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

Percentage Accurate: 94.3% → 91.4%

Time: 8.8s

Alternatives: 12

Speedup: 1.0×

• 20.2% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 94.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 91.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.6 \cdot 10^{-10}:\\ \;\;\;\;x \cdot \frac{y + t}{z}\\ \mathbf{elif}\;z \leq 13:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - \left(t + z \cdot t\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y + t\right) \cdot \frac{x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -1.6e-10)
   (* x (/ (+ y t) z))
   (if (<= z 13.0) (* x (- (/ y z) (+ t (* z t)))) (* (+ y t) (/ x z)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -1.6e-10) {
		tmp = x * ((y + t) / z);
	} else if (z <= 13.0) {
		tmp = x * ((y / z) - (t + (z * t)));
	} else {
		tmp = (y + t) * (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 (z <= (-1.6d-10)) then
        tmp = x * ((y + t) / z)
    else if (z <= 13.0d0) then
        tmp = x * ((y / z) - (t + (z * t)))
    else
        tmp = (y + t) * (x / z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -1.6e-10) {
		tmp = x * ((y + t) / z);
	} else if (z <= 13.0) {
		tmp = x * ((y / z) - (t + (z * t)));
	} else {
		tmp = (y + t) * (x / z);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= -1.6e-10:
		tmp = x * ((y + t) / z)
	elif z <= 13.0:
		tmp = x * ((y / z) - (t + (z * t)))
	else:
		tmp = (y + t) * (x / z)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -1.6e-10)
		tmp = Float64(x * Float64(Float64(y + t) / z));
	elseif (z <= 13.0)
		tmp = Float64(x * Float64(Float64(y / z) - Float64(t + Float64(z * t))));
	else
		tmp = Float64(Float64(y + t) * Float64(x / z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= -1.6e-10)
		tmp = x * ((y + t) / z);
	elseif (z <= 13.0)
		tmp = x * ((y / z) - (t + (z * t)));
	else
		tmp = (y + t) * (x / z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -1.6e-10], N[(x * N[(N[(y + t), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 13.0], N[(x * N[(N[(y / z), $MachinePrecision] - N[(t + N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y + t), $MachinePrecision] * N[(x / z), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.5999999999999999e-10

