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

Percentage Accurate: 95.0% → 95.0%

Time: 8.6s

Alternatives: 15

Speedup: 1.0×

• 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: 95.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 95.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 95.0%

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

3. Final simplification 95.0%

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

Alternative 2: 81.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t + y}{z} \cdot x\\ \mathbf{if}\;z \leq -7.5 \cdot 10^{-159}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 6.4 \cdot 10^{-79}:\\ \;\;\;\;\frac{y \cdot x}{z}\\ \mathbf{elif}\;z \leq 7.6 \cdot 10^{-29}:\\ \;\;\;\;\frac{t \cdot x}{z - 1}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (/ (+ t y) z) x)))
   (if (<= z -7.5e-159)
     t_1
     (if (<= z 6.4e-79)
       (/ (* y x) z)
       (if (<= z 7.6e-29) (/ (* t x) (- z 1.0)) t_1)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = ((t + y) / z) * x;
	double tmp;
	if (z <= -7.5e-159) {
		tmp = t_1;
	} else if (z <= 6.4e-79) {
		tmp = (y * x) / z;
	} else if (z <= 7.6e-29) {
		tmp = (t * x) / (z - 1.0);
	} 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 + y) / z) * x
    if (z <= (-7.5d-159)) then
        tmp = t_1
    else if (z <= 6.4d-79) then
        tmp = (y * x) / z
    else if (z <= 7.6d-29) then
        tmp = (t * x) / (z - 1.0d0)
    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 + y) / z) * x;
	double tmp;
	if (z <= -7.5e-159) {
		tmp = t_1;
	} else if (z <= 6.4e-79) {
		tmp = (y * x) / z;
	} else if (z <= 7.6e-29) {
		tmp = (t * x) / (z - 1.0);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = ((t + y) / z) * x
	tmp = 0
	if z <= -7.5e-159:
		tmp = t_1
	elif z <= 6.4e-79:
		tmp = (y * x) / z
	elif z <= 7.6e-29:
		tmp = (t * x) / (z - 1.0)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(Float64(t + y) / z) * x)
	tmp = 0.0
	if (z <= -7.5e-159)
		tmp = t_1;
	elseif (z <= 6.4e-79)
		tmp = Float64(Float64(y * x) / z);
	elseif (z <= 7.6e-29)
		tmp = Float64(Float64(t * x) / Float64(z - 1.0));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = ((t + y) / z) * x;
	tmp = 0.0;
	if (z <= -7.5e-159)
		tmp = t_1;
	elseif (z <= 6.4e-79)
		tmp = (y * x) / z;
	elseif (z <= 7.6e-29)
		tmp = (t * x) / (z - 1.0);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(N[(t + y), $MachinePrecision] / z), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[z, -7.5e-159], t$95$1, If[LessEqual[z, 6.4e-79], N[(N[(y * x), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[z, 7.6e-29], N[(N[(t * x), $MachinePrecision] / N[(z - 1.0), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if z < -7.5e-159 or 7.59999999999999951e-29 < z

   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 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. cancel-sign-sub-inv N/A

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

      3. metadata-eval N/A

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

      4. *-lft-identity N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 90.0

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

   5. Applied rewrites 90.0%

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

   if -7.5e-159 < z < 6.39999999999999975e-79

   1. Initial program 90.9%

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

   3. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 72.6

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

   5. Applied rewrites 72.6%

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

   if 6.39999999999999975e-79 < z < 7.59999999999999951e-29

   1. Initial program 100.0%

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

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{-1 \cdot \frac{t \cdot x}{1 - z}} \]
   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. sub-neg N/A

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

      6. mul-1-neg N/A

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

      7. +-commutative N/A

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

      8. distribute-neg-in N/A

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

      9. mul-1-neg N/A

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

      10. remove-double-neg N/A

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

      11. sub-neg N/A

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

      12. lower--.f64 83.9

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

   5. Applied rewrites 83.9%

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

4. Final simplification 84.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -7.5 \cdot 10^{-159}:\\ \;\;\;\;\frac{t + y}{z} \cdot x\\ \mathbf{elif}\;z \leq 6.4 \cdot 10^{-79}:\\ \;\;\;\;\frac{y \cdot x}{z}\\ \mathbf{elif}\;z \leq 7.6 \cdot 10^{-29}:\\ \;\;\;\;\frac{t \cdot x}{z - 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{t + y}{z} \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 95.2% accurate, 0.9× speedup

