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

Percentage Accurate: 94.6% → 98.1%

Time: 8.4s

Alternatives: 12

Speedup: 0.3×

• 21.3% 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: 98.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{y}{z} - \frac{t}{1 - z}\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\frac{y \cdot x}{z}\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+291}:\\ \;\;\;\;x \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (/ y z) (/ t (- 1.0 z)))))
   (if (<= t_1 (- INFINITY))
     (/ (* y x) z)
     (if (<= t_1 5e+291) (* x t_1) (* y (/ x z))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (y / z) - (t / (1.0 - z));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = (y * x) / z;
	} else if (t_1 <= 5e+291) {
		tmp = x * t_1;
	} else {
		tmp = y * (x / z);
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (y / z) - (t / (1.0 - z));
	double tmp;
	if (t_1 <= -Double.POSITIVE_INFINITY) {
		tmp = (y * x) / z;
	} else if (t_1 <= 5e+291) {
		tmp = x * t_1;
	} else {
		tmp = y * (x / z);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (y / z) - (t / (1.0 - z))
	tmp = 0
	if t_1 <= -math.inf:
		tmp = (y * x) / z
	elif t_1 <= 5e+291:
		tmp = x * t_1
	else:
		tmp = y * (x / z)
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(y / z) - Float64(t / Float64(1.0 - z)))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(Float64(y * x) / z);
	elseif (t_1 <= 5e+291)
		tmp = Float64(x * t_1);
	else
		tmp = Float64(y * Float64(x / z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (y / z) - (t / (1.0 - z));
	tmp = 0.0;
	if (t_1 <= -Inf)
		tmp = (y * x) / z;
	elseif (t_1 <= 5e+291)
		tmp = x * t_1;
	else
		tmp = y * (x / z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(y / z), $MachinePrecision] - N[(t / N[(1.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(y * x), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[t$95$1, 5e+291], N[(x * t$95$1), $MachinePrecision], N[(y * N[(x / z), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 56.9%

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

   3. Taylor expanded in y around inf

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 56.9

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

   5. Applied rewrites 56.9%

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

   7. Applied rewrites 99.9%

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

   if -inf.0 < (-.f64 (/.f64 y z) (/.f64 t (-.f64 #s(literal 1 binary64) z))) < 5.0000000000000001e291

   1. Initial program 97.3%

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

   if 5.0000000000000001e291 < (-.f64 (/.f64 y z) (/.f64 t (-.f64 #s(literal 1 binary64) z)))

   1. Initial program 61.9%

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

   3. Taylor expanded in y around inf

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 61.9

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

   5. Applied rewrites 61.9%

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

   7. Applied rewrites 99.8%

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

Alternative 2: 70.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t \cdot x}{-1 + z}\\ \mathbf{if}\;t \leq -4.8 \cdot 10^{+153}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq -4.1 \cdot 10^{-249}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{elif}\;t \leq 7.5 \cdot 10^{+131}:\\ \;\;\;\;\frac{y \cdot x}{z}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (* t x) (+ -1.0 z))))
   (if (<= t -4.8e+153)
     t_1
     (if (<= t -4.1e-249)
       (* y (/ x z))
       (if (<= t 7.5e+131) (/ (* y x) z) t_1)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (t * x) / (-1.0 + z);
	double tmp;
	if (t <= -4.8e+153) {
		tmp = t_1;
	} else if (t <= -4.1e-249) {
		tmp = y * (x / z);
	} else if (t <= 7.5e+131) {
		tmp = (y * x) / z;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (t * x) / ((-1.0d0) + z)
    if (t <= (-4.8d+153)) then
        tmp = t_1
    else if (t <= (-4.1d-249)) then
        tmp = y * (x / z)
    else if (t <= 7.5d+131) then
        tmp = (y * x) / z
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (t * x) / (-1.0 + z);
	double tmp;
	if (t <= -4.8e+153) {
		tmp = t_1;
	} else if (t <= -4.1e-249) {
		tmp = y * (x / z);
	} else if (t <= 7.5e+131) {
		tmp = (y * x) / z;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (t * x) / (-1.0 + z)
	tmp = 0
	if t <= -4.8e+153:
		tmp = t_1
	elif t <= -4.1e-249:
		tmp = y * (x / z)
	elif t <= 7.5e+131:
		tmp = (y * x) / z
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(t * x) / Float64(-1.0 + z))
	tmp = 0.0
	if (t <= -4.8e+153)
		tmp = t_1;
	elseif (t <= -4.1e-249)
		tmp = Float64(y * Float64(x / z));
	elseif (t <= 7.5e+131)
		tmp = Float64(Float64(y * x) / z);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (t * x) / (-1.0 + z);
	tmp = 0.0;
	if (t <= -4.8e+153)
		tmp = t_1;
	elseif (t <= -4.1e-249)
		tmp = y * (x / z);
	elseif (t <= 7.5e+131)
		tmp = (y * x) / z;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(t * x), $MachinePrecision] / N[(-1.0 + z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -4.8e+153], t$95$1, If[LessEqual[t, -4.1e-249], N[(y * N[(x / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 7.5e+131], N[(N[(y * x), $MachinePrecision] / z), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if t < -4.79999999999999985e153 or 7.4999999999999995e131 < t