   1. Initial program 99.6%

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

   3. Taylor expanded in z around inf 89.5%

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

      1. *-commutative 89.5%

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

      2. remove-double-neg 89.5%

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

      3. cancel-sign-sub-inv 89.5%

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

      4. metadata-eval 89.5%

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

      5. *-lft-identity 89.5%

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

      6. distribute-neg-out 89.5%

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

      7. neg-mul-1 89.5%

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

      8. sub-neg 89.5%

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

      9. distribute-lft-neg-in 89.5%

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

      10. *-commutative 89.5%

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

      11. distribute-neg-frac 89.5%

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

      12. associate-/l* 99.5%

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

      13. distribute-rgt-neg-in 99.5%

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

      14. distribute-neg-frac 99.5%

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

   5. Simplified 99.5%

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

   if -1.5999999999999999e-10 < z < 13

   1. Initial program 93.8%

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

   3. Taylor expanded in z around 0 93.2%

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

   if 13 < z

   1. Initial program 94.6%

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

   3. Taylor expanded in z around inf 81.0%

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

      1. *-commutative 81.0%

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

      2. associate-/l* 96.9%

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

      3. cancel-sign-sub-inv 96.9%

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

      4. metadata-eval 96.9%

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

      5. *-lft-identity 96.9%

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

      6. +-commutative 96.9%

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

   5. Simplified 96.9%

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

4. Final simplification 96.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.6 \cdot 10^{-10}:\\ \;\;\;\;x \cdot \frac{y + t}{z}\\ \mathbf{elif}\;z \leq 13:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - \left(t + z \cdot t\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y + t\right) \cdot \frac{x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 72.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4400000:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{elif}\;y \leq 9.8 \cdot 10^{-182}:\\ \;\;\;\;t \cdot \frac{x}{z + -1}\\ \mathbf{elif}\;y \leq 2.7 \cdot 10^{+118}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - t\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y -4400000.0)
   (* x (/ y z))
   (if (<= y 9.8e-182)
     (* t (/ x (+ z -1.0)))
     (if (<= y 2.7e+118) (* x (- (/ y z) t)) (/ (* x y) z)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -4400000.0) {
		tmp = x * (y / z);
	} else if (y <= 9.8e-182) {
		tmp = t * (x / (z + -1.0));
	} else if (y <= 2.7e+118) {
		tmp = x * ((y / z) - t);
	} else {
		tmp = (x * y) / 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 (y <= (-4400000.0d0)) then
        tmp = x * (y / z)
    else if (y <= 9.8d-182) then
        tmp = t * (x / (z + (-1.0d0)))
    else if (y <= 2.7d+118) then
        tmp = x * ((y / z) - t)
    else
        tmp = (x * y) / z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -4400000.0) {
		tmp = x * (y / z);
	} else if (y <= 9.8e-182) {
		tmp = t * (x / (z + -1.0));
	} else if (y <= 2.7e+118) {
		tmp = x * ((y / z) - t);
	} else {
		tmp = (x * y) / z;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if y <= -4400000.0:
		tmp = x * (y / z)
	elif y <= 9.8e-182:
		tmp = t * (x / (z + -1.0))
	elif y <= 2.7e+118:
		tmp = x * ((y / z) - t)
	else:
		tmp = (x * y) / z
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= -4400000.0)
		tmp = Float64(x * Float64(y / z));
	elseif (y <= 9.8e-182)
		tmp = Float64(t * Float64(x / Float64(z + -1.0)));
	elseif (y <= 2.7e+118)
		tmp = Float64(x * Float64(Float64(y / z) - t));
	else
		tmp = Float64(Float64(x * y) / z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (y <= -4400000.0)
		tmp = x * (y / z);
	elseif (y <= 9.8e-182)
		tmp = t * (x / (z + -1.0));
	elseif (y <= 2.7e+118)
		tmp = x * ((y / z) - t);
	else
		tmp = (x * y) / z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, -4400000.0], N[(x * N[(y / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 9.8e-182], N[(t * N[(x / N[(z + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.7e+118], N[(x * N[(N[(y / z), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision], N[(N[(x * y), $MachinePrecision] / z), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if y < -4.4e6