Math

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

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -1.0)
   (* (/ (+ t y) z) x)
   (if (<= z 1.0) (* (/ x z) (fma (- z) t y)) (/ x (/ z (+ t y))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -1.0) {
		tmp = ((t + y) / z) * x;
	} else if (z <= 1.0) {
		tmp = (x / z) * fma(-z, t, y);
	} else {
		tmp = x / (z / (t + y));
	}
	return tmp;
}

Julia

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

Wolfram

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

Derivation

1. Split input into 3 regimes

2. if z < -1

   1. Initial program 96.6%

      \[x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \]
   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. cancel-sign-sub-inv N/A

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

      3. metadata-eval N/A

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

      4. *-lft-identity N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 95.4

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

   5. Applied rewrites 95.4%

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

   if -1 < z < 1

   1. Initial program 94.0%

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

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

      9. times-frac N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

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

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-neg.f64 N/A

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

      17. lower-/.f64 95.3

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

   4. Applied rewrites 95.3%

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

   5. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. lower-fma.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 93.8

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

   7. Applied rewrites 93.8%

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

   if 1 < z

   1. Initial program 95.6%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. flip-- N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. clear-num N/A

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

      8. flip-- N/A

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

      9. lift--.f64 N/A

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

      10. lower-/.f64 95.5

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

   4. Applied rewrites 95.5%

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

   5. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. sub-neg N/A

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

      3. mul-1-neg N/A

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

      4. remove-double-neg N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 94.3

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

   7. Applied rewrites 94.3%

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

4. Final simplification 94.3%

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

Alternative 4: 95.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t + y}{z} \cdot x\\ \mathbf{if}\;z \leq -1:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1.8 \cdot 10^{-8}:\\ \;\;\;\;\frac{x}{z} \cdot \mathsf{fma}\left(-z, t, y\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (/ (+ t y) z) x)))
   (if (<= z -1.0) t_1 (if (<= z 1.8e-8) (* (/ x z) (fma (- z) t y)) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = ((t + y) / z) * x;
	double tmp;
	if (z <= -1.0) {
		tmp = t_1;
	} else if (z <= 1.8e-8) {
		tmp = (x / z) * fma(-z, t, y);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(Float64(t + y) / z) * x)
	tmp = 0.0
	if (z <= -1.0)
		tmp = t_1;
	elseif (z <= 1.8e-8)
		tmp = Float64(Float64(x / z) * fma(Float64(-z), t, y));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(N[(t + y), $MachinePrecision] / z), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[z, -1.0], t$95$1, If[LessEqual[z, 1.8e-8], N[(N[(x / z), $MachinePrecision] * N[((-z) * t + y), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -1 or 1.79999999999999991e-8 < z

   1. Initial program 96.1%

      \[x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \]
   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. cancel-sign-sub-inv N/A

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

      3. metadata-eval N/A

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

      4. *-lft-identity N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 94.8

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

   5. Applied rewrites 94.8%

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

   if -1 < z < 1.79999999999999991e-8

   1. Initial program 93.9%

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

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

      9. times-frac N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

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

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-neg.f64 N/A

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

      17. lower-/.f64 95.3

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

   4. Applied rewrites 95.3%

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

   5. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. lower-fma.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 93.7

