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

Percentage Accurate: 94.6% → 94.6%

Time: 8.5s

Alternatives: 14

Speedup: 1.0×

• 20.5% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 94.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 94.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 95.8%

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

Alternative 2: 93.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -0.75 \lor \neg \left(z \leq 0.001\right):\\ \;\;\;\;x \cdot \frac{t + y}{z}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(z, x, x\right), -t, \frac{y}{z} \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -0.75) (not (<= z 0.001)))
   (* x (/ (+ t y) z))
   (fma (fma z x x) (- t) (* (/ y z) x))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -0.75) || !(z <= 0.001)) {
		tmp = x * ((t + y) / z);
	} else {
		tmp = fma(fma(z, x, x), -t, ((y / z) * x));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((z <= -0.75) || !(z <= 0.001))
		tmp = Float64(x * Float64(Float64(t + y) / z));
	else
		tmp = fma(fma(z, x, x), Float64(-t), Float64(Float64(y / z) * x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[z, -0.75], N[Not[LessEqual[z, 0.001]], $MachinePrecision]], N[(x * N[(N[(t + y), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], N[(N[(z * x + x), $MachinePrecision] * (-t) + N[(N[(y / z), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -0.75 or 1e-3 < z

   1. Initial program 96.5%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

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

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

      3. metadata-eval N/A

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

      4. *-lft-identity N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 96.0

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

   5. Applied rewrites 96.0%

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

   if -0.75 < z < 1e-3

   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. +-commutative N/A

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

      2. div-add N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. distribute-lft-out N/A

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

      6. *-commutative N/A

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

      7. *-inverses N/A

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

      8. associate-*l* N/A

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

      9. metadata-eval N/A

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

      10. *-commutative N/A

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

      11. distribute-lft-out N/A

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

      12. associate-*r* N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

   5. Applied rewrites 95.0%

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

4. Final simplification 95.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -0.75 \lor \neg \left(z \leq 0.001\right):\\ \;\;\;\;x \cdot \frac{t + y}{z}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(z, x, x\right), -t, \frac{y}{z} \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 93.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -0.98 \lor \neg \left(z \leq 0.001\right):\\ \;\;\;\;x \cdot \frac{t + y}{z}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-t, x, \frac{y}{z} \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -0.98) (not (<= z 0.001)))
   (* x (/ (+ t y) z))
   (fma (- t) x (* (/ y z) x))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -0.98) || !(z <= 0.001)) {
		tmp = x * ((t + y) / z);
	} else {
		tmp = fma(-t, x, ((y / z) * x));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((z <= -0.98) || !(z <= 0.001))
		tmp = Float64(x * Float64(Float64(t + y) / z));
	else
		tmp = fma(Float64(-t), x, Float64(Float64(y / z) * x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[z, -0.98], N[Not[LessEqual[z, 0.001]], $MachinePrecision]], N[(x * N[(N[(t + y), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], N[((-t) * x + N[(N[(y / z), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -0.97999999999999998 or 1e-3 < z

