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

Percentage Accurate: 94.7% → 98.0%

Time: 12.3s

Alternatives: 8

Speedup: 0.9×

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 94.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{y}{z} + \frac{t}{z + -1}\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{elif}\;t\_1 \leq 3 \cdot 10^{+288}:\\ \;\;\;\;t\_1 \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (+ (/ y z) (/ t (+ z -1.0)))))
   (if (<= t_1 (- INFINITY))
     (* y (/ x z))
     (if (<= t_1 3e+288) (* t_1 x) (/ (* y x) z)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (y / z) + (t / (z + -1.0));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = y * (x / z);
	} else if (t_1 <= 3e+288) {
		tmp = t_1 * x;
	} else {
		tmp = (y * x) / z;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (y / z) + (t / (z + -1.0));
	double tmp;
	if (t_1 <= -Double.POSITIVE_INFINITY) {
		tmp = y * (x / z);
	} else if (t_1 <= 3e+288) {
		tmp = t_1 * x;
	} else {
		tmp = (y * x) / z;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (y / z) + (t / (z + -1.0))
	tmp = 0
	if t_1 <= -math.inf:
		tmp = y * (x / z)
	elif t_1 <= 3e+288:
		tmp = t_1 * x
	else:
		tmp = (y * x) / z
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(y / z) + Float64(t / Float64(z + -1.0)))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(y * Float64(x / z));
	elseif (t_1 <= 3e+288)
		tmp = Float64(t_1 * x);
	else
		tmp = Float64(Float64(y * x) / z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (y / z) + (t / (z + -1.0));
	tmp = 0.0;
	if (t_1 <= -Inf)
		tmp = y * (x / z);
	elseif (t_1 <= 3e+288)
		tmp = t_1 * x;
	else
		tmp = (y * x) / z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(y / z), $MachinePrecision] + N[(t / N[(z + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(y * N[(x / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 3e+288], N[(t$95$1 * x), $MachinePrecision], N[(N[(y * x), $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 68.2%

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

   3. Taylor expanded in y around inf

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

      1. +-rgt-identity N/A

         \[\leadsto \color{blue}{\frac{x \cdot y}{z} + 0} \]

      2. *-commutative N/A

         \[\leadsto \frac{\color{blue}{y \cdot x}}{z} + 0 \]

      3. associate-*r/ N/A

         \[\leadsto \color{blue}{y \cdot \frac{x}{z}} + 0 \]

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

         \[\leadsto \color{blue}{\mathsf{fma}\left(y, \frac{x}{z}, 0\right)} \]

      5. /-lowering-/.f64 99.9

         \[\leadsto \mathsf{fma}\left(y, \color{blue}{\frac{x}{z}}, 0\right) \]

   5. Simplified 99.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, \frac{x}{z}, 0\right)} \]
   6. Step-by-step derivation

      1. +-rgt-identity N/A

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

      2. *-commutative N/A

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

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

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

      4. /-lowering-/.f64 99.9

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

   7. Applied egg-rr 99.9%

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

   if -inf.0 < (-.f64 (/.f64 y z) (/.f64 t (-.f64 #s(literal 1 binary64) z))) < 2.9999999999999998e288

   1. Initial program 98.4%

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

   if 2.9999999999999998e288 < (-.f64 (/.f64 y z) (/.f64 t (-.f64 #s(literal 1 binary64) z)))