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

   7. Applied rewrites 93.7%

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

4. Final simplification 94.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1:\\ \;\;\;\;\frac{t + y}{z} \cdot x\\ \mathbf{elif}\;z \leq 1.8 \cdot 10^{-8}:\\ \;\;\;\;\frac{x}{z} \cdot \mathsf{fma}\left(-z, t, y\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t + y}{z} \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 94.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t + y}{z} \cdot x\\ \mathbf{if}\;z \leq -1:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1.8 \cdot 10^{-8}:\\ \;\;\;\;\frac{y - t \cdot z}{z} \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (/ (+ t y) z) x)))
   (if (<= z -1.0) t_1 (if (<= z 1.8e-8) (* (/ (- y (* t z)) z) x) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = ((t + y) / z) * x;
	double tmp;
	if (z <= -1.0) {
		tmp = t_1;
	} else if (z <= 1.8e-8) {
		tmp = ((y - (t * z)) / z) * x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = ((t + y) / z) * x
    if (z <= (-1.0d0)) then
        tmp = t_1
    else if (z <= 1.8d-8) then
        tmp = ((y - (t * z)) / z) * x
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = ((t + y) / z) * x;
	double tmp;
	if (z <= -1.0) {
		tmp = t_1;
	} else if (z <= 1.8e-8) {
		tmp = ((y - (t * z)) / z) * x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = ((t + y) / z) * x
	tmp = 0
	if z <= -1.0:
		tmp = t_1
	elif z <= 1.8e-8:
		tmp = ((y - (t * z)) / z) * x
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(Float64(t + y) / z) * x)
	tmp = 0.0
	if (z <= -1.0)
		tmp = t_1;
	elseif (z <= 1.8e-8)
		tmp = Float64(Float64(Float64(y - Float64(t * z)) / z) * x);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = ((t + y) / z) * x;
	tmp = 0.0;
	if (z <= -1.0)
		tmp = t_1;
	elseif (z <= 1.8e-8)
		tmp = ((y - (t * z)) / z) * x;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(N[(t + y), $MachinePrecision] / z), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[z, -1.0], t$95$1, If[LessEqual[z, 1.8e-8], N[(N[(N[(y - N[(t * z), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision] * x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -1 or 1.79999999999999991e-8 < z

   1. Initial program 96.1%

      \[x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \]
   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. cancel-sign-sub-inv N/A

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

      3. metadata-eval N/A

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

      4. *-lft-identity N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 94.8

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

   5. Applied rewrites 94.8%

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

   if -1 < z < 1.79999999999999991e-8

   1. Initial program 93.9%

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

   3. Taylor expanded in z around 0

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

      1. lower-/.f64 N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 92.3

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

   5. Applied rewrites 92.3%

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

4. Final simplification 93.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1:\\ \;\;\;\;\frac{t + y}{z} \cdot x\\ \mathbf{elif}\;z \leq 1.8 \cdot 10^{-8}:\\ \;\;\;\;\frac{y - t \cdot z}{z} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{t + y}{z} \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 77.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t}{z - 1} \cdot x\\ \mathbf{if}\;t \leq -1.08 \cdot 10^{+151}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 4.9 \cdot 10^{+95}:\\ \;\;\;\;\frac{\left(t + y\right) \cdot x}{z}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (/ t (- z 1.0)) x)))
   (if (<= t -1.08e+151) t_1 (if (<= t 4.9e+95) (/ (* (+ t y) x) z) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (t / (z - 1.0)) * x;
	double tmp;
	if (t <= -1.08e+151) {
		tmp = t_1;
	} else if (t <= 4.9e+95) {
		tmp = ((t + 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 / (z - 1.0d0)) * x
    if (t <= (-1.08d+151)) then
        tmp = t_1
    else if (t <= 4.9d+95) then
        tmp = ((t + 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 / (z - 1.0)) * x;
	double tmp;
	if (t <= -1.08e+151) {
		tmp = t_1;
	} else if (t <= 4.9e+95) {
		tmp = ((t + y) * x) / z;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (t / (z - 1.0)) * x
	tmp = 0
	if t <= -1.08e+151:
		tmp = t_1
	elif t <= 4.9e+95:
		tmp = ((t + y) * x) / z
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(t / Float64(z - 1.0)) * x)
	tmp = 0.0
	if (t <= -1.08e+151)
		tmp = t_1;
	elseif (t <= 4.9e+95)
		tmp = Float64(Float64(Float64(t + y) * x) / z);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (t / (z - 1.0)) * x;
	tmp = 0.0;
	if (t <= -1.08e+151)
		tmp = t_1;
	elseif (t <= 4.9e+95)
		tmp = ((t + 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 / N[(z - 1.0), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[t, -1.08e+151], t$95$1, If[LessEqual[t, 4.9e+95], N[(N[(N[(t + y), $MachinePrecision] * x), $MachinePrecision] / z), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -1.08000000000000003e151 or 4.8999999999999999e95 < t

