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

Percentage Accurate: 94.6% → 96.5%

Time: 10.5s

Alternatives: 11

Speedup: 0.5×

• 20.8% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 94.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 96.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{y}{z} + \frac{t}{z + -1}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+307}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (+ (/ y z) (/ t (+ z -1.0)))))
   (if (<= t_1 -2e+307) (* y (/ x z)) (* t_1 x))))

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 <= -2e+307) {
		tmp = y * (x / z);
	} else {
		tmp = t_1 * 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) :: t_1
    real(8) :: tmp
    t_1 = (y / z) + (t / (z + (-1.0d0)))
    if (t_1 <= (-2d+307)) then
        tmp = y * (x / z)
    else
        tmp = t_1 * x
    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 / (z + -1.0));
	double tmp;
	if (t_1 <= -2e+307) {
		tmp = y * (x / z);
	} else {
		tmp = t_1 * x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (y / z) + (t / (z + -1.0))
	tmp = 0
	if t_1 <= -2e+307:
		tmp = y * (x / z)
	else:
		tmp = t_1 * x
	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 <= -2e+307)
		tmp = Float64(y * Float64(x / z));
	else
		tmp = Float64(t_1 * x);
	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 <= -2e+307)
		tmp = y * (x / z);
	else
		tmp = t_1 * x;
	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, -2e+307], N[(y * N[(x / z), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (/.f64 y z) (/.f64 t (-.f64 #s(literal 1 binary64) z))) < -1.99999999999999997e307

   1. Initial program 63.6%

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

   3. Taylor expanded in y around inf 99.8%

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

      1. associate-*r/ 63.6%

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

   5. Simplified 63.6%

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

      1. *-commutative 63.6%

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

      2. associate-*l/ 99.8%

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

   7. Applied egg-rr 99.8%

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

      1. associate-/l* 99.9%

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

   9. Applied egg-rr 99.9%

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

   if -1.99999999999999997e307 < (-.f64 (/.f64 y z) (/.f64 t (-.f64 #s(literal 1 binary64) z)))

   1. Initial program 97.3%

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

4. Final simplification 97.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y}{z} + \frac{t}{z + -1} \leq -2 \cdot 10^{+307}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{y}{z} + \frac{t}{z + -1}\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 93.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -6800000 \lor \neg \left(z \leq 1\right):\\ \;\;\;\;x \cdot \frac{y + t}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -6800000.0) (not (<= z 1.0)))
   (* x (/ (+ y t) z))
   (* x (- (/ y z) t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -6800000.0) || !(z <= 1.0)) {
		tmp = x * ((y + t) / z);
	} else {
		tmp = x * ((y / z) - t);
	}
	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 <= (-6800000.0d0)) .or. (.not. (z <= 1.0d0))) then
        tmp = x * ((y + t) / z)
    else
        tmp = x * ((y / z) - t)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -6800000.0) || !(z <= 1.0)) {
		tmp = x * ((y + t) / z);
	} else {
		tmp = x * ((y / z) - t);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z <= -6800000.0) or not (z <= 1.0):
		tmp = x * ((y + t) / z)
	else:
		tmp = x * ((y / z) - t)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((z <= -6800000.0) || !(z <= 1.0))
		tmp = Float64(x * Float64(Float64(y + t) / z));
	else
		tmp = Float64(x * Float64(Float64(y / z) - t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((z <= -6800000.0) || ~((z <= 1.0)))
		tmp = x * ((y + t) / z);
	else
		tmp = x * ((y / z) - t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[z, -6800000.0], N[Not[LessEqual[z, 1.0]], $MachinePrecision]], N[(x * N[(N[(y + t), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], N[(x * N[(N[(y / z), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -6.8e6 or 1 < z