   1. Initial program 81.0%

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

   3. Taylor expanded in y around inf

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

      1. /-lowering-/.f64 81.0

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

   5. Simplified 81.0%

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

      1. associate-*r/ N/A

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

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

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

      3. *-commutative N/A

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

      4. *-lowering-*.f64 99.9

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

   7. Applied egg-rr 99.9%

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

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y}{z} + \frac{t}{z + -1} \leq -\infty:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{elif}\;\frac{y}{z} + \frac{t}{z + -1} \leq 3 \cdot 10^{+288}:\\ \;\;\;\;\left(\frac{y}{z} + \frac{t}{z + -1}\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 95.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \frac{y + t}{z}\\ \mathbf{if}\;z \leq -1.25 \cdot 10^{-18}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 0.00023:\\ \;\;\;\;\frac{x \cdot \left(y - z \cdot t\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* x (/ (+ y t) z))))
   (if (<= z -1.25e-18)
     t_1
     (if (<= z 0.00023) (/ (* x (- y (* z t))) z) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x * ((y + t) / z);
	double tmp;
	if (z <= -1.25e-18) {
		tmp = t_1;
	} else if (z <= 0.00023) {
		tmp = (x * (y - (z * 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 = x * ((y + t) / z)
    if (z <= (-1.25d-18)) then
        tmp = t_1
    else if (z <= 0.00023d0) then
        tmp = (x * (y - (z * 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 = x * ((y + t) / z);
	double tmp;
	if (z <= -1.25e-18) {
		tmp = t_1;
	} else if (z <= 0.00023) {
		tmp = (x * (y - (z * t))) / z;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = x * ((y + t) / z)
	tmp = 0
	if z <= -1.25e-18:
		tmp = t_1
	elif z <= 0.00023:
		tmp = (x * (y - (z * t))) / z
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(x * Float64(Float64(y + t) / z))
	tmp = 0.0
	if (z <= -1.25e-18)
		tmp = t_1;
	elseif (z <= 0.00023)
		tmp = Float64(Float64(x * Float64(y - Float64(z * t))) / z);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = x * ((y + t) / z);
	tmp = 0.0;
	if (z <= -1.25e-18)
		tmp = t_1;
	elseif (z <= 0.00023)
		tmp = (x * (y - (z * t))) / 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 + t), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -1.25e-18], t$95$1, If[LessEqual[z, 0.00023], N[(N[(x * N[(y - N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.25000000000000009e-18 or 2.3000000000000001e-4 < z

   1. Initial program 97.4%

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

   3. Taylor expanded in z around inf

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

      1. /-lowering-/.f64 N/A

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

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

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

      3. metadata-eval N/A

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

      4. *-lft-identity N/A

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

      5. +-commutative N/A

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

      6. +-lowering-+.f64 96.4

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

   5. Simplified 96.4%

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

   if -1.25000000000000009e-18 < z < 2.3000000000000001e-4

   1. Initial program 92.7%

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

      1. sub-neg N/A

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

      2. distribute-rgt-in N/A

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

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

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

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

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

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

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

      6. distribute-neg-frac2 N/A

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

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

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

      8. sub-neg N/A

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

      9. distribute-neg-in N/A

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

      10. metadata-eval N/A

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

      11. remove-double-neg N/A

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

      12. +-lowering-+.f64 90.4

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

   4. Applied egg-rr 90.4%

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

   5. Taylor expanded in z around 0

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

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

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

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

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

      11. *-lowering-*.f64 96.0

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

   7. Simplified 96.0%

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

4. Final simplification 96.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.25 \cdot 10^{-18}:\\ \;\;\;\;x \cdot \frac{y + t}{z}\\ \mathbf{elif}\;z \leq 0.00023:\\ \;\;\;\;\frac{x \cdot \left(y - z \cdot t\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y + t}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 93.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \frac{y + t}{z}\\ \mathbf{if}\;z \leq -1:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 8.5 \cdot 10^{-11}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* x (/ (+ y t) z))))
   (if (<= z -1.0) t_1 (if (<= z 8.5e-11) (* x (- (/ y z) t)) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x * ((y + t) / z);
	double tmp;
	if (z <= -1.0) {
		tmp = t_1;
	} else if (z <= 8.5e-11) {
		tmp = x * ((y / z) - 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 + t) / z)
    if (z <= (-1.0d0)) then
        tmp = t_1
    else if (z <= 8.5d-11) then
        tmp = x * ((y / z) - 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 + t) / z);
	double tmp;
	if (z <= -1.0) {
		tmp = t_1;
	} else if (z <= 8.5e-11) {
		tmp = x * ((y / z) - t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = x * ((y + t) / z)
	tmp = 0
	if z <= -1.0:
		tmp = t_1
	elif z <= 8.5e-11:
		tmp = x * ((y / z) - t)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(x * Float64(Float64(y + t) / z))
	tmp = 0.0
	if (z <= -1.0)
		tmp = t_1;
	elseif (z <= 8.5e-11)
		tmp = Float64(x * Float64(Float64(y / z) - t));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = x * ((y + t) / z);
	tmp = 0.0;
	if (z <= -1.0)
		tmp = t_1;
	elseif (z <= 8.5e-11)
		tmp = x * ((y / z) - t);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x * N[(N[(y + t), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -1.0], t$95$1, If[LessEqual[z, 8.5e-11], N[(x * N[(N[(y / z), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -1 or 8.50000000000000037e-11 < z