   1. Initial program 95.2%

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

   3. Taylor expanded in t around inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. lower-/.f64 N/A

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

      4. sub-neg N/A

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

      5. mul-1-neg N/A

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

      6. +-commutative N/A

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

      7. distribute-neg-in N/A

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

      8. mul-1-neg N/A

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

      9. remove-double-neg N/A

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

      10. sub-neg N/A

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

      11. lower--.f64 77.5

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

   5. Applied rewrites 77.5%

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

   if -1.08000000000000003e151 < t < 4.8999999999999999e95

   1. Initial program 94.9%

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

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

      3. flip-- N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. clear-num N/A

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

      8. flip-- N/A

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

      9. lift--.f64 N/A

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

      10. lower-/.f64 94.9

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

   4. Applied rewrites 94.9%

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

   5. Taylor expanded in z around inf

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

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

      5. mul-1-neg N/A

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

      6. remove-double-neg 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 82.9

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

   7. Applied rewrites 82.9%

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

4. Final simplification 81.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.08 \cdot 10^{+151}:\\ \;\;\;\;\frac{t}{z - 1} \cdot x\\ \mathbf{elif}\;t \leq 4.9 \cdot 10^{+95}:\\ \;\;\;\;\frac{\left(t + y\right) \cdot x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{t}{z - 1} \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 71.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t \cdot x}{z - 1}\\ \mathbf{if}\;t \leq -3 \cdot 10^{-50}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 7.8 \cdot 10^{+87}:\\ \;\;\;\;\frac{y}{z} \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (* t x) (- z 1.0))))
   (if (<= t -3e-50) t_1 (if (<= t 7.8e+87) (* (/ y z) x) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (t * x) / (z - 1.0);
	double tmp;
	if (t <= -3e-50) {
		tmp = t_1;
	} else if (t <= 7.8e+87) {
		tmp = (y / z) * x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (t * x) / (z - 1.0d0)
    if (t <= (-3d-50)) then
        tmp = t_1
    else if (t <= 7.8d+87) then
        tmp = (y / z) * x
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (t * x) / (z - 1.0);
	double tmp;
	if (t <= -3e-50) {
		tmp = t_1;
	} else if (t <= 7.8e+87) {
		tmp = (y / z) * x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (t * x) / (z - 1.0)
	tmp = 0
	if t <= -3e-50:
		tmp = t_1
	elif t <= 7.8e+87:
		tmp = (y / z) * x
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(t * x) / Float64(z - 1.0))
	tmp = 0.0
	if (t <= -3e-50)
		tmp = t_1;
	elseif (t <= 7.8e+87)
		tmp = Float64(Float64(y / z) * x);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (t * x) / (z - 1.0);
	tmp = 0.0;
	if (t <= -3e-50)
		tmp = t_1;
	elseif (t <= 7.8e+87)
		tmp = (y / z) * x;
	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[(z - 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -3e-50], t$95$1, If[LessEqual[t, 7.8e+87], N[(N[(y / z), $MachinePrecision] * x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -2.9999999999999999e-50 or 7.80000000000000039e87 < t

   1. Initial program 95.7%

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

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{-1 \cdot \frac{t \cdot x}{1 - z}} \]
   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. sub-neg N/A