   1. Initial program 95.9%

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

   3. Taylor expanded in z around inf 84.8%

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

      1. associate-/l* 95.9%

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

      2. cancel-sign-sub-inv 95.9%

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

      3. metadata-eval 95.9%

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

      4. *-lft-identity 95.9%

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

      5. +-commutative 95.9%

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

   5. Simplified 95.9%

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

   if -6.8e6 < z < 1

   1. Initial program 94.0%

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

   3. Taylor expanded in z around 0 92.6%

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

4. Final simplification 94.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -6800000 \lor \neg \left(z \leq 1\right):\\ \;\;\;\;x \cdot \frac{y + t}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 71.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.52 \cdot 10^{-18} \lor \neg \left(t \leq 4.4 \cdot 10^{-63}\right):\\ \;\;\;\;x \cdot \frac{t}{z + -1}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= t -1.52e-18) (not (<= t 4.4e-63)))
   (* x (/ t (+ z -1.0)))
   (* y (/ x z))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((t <= -1.52e-18) || !(t <= 4.4e-63)) {
		tmp = x * (t / (z + -1.0));
	} else {
		tmp = y * (x / 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 ((t <= (-1.52d-18)) .or. (.not. (t <= 4.4d-63))) then
        tmp = x * (t / (z + (-1.0d0)))
    else
        tmp = y * (x / z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((t <= -1.52e-18) || !(t <= 4.4e-63)) {
		tmp = x * (t / (z + -1.0));
	} else {
		tmp = y * (x / z);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (t <= -1.52e-18) or not (t <= 4.4e-63):
		tmp = x * (t / (z + -1.0))
	else:
		tmp = y * (x / z)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((t <= -1.52e-18) || !(t <= 4.4e-63))
		tmp = Float64(x * Float64(t / Float64(z + -1.0)));
	else
		tmp = Float64(y * Float64(x / z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((t <= -1.52e-18) || ~((t <= 4.4e-63)))
		tmp = x * (t / (z + -1.0));
	else
		tmp = y * (x / z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[t, -1.52e-18], N[Not[LessEqual[t, 4.4e-63]], $MachinePrecision]], N[(x * N[(t / N[(z + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * N[(x / z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -1.52e-18 or 4.3999999999999999e-63 < t

   1. Initial program 96.9%

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

   3. Taylor expanded in y around inf 84.6%

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

   4. Taylor expanded in y around 0 72.3%

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

      1. mul-1-neg 72.3%

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

      2. distribute-neg-frac2 72.3%

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

      3. sub-neg 72.3%

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

      4. distribute-neg-in 72.3%

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

      5. metadata-eval 72.3%

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

      6. remove-double-neg 72.3%

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

   6. Simplified 72.3%

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

   if -1.52e-18 < t < 4.3999999999999999e-63

   1. Initial program 92.7%

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

   3. Taylor expanded in y around inf 86.2%

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

      1. associate-*r/ 88.2%

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

   5. Simplified 88.2%

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

      1. *-commutative 88.2%

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

      2. associate-*l/ 86.2%

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

   7. Applied egg-rr 86.2%

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

      1. associate-/l* 89.5%

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

   9. Applied egg-rr 89.5%

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

4. Final simplification 80.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.52 \cdot 10^{-18} \lor \neg \left(t \leq 4.4 \cdot 10^{-63}\right):\\ \;\;\;\;x \cdot \frac{t}{z + -1}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 68.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -6.7 \cdot 10^{-18} \lor \neg \left(t \leq 4.4 \cdot 10^{-63}\right):\\ \;\;\;\;t \cdot \frac{x}{z + -1}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= t -6.7e-18) (not (<= t 4.4e-63)))
   (* t (/ x (+ z -1.0)))
   (* y (/ x z))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((t <= -6.7e-18) || !(t <= 4.4e-63)) {
		tmp = t * (x / (z + -1.0));
	} else {
		tmp = y * (x / 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 ((t <= (-6.7d-18)) .or. (.not. (t <= 4.4d-63))) then
        tmp = t * (x / (z + (-1.0d0)))
    else
        tmp = y * (x / z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((t <= -6.7e-18) || !(t <= 4.4e-63)) {
		tmp = t * (x / (z + -1.0));
	} else {
		tmp = y * (x / z);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (t <= -6.7e-18) or not (t <= 4.4e-63):
		tmp = t * (x / (z + -1.0))
	else:
		tmp = y * (x / z)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((t <= -6.7e-18) || !(t <= 4.4e-63))
		tmp = Float64(t * Float64(x / Float64(z + -1.0)));
	else
		tmp = Float64(y * Float64(x / z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((t <= -6.7e-18) || ~((t <= 4.4e-63)))
		tmp = t * (x / (z + -1.0));
	else
		tmp = y * (x / z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[t, -6.7e-18], N[Not[LessEqual[t, 4.4e-63]], $MachinePrecision]], N[(t * N[(x / N[(z + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * N[(x / z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -6.6999999999999998e-18 or 4.3999999999999999e-63 < t