   1. Initial program 97.3%

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

   3. Taylor expanded in z around inf

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

      1. /-lowering-/.f64 N/A

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

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

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

      3. metadata-eval N/A

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

      4. *-lft-identity N/A

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

      5. +-commutative N/A

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

      6. +-lowering-+.f64 96.3

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

   5. Simplified 96.3%

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

   if -1 < z < 8.50000000000000037e-11

   1. Initial program 92.9%

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

   3. Taylor expanded in z around 0

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

   5. Simplified 92.8%

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

4. Final simplification 94.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1:\\ \;\;\;\;x \cdot \frac{y + t}{z}\\ \mathbf{elif}\;z \leq 8.5 \cdot 10^{-11}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - t\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y + t}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 76.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \frac{t}{z + -1}\\ \mathbf{if}\;t \leq -2550000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 4 \cdot 10^{+55}:\\ \;\;\;\;\frac{y}{z} \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* x (/ t (+ z -1.0)))))
   (if (<= t -2550000.0) t_1 (if (<= t 4e+55) (* (/ y z) x) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x * (t / (z + -1.0));
	double tmp;
	if (t <= -2550000.0) {
		tmp = t_1;
	} else if (t <= 4e+55) {
		tmp = (y / 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 = x * (t / (z + (-1.0d0)))
    if (t <= (-2550000.0d0)) then
        tmp = t_1
    else if (t <= 4d+55) then
        tmp = (y / 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 = x * (t / (z + -1.0));
	double tmp;
	if (t <= -2550000.0) {
		tmp = t_1;
	} else if (t <= 4e+55) {
		tmp = (y / z) * x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = x * (t / (z + -1.0))
	tmp = 0
	if t <= -2550000.0:
		tmp = t_1
	elif t <= 4e+55:
		tmp = (y / z) * x
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(x * Float64(t / Float64(z + -1.0)))
	tmp = 0.0
	if (t <= -2550000.0)
		tmp = t_1;
	elseif (t <= 4e+55)
		tmp = Float64(Float64(y / z) * x);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = x * (t / (z + -1.0));
	tmp = 0.0;
	if (t <= -2550000.0)
		tmp = t_1;
	elseif (t <= 4e+55)
		tmp = (y / z) * x;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x * N[(t / N[(z + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -2550000.0], t$95$1, If[LessEqual[t, 4e+55], N[(N[(y / z), $MachinePrecision] * x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -2.55e6 or 4.00000000000000004e55 < t

   1. Initial program 94.3%

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

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

      7. metadata-eval N/A

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

      8. mul-1-neg N/A

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

      9. remove-double-neg N/A

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

      10. +-lowering-+.f64 69.8

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

   5. Simplified 69.8%

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

   if -2.55e6 < t < 4.00000000000000004e55

   1. Initial program 95.5%

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

   3. Taylor expanded in y around inf

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

      1. /-lowering-/.f64 90.1

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

   5. Simplified 90.1%

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

4. Final simplification 80.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2550000:\\ \;\;\;\;x \cdot \frac{t}{z + -1}\\ \mathbf{elif}\;t \leq 4 \cdot 10^{+55}:\\ \;\;\;\;\frac{y}{z} \cdot x\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{t}{z + -1}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 74.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1050000000000:\\ \;\;\;\;x \cdot \frac{t}{z}\\ \mathbf{elif}\;z \leq 8.5 \cdot 10^{-11}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - t\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z} \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -1050000000000.0)
   (* x (/ t z))
   (if (<= z 8.5e-11) (* x (- (/ y z) t)) (* (/ y z) x))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -1050000000000.0) {
		tmp = x * (t / z);
	} else if (z <= 8.5e-11) {
		tmp = x * ((y / z) - t);
	} else {
		tmp = (y / z) * x;
	}
	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 <= (-1050000000000.0d0)) then
        tmp = x * (t / z)
    else if (z <= 8.5d-11) then
        tmp = x * ((y / z) - t)
    else
        tmp = (y / z) * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -1050000000000.0) {
		tmp = x * (t / z);
	} else if (z <= 8.5e-11) {
		tmp = x * ((y / z) - t);
	} else {
		tmp = (y / z) * x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= -1050000000000.0:
		tmp = x * (t / z)
	elif z <= 8.5e-11:
		tmp = x * ((y / z) - t)
	else:
		tmp = (y / z) * x
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -1050000000000.0)
		tmp = Float64(x * Float64(t / z));
	elseif (z <= 8.5e-11)
		tmp = Float64(x * Float64(Float64(y / z) - t));
	else
		tmp = Float64(Float64(y / z) * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= -1050000000000.0)
		tmp = x * (t / z);
	elseif (z <= 8.5e-11)
		tmp = x * ((y / z) - t);
	else
		tmp = (y / z) * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -1050000000000.0], N[(x * N[(t / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 8.5e-11], N[(x * N[(N[(y / z), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision], N[(N[(y / z), $MachinePrecision] * x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.05e12