   1. Initial program 96.5%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

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

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

      3. metadata-eval N/A

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

      4. *-lft-identity N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 96.0

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

   5. Applied rewrites 96.0%

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

   if -0.97999999999999998 < z < 1e-3

   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

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. associate-*r* N/A

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

      8. distribute-rgt-out N/A

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

      9. lower-*.f64 N/A

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

      10. associate-*r* N/A

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

      11. mul-1-neg N/A

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

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

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

      13. lower--.f64 N/A

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

      14. lower-*.f64 94.0

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

   5. Applied rewrites 94.0%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 94.8%

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

4. Final simplification 95.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -0.98 \lor \neg \left(z \leq 0.001\right):\\ \;\;\;\;x \cdot \frac{t + y}{z}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-t, x, \frac{y}{z} \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 93.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1 \lor \neg \left(z \leq 0.001\right):\\ \;\;\;\;x \cdot \frac{t + y}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -1.0) (not (<= z 0.001)))
   (* x (/ (+ t y) z))
   (* x (- (/ y z) t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -1.0) || !(z <= 0.001)) {
		tmp = x * ((t + y) / z);
	} else {
		tmp = x * ((y / z) - t);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((z <= (-1.0d0)) .or. (.not. (z <= 0.001d0))) then
        tmp = x * ((t + y) / z)
    else
        tmp = x * ((y / z) - t)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -1.0) || !(z <= 0.001)) {
		tmp = x * ((t + y) / z);
	} else {
		tmp = x * ((y / z) - t);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z <= -1.0) or not (z <= 0.001):
		tmp = x * ((t + y) / z)
	else:
		tmp = x * ((y / z) - t)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((z <= -1.0) || !(z <= 0.001))
		tmp = Float64(x * Float64(Float64(t + y) / z));
	else
		tmp = Float64(x * Float64(Float64(y / z) - t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((z <= -1.0) || ~((z <= 0.001)))
		tmp = x * ((t + y) / z);
	else
		tmp = x * ((y / z) - t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[z, -1.0], N[Not[LessEqual[z, 0.001]], $MachinePrecision]], N[(x * N[(N[(t + y), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], N[(x * N[(N[(y / z), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1 or 1e-3 < z

   1. Initial program 96.5%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

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

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

      3. metadata-eval N/A

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

      4. *-lft-identity N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 96.0

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

   5. Applied rewrites 96.0%

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

   if -1 < z < 1e-3

   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

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

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

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

      4. div-sub N/A

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

      5. associate-/l* N/A

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

      6. *-inverses N/A

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

      7. *-rgt-identity N/A

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

      8. lower--.f64 N/A

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

      9. lower-/.f64 94.7

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

   5. Applied rewrites 94.7%

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

4. Final simplification 95.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1 \lor \neg \left(z \leq 0.001\right):\\ \;\;\;\;x \cdot \frac{t + y}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 78.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -4.7 \cdot 10^{-79} \lor \neg \left(z \leq 0.001\right):\\ \;\;\;\;\frac{x}{z} \cdot \left(y + t\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -4.7e-79) (not (<= z 0.001)))
   (* (/ x z) (+ y t))
   (* y (/ x z))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -4.7e-79) || !(z <= 0.001)) {
		tmp = (x / z) * (y + t);
	} else {
		tmp = y * (x / z);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((z <= (-4.7d-79)) .or. (.not. (z <= 0.001d0))) then
        tmp = (x / z) * (y + t)
    else
        tmp = y * (x / z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -4.7e-79) || !(z <= 0.001)) {
		tmp = (x / z) * (y + t);
	} else {
		tmp = y * (x / z);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z <= -4.7e-79) or not (z <= 0.001):
		tmp = (x / z) * (y + t)
	else:
		tmp = y * (x / z)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((z <= -4.7e-79) || !(z <= 0.001))
		tmp = Float64(Float64(x / z) * Float64(y + t));
	else
		tmp = Float64(y * Float64(x / z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((z <= -4.7e-79) || ~((z <= 0.001)))
		tmp = (x / z) * (y + t);
	else
		tmp = y * (x / z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[z, -4.7e-79], N[Not[LessEqual[z, 0.001]], $MachinePrecision]], N[(N[(x / z), $MachinePrecision] * N[(y + t), $MachinePrecision]), $MachinePrecision], N[(y * N[(x / z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -4.7000000000000002e-79 or 1e-3 < z