   1. Initial program 98.2%

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

   3. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. *-lft-identity N/A

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

      6. metadata-eval N/A

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

      7. fp-cancel-sign-sub-inv N/A

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

      8. mul-1-neg N/A

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

      9. distribute-neg-in N/A

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

      10. metadata-eval N/A

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

      11. remove-double-neg N/A

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

      12. lower-+.f64 88.0

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

   5. Applied rewrites 88.0%

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

   if -4.79999999999999985e153 < t < -4.10000000000000004e-249

   1. Initial program 89.0%

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

   3. Taylor expanded in y around inf

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 69.0

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

   5. Applied rewrites 69.0%

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

   7. Applied rewrites 73.5%

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

   if -4.10000000000000004e-249 < t < 7.4999999999999995e131

   1. Initial program 91.5%

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

   3. Taylor expanded in y around inf

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 70.3

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

   5. Applied rewrites 70.3%

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

   7. Applied rewrites 75.1%

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

Alternative 3: 92.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -550000000:\\ \;\;\;\;\frac{\left(t + y\right) \cdot x}{z}\\ \mathbf{elif}\;z \leq 3.3 \cdot 10^{-7}:\\ \;\;\;\;\frac{x \cdot \left(y - t \cdot z\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{t + y}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -550000000.0)
   (/ (* (+ t y) x) z)
   (if (<= z 3.3e-7) (/ (* x (- y (* t z))) z) (* x (/ (+ t y) z)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -550000000.0) {
		tmp = ((t + y) * x) / z;
	} else if (z <= 3.3e-7) {
		tmp = (x * (y - (t * z))) / z;
	} else {
		tmp = x * ((t + 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 (z <= (-550000000.0d0)) then
        tmp = ((t + y) * x) / z
    else if (z <= 3.3d-7) then
        tmp = (x * (y - (t * z))) / z
    else
        tmp = x * ((t + y) / z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -550000000.0) {
		tmp = ((t + y) * x) / z;
	} else if (z <= 3.3e-7) {
		tmp = (x * (y - (t * z))) / z;
	} else {
		tmp = x * ((t + y) / z);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= -550000000.0:
		tmp = ((t + y) * x) / z
	elif z <= 3.3e-7:
		tmp = (x * (y - (t * z))) / z
	else:
		tmp = x * ((t + y) / z)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -550000000.0)
		tmp = Float64(Float64(Float64(t + y) * x) / z);
	elseif (z <= 3.3e-7)
		tmp = Float64(Float64(x * Float64(y - Float64(t * z))) / z);
	else
		tmp = Float64(x * Float64(Float64(t + y) / z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= -550000000.0)
		tmp = ((t + y) * x) / z;
	elseif (z <= 3.3e-7)
		tmp = (x * (y - (t * z))) / z;
	else
		tmp = x * ((t + y) / z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -550000000.0], N[(N[(N[(t + y), $MachinePrecision] * x), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[z, 3.3e-7], N[(N[(x * N[(y - N[(t * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], N[(x * N[(N[(t + y), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -5.5e8

   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 inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. fp-cancel-sub-sign-inv N/A