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

      6. mul-1-neg N/A

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

      7. +-commutative N/A

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

      8. distribute-neg-in N/A

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

      9. mul-1-neg N/A

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

      10. remove-double-neg N/A

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

      11. sub-neg N/A

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

      12. lower--.f64 71.7

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

   5. Applied rewrites 71.7%

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

   if -2.9999999999999999e-50 < t < 7.80000000000000039e87

   1. Initial program 94.4%

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

   3. Taylor expanded in t around 0

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

      1. lower-/.f64 84.4

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

   5. Applied rewrites 84.4%

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

4. Final simplification 78.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -3 \cdot 10^{-50}:\\ \;\;\;\;\frac{t \cdot x}{z - 1}\\ \mathbf{elif}\;t \leq 7.8 \cdot 10^{+87}:\\ \;\;\;\;\frac{y}{z} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{t \cdot x}{z - 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 63.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.8 \cdot 10^{-49}:\\ \;\;\;\;\frac{t}{z} \cdot x\\ \mathbf{elif}\;t \leq 1.75 \cdot 10^{+107}:\\ \;\;\;\;\frac{y}{z} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-1 - z\right) \cdot t\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t -1.8e-49)
   (* (/ t z) x)
   (if (<= t 1.75e+107) (* (/ y z) x) (* (* (- -1.0 z) t) x))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -1.8e-49) {
		tmp = (t / z) * x;
	} else if (t <= 1.75e+107) {
		tmp = (y / z) * x;
	} else {
		tmp = ((-1.0 - z) * t) * x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-1.8d-49)) then
        tmp = (t / z) * x
    else if (t <= 1.75d+107) then
        tmp = (y / z) * x
    else
        tmp = (((-1.0d0) - z) * t) * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -1.8e-49) {
		tmp = (t / z) * x;
	} else if (t <= 1.75e+107) {
		tmp = (y / z) * x;
	} else {
		tmp = ((-1.0 - z) * t) * x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if t <= -1.8e-49:
		tmp = (t / z) * x
	elif t <= 1.75e+107:
		tmp = (y / z) * x
	else:
		tmp = ((-1.0 - z) * t) * x
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= -1.8e-49)
		tmp = Float64(Float64(t / z) * x);
	elseif (t <= 1.75e+107)
		tmp = Float64(Float64(y / z) * x);
	else
		tmp = Float64(Float64(Float64(-1.0 - z) * t) * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (t <= -1.8e-49)
		tmp = (t / z) * x;
	elseif (t <= 1.75e+107)
		tmp = (y / z) * x;
	else
		tmp = ((-1.0 - z) * t) * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, -1.8e-49], N[(N[(t / z), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[t, 1.75e+107], N[(N[(y / z), $MachinePrecision] * x), $MachinePrecision], N[(N[(N[(-1.0 - z), $MachinePrecision] * t), $MachinePrecision] * x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -1.79999999999999985e-49

   1. Initial program 94.3%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

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

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

      3. metadata-eval N/A

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

      4. *-lft-identity N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 71.5

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

   5. Applied rewrites 71.5%

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

   6. Taylor expanded in t around inf

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

   8. Applied rewrites 58.9%

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

   if -1.79999999999999985e-49 < t < 1.7499999999999999e107

   1. Initial program 94.6%

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

   3. Taylor expanded in t around 0

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

      1. lower-/.f64 82.7

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

   5. Applied rewrites 82.7%

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

   if 1.7499999999999999e107 < t

   1. Initial program 97.6%

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

   3. Taylor expanded in z around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. distribute-lft-out N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. distribute-rgt-out N/A

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

      11. lower-*.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. mul-1-neg N/A

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

      15. lower-neg.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 64.6

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

   5. Applied rewrites 64.6%

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

   6. Taylor expanded in t around inf

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

   8. Applied rewrites 55.4%

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

4. Final simplification 71.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.8 \cdot 10^{-49}:\\ \;\;\;\;\frac{t}{z} \cdot x\\ \mathbf{elif}\;t \leq 1.75 \cdot 10^{+107}:\\ \;\;\;\;\frac{y}{z} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-1 - z\right) \cdot t\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 61.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.66 \cdot 10^{-49}:\\ \;\;\;\;\frac{t}{z} \cdot x\\ \mathbf{elif}\;t \leq 1.75 \cdot 10^{+107}:\\ \;\;\;\;\frac{y \cdot x}{z}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-1 - z\right) \cdot t\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t -1.66e-49)
   (* (/ t z) x)
   (if (<= t 1.75e+107) (/ (* y x) z) (* (* (- -1.0 z) t) x))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -1.66e-49) {
		tmp = (t / z) * x;
	} else if (t <= 1.75e+107) {
		tmp = (y * x) / z;
	} else {
		tmp = ((-1.0 - z) * t) * x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-1.66d-49)) then
        tmp = (t / z) * x
    else if (t <= 1.75d+107) then
        tmp = (y * x) / z
    else
        tmp = (((-1.0d0) - z) * t) * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -1.66e-49) {
		tmp = (t / z) * x;
	} else if (t <= 1.75e+107) {
		tmp = (y * x) / z;
	} else {
		tmp = ((-1.0 - z) * t) * x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if t <= -1.66e-49:
		tmp = (t / z) * x
	elif t <= 1.75e+107:
		tmp = (y * x) / z
	else:
		tmp = ((-1.0 - z) * t) * x
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= -1.66e-49)
		tmp = Float64(Float64(t / z) * x);
	elseif (t <= 1.75e+107)
		tmp = Float64(Float64(y * x) / z);
	else
		tmp = Float64(Float64(Float64(-1.0 - z) * t) * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (t <= -1.66e-49)
		tmp = (t / z) * x;
	elseif (t <= 1.75e+107)
		tmp = (y * x) / z;
	else
		tmp = ((-1.0 - z) * t) * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, -1.66e-49], N[(N[(t / z), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[t, 1.75e+107], N[(N[(y * x), $MachinePrecision] / z), $MachinePrecision], N[(N[(N[(-1.0 - z), $MachinePrecision] * t), $MachinePrecision] * x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -1.6599999999999999e-49