   1. Initial program 96.9%

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

   3. Taylor expanded in t around inf

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

      1. distribute-rgt-in N/A

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

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

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

      3. mul-1-neg N/A

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

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

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

      5. *-commutative N/A

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

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

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

      7. *-commutative N/A

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

      8. mul-1-neg N/A

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

      9. distribute-rgt-in N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

   5. Applied rewrites 72.2%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 91.4%

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

   9. Taylor expanded in z around inf

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

       1. lower-/.f64 N/A

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

       2. *-commutative N/A

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

       3. lower-*.f64 N/A

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

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

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

       5. metadata-eval N/A

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

       6. *-lft-identity N/A

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

       7. lower-+.f64 86.4

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

   11. Applied rewrites 86.4%

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

   13. Applied rewrites 84.8%

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

   if -4.7000000000000002e-79 < z < 1e-3

   1. Initial program 94.2%

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

   3. Taylor expanded in y around inf

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 70.0

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

   5. Applied rewrites 70.0%

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

   7. Applied rewrites 71.8%

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

4. Final simplification 79.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4.7 \cdot 10^{-79} \lor \neg \left(z \leq 0.001\right):\\ \;\;\;\;\frac{x}{z} \cdot \left(y + t\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 88.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1:\\ \;\;\;\;\frac{\left(t + y\right) \cdot x}{z}\\ \mathbf{elif}\;z \leq 0.001:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - t\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \left(y + t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -1.0)
   (/ (* (+ t y) x) z)
   (if (<= z 0.001) (* x (- (/ y z) t)) (* (/ x z) (+ y t)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -1.0) {
		tmp = ((t + y) * x) / z;
	} else if (z <= 0.001) {
		tmp = x * ((y / z) - t);
	} else {
		tmp = (x / z) * (y + t);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (z <= (-1.0d0)) then
        tmp = ((t + y) * x) / z
    else if (z <= 0.001d0) then
        tmp = x * ((y / z) - t)
    else
        tmp = (x / z) * (y + t)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -1.0) {
		tmp = ((t + y) * x) / z;
	} else if (z <= 0.001) {
		tmp = x * ((y / z) - t);
	} else {
		tmp = (x / z) * (y + t);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= -1.0:
		tmp = ((t + y) * x) / z
	elif z <= 0.001:
		tmp = x * ((y / z) - t)
	else:
		tmp = (x / z) * (y + t)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -1.0)
		tmp = Float64(Float64(Float64(t + y) * x) / z);
	elseif (z <= 0.001)
		tmp = Float64(x * Float64(Float64(y / z) - t));
	else
		tmp = Float64(Float64(x / z) * Float64(y + t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= -1.0)
		tmp = ((t + y) * x) / z;
	elseif (z <= 0.001)
		tmp = x * ((y / z) - t);
	else
		tmp = (x / z) * (y + t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -1.0], N[(N[(N[(t + y), $MachinePrecision] * x), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[z, 0.001], N[(x * N[(N[(y / z), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision], N[(N[(x / z), $MachinePrecision] * N[(y + t), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1