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

      5. metadata-eval N/A

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

      6. *-lft-identity N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 94.1

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

   5. Applied rewrites 94.1%

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

   if -5.5e8 < z < 3.3000000000000002e-7

   1. Initial program 88.4%

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

   3. Taylor expanded in z around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. associate-*r* N/A

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

      8. distribute-rgt-out N/A

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

      9. lower-*.f64 N/A

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

      10. associate-*r* N/A

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

      11. mul-1-neg N/A

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

      12. fp-cancel-sub-sign N/A

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

      13. lower--.f64 N/A

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

      14. lower-*.f64 92.2

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

   5. Applied rewrites 92.2%

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

   if 3.3000000000000002e-7 < z

   1. Initial program 97.0%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

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

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

      3. metadata-eval N/A

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

      4. *-lft-identity N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 97.0

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

   5. Applied rewrites 97.0%

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

Alternative 4: 63.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -6.5 \cdot 10^{+174}:\\ \;\;\;\;\frac{x \cdot t}{z}\\ \mathbf{elif}\;t \leq -4.1 \cdot 10^{-249}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{elif}\;t \leq 3.2 \cdot 10^{+132}:\\ \;\;\;\;\frac{y \cdot x}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(-t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t -6.5e+174)
   (/ (* x t) z)
   (if (<= t -4.1e-249)
     (* y (/ x z))
     (if (<= t 3.2e+132) (/ (* y x) z) (* x (- t))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -6.5e+174) {
		tmp = (x * t) / z;
	} else if (t <= -4.1e-249) {
		tmp = y * (x / z);
	} else if (t <= 3.2e+132) {
		tmp = (y * 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 (t <= (-6.5d+174)) then
        tmp = (x * t) / z
    else if (t <= (-4.1d-249)) then
        tmp = y * (x / z)
    else if (t <= 3.2d+132) then
        tmp = (y * 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 (t <= -6.5e+174) {
		tmp = (x * t) / z;
	} else if (t <= -4.1e-249) {
		tmp = y * (x / z);
	} else if (t <= 3.2e+132) {
		tmp = (y * x) / z;
	} else {
		tmp = x * -t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if t <= -6.5e+174:
		tmp = (x * t) / z
	elif t <= -4.1e-249:
		tmp = y * (x / z)
	elif t <= 3.2e+132:
		tmp = (y * x) / z
	else:
		tmp = x * -t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= -6.5e+174)
		tmp = Float64(Float64(x * t) / z);
	elseif (t <= -4.1e-249)
		tmp = Float64(y * Float64(x / z));
	elseif (t <= 3.2e+132)
		tmp = Float64(Float64(y * 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 (t <= -6.5e+174)
		tmp = (x * t) / z;
	elseif (t <= -4.1e-249)
		tmp = y * (x / z);
	elseif (t <= 3.2e+132)
		tmp = (y * x) / z;
	else
		tmp = x * -t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, -6.5e+174], N[(N[(x * t), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[t, -4.1e-249], N[(y * N[(x / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 3.2e+132], N[(N[(y * x), $MachinePrecision] / z), $MachinePrecision], N[(x * (-t)), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if t < -6.5000000000000001e174

   1. Initial program 99.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift--.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. frac-sub N/A

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

      7. associate-*l/ N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

      10. fp-cancel-sub-sign-inv N/A

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

      11. +-commutative N/A

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

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

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

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

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

      14. *-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. lower-neg.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 N/A

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

      19. lower-/.f64 N/A

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

      20. *-commutative N/A

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

      21. lower-*.f64 33.0

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

   4. Applied rewrites 33.0%

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

   5. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. fp-cancel-sign-sub-inv N/A