   1. Initial program 94.6%

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

   3. Taylor expanded in y around inf 82.0%

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

      1. associate-*r/ 85.7%

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

   5. Simplified 85.7%

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

   if -4.4e6 < y < 9.8000000000000006e-182

   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 74.9%

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

      1. mul-1-neg 74.9%

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

      2. associate-/l* 78.7%

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

      3. distribute-rgt-neg-in 78.7%

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

      4. distribute-neg-frac2 78.7%

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

      5. neg-sub0 78.7%

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

      6. associate--r- 78.7%

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

      7. metadata-eval 78.7%

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

   5. Simplified 78.7%

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

   if 9.8000000000000006e-182 < y < 2.7e118

   1. Initial program 98.4%

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

   3. Taylor expanded in z around 0 72.2%

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

      1. mul-1-neg 72.2%

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

      2. unsub-neg 72.2%

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

      3. div-sub 72.2%

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

      4. associate-/l* 72.3%

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

      5. *-inverses 72.3%

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

      6. *-rgt-identity 72.3%

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

   5. Simplified 72.3%

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

   if 2.7e118 < y

   1. Initial program 91.4%

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

   3. Taylor expanded in y around inf 95.4%

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

      1. associate-*r/ 91.2%

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

   5. Simplified 91.2%

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

      1. associate-*r/ 95.4%

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

      2. clear-num 95.3%

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

   7. Applied egg-rr 95.3%

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

      1. associate-/r/ 95.3%

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

   9. Simplified 95.3%

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

       1. associate-*l/ 95.4%

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

       2. *-un-lft-identity 95.4%

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

       3. *-commutative 95.4%

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

   11. Applied egg-rr 95.4%

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

4. Final simplification 81.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4400000:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{elif}\;y \leq 9.8 \cdot 10^{-182}:\\ \;\;\;\;t \cdot \frac{x}{z + -1}\\ \mathbf{elif}\;y \leq 2.7 \cdot 10^{+118}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - t\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 93.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.6 \cdot 10^{-10} \lor \neg \left(z \leq 13\right):\\ \;\;\;\;x \cdot \frac{y + t}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -1.6e-10) (not (<= z 13.0)))
   (* x (/ (+ y t) z))
   (* x (- (/ y z) t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -1.6e-10) || !(z <= 13.0)) {
		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 <= (-1.6d-10)) .or. (.not. (z <= 13.0d0))) 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 <= -1.6e-10) || !(z <= 13.0)) {
		tmp = x * ((y + t) / z);
	} else {
		tmp = x * ((y / z) - t);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z <= -1.6e-10) or not (z <= 13.0):
		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 <= -1.6e-10) || !(z <= 13.0))
		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 <= -1.6e-10) || ~((z <= 13.0)))
		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, -1.6e-10], N[Not[LessEqual[z, 13.0]], $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 3 regimes

2. if z < -1.5999999999999999e-10

   1. Initial program 99.6%

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

   3. Taylor expanded in z around inf 89.5%

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

      1. *-commutative 89.5%

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

      2. remove-double-neg 89.5%

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

      3. cancel-sign-sub-inv 89.5%

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

      4. metadata-eval 89.5%

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

      5. *-lft-identity 89.5%

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

      6. distribute-neg-out 89.5%

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

      7. neg-mul-1 89.5%

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

      8. sub-neg 89.5%

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

      9. distribute-lft-neg-in 89.5%

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

      10. *-commutative 89.5%

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

      11. distribute-neg-frac 89.5%

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

      12. associate-/l* 99.5%

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

      13. distribute-rgt-neg-in 99.5%

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

      14. distribute-neg-frac 99.5%

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

   5. Simplified 99.5%

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

   if -1.5999999999999999e-10 < z < 13

   1. Initial program 93.8%

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

   3. Taylor expanded in z around 0 92.5%

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

      1. mul-1-neg 92.5%

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

      2. unsub-neg 92.5%

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

      3. div-sub 92.5%

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

      4. associate-/l* 92.6%

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

      5. *-inverses 92.6%

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

      6. *-rgt-identity 92.6%

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

   5. Simplified 92.6%

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

   if 13 < z

   1. Initial program 94.6%

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

      1. clear-num 93.6%

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

      2. associate-/r/ 94.6%

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

   4. Applied egg-rr 94.6%

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

   5. Taylor expanded in z around inf 81.0%

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

      1. associate-/l* 94.6%

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

      2. cancel-sign-sub-inv 94.6%

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

      3. metadata-eval 94.6%

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

      4. *-lft-identity 94.6%

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

   7. Simplified 94.6%

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

4. Final simplification 95.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.6 \cdot 10^{-10} \lor \neg \left(z \leq 13\right):\\ \;\;\;\;x \cdot \frac{y + t}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 75.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.75 \cdot 10^{+40} \lor \neg \left(t \leq 4500000\right):\\ \;\;\;\;x \cdot \frac{t}{z + -1}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= t -1.75e+40) (not (<= t 4500000.0)))
   (* x (/ t (+ z -1.0)))
   (* x (/ y z))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((t <= -1.75e+40) || !(t <= 4500000.0)) {
		tmp = x * (t / (z + -1.0));
	} else {
		tmp = x * (y / 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 <= (-1.75d+40)) .or. (.not. (t <= 4500000.0d0))) then
        tmp = x * (t / (z + (-1.0d0)))
    else
        tmp = x * (y / z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((t <= -1.75e+40) || !(t <= 4500000.0)) {
		tmp = x * (t / (z + -1.0));
	} else {
		tmp = x * (y / z);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (t <= -1.75e+40) or not (t <= 4500000.0):
		tmp = x * (t / (z + -1.0))
	else:
		tmp = x * (y / z)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((t <= -1.75e+40) || !(t <= 4500000.0))
		tmp = Float64(x * Float64(t / Float64(z + -1.0)));
	else
		tmp = Float64(x * Float64(y / z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((t <= -1.75e+40) || ~((t <= 4500000.0)))
		tmp = x * (t / (z + -1.0));
	else
		tmp = x * (y / z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[t, -1.75e+40], N[Not[LessEqual[t, 4500000.0]], $MachinePrecision]], N[(x * N[(t / N[(z + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(y / z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -1.75e40 or 4.5e6 < t