   1. Initial program 94.3%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

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

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

      3. metadata-eval N/A

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

      4. *-lft-identity N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 71.5

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

   5. Applied rewrites 71.5%

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

   6. Taylor expanded in t around inf

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

   8. Applied rewrites 58.9%

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

   if -1.6599999999999999e-49 < t < 1.7499999999999999e107

   1. Initial program 94.6%

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

   3. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 80.7

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

   5. Applied rewrites 80.7%

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

   if 1.7499999999999999e107 < t

   1. Initial program 97.6%

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

   3. Taylor expanded in z around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. distribute-lft-out N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. distribute-rgt-out N/A

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

      11. lower-*.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. mul-1-neg N/A

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

      15. lower-neg.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 64.6

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

   5. Applied rewrites 64.6%

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

   6. Taylor expanded in t around inf

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

   8. Applied rewrites 55.4%

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

4. Final simplification 70.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.66 \cdot 10^{-49}:\\ \;\;\;\;\frac{t}{z} \cdot x\\ \mathbf{elif}\;t \leq 1.75 \cdot 10^{+107}:\\ \;\;\;\;\frac{y \cdot x}{z}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-1 - z\right) \cdot t\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 73.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 1.2 \cdot 10^{+88}:\\ \;\;\;\;\frac{\left(t + y\right) \cdot x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{t \cdot x}{z - 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t 1.2e+88) (/ (* (+ t y) x) z) (/ (* t x) (- z 1.0))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= 1.2e+88) {
		tmp = ((t + y) * x) / z;
	} else {
		tmp = (t * x) / (z - 1.0);
	}
	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 <= 1.2d+88) then
        tmp = ((t + y) * x) / z
    else
        tmp = (t * x) / (z - 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= 1.2e+88) {
		tmp = ((t + y) * x) / z;
	} else {
		tmp = (t * x) / (z - 1.0);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if t <= 1.2e+88:
		tmp = ((t + y) * x) / z
	else:
		tmp = (t * x) / (z - 1.0)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= 1.2e+88)
		tmp = Float64(Float64(Float64(t + y) * x) / z);
	else
		tmp = Float64(Float64(t * x) / Float64(z - 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (t <= 1.2e+88)
		tmp = ((t + y) * x) / z;
	else
		tmp = (t * x) / (z - 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, 1.2e+88], N[(N[(N[(t + y), $MachinePrecision] * x), $MachinePrecision] / z), $MachinePrecision], N[(N[(t * x), $MachinePrecision] / N[(z - 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 1.2e88

   1. Initial program 94.4%

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

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

      3. flip-- N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. clear-num N/A

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

      8. flip-- N/A

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

      9. lift--.f64 N/A

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

      10. lower-/.f64 94.3

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

   4. Applied rewrites 94.3%

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

   5. Taylor expanded in z around inf

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

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

      5. mul-1-neg N/A

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

      6. remove-double-neg 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 78.4

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

   7. Applied rewrites 78.4%

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

   if 1.2e88 < t

   1. Initial program 97.8%

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

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{-1 \cdot \frac{t \cdot x}{1 - z}} \]
   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. sub-neg N/A