   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

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

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

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

      5. metadata-eval N/A

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

      6. *-lft-identity N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 89.1

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

   5. Applied rewrites 89.1%

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

   if -1 < z < 1e-3

   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

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

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

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

      4. div-sub N/A

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

      5. associate-/l* N/A

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

      6. *-inverses N/A

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

      7. *-rgt-identity N/A

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

      8. lower--.f64 N/A

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

      9. lower-/.f64 94.7

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

   5. Applied rewrites 94.7%

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

   if 1e-3 < z

   1. Initial program 96.1%

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

   3. Taylor expanded in t around inf

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

      1. distribute-rgt-in N/A

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

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

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

      3. mul-1-neg N/A

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

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

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

      5. *-commutative N/A

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

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

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

      7. *-commutative N/A

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

      8. mul-1-neg N/A

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

      9. distribute-rgt-in N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

   5. Applied rewrites 72.7%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 95.0%

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

   9. Taylor expanded in z around inf

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

       1. lower-/.f64 N/A

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

       2. *-commutative N/A

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

       3. lower-*.f64 N/A

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

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

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

       5. metadata-eval N/A

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

       6. *-lft-identity N/A

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

       7. lower-+.f64 88.7

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

   11. Applied rewrites 88.7%

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

   13. Applied rewrites 90.0%

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

Alternative 7: 78.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -4.6 \cdot 10^{-79}:\\ \;\;\;\;\frac{\left(t + y\right) \cdot x}{z}\\ \mathbf{elif}\;z \leq 0.001:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \left(y + t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -4.6e-79)
   (/ (* (+ t y) x) z)
   (if (<= z 0.001) (* y (/ x z)) (* (/ x z) (+ y t)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -4.6e-79) {
		tmp = ((t + y) * x) / z;
	} else if (z <= 0.001) {
		tmp = y * (x / z);
	} else {
		tmp = (x / z) * (y + t);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (z <= (-4.6d-79)) then
        tmp = ((t + y) * x) / z
    else if (z <= 0.001d0) then
        tmp = y * (x / z)
    else
        tmp = (x / z) * (y + t)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -4.6e-79) {
		tmp = ((t + y) * x) / z;
	} else if (z <= 0.001) {
		tmp = y * (x / z);
	} else {
		tmp = (x / z) * (y + t);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= -4.6e-79:
		tmp = ((t + y) * x) / z
	elif z <= 0.001:
		tmp = y * (x / z)
	else:
		tmp = (x / z) * (y + t)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -4.6e-79)
		tmp = Float64(Float64(Float64(t + y) * x) / z);
	elseif (z <= 0.001)
		tmp = Float64(y * Float64(x / z));
	else
		tmp = Float64(Float64(x / z) * Float64(y + t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= -4.6e-79)
		tmp = ((t + y) * x) / z;
	elseif (z <= 0.001)
		tmp = y * (x / z);
	else
		tmp = (x / z) * (y + t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -4.6e-79], N[(N[(N[(t + y), $MachinePrecision] * x), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[z, 0.001], N[(y * N[(x / z), $MachinePrecision]), $MachinePrecision], N[(N[(x / z), $MachinePrecision] * N[(y + t), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -4.60000000000000023e-79

   1. Initial program 97.3%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

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

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

      5. metadata-eval N/A

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

      6. *-lft-identity N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 84.9

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

   5. Applied rewrites 84.9%

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

   if -4.60000000000000023e-79 < z < 1e-3

   1. Initial program 94.2%

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

   3. Taylor expanded in y around inf

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 70.0

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

   5. Applied rewrites 70.0%

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

   7. Applied rewrites 71.8%

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

   if 1e-3 < z

   1. Initial program 96.1%

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

   3. Taylor expanded in t around inf

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

      1. distribute-rgt-in N/A

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

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

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

      3. mul-1-neg N/A

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

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

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

      5. *-commutative N/A

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

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

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

      7. *-commutative N/A

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

      8. mul-1-neg N/A

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

      9. distribute-rgt-in N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

   5. Applied rewrites 72.7%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 95.0%

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

   9. Taylor expanded in z around inf

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

       1. lower-/.f64 N/A

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

       2. *-commutative N/A

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

       3. lower-*.f64 N/A

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

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

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

       5. metadata-eval N/A

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

       6. *-lft-identity N/A

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

       7. lower-+.f64 88.7

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

   11. Applied rewrites 88.7%

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

   13. Applied rewrites 90.0%

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

Alternative 8: 72.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3 \cdot 10^{-152}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{elif}\;y \leq 2.35 \cdot 10^{-107}:\\ \;\;\;\;\frac{x}{z - 1} \cdot t\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z} \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y -3e-152)
   (* y (/ x z))
   (if (<= y 2.35e-107) (* (/ x (- z 1.0)) t) (* (/ y z) x))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -3e-152) {
		tmp = y * (x / z);
	} else if (y <= 2.35e-107) {
		tmp = (x / (z - 1.0)) * t;
	} else {
		tmp = (y / z) * x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (y <= (-3d-152)) then
        tmp = y * (x / z)
    else if (y <= 2.35d-107) then
        tmp = (x / (z - 1.0d0)) * t
    else
        tmp = (y / z) * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -3e-152) {
		tmp = y * (x / z);
	} else if (y <= 2.35e-107) {
		tmp = (x / (z - 1.0)) * t;
	} else {
		tmp = (y / z) * x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if y <= -3e-152:
		tmp = y * (x / z)
	elif y <= 2.35e-107:
		tmp = (x / (z - 1.0)) * t
	else:
		tmp = (y / z) * x
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= -3e-152)
		tmp = Float64(y * Float64(x / z));
	elseif (y <= 2.35e-107)
		tmp = Float64(Float64(x / Float64(z - 1.0)) * t);
	else
		tmp = Float64(Float64(y / z) * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (y <= -3e-152)
		tmp = y * (x / z);
	elseif (y <= 2.35e-107)
		tmp = (x / (z - 1.0)) * t;
	else
		tmp = (y / z) * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, -3e-152], N[(y * N[(x / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.35e-107], N[(N[(x / N[(z - 1.0), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision], N[(N[(y / z), $MachinePrecision] * x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -3e-152