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

      5. metadata-eval N/A

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

      6. *-lft-identity N/A

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

      7. lower--.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 33.1

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

   7. Applied rewrites 33.1%

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

   8. Taylor expanded in z around -inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 79.3

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

   10. Applied rewrites 79.3%

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

   11. Taylor expanded in y around 0

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

   13. Applied rewrites 73.8%

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

   if -6.5000000000000001e174 < t < -4.10000000000000004e-249

   1. Initial program 89.8%

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

   3. Taylor expanded in y around inf

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 67.3

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

   5. Applied rewrites 67.3%

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

   7. Applied rewrites 71.5%

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

   if -4.10000000000000004e-249 < t < 3.1999999999999997e132

   1. Initial program 91.5%

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

   3. Taylor expanded in y around inf

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 70.3

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

   5. Applied rewrites 70.3%

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

   7. Applied rewrites 75.1%

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

   if 3.1999999999999997e132 < t

   1. Initial program 96.4%

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

   3. Taylor expanded in z around 0

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. fp-cancel-sub-sign N/A

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

      4. div-sub N/A

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

      5. associate-/l* N/A

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

      6. *-inverses N/A

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

      7. *-rgt-identity N/A

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

      8. lower--.f64 N/A

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

      9. lower-/.f64 65.6

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

   5. Applied rewrites 65.6%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 61.2%

      \[\leadsto x \cdot \left(-t\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 72.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -6.5 \cdot 10^{+174}:\\ \;\;\;\;\frac{x \cdot t}{z}\\ \mathbf{elif}\;t \leq -4.1 \cdot 10^{-249}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{elif}\;t \leq 3.2 \cdot 10^{+132}:\\ \;\;\;\;\frac{y \cdot x}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(-t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 63.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -5.6 \cdot 10^{+174}:\\ \;\;\;\;\frac{x}{z} \cdot t\\ \mathbf{elif}\;t \leq -4.1 \cdot 10^{-249}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{elif}\;t \leq 3.2 \cdot 10^{+132}:\\ \;\;\;\;\frac{y \cdot x}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(-t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t -5.6e+174)
   (* (/ x z) t)
   (if (<= t -4.1e-249)
     (* y (/ x z))
     (if (<= t 3.2e+132) (/ (* y x) z) (* x (- t))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -5.6e+174) {
		tmp = (x / z) * t;
	} else if (t <= -4.1e-249) {
		tmp = y * (x / z);
	} else if (t <= 3.2e+132) {
		tmp = (y * 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 (t <= (-5.6d+174)) then
        tmp = (x / z) * t
    else if (t <= (-4.1d-249)) then
        tmp = y * (x / z)
    else if (t <= 3.2d+132) then
        tmp = (y * 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 (t <= -5.6e+174) {
		tmp = (x / z) * t;
	} else if (t <= -4.1e-249) {
		tmp = y * (x / z);
	} else if (t <= 3.2e+132) {
		tmp = (y * x) / z;
	} else {
		tmp = x * -t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if t <= -5.6e+174:
		tmp = (x / z) * t
	elif t <= -4.1e-249:
		tmp = y * (x / z)
	elif t <= 3.2e+132:
		tmp = (y * x) / z
	else:
		tmp = x * -t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= -5.6e+174)
		tmp = Float64(Float64(x / z) * t);
	elseif (t <= -4.1e-249)
		tmp = Float64(y * Float64(x / z));
	elseif (t <= 3.2e+132)
		tmp = Float64(Float64(y * 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 (t <= -5.6e+174)
		tmp = (x / z) * t;
	elseif (t <= -4.1e-249)
		tmp = y * (x / z);
	elseif (t <= 3.2e+132)
		tmp = (y * x) / z;
	else
		tmp = x * -t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, -5.6e+174], N[(N[(x / z), $MachinePrecision] * t), $MachinePrecision], If[LessEqual[t, -4.1e-249], N[(y * N[(x / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 3.2e+132], N[(N[(y * x), $MachinePrecision] / z), $MachinePrecision], N[(x * (-t)), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if t < -5.5999999999999999e174

   1. Initial program 99.5%

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

   3. Taylor expanded in t around inf

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

      1. distribute-rgt-in N/A

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

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

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

      3. mul-1-neg N/A

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

      4. distribute-rgt-out-- N/A

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

      5. *-commutative N/A

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

      6. fp-cancel-sub-sign-inv N/A

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

      7. *-commutative N/A

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

      8. mul-1-neg N/A

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

      9. distribute-rgt-in N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

   5. Applied rewrites 69.5%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 69.5%

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

   9. Taylor expanded in z around inf

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

   11. Applied rewrites 55.8%

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

   if -5.5999999999999999e174 < t < -4.10000000000000004e-249

   1. Initial program 89.8%

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

   3. Taylor expanded in y around inf

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 67.3

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

   5. Applied rewrites 67.3%

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

   7. Applied rewrites 71.5%

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

   if -4.10000000000000004e-249 < t < 3.1999999999999997e132

   1. Initial program 91.5%

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

   3. Taylor expanded in y around inf

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 70.3

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

   5. Applied rewrites 70.3%

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

   7. Applied rewrites 75.1%

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

   if 3.1999999999999997e132 < t

   1. Initial program 96.4%

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

   3. Taylor expanded in z around 0

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. fp-cancel-sub-sign N/A

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

      4. div-sub N/A

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

      5. associate-/l* N/A

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

      6. *-inverses N/A

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

      7. *-rgt-identity N/A

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

      8. lower--.f64 N/A

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

      9. lower-/.f64 65.6

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

   5. Applied rewrites 65.6%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 61.2%

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

Alternative 6: 88.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -550000000 \lor \neg \left(z \leq 2.6 \cdot 10^{-8}\right):\\ \;\;\;\;\frac{\left(t + y\right) \cdot x}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -550000000.0) (not (<= z 2.6e-8)))
   (/ (* (+ t y) x) z)
   (* x (- (/ y z) t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -550000000.0) || !(z <= 2.6e-8)) {
		tmp = ((t + y) * x) / 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 <= (-550000000.0d0)) .or. (.not. (z <= 2.6d-8))) then
        tmp = ((t + y) * x) / 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 <= -550000000.0) || !(z <= 2.6e-8)) {
		tmp = ((t + y) * x) / z;
	} else {
		tmp = x * ((y / z) - t);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z <= -550000000.0) or not (z <= 2.6e-8):
		tmp = ((t + y) * x) / z
	else:
		tmp = x * ((y / z) - t)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((z <= -550000000.0) || !(z <= 2.6e-8))
		tmp = Float64(Float64(Float64(t + y) * x) / 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 <= -550000000.0) || ~((z <= 2.6e-8)))
		tmp = ((t + y) * x) / z;
	else
		tmp = x * ((y / z) - t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[z, -550000000.0], N[Not[LessEqual[z, 2.6e-8]], $MachinePrecision]], N[(N[(N[(t + y), $MachinePrecision] * x), $MachinePrecision] / z), $MachinePrecision], N[(x * N[(N[(y / z), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -5.5e8 or 2.6000000000000001e-8 < z