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

      6. mul-1-neg N/A

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

      7. +-commutative N/A

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

      8. distribute-neg-in N/A

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

      9. mul-1-neg N/A

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

      10. remove-double-neg N/A

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

      11. sub-neg N/A

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

      12. lower--.f64 81.4

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

   5. Applied rewrites 81.4%

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

Alternative 11: 60.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 1.75 \cdot 10^{+107}:\\ \;\;\;\;\frac{y \cdot x}{z}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-1 - z\right) \cdot t\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t 1.75e+107) (/ (* y x) z) (* (* (- -1.0 z) t) x)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= 1.75e+107) {
		tmp = (y * x) / z;
	} else {
		tmp = ((-1.0 - z) * t) * x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= 1.75d+107) then
        tmp = (y * x) / z
    else
        tmp = (((-1.0d0) - z) * t) * x
    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+107) {
		tmp = (y * x) / z;
	} else {
		tmp = ((-1.0 - z) * t) * x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if t <= 1.75e+107:
		tmp = (y * x) / z
	else:
		tmp = ((-1.0 - z) * t) * x
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= 1.75e+107)
		tmp = Float64(Float64(y * x) / z);
	else
		tmp = Float64(Float64(Float64(-1.0 - z) * t) * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (t <= 1.75e+107)
		tmp = (y * x) / z;
	else
		tmp = ((-1.0 - z) * t) * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, 1.75e+107], N[(N[(y * x), $MachinePrecision] / z), $MachinePrecision], N[(N[(N[(-1.0 - z), $MachinePrecision] * t), $MachinePrecision] * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 1.7499999999999999e107

   1. Initial program 94.5%

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

   3. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 67.0

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

   5. Applied rewrites 67.0%

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

   if 1.7499999999999999e107 < t

   1. Initial program 97.6%

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

   3. Taylor expanded in z around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. distribute-lft-out N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. distribute-rgt-out N/A

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

      11. lower-*.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. mul-1-neg N/A

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

      15. lower-neg.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 64.6

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

   5. Applied rewrites 64.6%

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

   6. Taylor expanded in t around inf

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

   8. Applied rewrites 55.4%

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

4. Final simplification 65.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.75 \cdot 10^{+107}:\\ \;\;\;\;\frac{y \cdot x}{z}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-1 - z\right) \cdot t\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 61.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 1.3 \cdot 10^{+107}:\\ \;\;\;\;\frac{x}{z} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-1 - z\right) \cdot t\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t 1.3e+107) (* (/ x z) y) (* (* (- -1.0 z) t) x)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= 1.3e+107) {
		tmp = (x / z) * y;
	} else {
		tmp = ((-1.0 - z) * t) * x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= 1.3d+107) then
        tmp = (x / z) * y
    else
        tmp = (((-1.0d0) - z) * t) * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= 1.3e+107) {
		tmp = (x / z) * y;
	} else {
		tmp = ((-1.0 - z) * t) * x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if t <= 1.3e+107:
		tmp = (x / z) * y
	else:
		tmp = ((-1.0 - z) * t) * x
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= 1.3e+107)
		tmp = Float64(Float64(x / z) * y);
	else
		tmp = Float64(Float64(Float64(-1.0 - z) * t) * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (t <= 1.3e+107)
		tmp = (x / z) * y;
	else
		tmp = ((-1.0 - z) * t) * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, 1.3e+107], N[(N[(x / z), $MachinePrecision] * y), $MachinePrecision], N[(N[(N[(-1.0 - z), $MachinePrecision] * t), $MachinePrecision] * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 1.3000000000000001e107

   1. Initial program 94.5%

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

   3. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 67.0

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

   5. Applied rewrites 67.0%

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

   7. Applied rewrites 65.0%

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

   if 1.3000000000000001e107 < t

   1. Initial program 97.6%

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

   3. Taylor expanded in z around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. distribute-lft-out N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. distribute-rgt-out N/A