   1. Initial program 96.9%

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

   3. Taylor expanded in y around 0 63.9%

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

      1. mul-1-neg 63.9%

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

      2. associate-/l* 68.6%

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

      3. distribute-rgt-neg-in 68.6%

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

      4. distribute-neg-frac2 68.6%

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

      5. neg-sub0 68.6%

         \[\leadsto t \cdot \frac{x}{\color{blue}{0 - \left(1 - z\right)}} \]

      6. associate--r- 68.6%

         \[\leadsto t \cdot \frac{x}{\color{blue}{\left(0 - 1\right) + z}} \]

      7. metadata-eval 68.6%

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

   5. Simplified 68.6%

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

   if -6.6999999999999998e-18 < t < 4.3999999999999999e-63

   1. Initial program 92.7%

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

   3. Taylor expanded in y around inf 86.2%

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

      1. associate-*r/ 88.2%

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

   5. Simplified 88.2%

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

      1. *-commutative 88.2%

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

      2. associate-*l/ 86.2%

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

   7. Applied egg-rr 86.2%

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

      1. associate-/l* 89.5%

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

   9. Applied egg-rr 89.5%

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

4. Final simplification 78.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -6.7 \cdot 10^{-18} \lor \neg \left(t \leq 4.4 \cdot 10^{-63}\right):\\ \;\;\;\;t \cdot \frac{x}{z + -1}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 74.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.05 \cdot 10^{+18}:\\ \;\;\;\;\frac{x}{\frac{z}{t}}\\ \mathbf{elif}\;z \leq 2.25 \cdot 10^{+21}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - t\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{t}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -2.05e+18)
   (/ x (/ z t))
   (if (<= z 2.25e+21) (* x (- (/ y z) t)) (* x (/ t z)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -2.05e+18) {
		tmp = x / (z / t);
	} else if (z <= 2.25e+21) {
		tmp = x * ((y / z) - t);
	} 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 <= (-2.05d+18)) then
        tmp = x / (z / t)
    else if (z <= 2.25d+21) then
        tmp = x * ((y / z) - t)
    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 <= -2.05e+18) {
		tmp = x / (z / t);
	} else if (z <= 2.25e+21) {
		tmp = x * ((y / z) - t);
	} else {
		tmp = x * (t / z);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= -2.05e+18:
		tmp = x / (z / t)
	elif z <= 2.25e+21:
		tmp = x * ((y / z) - t)
	else:
		tmp = x * (t / z)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -2.05e+18)
		tmp = Float64(x / Float64(z / t));
	elseif (z <= 2.25e+21)
		tmp = Float64(x * Float64(Float64(y / z) - t));
	else
		tmp = Float64(x * Float64(t / z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= -2.05e+18)
		tmp = x / (z / t);
	elseif (z <= 2.25e+21)
		tmp = x * ((y / z) - t);
	else
		tmp = x * (t / z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -2.05e+18], N[(x / N[(z / t), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 2.25e+21], N[(x * N[(N[(y / z), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision], N[(x * N[(t / z), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -2.05e18