   1. Initial program 95.8%

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

   3. Taylor expanded in y around 0 73.3%

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

      1. mul-1-neg 73.3%

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

      2. distribute-neg-frac2 73.3%

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

      3. neg-sub0 73.3%

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

      4. associate--r- 73.3%

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

      5. metadata-eval 73.3%

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

   5. Simplified 73.3%

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

   if -1.75e40 < t < 4.5e6

   1. Initial program 95.9%

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

   3. Taylor expanded in y around inf 82.3%

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

      1. associate-*r/ 85.2%

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

   5. Simplified 85.2%

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

4. Final simplification 79.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.75 \cdot 10^{+40} \lor \neg \left(t \leq 4500000\right):\\ \;\;\;\;x \cdot \frac{t}{z + -1}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 91.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.6 \cdot 10^{-10}:\\ \;\;\;\;x \cdot \frac{y + t}{z}\\ \mathbf{elif}\;z \leq 13:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - t\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y + t\right) \cdot \frac{x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -1.6e-10)
   (* x (/ (+ y t) z))
   (if (<= z 13.0) (* x (- (/ y z) t)) (* (+ y t) (/ x z)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -1.6e-10) {
		tmp = x * ((y + t) / z);
	} else if (z <= 13.0) {
		tmp = x * ((y / z) - t);
	} else {
		tmp = (y + t) * (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 (z <= (-1.6d-10)) then
        tmp = x * ((y + t) / z)
    else if (z <= 13.0d0) then
        tmp = x * ((y / z) - t)
    else
        tmp = (y + t) * (x / z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -1.6e-10) {
		tmp = x * ((y + t) / z);
	} else if (z <= 13.0) {
		tmp = x * ((y / z) - t);
	} else {
		tmp = (y + t) * (x / z);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= -1.6e-10:
		tmp = x * ((y + t) / z)
	elif z <= 13.0:
		tmp = x * ((y / z) - t)
	else:
		tmp = (y + t) * (x / z)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -1.6e-10)
		tmp = Float64(x * Float64(Float64(y + t) / z));
	elseif (z <= 13.0)
		tmp = Float64(x * Float64(Float64(y / z) - t));
	else
		tmp = Float64(Float64(y + t) * Float64(x / z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= -1.6e-10)
		tmp = x * ((y + t) / z);
	elseif (z <= 13.0)
		tmp = x * ((y / z) - t);
	else
		tmp = (y + t) * (x / z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -1.6e-10], N[(x * N[(N[(y + t), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 13.0], N[(x * N[(N[(y / z), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision], N[(N[(y + t), $MachinePrecision] * N[(x / z), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.5999999999999999e-10