   1. Initial program 95.0%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. fp-cancel-sub-sign-inv N/A

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

      5. metadata-eval N/A

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

      6. *-lft-identity N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 91.8

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

   5. Applied rewrites 91.8%

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

   if -5.5e8 < z < 2.6000000000000001e-8

   1. Initial program 89.1%

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

   3. Taylor expanded in z around 0

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. fp-cancel-sub-sign N/A

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

      4. div-sub N/A

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

      5. associate-/l* N/A

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

      6. *-inverses N/A

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

      7. *-rgt-identity N/A

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

      8. lower--.f64 N/A

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

      9. lower-/.f64 87.7

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

   5. Applied rewrites 87.7%

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

4. Final simplification 89.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -550000000 \lor \neg \left(z \leq 2.6 \cdot 10^{-8}\right):\\ \;\;\;\;\frac{\left(t + y\right) \cdot x}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 75.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4.4 \cdot 10^{-204} \lor \neg \left(y \leq 3.1 \cdot 10^{-137}\right):\\ \;\;\;\;\frac{\left(t + y\right) \cdot x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{t \cdot x}{-1 + z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= y -4.4e-204) (not (<= y 3.1e-137)))
   (/ (* (+ t y) x) z)
   (/ (* t x) (+ -1.0 z))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((y <= -4.4e-204) || !(y <= 3.1e-137)) {
		tmp = ((t + y) * x) / z;
	} else {
		tmp = (t * x) / (-1.0 + 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 <= (-4.4d-204)) .or. (.not. (y <= 3.1d-137))) then
        tmp = ((t + y) * x) / z
    else
        tmp = (t * x) / ((-1.0d0) + z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((y <= -4.4e-204) || !(y <= 3.1e-137)) {
		tmp = ((t + y) * x) / z;
	} else {
		tmp = (t * x) / (-1.0 + z);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (y <= -4.4e-204) or not (y <= 3.1e-137):
		tmp = ((t + y) * x) / z
	else:
		tmp = (t * x) / (-1.0 + z)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((y <= -4.4e-204) || !(y <= 3.1e-137))
		tmp = Float64(Float64(Float64(t + y) * x) / z);
	else
		tmp = Float64(Float64(t * x) / Float64(-1.0 + z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((y <= -4.4e-204) || ~((y <= 3.1e-137)))
		tmp = ((t + y) * x) / z;
	else
		tmp = (t * x) / (-1.0 + z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[y, -4.4e-204], N[Not[LessEqual[y, 3.1e-137]], $MachinePrecision]], N[(N[(N[(t + y), $MachinePrecision] * x), $MachinePrecision] / z), $MachinePrecision], N[(N[(t * x), $MachinePrecision] / N[(-1.0 + z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -4.3999999999999997e-204 or 3.09999999999999978e-137 < y

   1. Initial program 92.1%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. fp-cancel-sub-sign-inv N/A

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

      5. metadata-eval N/A

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

      6. *-lft-identity N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 82.6

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

   5. Applied rewrites 82.6%

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

   if -4.3999999999999997e-204 < y < 3.09999999999999978e-137

   1. Initial program 92.6%

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

   3. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. *-lft-identity N/A

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

      6. metadata-eval N/A

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

      7. fp-cancel-sign-sub-inv N/A

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

      8. mul-1-neg N/A

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

      9. distribute-neg-in N/A

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

      10. metadata-eval N/A

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

      11. remove-double-neg N/A

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

      12. lower-+.f64 91.1

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

   5. Applied rewrites 91.1%

      \[\leadsto \color{blue}{\frac{t \cdot x}{-1 + z}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 84.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.4 \cdot 10^{-204} \lor \neg \left(y \leq 3.1 \cdot 10^{-137}\right):\\ \;\;\;\;\frac{\left(t + y\right) \cdot x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{t \cdot x}{-1 + z}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 71.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.6 \cdot 10^{-168} \lor \neg \left(y \leq 1.02 \cdot 10^{-81}\right):\\ \;\;\;\;\frac{y \cdot x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z - 1} \cdot t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= y -2.6e-168) (not (<= y 1.02e-81)))
   (/ (* y x) z)
   (* (/ x (- z 1.0)) t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((y <= -2.6e-168) || !(y <= 1.02e-81)) {
		tmp = (y * x) / z;
	} else {
		tmp = (x / (z - 1.0)) * 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 ((y <= (-2.6d-168)) .or. (.not. (y <= 1.02d-81))) then
        tmp = (y * x) / z
    else
        tmp = (x / (z - 1.0d0)) * t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((y <= -2.6e-168) || !(y <= 1.02e-81)) {
		tmp = (y * x) / z;
	} else {
		tmp = (x / (z - 1.0)) * t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (y <= -2.6e-168) or not (y <= 1.02e-81):
		tmp = (y * x) / z
	else:
		tmp = (x / (z - 1.0)) * t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((y <= -2.6e-168) || !(y <= 1.02e-81))
		tmp = Float64(Float64(y * x) / z);
	else
		tmp = Float64(Float64(x / Float64(z - 1.0)) * t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((y <= -2.6e-168) || ~((y <= 1.02e-81)))
		tmp = (y * x) / z;
	else
		tmp = (x / (z - 1.0)) * t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[y, -2.6e-168], N[Not[LessEqual[y, 1.02e-81]], $MachinePrecision]], N[(N[(y * x), $MachinePrecision] / z), $MachinePrecision], N[(N[(x / N[(z - 1.0), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -2.6000000000000001e-168 or 1.01999999999999998e-81 < y

