 1.82e+245) {
		tmp = (t / z) * 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 = (y / z) * x
    if (z <= (-3.7d+121)) then
        tmp = t_1
    else if (z <= (-800.0d0)) then
        tmp = (x * t) / (z - 1.0d0)
    else if (z <= 7d+34) then
        tmp = ((y / z) - t) * x
    else if (z <= 1.82d+245) then
        tmp = (t / z) * 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 = (y / z) * x;
	double tmp;
	if (z <= -3.7e+121) {
		tmp = t_1;
	} else if (z <= -800.0) {
		tmp = (x * t) / (z - 1.0);
	} else if (z <= 7e+34) {
		tmp = ((y / z) - t) * x;
	} else if (z <= 1.82e+245) {
		tmp = (t / z) * x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (y / z) * x
	tmp = 0
	if z <= -3.7e+121:
		tmp = t_1
	elif z <= -800.0:
		tmp = (x * t) / (z - 1.0)
	elif z <= 7e+34:
		tmp = ((y / z) - t) * x
	elif z <= 1.82e+245:
		tmp = (t / z) * x
	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 <= -3.7e+121)
		tmp = t_1;
	elseif (z <= -800.0)
		tmp = Float64(Float64(x * t) / Float64(z - 1.0));
	elseif (z <= 7e+34)
		tmp = Float64(Float64(Float64(y / z) - t) * x);
	elseif (z <= 1.82e+245)
		tmp = Float64(Float64(t / z) * x);
	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 <= -3.7e+121)
		tmp = t_1;
	elseif (z <= -800.0)
		tmp = (x * t) / (z - 1.0);
	elseif (z <= 7e+34)
		tmp = ((y / z) - t) * x;
	elseif (z <= 1.82e+245)
		tmp = (t / z) * x;
	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, -3.7e+121], t$95$1, If[LessEqual[z, -800.0], N[(N[(x * t), $MachinePrecision] / N[(z - 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 7e+34], N[(N[(N[(y / z), $MachinePrecision] - t), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[z, 1.82e+245], N[(N[(t / z), $MachinePrecision] * x), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 4 regimes

2. if z < -3.70000000000000013e121 or 1.82000000000000003e245 < z

   1. Initial program 95.0%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 58.6

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

   5. Applied rewrites 58.6%

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

   if -3.70000000000000013e121 < z < -800

   1. Initial program 99.3%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. distribute-neg-frac2 N/A

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

      6. div-inv N/A

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

      7. lower-fma.f64 N/A

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

      8. frac-2neg N/A

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

      9. metadata-eval N/A

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

      10. remove-double-neg N/A

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

      11. lower-/.f64 99.2

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

   4. Applied rewrites 99.2%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. distribute-lft-in N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. metadata-eval N/A

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

      10. lift--.f64 N/A

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

      11. sub-neg N/A

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

      12. metadata-eval N/A

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

      13. distribute-neg-in N/A

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

      14. lift-+.f64 N/A

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

      15. frac-2neg N/A

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

      16. lower-/.f64 N/A

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 N/A

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

      20. lower-*.f64 96.7

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

   6. Applied rewrites 96.7%

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

   7. Taylor expanded in t around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 50.5

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

   9. Applied rewrites 50.5%

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

   if -800 < z < 6.99999999999999996e34

   1. Initial program 92.3%

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

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

      14. associate-*r* N/A

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

      15. lower-fma.f64 N/A

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

      16. mul-1-neg N/A

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

      17. lower-neg.f64 90.7

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

   5. Applied rewrites 90.7%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 88.8%

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

   if 6.99999999999999996e34 < z < 1.82000000000000003e245

   1. Initial program 97.1%

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

   3. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. lower-/.f64 N/A

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

      4. sub-neg N/A

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

      5. mul-1-neg N/A

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

      6. +-commutative N/A

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

      7. distribute-neg-in N/A

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

      8. mul-1-neg N/A

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

      9. remove-double-neg N/A

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

      10. sub-neg N/A

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

      11. lower--.f64 55.1

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

   5. Applied rewrites 55.1%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 55.1%

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

4. Final simplification 73.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.7 \cdot 10^{+121}:\\ \;\;\;\;\frac{y}{z} \cdot x\\ \mathbf{elif}\;z \leq -800:\\ \;\;\;\;\frac{x \cdot t}{z - 1}\\ \mathbf{elif}\;z \leq 7 \cdot 10^{+34}:\\ \;\;\;\;\left(\frac{y}{z} - t\right) \cdot x\\ \mathbf{elif}\;z \leq 1.82 \cdot 10^{+245}:\\ \;\;\;\;\frac{t}{z} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z} \cdot x\\ \end{array} \]
5. Add Preprocessing

Developer Target 1: 94.7% 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 2024230 
(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)))))