   1. Initial program 99.6%

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

   3. Taylor expanded in z around inf 89.5%

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

      1. *-commutative 89.5%

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

      2. remove-double-neg 89.5%

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

      3. cancel-sign-sub-inv 89.5%

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

      4. metadata-eval 89.5%

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

      5. *-lft-identity 89.5%

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

      6. distribute-neg-out 89.5%

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

      7. neg-mul-1 89.5%

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

      8. sub-neg 89.5%

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

      9. distribute-lft-neg-in 89.5%

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

      10. *-commutative 89.5%

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

      11. distribute-neg-frac 89.5%

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

      12. associate-/l* 99.5%

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

      13. distribute-rgt-neg-in 99.5%

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

      14. distribute-neg-frac 99.5%

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

   5. Simplified 99.5%

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

   if -1.5999999999999999e-10 < z < 13

   1. Initial program 93.8%

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

   3. Taylor expanded in z around 0 92.5%

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

      1. mul-1-neg 92.5%

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

      2. unsub-neg 92.5%

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

      3. div-sub 92.5%

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

      4. associate-/l* 92.6%

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

      5. *-inverses 92.6%

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

      6. *-rgt-identity 92.6%

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

   5. Simplified 92.6%

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

   if 13 < z

   1. Initial program 94.6%

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

   3. Taylor expanded in z around inf 81.0%

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

      1. *-commutative 81.0%

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

      2. associate-/l* 96.9%

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

      3. cancel-sign-sub-inv 96.9%

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

      4. metadata-eval 96.9%

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

      5. *-lft-identity 96.9%

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

      6. +-commutative 96.9%

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

   5. Simplified 96.9%

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

4. Final simplification 95.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.6 \cdot 10^{-10}:\\ \;\;\;\;x \cdot \frac{y + t}{z}\\ \mathbf{elif}\;z \leq 13:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - t\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y + t\right) \cdot \frac{x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 72.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -28000000000000:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{elif}\;y \leq 8.2 \cdot 10^{-141}:\\ \;\;\;\;t \cdot \frac{x}{z + -1}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y -28000000000000.0)
   (* x (/ y z))
   (if (<= y 8.2e-141) (* t (/ x (+ z -1.0))) (* y (/ x z)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -28000000000000.0) {
		tmp = x * (y / z);
	} else if (y <= 8.2e-141) {
		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 (y <= (-28000000000000.0d0)) then
        tmp = x * (y / z)
    else if (y <= 8.2d-141) 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 (y <= -28000000000000.0) {
		tmp = x * (y / z);
	} else if (y <= 8.2e-141) {
		tmp = t * (x / (z + -1.0));
	} else {
		tmp = y * (x / z);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if y <= -28000000000000.0:
		tmp = x * (y / z)
	elif y <= 8.2e-141:
		tmp = t * (x / (z + -1.0))
	else:
		tmp = y * (x / z)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= -28000000000000.0)
		tmp = Float64(x * Float64(y / z));
	elseif (y <= 8.2e-141)
		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 (y <= -28000000000000.0)
		tmp = x * (y / z);
	elseif (y <= 8.2e-141)
		tmp = t * (x / (z + -1.0));
	else
		tmp = y * (x / z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, -28000000000000.0], N[(x * N[(y / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 8.2e-141], N[(t * N[(x / N[(z + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * N[(x / z), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.8e13

   1. Initial program 94.6%

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

   3. Taylor expanded in y around inf 82.0%

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

      1. associate-*r/ 85.7%

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

   5. Simplified 85.7%

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

   if -2.8e13 < y < 8.20000000000000005e-141

   1. Initial program 97.2%

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

   3. Taylor expanded in y around 0 73.8%

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

      1. mul-1-neg 73.8%

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

      2. associate-/l* 76.2%

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

      3. distribute-rgt-neg-in 76.2%

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

      4. distribute-neg-frac2 76.2%

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

      5. neg-sub0 76.2%

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

      6. associate--r- 76.2%

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

      7. metadata-eval 76.2%

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

   5. Simplified 76.2%

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

   if 8.20000000000000005e-141 < y

   1. Initial program 95.3%

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

   3. Taylor expanded in y around inf 76.2%

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

      1. associate-*r/ 75.7%

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

   5. Simplified 75.7%

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

      1. clear-num 75.7%

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

      2. un-div-inv 75.7%

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

   7. Applied egg-rr 75.7%

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

      1. associate-/r/ 76.6%

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

   9. Simplified 76.6%

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

4. Final simplification 78.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -28000000000000:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{elif}\;y \leq 8.2 \cdot 10^{-141}:\\ \;\;\;\;t \cdot \frac{x}{z + -1}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 67.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -3.8 \cdot 10^{+46} \lor \neg \left(t \leq 1.8 \cdot 10^{+139}\right):\\ \;\;\;\;x \cdot \frac{t}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= t -3.8e+46) (not (<= t 1.8e+139))) (* x (/ t z)) (* x (/ y z))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((t <= -3.8e+46) || !(t <= 1.8e+139)) {
		tmp = x * (t / z);
	} else {
		tmp = x * (y / 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 <= (-3.8d+46)) .or. (.not. (t <= 1.8d+139))) then
        tmp = x * (t / z)
    else
        tmp = x * (y / z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((t <= -3.8e+46) || !(t <= 1.8e+139)) {
		tmp = x * (t / z);
	} else {
		tmp = x * (y / z);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (t <= -3.8e+46) or not (t <= 1.8e+139):
		tmp = x * (t / z)
	else:
		tmp = x * (y / z)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((t <= -3.8e+46) || !(t <= 1.8e+139))
		tmp = Float64(x * Float64(t / z));
	else
		tmp = Float64(x * Float64(y / z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((t <= -3.8e+46) || ~((t <= 1.8e+139)))
		tmp = x * (t / z);
	else
		tmp = x * (y / z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[t, -3.8e+46], N[Not[LessEqual[t, 1.8e+139]], $MachinePrecision]], N[(x * N[(t / z), $MachinePrecision]), $MachinePrecision], N[(x * N[(y / z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -3.7999999999999999e46 or 1.79999999999999993e139 < t