   1. Initial program 96.7%

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

      1. clear-num 96.6%

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

      2. associate-/r/ 96.6%

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

   4. Applied egg-rr 96.6%

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

   5. Taylor expanded in z around inf 88.3%

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

      1. associate-/l* 96.7%

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

      2. sub-neg 96.7%

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

      3. mul-1-neg 96.7%

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

      4. remove-double-neg 96.7%

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

   7. Simplified 96.7%

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

      1. clear-num 96.5%

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

      2. un-div-inv 96.7%

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

   9. Applied egg-rr 96.7%

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

   10. Taylor expanded in y around 0 67.4%

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

   if -2.05e18 < z < 2.25e21

   1. Initial program 94.0%

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

   3. Taylor expanded in z around 0 92.6%

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

   if 2.25e21 < z

   1. Initial program 95.0%

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

      1. clear-num 94.9%

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

      2. associate-/r/ 94.9%

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

   4. Applied egg-rr 94.9%

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

   5. Taylor expanded in z around inf 80.7%

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

      1. associate-/l* 95.0%

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

      2. sub-neg 95.0%

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

      3. mul-1-neg 95.0%

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

      4. remove-double-neg 95.0%

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

   7. Simplified 95.0%

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

   8. Taylor expanded in y around 0 61.6%

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

Alternative 6: 67.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -7.2 \cdot 10^{+65} \lor \neg \left(t \leq 1.05 \cdot 10^{+141}\right):\\ \;\;\;\;x \cdot \frac{t}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= t -7.2e+65) (not (<= t 1.05e+141))) (* x (/ t z)) (* y (/ x z))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((t <= -7.2e+65) || !(t <= 1.05e+141)) {
		tmp = x * (t / z);
	} else {
		tmp = y * (x / 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 ((t <= (-7.2d+65)) .or. (.not. (t <= 1.05d+141))) then
        tmp = x * (t / z)
    else
        tmp = y * (x / z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((t <= -7.2e+65) || !(t <= 1.05e+141)) {
		tmp = x * (t / z);
	} else {
		tmp = y * (x / z);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (t <= -7.2e+65) or not (t <= 1.05e+141):
		tmp = x * (t / z)
	else:
		tmp = y * (x / z)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((t <= -7.2e+65) || !(t <= 1.05e+141))
		tmp = Float64(x * Float64(t / z));
	else
		tmp = Float64(y * Float64(x / z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((t <= -7.2e+65) || ~((t <= 1.05e+141)))
		tmp = x * (t / z);
	else
		tmp = y * (x / z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[t, -7.2e+65], N[Not[LessEqual[t, 1.05e+141]], $MachinePrecision]], N[(x * N[(t / z), $MachinePrecision]), $MachinePrecision], N[(y * N[(x / z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -7.19999999999999957e65 or 1.0499999999999999e141 < t

   1. Initial program 97.3%

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

      1. clear-num 97.3%

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

      2. associate-/r/ 97.3%

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

   4. Applied egg-rr 97.3%

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

   5. Taylor expanded in z around inf 54.1%

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

      1. associate-/l* 66.3%

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

      2. sub-neg 66.3%

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

      3. mul-1-neg 66.3%

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

      4. remove-double-neg 66.3%

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

   7. Simplified 66.3%

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

   8. Taylor expanded in y around 0 57.3%

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

   if -7.19999999999999957e65 < t < 1.0499999999999999e141

   1. Initial program 93.9%

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

   3. Taylor expanded in y around inf 72.1%

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

      1. associate-*r/ 74.6%

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

   5. Simplified 74.6%

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

      1. *-commutative 74.6%

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

      2. associate-*l/ 72.1%

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

   7. Applied egg-rr 72.1%

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

      1. associate-/l* 75.5%

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

   9. Applied egg-rr 75.5%

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

4. Final simplification 70.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -7.2 \cdot 10^{+65} \lor \neg \left(t \leq 1.05 \cdot 10^{+141}\right):\\ \;\;\;\;x \cdot \frac{t}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 68.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -4.8 \cdot 10^{+65} \lor \neg \left(t \leq 3.9 \cdot 10^{+43}\right):\\ \;\;\;\;x \cdot \frac{t}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z} \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= t -4.8e+65) (not (<= t 3.9e+43))) (* x (/ t z)) (* (/ y z) x)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((t <= -4.8e+65) || !(t <= 3.9e+43)) {
		tmp = x * (t / z);
	} 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 ((t <= (-4.8d+65)) .or. (.not. (t <= 3.9d+43))) then
        tmp = x * (t / z)
    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 ((t <= -4.8e+65) || !(t <= 3.9e+43)) {
		tmp = x * (t / z);
	} else {
		tmp = (y / z) * x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (t <= -4.8e+65) or not (t <= 3.9e+43):
		tmp = x * (t / z)
	else:
		tmp = (y / z) * x
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((t <= -4.8e+65) || !(t <= 3.9e+43))
		tmp = Float64(x * Float64(t / z));
	else
		tmp = Float64(Float64(y / z) * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((t <= -4.8e+65) || ~((t <= 3.9e+43)))
		tmp = x * (t / z);
	else
		tmp = (y / z) * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[t, -4.8e+65], N[Not[LessEqual[t, 3.9e+43]], $MachinePrecision]], N[(x * N[(t / z), $MachinePrecision]), $MachinePrecision], N[(N[(y / z), $MachinePrecision] * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -4.8000000000000003e65 or 3.9000000000000001e43 < t