   1. Initial program 93.3%

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

   3. Taylor expanded in y around inf

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 71.8

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

   5. Applied rewrites 71.8%

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

   7. Applied rewrites 75.4%

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

   if -3e-152 < y < 2.34999999999999999e-107

   1. Initial program 97.6%

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

   3. Taylor expanded in t around inf

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

      1. distribute-rgt-in N/A

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

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

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

      3. mul-1-neg N/A

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

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

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

      5. *-commutative N/A

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

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

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

      7. *-commutative N/A

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

      8. mul-1-neg N/A

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

      9. distribute-rgt-in N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

   5. Applied rewrites 81.7%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 77.8%

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

   if 2.34999999999999999e-107 < y

   1. Initial program 97.5%

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

   3. Taylor expanded in y around inf

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 74.6

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

   5. Applied rewrites 74.6%

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

Alternative 9: 42.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -0.75 \lor \neg \left(z \leq 1.2 \cdot 10^{+20}\right):\\ \;\;\;\;\frac{x}{z} \cdot t\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(-\mathsf{fma}\left(t, z, t\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -0.75) (not (<= z 1.2e+20)))
   (* (/ x z) t)
   (* x (- (fma t z t)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -0.75) || !(z <= 1.2e+20)) {
		tmp = (x / z) * t;
	} else {
		tmp = x * -fma(t, z, t);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((z <= -0.75) || !(z <= 1.2e+20))
		tmp = Float64(Float64(x / z) * t);
	else
		tmp = Float64(x * Float64(-fma(t, z, t)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[z, -0.75], N[Not[LessEqual[z, 1.2e+20]], $MachinePrecision]], N[(N[(x / z), $MachinePrecision] * t), $MachinePrecision], N[(x * (-N[(t * z + t), $MachinePrecision])), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -0.75 or 1.2e20 < z