   1. Initial program 94.6%

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

   3. Taylor expanded in y around 0 75.4%

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

      1. mul-1-neg 75.4%

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

      2. distribute-neg-frac2 75.4%

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

      3. neg-sub0 75.4%

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

      4. associate--r- 75.4%

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

      5. metadata-eval 75.4%

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

   5. Simplified 75.4%

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

   6. Taylor expanded in z around inf 55.5%

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

   if -3.7999999999999999e46 < t < 1.79999999999999993e139

   1. Initial program 96.6%

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

   3. Taylor expanded in y around inf 77.1%

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

      1. associate-*r/ 80.7%

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

   5. Simplified 80.7%

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

4. Final simplification 71.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -3.8 \cdot 10^{+46} \lor \neg \left(t \leq 1.8 \cdot 10^{+139}\right):\\ \;\;\;\;x \cdot \frac{t}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 43.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.6 \cdot 10^{-10} \lor \neg \left(z \leq 13\right):\\ \;\;\;\;t \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(-t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -1.6e-10) (not (<= z 13.0))) (* t (/ x z)) (* x (- t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -1.6e-10) || !(z <= 13.0)) {
		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 <= (-1.6d-10)) .or. (.not. (z <= 13.0d0))) 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 <= -1.6e-10) || !(z <= 13.0)) {
		tmp = t * (x / z);
	} else {
		tmp = x * -t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z <= -1.6e-10) or not (z <= 13.0):
		tmp = t * (x / z)
	else:
		tmp = x * -t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((z <= -1.6e-10) || !(z <= 13.0))
		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 <= -1.6e-10) || ~((z <= 13.0)))
		tmp = t * (x / z);
	else
		tmp = x * -t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[z, -1.6e-10], N[Not[LessEqual[z, 13.0]], $MachinePrecision]], N[(t * N[(x / z), $MachinePrecision]), $MachinePrecision], N[(x * (-t)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.5999999999999999e-10 or 13 < z