   1. Initial program 96.3%

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

   3. Taylor expanded in z around inf

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

      1. /-lowering-/.f64 N/A

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

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

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

      3. metadata-eval N/A

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

      4. *-lft-identity N/A

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

      5. +-commutative N/A

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

      6. +-lowering-+.f64 96.0

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

   5. Simplified 96.0%

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

   6. Taylor expanded in t around inf

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

      1. /-lowering-/.f64 55.4

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

   8. Simplified 55.4%

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

   if -1.05e12 < z < 8.50000000000000037e-11

   1. Initial program 92.9%

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

   3. Taylor expanded in z around 0

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

   5. Simplified 92.8%

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

   if 8.50000000000000037e-11 < z

   1. Initial program 98.2%

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

   3. Taylor expanded in y around inf

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

      1. /-lowering-/.f64 65.9

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

   5. Simplified 65.9%

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

4. Final simplification 77.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1050000000000:\\ \;\;\;\;x \cdot \frac{t}{z}\\ \mathbf{elif}\;z \leq 8.5 \cdot 10^{-11}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - t\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z} \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 68.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \frac{t}{z}\\ \mathbf{if}\;t \leq -1.6 \cdot 10^{+194}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 3.8 \cdot 10^{+78}:\\ \;\;\;\;\frac{y}{z} \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* x (/ t z))))
   (if (<= t -1.6e+194) t_1 (if (<= t 3.8e+78) (* (/ y z) x) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x * (t / z);
	double tmp;
	if (t <= -1.6e+194) {
		tmp = t_1;
	} else if (t <= 3.8e+78) {
		tmp = (y / 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 = x * (t / z)
    if (t <= (-1.6d+194)) then
        tmp = t_1
    else if (t <= 3.8d+78) then
        tmp = (y / 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 = x * (t / z);
	double tmp;
	if (t <= -1.6e+194) {
		tmp = t_1;
	} else if (t <= 3.8e+78) {
		tmp = (y / z) * x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = x * (t / z)
	tmp = 0
	if t <= -1.6e+194:
		tmp = t_1
	elif t <= 3.8e+78:
		tmp = (y / z) * x
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(x * Float64(t / z))
	tmp = 0.0
	if (t <= -1.6e+194)
		tmp = t_1;
	elseif (t <= 3.8e+78)
		tmp = Float64(Float64(y / z) * x);
	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 <= -1.6e+194)
		tmp = t_1;
	elseif (t <= 3.8e+78)
		tmp = (y / z) * x;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x * N[(t / z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -1.6e+194], t$95$1, If[LessEqual[t, 3.8e+78], N[(N[(y / z), $MachinePrecision] * x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -1.60000000000000011e194 or 3.7999999999999999e78 < t

   1. Initial program 95.0%

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

   3. Taylor expanded in z around inf

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

      1. /-lowering-/.f64 N/A

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

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

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

      3. metadata-eval N/A

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

      4. *-lft-identity N/A

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

      5. +-commutative N/A

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

      6. +-lowering-+.f64 69.7

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

   5. Simplified 69.7%

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

   6. Taylor expanded in t around inf

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

      1. /-lowering-/.f64 58.7

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

   8. Simplified 58.7%

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

   if -1.60000000000000011e194 < t < 3.7999999999999999e78

   1. Initial program 94.9%

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

   3. Taylor expanded in y around inf

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

      1. /-lowering-/.f64 77.9

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

   5. Simplified 77.9%

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

4. Final simplification 71.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.6 \cdot 10^{+194}:\\ \;\;\;\;x \cdot \frac{t}{z}\\ \mathbf{elif}\;t \leq 3.8 \cdot 10^{+78}:\\ \;\;\;\;\frac{y}{z} \cdot x\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{t}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 44.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \frac{t}{z}\\ \mathbf{if}\;z \leq -0.75:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1:\\ \;\;\;\;t \cdot \left(x \cdot \left(-1 - z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* x (/ t z))))
   (if (<= z -0.75) t_1 (if (<= z 1.0) (* t (* x (- -1.0 z))) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x * (t / z);
	double tmp;
	if (z <= -0.75) {
		tmp = t_1;
	} else if (z <= 1.0) {
		tmp = 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) :: tmp
    t_1 = x * (t / z)
    if (z <= (-0.75d0)) then
        tmp = t_1
    else if (z <= 1.0d0) then
        tmp = 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 * (t / z);
	double tmp;
	if (z <= -0.75) {
		tmp = t_1;
	} else if (z <= 1.0) {
		tmp = t * (x * (-1.0 - z));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = x * (t / z)
	tmp = 0
	if z <= -0.75:
		tmp = t_1
	elif z <= 1.0:
		tmp = t * (x * (-1.0 - z))
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(x * Float64(t / z))
	tmp = 0.0
	if (z <= -0.75)
		tmp = t_1;
	elseif (z <= 1.0)
		tmp = Float64(t * Float64(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 * (t / z);
	tmp = 0.0;
	if (z <= -0.75)
		tmp = t_1;
	elseif (z <= 1.0)
		tmp = 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[(t / z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -0.75], t$95$1, If[LessEqual[z, 1.0], N[(t * N[(x * N[(-1.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -0.75 or 1 < z