   1. Initial program 96.4%

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

   3. Taylor expanded in t around inf

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

      1. distribute-rgt-in N/A

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

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

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

      3. mul-1-neg N/A

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

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

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

      5. *-commutative N/A

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

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

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

      7. *-commutative N/A

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

      8. mul-1-neg N/A

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

      9. distribute-rgt-in N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

   5. Applied rewrites 70.5%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 93.1%

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

   9. Taylor expanded in z around inf

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

       1. lower-/.f64 N/A

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

       2. *-commutative N/A

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

       3. lower-*.f64 N/A

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

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

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

       5. metadata-eval N/A

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

       6. *-lft-identity N/A

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

       7. lower-+.f64 88.7

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

   11. Applied rewrites 88.7%

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

   12. Taylor expanded in y around 0

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

   14. Applied rewrites 45.8%

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

   if -0.75 < z < 1.2e20

   1. Initial program 95.1%

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

   3. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. lower-/.f64 N/A

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

      4. *-lft-identity N/A

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

      5. metadata-eval N/A

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

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

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

      7. mul-1-neg N/A

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

      8. distribute-neg-in N/A

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

      9. metadata-eval N/A

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

      10. remove-double-neg N/A

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

      11. lower-+.f64 29.3

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

   5. Applied rewrites 29.3%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 29.3%

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

4. Final simplification 38.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -0.75 \lor \neg \left(z \leq 1.2 \cdot 10^{+20}\right):\\ \;\;\;\;\frac{x}{z} \cdot t\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(-\mathsf{fma}\left(t, z, t\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 65.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -9 \cdot 10^{+216}:\\ \;\;\;\;x \cdot \left(-t\right)\\ \mathbf{elif}\;t \leq 3.5 \cdot 10^{+200}:\\ \;\;\;\;\frac{y}{z} \cdot x\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{t}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t -9e+216)
   (* x (- t))
   (if (<= t 3.5e+200) (* (/ y z) x) (* x (/ t z)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -9e+216) {
		tmp = x * -t;
	} else if (t <= 3.5e+200) {
		tmp = (y / z) * 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 (t <= (-9d+216)) then
        tmp = x * -t
    else if (t <= 3.5d+200) then
        tmp = (y / z) * 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 (t <= -9e+216) {
		tmp = x * -t;
	} else if (t <= 3.5e+200) {
		tmp = (y / z) * x;
	} else {
		tmp = x * (t / z);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if t <= -9e+216:
		tmp = x * -t
	elif t <= 3.5e+200:
		tmp = (y / z) * x
	else:
		tmp = x * (t / z)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= -9e+216)
		tmp = Float64(x * Float64(-t));
	elseif (t <= 3.5e+200)
		tmp = Float64(Float64(y / z) * x);
	else
		tmp = Float64(x * Float64(t / z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (t <= -9e+216)
		tmp = x * -t;
	elseif (t <= 3.5e+200)
		tmp = (y / z) * x;
	else
		tmp = x * (t / z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, -9e+216], N[(x * (-t)), $MachinePrecision], If[LessEqual[t, 3.5e+200], N[(N[(y / z), $MachinePrecision] * x), $MachinePrecision], N[(x * N[(t / z), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -9.0000000000000005e216