   1. Initial program 96.9%

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

      1. clear-num 96.9%

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

      2. associate-/r/ 96.9%

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

   4. Applied egg-rr 96.9%

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

   5. Taylor expanded in z around inf 57.7%

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

      1. associate-/l* 66.2%

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

      2. sub-neg 66.2%

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

      3. mul-1-neg 66.2%

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

      4. remove-double-neg 66.2%

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

   7. Simplified 66.2%

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

   8. Taylor expanded in y around 0 54.3%

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

   if -4.8000000000000003e65 < t < 3.9000000000000001e43

   1. Initial program 93.7%

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

   3. Taylor expanded in y around inf 76.6%

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

      1. associate-*r/ 79.4%

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

   5. Simplified 79.4%

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

4. Final simplification 69.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -4.8 \cdot 10^{+65} \lor \neg \left(t \leq 3.9 \cdot 10^{+43}\right):\\ \;\;\;\;x \cdot \frac{t}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z} \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 44.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -6800000 \lor \neg \left(z \leq 2.6 \cdot 10^{+20}\right):\\ \;\;\;\;x \cdot \frac{t}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(-t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -6800000.0) (not (<= z 2.6e+20))) (* x (/ t z)) (* x (- t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -6800000.0) || !(z <= 2.6e+20)) {
		tmp = x * (t / z);
	} else {
		tmp = x * -t;
	}
	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 <= (-6800000.0d0)) .or. (.not. (z <= 2.6d+20))) then
        tmp = x * (t / z)
    else
        tmp = x * -t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -6800000.0) || !(z <= 2.6e+20)) {
		tmp = x * (t / z);
	} else {
		tmp = x * -t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z <= -6800000.0) or not (z <= 2.6e+20):
		tmp = x * (t / z)
	else:
		tmp = x * -t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((z <= -6800000.0) || !(z <= 2.6e+20))
		tmp = Float64(x * Float64(t / z));
	else
		tmp = Float64(x * Float64(-t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((z <= -6800000.0) || ~((z <= 2.6e+20)))
		tmp = x * (t / z);
	else
		tmp = x * -t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[z, -6800000.0], N[Not[LessEqual[z, 2.6e+20]], $MachinePrecision]], N[(x * N[(t / z), $MachinePrecision]), $MachinePrecision], N[(x * (-t)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -6.8e6 or 2.6e20 < z

   1. Initial program 95.9%

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

      1. clear-num 95.8%

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

      2. associate-/r/ 95.8%

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

   4. Applied egg-rr 95.8%

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

   5. Taylor expanded in z around inf 84.7%

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

      1. associate-/l* 95.9%

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

      2. sub-neg 95.9%

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

      3. mul-1-neg 95.9%

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

      4. remove-double-neg 95.9%

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

   7. Simplified 95.9%

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

   8. Taylor expanded in y around 0 64.6%

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

   if -6.8e6 < z < 2.6e20

   1. Initial program 94.0%

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

   3. Taylor expanded in z around 0 92.6%

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

   4. Taylor expanded in y around 0 34.0%

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

      1. associate-*r* 34.0%

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

      2. mul-1-neg 34.0%

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

   6. Simplified 34.0%

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

4. Final simplification 49.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -6800000 \lor \neg \left(z \leq 2.6 \cdot 10^{+20}\right):\\ \;\;\;\;x \cdot \frac{t}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(-t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 42.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -6800000 \lor \neg \left(z \leq 2.6 \cdot 10^{+20}\right):\\ \;\;\;\;t \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(-t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -6800000.0) (not (<= z 2.6e+20))) (* t (/ x z)) (* x (- t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -6800000.0) || !(z <= 2.6e+20)) {
		tmp = t * (x / z);
	} else {
		tmp = x * -t;
	}
	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 <= (-6800000.0d0)) .or. (.not. (z <= 2.6d+20))) then
        tmp = t * (x / z)
    else
        tmp = x * -t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -6800000.0) || !(z <= 2.6e+20)) {
		tmp = t * (x / z);
	} else {
		tmp = x * -t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z <= -6800000.0) or not (z <= 2.6e+20):
		tmp = t * (x / z)
	else:
		tmp = x * -t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((z <= -6800000.0) || !(z <= 2.6e+20))
		tmp = Float64(t * Float64(x / z));
	else
		tmp = Float64(x * Float64(-t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((z <= -6800000.0) || ~((z <= 2.6e+20)))
		tmp = t * (x / z);
	else
		tmp = x * -t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[z, -6800000.0], N[Not[LessEqual[z, 2.6e+20]], $MachinePrecision]], N[(t * N[(x / z), $MachinePrecision]), $MachinePrecision], N[(x * (-t)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -6.8e6 or 2.6e20 < z