   1. Initial program 91.3%

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

   3. Taylor expanded in y around inf

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 66.6

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

   5. Applied rewrites 66.6%

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

   7. Applied rewrites 70.3%

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

   if -2.6000000000000001e-168 < y < 1.01999999999999998e-81

   1. Initial program 94.3%

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

   3. Taylor expanded in t around inf

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

      1. distribute-rgt-in N/A

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

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

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

      3. mul-1-neg N/A

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

      4. distribute-rgt-out-- N/A

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

      5. *-commutative N/A

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

      6. fp-cancel-sub-sign-inv N/A

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

      7. *-commutative N/A

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

      8. mul-1-neg N/A

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

      9. distribute-rgt-in N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

   5. Applied rewrites 83.6%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 84.7%

      \[\leadsto \frac{x}{z - 1} \cdot \color{blue}{t} \]
3. Recombined 2 regimes into one program.

4. Final simplification 74.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.6 \cdot 10^{-168} \lor \neg \left(y \leq 1.02 \cdot 10^{-81}\right):\\ \;\;\;\;\frac{y \cdot x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z - 1} \cdot t\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 90.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -550000000:\\ \;\;\;\;\frac{\left(t + y\right) \cdot x}{z}\\ \mathbf{elif}\;z \leq 2.6 \cdot 10^{-8}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - t\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{t + y}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -550000000.0)
   (/ (* (+ t y) x) z)
   (if (<= z 2.6e-8) (* x (- (/ y z) t)) (* x (/ (+ t y) z)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -550000000.0) {
		tmp = ((t + y) * x) / z;
	} else if (z <= 2.6e-8) {
		tmp = x * ((y / z) - t);
	} else {
		tmp = x * ((t + 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 (z <= (-550000000.0d0)) then
        tmp = ((t + y) * x) / z
    else if (z <= 2.6d-8) then
        tmp = x * ((y / z) - t)
    else
        tmp = x * ((t + y) / z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -550000000.0) {
		tmp = ((t + y) * x) / z;
	} else if (z <= 2.6e-8) {
		tmp = x * ((y / z) - t);
	} else {
		tmp = x * ((t + y) / z);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= -550000000.0:
		tmp = ((t + y) * x) / z
	elif z <= 2.6e-8:
		tmp = x * ((y / z) - t)
	else:
		tmp = x * ((t + y) / z)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -550000000.0)
		tmp = Float64(Float64(Float64(t + y) * x) / z);
	elseif (z <= 2.6e-8)
		tmp = Float64(x * Float64(Float64(y / z) - t));
	else
		tmp = Float64(x * Float64(Float64(t + y) / z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= -550000000.0)
		tmp = ((t + y) * x) / z;
	elseif (z <= 2.6e-8)
		tmp = x * ((y / z) - t);
	else
		tmp = x * ((t + y) / z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -550000000.0], N[(N[(N[(t + y), $MachinePrecision] * x), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[z, 2.6e-8], N[(x * N[(N[(y / z), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision], N[(x * N[(N[(t + y), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -5.5e8

   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 inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. fp-cancel-sub-sign-inv N/A

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

      5. metadata-eval N/A

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

      6. *-lft-identity N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 94.1

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

   5. Applied rewrites 94.1%

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

   if -5.5e8 < z < 2.6000000000000001e-8

   1. Initial program 89.1%

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

   3. Taylor expanded in z around 0

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. fp-cancel-sub-sign N/A

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

      4. div-sub N/A

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

      5. associate-/l* N/A

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

      6. *-inverses N/A

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

      7. *-rgt-identity N/A

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

      8. lower--.f64 N/A

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

      9. lower-/.f64 87.7

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

   5. Applied rewrites 87.7%

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

   if 2.6000000000000001e-8 < z

   1. Initial program 95.8%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

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

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

      3. metadata-eval N/A

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

      4. *-lft-identity N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 95.9

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

   5. Applied rewrites 95.9%

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

Alternative 10: 43.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -550000000 \lor \neg \left(z \leq 2.6 \cdot 10^{-8}\right):\\ \;\;\;\;\frac{x}{z} \cdot t\\ \mathbf{else}:\\ \;\;\;\;\left(-\mathsf{fma}\left(z, x, x\right)\right) \cdot t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -550000000.0) (not (<= z 2.6e-8)))
   (* (/ x z) t)
   (* (- (fma z x x)) t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -550000000.0) || !(z <= 2.6e-8)) {
		tmp = (x / z) * t;
	} else {
		tmp = -fma(z, x, x) * t;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((z <= -550000000.0) || !(z <= 2.6e-8))
		tmp = Float64(Float64(x / z) * t);
	else
		tmp = Float64(Float64(-fma(z, x, x)) * t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[z, -550000000.0], N[Not[LessEqual[z, 2.6e-8]], $MachinePrecision]], N[(N[(x / z), $MachinePrecision] * t), $MachinePrecision], N[((-N[(z * x + x), $MachinePrecision]) * t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -5.5e8 or 2.6000000000000001e-8 < z