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

      11. lower-*.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. mul-1-neg N/A

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

      15. lower-neg.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 64.6

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

   5. Applied rewrites 64.6%

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

   6. Taylor expanded in t around inf

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

   8. Applied rewrites 55.4%

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

4. Final simplification 63.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.3 \cdot 10^{+107}:\\ \;\;\;\;\frac{x}{z} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-1 - z\right) \cdot t\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 20.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 75:\\ \;\;\;\;\left(\left(-1 - z\right) \cdot t\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-x\right) \cdot t\right) \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z 75.0) (* (* (- -1.0 z) t) x) (* (* (- x) t) z)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 75.0) {
		tmp = ((-1.0 - z) * t) * x;
	} else {
		tmp = (-x * t) * z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (z <= 75.0d0) then
        tmp = (((-1.0d0) - z) * t) * x
    else
        tmp = (-x * t) * z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 75.0) {
		tmp = ((-1.0 - z) * t) * x;
	} else {
		tmp = (-x * t) * z;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= 75.0:
		tmp = ((-1.0 - z) * t) * x
	else:
		tmp = (-x * t) * z
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= 75.0)
		tmp = Float64(Float64(Float64(-1.0 - z) * t) * x);
	else
		tmp = Float64(Float64(Float64(-x) * t) * z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= 75.0)
		tmp = ((-1.0 - z) * t) * x;
	else
		tmp = (-x * t) * z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, 75.0], N[(N[(N[(-1.0 - z), $MachinePrecision] * t), $MachinePrecision] * x), $MachinePrecision], N[(N[((-x) * t), $MachinePrecision] * z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 75

   1. Initial program 94.8%

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

   3. Taylor expanded in z around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. distribute-lft-out N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. distribute-rgt-out N/A

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

      11. lower-*.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. mul-1-neg N/A

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

      15. lower-neg.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 71.8

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

   5. Applied rewrites 71.8%

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

   6. Taylor expanded in t around inf

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

   8. Applied rewrites 27.9%

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

   if 75 < z

   1. Initial program 95.6%

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

   3. Taylor expanded in z around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. distribute-lft-out N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. distribute-rgt-out N/A

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

      11. lower-*.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. mul-1-neg N/A

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

      15. lower-neg.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 14.6

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

   5. Applied rewrites 14.6%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 6.5%

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

   10. Applied rewrites 14.5%

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

4. Final simplification 24.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 75:\\ \;\;\;\;\left(\left(-1 - z\right) \cdot t\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-x\right) \cdot t\right) \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 10.3% accurate, 2.6× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z, double t) {
	return (-x * t) * 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 * t) * z
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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 0

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

   1. lower-/.f64 N/A

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

   2. +-commutative N/A

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

   3. *-commutative N/A

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

   4. distribute-lft-out N/A

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

   5. *-commutative N/A

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

   6. associate-*l* N/A

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

   7. lower-fma.f64 N/A

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

   8. *-commutative N/A

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

   9. associate-*r* N/A

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

   10. distribute-rgt-out N/A

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

   11. lower-*.f64 N/A

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

   12. +-commutative N/A

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

   13. lower-fma.f64 N/A

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

   14. mul-1-neg N/A

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

   15. lower-neg.f64 N/A

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

   16. *-commutative N/A

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

   17. lower-*.f64 56.0

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

5. Applied rewrites 56.0%

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

6. Taylor expanded in z around inf

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

8. Applied rewrites 6.6%

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

10. Applied rewrites 10.6%

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

Alternative 15: 5.9% accurate, 2.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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 0

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

   1. lower-/.f64 N/A

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

   2. +-commutative N/A

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

   3. *-commutative N/A

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

   4. distribute-lft-out N/A

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

   5. *-commutative N/A

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

   6. associate-*l* N/A

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

   7. lower-fma.f64 N/A

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

   8. *-commutative N/A

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

   9. associate-*r* N/A

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

   10. distribute-rgt-out N/A

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

   11. lower-*.f64 N/A

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

   12. +-commutative N/A

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

   13. lower-fma.f64 N/A

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

   14. mul-1-neg N/A

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

   15. lower-neg.f64 N/A

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

   16. *-commutative N/A

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

   17. lower-*.f64 56.0

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

5. Applied rewrites 56.0%

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

6. Taylor expanded in z around inf

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

8. Applied rewrites 6.6%

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

10. Applied rewrites 6.6%

    \[\leadsto \left(t \cdot z\right) \cdot \left(-x\right) \]

11. Final simplification 6.6%

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

Developer Target 1: 95.5% 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 2024254 
(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)))))