   1. Initial program 95.9%

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

      1. clear-num 95.8%

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

      2. associate-/r/ 95.8%

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

   4. Applied egg-rr 95.8%

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

   5. Taylor expanded in z around inf 84.7%

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

      1. associate-/l* 95.9%

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

      2. sub-neg 95.9%

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

      3. mul-1-neg 95.9%

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

      4. remove-double-neg 95.9%

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

   7. Simplified 95.9%

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

   8. Taylor expanded in y around 0 56.3%

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

      1. associate-/l* 61.4%

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

   10. Simplified 61.4%

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

   if -6.8e6 < z < 2.6e20

   1. Initial program 94.0%

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

   3. Taylor expanded in z around 0 92.6%

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

   4. Taylor expanded in y around 0 34.0%

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

      1. associate-*r* 34.0%

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

      2. mul-1-neg 34.0%

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

   6. Simplified 34.0%

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

4. Final simplification 47.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -6800000 \lor \neg \left(z \leq 2.6 \cdot 10^{+20}\right):\\ \;\;\;\;t \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(-t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 67.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -3.7 \cdot 10^{+65}:\\ \;\;\;\;x \cdot \frac{t}{z}\\ \mathbf{elif}\;t \leq 6.8 \cdot 10^{+140}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{t}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t -3.7e+65)
   (* x (/ t z))
   (if (<= t 6.8e+140) (* y (/ x z)) (/ x (/ z t)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -3.7e+65) {
		tmp = x * (t / z);
	} else if (t <= 6.8e+140) {
		tmp = y * (x / z);
	} else {
		tmp = x / (z / t);
	}
	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 (t <= (-3.7d+65)) then
        tmp = x * (t / z)
    else if (t <= 6.8d+140) then
        tmp = y * (x / z)
    else
        tmp = x / (z / t)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -3.7e+65) {
		tmp = x * (t / z);
	} else if (t <= 6.8e+140) {
		tmp = y * (x / z);
	} else {
		tmp = x / (z / t);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if t <= -3.7e+65:
		tmp = x * (t / z)
	elif t <= 6.8e+140:
		tmp = y * (x / z)
	else:
		tmp = x / (z / t)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= -3.7e+65)
		tmp = Float64(x * Float64(t / z));
	elseif (t <= 6.8e+140)
		tmp = Float64(y * Float64(x / z));
	else
		tmp = Float64(x / Float64(z / t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (t <= -3.7e+65)
		tmp = x * (t / z);
	elseif (t <= 6.8e+140)
		tmp = y * (x / z);
	else
		tmp = x / (z / t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, -3.7e+65], N[(x * N[(t / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 6.8e+140], N[(y * N[(x / z), $MachinePrecision]), $MachinePrecision], N[(x / N[(z / t), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -3.69999999999999995e65

   1. Initial program 98.0%

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

      1. clear-num 98.0%

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

      2. associate-/r/ 98.1%

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

   4. Applied egg-rr 98.1%

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

   5. Taylor expanded in z around inf 50.4%

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

      1. associate-/l* 64.6%

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

      2. sub-neg 64.6%

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

      3. mul-1-neg 64.6%

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

      4. remove-double-neg 64.6%

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

   7. Simplified 64.6%

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

   8. Taylor expanded in y around 0 55.3%

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

   if -3.69999999999999995e65 < t < 6.8e140

   1. Initial program 93.9%

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

   3. Taylor expanded in y around inf 72.1%

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

      1. associate-*r/ 74.6%

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

   5. Simplified 74.6%

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

      1. *-commutative 74.6%

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

      2. associate-*l/ 72.1%

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

   7. Applied egg-rr 72.1%

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

      1. associate-/l* 75.5%

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

   9. Applied egg-rr 75.5%

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

   if 6.8e140 < t

   1. Initial program 95.6%

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

      1. clear-num 95.6%

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

      2. associate-/r/ 95.6%

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

   4. Applied egg-rr 95.6%

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

   5. Taylor expanded in z around inf 62.6%

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

      1. associate-/l* 70.4%

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

      2. sub-neg 70.4%

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

      3. mul-1-neg 70.4%

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

      4. remove-double-neg 70.4%

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

   7. Simplified 70.4%

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

      1. clear-num 70.3%

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

      2. un-div-inv 70.6%

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

   9. Applied egg-rr 70.6%

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

   10. Taylor expanded in y around 0 62.0%

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

Alternative 11: 23.5% accurate, 2.8× speedup

Math

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

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(x * Float64(-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 64.4%

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

4. Taylor expanded in y around 0 25.0%

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

   1. associate-*r* 25.0%

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

   2. mul-1-neg 25.0%

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

6. Simplified 25.0%

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

7. Final simplification 25.0%

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

Developer Target 1: 95.3% accurate, 0.2× 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 2024180 
(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)))))