   1. Initial program 97.2%

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

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

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

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

      3. metadata-eval N/A

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

      4. *-lft-identity N/A

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

      5. +-commutative N/A

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

      6. +-lowering-+.f64 96.3

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

   5. Simplified 96.3%

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

   6. Taylor expanded in t around inf

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

      1. /-lowering-/.f64 54.0

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

   8. Simplified 54.0%

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

   if -0.75 < z < 1

   1. Initial program 93.0%

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

      1. clear-num N/A

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

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

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

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

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

      4. --lowering--.f64 93.0

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

   4. Applied egg-rr 93.0%

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

   5. Taylor expanded in y around 0

      \[\leadsto x \cdot \color{blue}{\left(-1 \cdot \frac{t}{1 - z}\right)} \]
   6. 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. *-lft-identity N/A

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

      3. associate-*l/ N/A

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

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

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

      5. mul-1-neg N/A

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

      6. sub-neg N/A

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

      7. metadata-eval N/A

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

      8. distribute-neg-in N/A

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

      9. +-commutative N/A

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

      10. metadata-eval N/A

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

      11. sub-neg N/A

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

      12. distribute-neg-frac2 N/A

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

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

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

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

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

      15. mul-1-neg N/A

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

      16. remove-double-neg N/A

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

      17. associate-*l/ N/A

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

      18. *-lft-identity N/A

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

      19. /-lowering-/.f64 N/A

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

      20. sub-neg N/A

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

      21. metadata-eval N/A

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

      22. +-lowering-+.f64 33.8

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

   7. Simplified 33.8%

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

   8. Taylor expanded in z around 0

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

      1. distribute-lft-out N/A

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

      2. mul-1-neg N/A

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

      3. distribute-lft-out N/A

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

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

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

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

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

      6. distribute-neg-in N/A

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

      7. mul-1-neg N/A

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

      8. *-commutative N/A

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

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

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

      10. mul-1-neg N/A

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

      11. distribute-rgt-out N/A

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

      12. metadata-eval N/A

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

      13. mul-1-neg N/A

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

      14. distribute-neg-in N/A

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

      15. +-commutative N/A

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

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

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

      17. +-commutative N/A

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

      18. distribute-neg-in N/A

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

      19. metadata-eval N/A

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

      20. sub-neg N/A

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

      21. --lowering--.f64 33.8

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

   10. Simplified 33.8%

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

Alternative 8: 23.2% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ t \cdot \left(0 - x\right) \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (* t (- 0.0 x)))

C

double code(double x, double y, double z, double t) {
	return t * (0.0 - x);
}

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 = t * (0.0d0 - x)
end function

Java

public static double code(double x, double y, double z, double t) {
	return t * (0.0 - x);
}

Python

def code(x, y, z, t):
	return t * (0.0 - x)

Julia

function code(x, y, z, t)
	return Float64(t * Float64(0.0 - x))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = t * (0.0 - x);
end

Wolfram

code[x_, y_, z_, t_] := N[(t * N[(0.0 - x), $MachinePrecision]), $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

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

5. Simplified 68.4%

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

6. Taylor expanded in y around 0

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

   1. mul-1-neg N/A

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

   2. neg-sub0 N/A

      \[\leadsto x \cdot \color{blue}{\left(0 - t\right)} \]

   3. --lowering--.f64 23.6

      \[\leadsto x \cdot \color{blue}{\left(0 - t\right)} \]

8. Simplified 23.6%

   \[\leadsto x \cdot \color{blue}{\left(0 - t\right)} \]

9. Final simplification 23.6%

   \[\leadsto t \cdot \left(0 - x\right) \]
10. Add Preprocessing

Developer Target 1: 94.8% 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 2024196 
(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)))))