   1. Initial program 92.6%

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

   3. Taylor expanded in y around 0

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

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

      5. metadata-eval N/A

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

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

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

      7. mul-1-neg N/A

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

      8. distribute-neg-in N/A

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

      9. metadata-eval N/A

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

      10. remove-double-neg N/A

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

      11. lower-+.f64 76.9

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

   5. Applied rewrites 76.9%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 58.4%

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

   if -9.0000000000000005e216 < t < 3.50000000000000006e200

   1. Initial program 96.7%

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

   3. Taylor expanded in y around inf

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 75.3

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

   5. Applied rewrites 75.3%

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

   if 3.50000000000000006e200 < t

   1. Initial program 91.7%

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

   3. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. lower-/.f64 N/A

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

      4. *-lft-identity N/A

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

      5. metadata-eval N/A

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

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

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

      7. mul-1-neg N/A

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

      8. distribute-neg-in N/A

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

      9. metadata-eval N/A

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

      10. remove-double-neg N/A

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

      11. lower-+.f64 78.7

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

   5. Applied rewrites 78.7%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 57.9%

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

Alternative 11: 65.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -9 \cdot 10^{+216}:\\ \;\;\;\;x \cdot \left(-t\right)\\ \mathbf{elif}\;t \leq 3.5 \cdot 10^{+200}:\\ \;\;\;\;\frac{y}{z} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{t \cdot x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t -9e+216)
   (* x (- t))
   (if (<= t 3.5e+200) (* (/ y z) x) (/ (* t x) z))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -9e+216) {
		tmp = x * -t;
	} else if (t <= 3.5e+200) {
		tmp = (y / z) * x;
	} else {
		tmp = (t * x) / z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-9d+216)) then
        tmp = x * -t
    else if (t <= 3.5d+200) then
        tmp = (y / z) * x
    else
        tmp = (t * x) / z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -9e+216) {
		tmp = x * -t;
	} else if (t <= 3.5e+200) {
		tmp = (y / z) * x;
	} else {
		tmp = (t * x) / z;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if t <= -9e+216:
		tmp = x * -t
	elif t <= 3.5e+200:
		tmp = (y / z) * x
	else:
		tmp = (t * x) / z
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= -9e+216)
		tmp = Float64(x * Float64(-t));
	elseif (t <= 3.5e+200)
		tmp = Float64(Float64(y / z) * x);
	else
		tmp = Float64(Float64(t * x) / z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (t <= -9e+216)
		tmp = x * -t;
	elseif (t <= 3.5e+200)
		tmp = (y / z) * x;
	else
		tmp = (t * x) / z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, -9e+216], N[(x * (-t)), $MachinePrecision], If[LessEqual[t, 3.5e+200], N[(N[(y / z), $MachinePrecision] * x), $MachinePrecision], N[(N[(t * x), $MachinePrecision] / z), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -9.0000000000000005e216

   1. Initial program 92.6%

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

   3. Taylor expanded in y around 0

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

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

      5. metadata-eval N/A

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

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

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

      7. mul-1-neg N/A

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

      8. distribute-neg-in N/A

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

      9. metadata-eval N/A

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

      10. remove-double-neg N/A

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

      11. lower-+.f64 76.9

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

   5. Applied rewrites 76.9%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 58.4%

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

   if -9.0000000000000005e216 < t < 3.50000000000000006e200

   1. Initial program 96.7%

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

   3. Taylor expanded in y around inf

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 75.3

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

   5. Applied rewrites 75.3%

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

   if 3.50000000000000006e200 < t

   1. Initial program 91.7%

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

   3. Taylor expanded in t around inf

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

      1. distribute-rgt-in N/A

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

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

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

      3. mul-1-neg N/A

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

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

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

      5. *-commutative N/A

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

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

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

      7. *-commutative N/A

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

      8. mul-1-neg N/A

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

      9. distribute-rgt-in N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

   5. Applied rewrites 66.6%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 74.8%

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

   9. Taylor expanded in z around inf

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

       1. lower-/.f64 N/A

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

       2. *-commutative N/A

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

       3. lower-*.f64 N/A

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

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

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

       5. metadata-eval N/A

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

       6. *-lft-identity N/A

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

       7. lower-+.f64 54.3

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

   11. Applied rewrites 54.3%

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

   12. Taylor expanded in y around 0

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

   14. Applied rewrites 46.0%

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

Alternative 12: 65.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -9 \cdot 10^{+216}:\\ \;\;\;\;x \cdot \left(-t\right)\\ \mathbf{elif}\;t \leq 3.6 \cdot 10^{+200}:\\ \;\;\;\;\frac{y}{z} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t -9e+216)
   (* x (- t))
   (if (<= t 3.6e+200) (* (/ y z) x) (* (/ x z) t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -9e+216) {
		tmp = x * -t;
	} else if (t <= 3.6e+200) {
		tmp = (y / z) * x;
	} else {
		tmp = (x / z) * t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-9d+216)) then
        tmp = x * -t
    else if (t <= 3.6d+200) then
        tmp = (y / z) * x
    else
        tmp = (x / z) * t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -9e+216) {
		tmp = x * -t;
	} else if (t <= 3.6e+200) {
		tmp = (y / z) * x;
	} else {
		tmp = (x / z) * t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if t <= -9e+216:
		tmp = x * -t
	elif t <= 3.6e+200:
		tmp = (y / z) * x
	else:
		tmp = (x / z) * t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= -9e+216)
		tmp = Float64(x * Float64(-t));
	elseif (t <= 3.6e+200)
		tmp = Float64(Float64(y / z) * x);
	else
		tmp = Float64(Float64(x / z) * t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (t <= -9e+216)
		tmp = x * -t;
	elseif (t <= 3.6e+200)
		tmp = (y / z) * x;
	else
		tmp = (x / z) * t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, -9e+216], N[(x * (-t)), $MachinePrecision], If[LessEqual[t, 3.6e+200], N[(N[(y / z), $MachinePrecision] * x), $MachinePrecision], N[(N[(x / z), $MachinePrecision] * t), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -9.0000000000000005e216