   1. Initial program 95.0%

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

   3. Taylor expanded in t around inf

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

      1. distribute-rgt-in N/A

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

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

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

      3. mul-1-neg N/A

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

      4. distribute-rgt-out-- N/A

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

      5. *-commutative N/A

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

      6. fp-cancel-sub-sign-inv N/A

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

      7. *-commutative N/A

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

      8. mul-1-neg N/A

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

      9. distribute-rgt-in N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

   5. Applied rewrites 72.9%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 51.1%

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

   9. Taylor expanded in z around inf

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

   11. Applied rewrites 50.8%

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

   if -5.5e8 < z < 2.6000000000000001e-8

   1. Initial program 89.1%

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

   3. Taylor expanded in t around inf

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

      1. distribute-rgt-in N/A

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

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

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

      3. mul-1-neg N/A

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

      4. distribute-rgt-out-- N/A

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

      5. *-commutative N/A

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

      6. fp-cancel-sub-sign-inv N/A

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

      7. *-commutative N/A

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

      8. mul-1-neg N/A

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

      9. distribute-rgt-in N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

   5. Applied rewrites 85.6%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 41.4%

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

   9. Taylor expanded in z around 0

      \[\leadsto \left(-1 \cdot x + -1 \cdot \left(x \cdot z\right)\right) \cdot t \]
   10. Step-by-step derivation

   11. Applied rewrites 40.5%

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

4. Final simplification 46.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -550000000 \lor \neg \left(z \leq 2.6 \cdot 10^{-8}\right):\\ \;\;\;\;\frac{x}{z} \cdot t\\ \mathbf{else}:\\ \;\;\;\;\left(-\mathsf{fma}\left(z, x, x\right)\right) \cdot t\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 63.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -5.6 \cdot 10^{+174}:\\ \;\;\;\;\frac{x}{z} \cdot t\\ \mathbf{elif}\;t \leq 1.26 \cdot 10^{+113}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(-t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t -5.6e+174)
   (* (/ x z) t)
   (if (<= t 1.26e+113) (* y (/ x z)) (* x (- t)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -5.6e+174) {
		tmp = (x / z) * t;
	} else if (t <= 1.26e+113) {
		tmp = y * (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 (t <= (-5.6d+174)) then
        tmp = (x / z) * t
    else if (t <= 1.26d+113) then
        tmp = y * (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 (t <= -5.6e+174) {
		tmp = (x / z) * t;
	} else if (t <= 1.26e+113) {
		tmp = y * (x / z);
	} else {
		tmp = x * -t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if t <= -5.6e+174:
		tmp = (x / z) * t
	elif t <= 1.26e+113:
		tmp = y * (x / z)
	else:
		tmp = x * -t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= -5.6e+174)
		tmp = Float64(Float64(x / z) * t);
	elseif (t <= 1.26e+113)
		tmp = Float64(y * 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 (t <= -5.6e+174)
		tmp = (x / z) * t;
	elseif (t <= 1.26e+113)
		tmp = y * (x / z);
	else
		tmp = x * -t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, -5.6e+174], N[(N[(x / z), $MachinePrecision] * t), $MachinePrecision], If[LessEqual[t, 1.26e+113], N[(y * N[(x / z), $MachinePrecision]), $MachinePrecision], N[(x * (-t)), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -5.5999999999999999e174

   1. Initial program 99.5%

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

   3. Taylor expanded in t around inf

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

      1. distribute-rgt-in N/A

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

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

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

      3. mul-1-neg N/A

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

      4. distribute-rgt-out-- N/A

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

      5. *-commutative N/A

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

      6. fp-cancel-sub-sign-inv N/A

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

      7. *-commutative N/A

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

      8. mul-1-neg N/A

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

      9. distribute-rgt-in N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

   5. Applied rewrites 69.5%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 69.5%

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

   9. Taylor expanded in z around inf

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

   11. Applied rewrites 55.8%

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

   if -5.5999999999999999e174 < t < 1.2599999999999999e113

   1. Initial program 90.5%

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

   3. Taylor expanded in y around inf

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 68.7

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

   5. Applied rewrites 68.7%

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

   7. Applied rewrites 71.0%

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

   if 1.2599999999999999e113 < t

   1. Initial program 96.8%

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

   3. Taylor expanded in z around 0

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. fp-cancel-sub-sign N/A

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

      4. div-sub N/A

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

      5. associate-/l* N/A

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

      6. *-inverses N/A

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

      7. *-rgt-identity N/A

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

      8. lower--.f64 N/A

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

      9. lower-/.f64 65.7

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

   5. Applied rewrites 65.7%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 60.1%

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

Alternative 12: 23.9% accurate, 4.3× 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 92.2%