   1. Initial program 97.4%

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

   3. Taylor expanded in y around 0 57.1%

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

      1. mul-1-neg 57.1%

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

      2. distribute-neg-frac2 57.1%

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

      3. neg-sub0 57.1%

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

      4. associate--r- 57.1%

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

      5. metadata-eval 57.1%

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

   5. Simplified 57.1%

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

   6. Taylor expanded in z around inf 49.9%

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

      1. associate-/l* 52.6%

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

   8. Simplified 52.6%

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

   if -1.5999999999999999e-10 < z < 13

   1. Initial program 93.8%

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

   3. Taylor expanded in y around 0 37.9%

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

      1. mul-1-neg 37.9%

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

      2. distribute-neg-frac2 37.9%

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

      3. neg-sub0 37.9%

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

      4. associate--r- 37.9%

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

      5. metadata-eval 37.9%

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

   5. Simplified 37.9%

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

   6. Taylor expanded in z around 0 36.7%

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

      1. *-commutative 36.7%

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

      2. neg-mul-1 36.7%

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

      3. distribute-rgt-neg-in 36.7%

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

   8. Simplified 36.7%

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

4. Final simplification 45.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.6 \cdot 10^{-10} \lor \neg \left(z \leq 13\right):\\ \;\;\;\;t \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(-t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 44.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.6 \cdot 10^{-10}:\\ \;\;\;\;x \cdot \frac{t}{z}\\ \mathbf{elif}\;z \leq 13:\\ \;\;\;\;x \cdot \left(-t\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -1.6e-10) (* x (/ t z)) (if (<= z 13.0) (* x (- t)) (* t (/ x z)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -1.6e-10) {
		tmp = x * (t / z);
	} else if (z <= 13.0) {
		tmp = x * -t;
	} else {
		tmp = t * (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 (z <= (-1.6d-10)) then
        tmp = x * (t / z)
    else if (z <= 13.0d0) then
        tmp = x * -t
    else
        tmp = t * (x / z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -1.6e-10) {
		tmp = x * (t / z);
	} else if (z <= 13.0) {
		tmp = x * -t;
	} else {
		tmp = t * (x / z);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= -1.6e-10:
		tmp = x * (t / z)
	elif z <= 13.0:
		tmp = x * -t
	else:
		tmp = t * (x / z)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -1.6e-10)
		tmp = Float64(x * Float64(t / z));
	elseif (z <= 13.0)
		tmp = Float64(x * Float64(-t));
	else
		tmp = Float64(t * Float64(x / z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= -1.6e-10)
		tmp = x * (t / z);
	elseif (z <= 13.0)
		tmp = x * -t;
	else
		tmp = t * (x / z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -1.6e-10], N[(x * N[(t / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 13.0], N[(x * (-t)), $MachinePrecision], N[(t * N[(x / z), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.5999999999999999e-10

   1. Initial program 99.6%

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

   3. Taylor expanded in y around 0 66.4%

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

      1. mul-1-neg 66.4%

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

      2. distribute-neg-frac2 66.4%

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

      3. neg-sub0 66.4%

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

      4. associate--r- 66.4%

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

      5. metadata-eval 66.4%

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

   5. Simplified 66.4%

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

   6. Taylor expanded in z around inf 66.2%

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

   if -1.5999999999999999e-10 < z < 13

   1. Initial program 93.8%

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

   3. Taylor expanded in y around 0 37.9%

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

      1. mul-1-neg 37.9%

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

      2. distribute-neg-frac2 37.9%

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

      3. neg-sub0 37.9%

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

      4. associate--r- 37.9%

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

      5. metadata-eval 37.9%

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

   5. Simplified 37.9%

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

   6. Taylor expanded in z around 0 36.7%

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

      1. *-commutative 36.7%

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

      2. neg-mul-1 36.7%

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

      3. distribute-rgt-neg-in 36.7%

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

   8. Simplified 36.7%

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

   if 13 < z

   1. Initial program 94.6%

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

   3. Taylor expanded in y around 0 45.2%

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

      1. mul-1-neg 45.2%

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

      2. distribute-neg-frac2 45.2%

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

      3. neg-sub0 45.2%

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

      4. associate--r- 45.2%

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

      5. metadata-eval 45.2%

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

   5. Simplified 45.2%

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

   6. Taylor expanded in z around inf 36.0%

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

      1. associate-/l* 48.9%

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

   8. Simplified 48.9%

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

Alternative 10: 94.3% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

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 + (-1.0d0)))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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 95.6%

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

   2. associate-/r/ 95.9%

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

4. Applied egg-rr 95.9%

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

5. Final simplification 95.9%

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

Alternative 11: 94.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

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 / (z + (-1.0d0))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 95.9%

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

3. Final simplification 95.9%

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

Alternative 12: 23.6% 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 95.9%

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

3. Taylor expanded in y around 0 49.0%

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

   1. mul-1-neg 49.0%

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

   2. distribute-neg-frac2 49.0%

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

   3. neg-sub0 49.0%

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

   4. associate--r- 49.0%

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

   5. metadata-eval 49.0%

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

5. Simplified 49.0%

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

6. Taylor expanded in z around 0 24.3%

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

   1. *-commutative 24.3%

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

   2. neg-mul-1 24.3%

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

   3. distribute-rgt-neg-in 24.3%

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

8. Simplified 24.3%

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

Developer target: 94.8% 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 2024103 
(FPCore (x y z t)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, C"
  :precision binary64

  :alt
  (if (< (* x (- (/ y z) (/ t (- 1.0 z)))) -7.623226303312042e-196) (* x (- (/ y z) (* t (/ 1.0 (- 1.0 z))))) (if (< (* x (- (/ y z) (/ t (- 1.0 z)))) 1.4133944927702302e-211) (+ (/ (* y x) z) (- (/ (* t x) (- 1.0 z)))) (* x (- (/ y z) (* t (/ 1.0 (- 1.0 z)))))))

  (* x (- (/ y z) (/ t (- 1.0 z)))))