   1. Initial program 92.6%

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

   3. Taylor expanded in y around 0

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

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

      5. metadata-eval N/A

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

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

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

      7. mul-1-neg N/A

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

      8. distribute-neg-in N/A

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

      9. metadata-eval N/A

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

      10. remove-double-neg N/A

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

      11. lower-+.f64 76.9

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

   5. Applied rewrites 76.9%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 58.4%

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

   if -9.0000000000000005e216 < t < 3.5999999999999998e200

   1. Initial program 96.7%

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

   3. Taylor expanded in y around inf

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 75.3

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

   5. Applied rewrites 75.3%

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

   if 3.5999999999999998e200 < t

   1. Initial program 91.7%

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

   3. Taylor expanded in t around inf

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

      1. distribute-rgt-in N/A

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

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

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

      3. mul-1-neg N/A

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

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

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

      5. *-commutative N/A

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

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

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

      7. *-commutative N/A

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

      8. mul-1-neg N/A

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

      9. distribute-rgt-in N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

   5. Applied rewrites 66.6%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 74.8%

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

   9. Taylor expanded in z around inf

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

       1. lower-/.f64 N/A

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

       2. *-commutative N/A

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

       3. lower-*.f64 N/A

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

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

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

       5. metadata-eval N/A

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

       6. *-lft-identity N/A

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

       7. lower-+.f64 54.3

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

   11. Applied rewrites 54.3%

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

   12. Taylor expanded in y around 0

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

   14. Applied rewrites 41.7%

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

Alternative 13: 62.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -9 \cdot 10^{+216}:\\ \;\;\;\;x \cdot \left(-t\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t -9e+216) (* x (- t)) (* y (/ x z))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -9e+216) {
		tmp = x * -t;
	} else {
		tmp = y * (x / z);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-9d+216)) then
        tmp = x * -t
    else
        tmp = y * (x / z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -9e+216) {
		tmp = x * -t;
	} else {
		tmp = y * (x / z);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if t <= -9e+216:
		tmp = x * -t
	else:
		tmp = y * (x / z)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= -9e+216)
		tmp = Float64(x * Float64(-t));
	else
		tmp = Float64(y * Float64(x / z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (t <= -9e+216)
		tmp = x * -t;
	else
		tmp = y * (x / z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, -9e+216], N[(x * (-t)), $MachinePrecision], N[(y * N[(x / z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -9.0000000000000005e216

   1. Initial program 92.6%

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

   3. Taylor expanded in y around 0

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

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

      5. metadata-eval N/A

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

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

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

      7. mul-1-neg N/A

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

      8. distribute-neg-in N/A

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

      9. metadata-eval N/A

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

      10. remove-double-neg N/A

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

      11. lower-+.f64 76.9

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

   5. Applied rewrites 76.9%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 58.4%

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

   if -9.0000000000000005e216 < t

   1. Initial program 96.2%

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

   3. Taylor expanded in y around inf

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 69.8

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

   5. Applied rewrites 69.8%

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

   7. Applied rewrites 69.5%

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

Alternative 14: 22.9% accurate, 4.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 95.8%

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

3. Taylor expanded in y around 0

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

   1. mul-1-neg N/A

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

   2. distribute-neg-frac2 N/A

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

   3. lower-/.f64 N/A

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

   4. *-lft-identity N/A

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

   5. metadata-eval N/A

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

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

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

   7. mul-1-neg N/A

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

   8. distribute-neg-in N/A

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

   9. metadata-eval N/A

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

   10. remove-double-neg N/A

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

   11. lower-+.f64 42.8

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

5. Applied rewrites 42.8%

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

6. Taylor expanded in z around 0

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

8. Applied rewrites 22.7%

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

Developer Target 1: 94.9% 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 2024329 
(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)))))