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

3. Taylor expanded in z around 0

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

   1. associate-*r* N/A

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

   2. mul-1-neg N/A

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

   3. fp-cancel-sub-sign N/A

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

   4. div-sub N/A

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

   5. associate-/l* N/A

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

   6. *-inverses N/A

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

   7. *-rgt-identity N/A

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

   8. lower--.f64 N/A

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

   9. lower-/.f64 59.9

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

5. Applied rewrites 59.9%

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

6. Taylor expanded in y around 0

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

8. Applied rewrites 23.9%

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

Developer Target 1: 95.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \left(\frac{y}{z} - t \cdot \frac{1}{1 - z}\right)\\ t_2 := x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right)\\ \mathbf{if}\;t\_2 < -7.623226303312042 \cdot 10^{-196}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 < 1.4133944927702302 \cdot 10^{-211}:\\ \;\;\;\;\frac{y \cdot x}{z} + \left(-\frac{t \cdot x}{1 - z}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* x (- (/ y z) (* t (/ 1.0 (- 1.0 z))))))
        (t_2 (* x (- (/ y z) (/ t (- 1.0 z))))))
   (if (< t_2 -7.623226303312042e-196)
     t_1
     (if (< t_2 1.4133944927702302e-211)
       (+ (/ (* y x) z) (- (/ (* t x) (- 1.0 z))))
       t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x * ((y / z) - (t * (1.0 / (1.0 - z))));
	double t_2 = x * ((y / z) - (t / (1.0 - z)));
	double tmp;
	if (t_2 < -7.623226303312042e-196) {
		tmp = t_1;
	} else if (t_2 < 1.4133944927702302e-211) {
		tmp = ((y * x) / z) + -((t * x) / (1.0 - z));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = x * ((y / z) - (t * (1.0d0 / (1.0d0 - z))))
    t_2 = x * ((y / z) - (t / (1.0d0 - z)))
    if (t_2 < (-7.623226303312042d-196)) then
        tmp = t_1
    else if (t_2 < 1.4133944927702302d-211) then
        tmp = ((y * x) / z) + -((t * x) / (1.0d0 - z))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = x * ((y / z) - (t * (1.0 / (1.0 - z))));
	double t_2 = x * ((y / z) - (t / (1.0 - z)));
	double tmp;
	if (t_2 < -7.623226303312042e-196) {
		tmp = t_1;
	} else if (t_2 < 1.4133944927702302e-211) {
		tmp = ((y * x) / z) + -((t * x) / (1.0 - z));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = x * ((y / z) - (t * (1.0 / (1.0 - z))))
	t_2 = x * ((y / z) - (t / (1.0 - z)))
	tmp = 0
	if t_2 < -7.623226303312042e-196:
		tmp = t_1
	elif t_2 < 1.4133944927702302e-211:
		tmp = ((y * x) / z) + -((t * x) / (1.0 - z))
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(x * Float64(Float64(y / z) - Float64(t * Float64(1.0 / Float64(1.0 - z)))))
	t_2 = Float64(x * Float64(Float64(y / z) - Float64(t / Float64(1.0 - z))))
	tmp = 0.0
	if (t_2 < -7.623226303312042e-196)
		tmp = t_1;
	elseif (t_2 < 1.4133944927702302e-211)
		tmp = Float64(Float64(Float64(y * x) / z) + Float64(-Float64(Float64(t * x) / Float64(1.0 - z))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = x * ((y / z) - (t * (1.0 / (1.0 - z))));
	t_2 = x * ((y / z) - (t / (1.0 - z)));
	tmp = 0.0;
	if (t_2 < -7.623226303312042e-196)
		tmp = t_1;
	elseif (t_2 < 1.4133944927702302e-211)
		tmp = ((y * x) / z) + -((t * x) / (1.0 - z));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x * N[(N[(y / z), $MachinePrecision] - N[(t * N[(1.0 / N[(1.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(x * N[(N[(y / z), $MachinePrecision] - N[(t / N[(1.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Less[t$95$2, -7.623226303312042e-196], t$95$1, If[Less[t$95$2, 1.4133944927702302e-211], N[(N[(N[(y * x), $MachinePrecision] / z), $MachinePrecision] + (-N[(N[(t * x), $MachinePrecision] / N[(1.0 - z), $MachinePrecision]), $MachinePrecision])), $MachinePrecision], t$95$1]]]]

Reproduce

herbie shell --seed 2024338 
(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)))))