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

Percentage Accurate: 94.5% → 98.0%

Time: 8.9s

Alternatives: 15

Speedup: 0.3×

• 20.4% 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: 94.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{y}{z} - \frac{t}{1 - z}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+289}:\\ \;\;\;\;\frac{1}{\left(1 - z\right) \cdot z} \cdot \left(x \cdot \mathsf{fma}\left(1 - z, y, \left(-z\right) \cdot t\right)\right)\\ \mathbf{elif}\;t\_1 \leq 10^{+299}:\\ \;\;\;\;x \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (/ y z) (/ t (- 1.0 z)))))
   (if (<= t_1 -2e+289)
     (* (/ 1.0 (* (- 1.0 z) z)) (* x (fma (- 1.0 z) y (* (- z) t))))
     (if (<= t_1 1e+299) (* x t_1) (/ (* x y) z)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (y / z) - (t / (1.0 - z));
	double tmp;
	if (t_1 <= -2e+289) {
		tmp = (1.0 / ((1.0 - z) * z)) * (x * fma((1.0 - z), y, (-z * t)));
	} else if (t_1 <= 1e+299) {
		tmp = x * t_1;
	} else {
		tmp = (x * y) / z;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(y / z) - Float64(t / Float64(1.0 - z)))
	tmp = 0.0
	if (t_1 <= -2e+289)
		tmp = Float64(Float64(1.0 / Float64(Float64(1.0 - z) * z)) * Float64(x * fma(Float64(1.0 - z), y, Float64(Float64(-z) * t))));
	elseif (t_1 <= 1e+299)
		tmp = Float64(x * t_1);
	else
		tmp = Float64(Float64(x * y) / z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(y / z), $MachinePrecision] - N[(t / N[(1.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+289], N[(N[(1.0 / N[(N[(1.0 - z), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision] * N[(x * N[(N[(1.0 - z), $MachinePrecision] * y + N[((-z) * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e+299], N[(x * t$95$1), $MachinePrecision], N[(N[(x * y), $MachinePrecision] / z), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (/.f64 y z) (/.f64 t (-.f64 #s(literal 1 binary64) z))) < -2.0000000000000001e289

   1. Initial program 73.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift--.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. frac-sub N/A

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

      7. associate-*l/ N/A

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

      8. div-inv N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

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

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-neg.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 99.9

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

   4. Applied rewrites 99.9%

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

   if -2.0000000000000001e289 < (-.f64 (/.f64 y z) (/.f64 t (-.f64 #s(literal 1 binary64) z))) < 1.0000000000000001e299

   1. Initial program 98.8%

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

   if 1.0000000000000001e299 < (-.f64 (/.f64 y z) (/.f64 t (-.f64 #s(literal 1 binary64) z)))

   1. Initial program 77.0%

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

   3. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 99.9

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

   5. Applied rewrites 99.9%

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

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y}{z} - \frac{t}{1 - z} \leq -2 \cdot 10^{+289}:\\ \;\;\;\;\frac{1}{\left(1 - z\right) \cdot z} \cdot \left(x \cdot \mathsf{fma}\left(1 - z, y, \left(-z\right) \cdot t\right)\right)\\ \mathbf{elif}\;\frac{y}{z} - \frac{t}{1 - z} \leq 10^{+299}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{y}{z} - \frac{t}{1 - z}\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\frac{x}{z} \cdot y\\ \mathbf{elif}\;t\_1 \leq 10^{+299}:\\ \;\;\;\;x \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (/ y z) (/ t (- 1.0 z)))))
   (if (<= t_1 (- INFINITY))
     (* (/ x z) y)
     (if (<= t_1 1e+299) (* x t_1) (/ (* x y) z)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (y / z) - (t / (1.0 - z));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = (x / z) * y;
	} else if (t_1 <= 1e+299) {
		tmp = x * t_1;
	} else {
		tmp = (x * y) / z;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (y / z) - (t / (1.0 - z));
	double tmp;
	if (t_1 <= -Double.POSITIVE_INFINITY) {
		tmp = (x / z) * y;
	} else if (t_1 <= 1e+299) {
		tmp = x * t_1;
	} else {
		tmp = (x * y) / z;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (y / z) - (t / (1.0 - z))
	tmp = 0
	if t_1 <= -math.inf:
		tmp = (x / z) * y
	elif t_1 <= 1e+299:
		tmp = x * t_1
	else:
		tmp = (x * y) / z
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(y / z) - Float64(t / Float64(1.0 - z)))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(Float64(x / z) * y);
	elseif (t_1 <= 1e+299)
		tmp = Float64(x * t_1);
	else
		tmp = Float64(Float64(x * y) / z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (y / z) - (t / (1.0 - z));
	tmp = 0.0;
	if (t_1 <= -Inf)
		tmp = (x / z) * y;
	elseif (t_1 <= 1e+299)
		tmp = x * t_1;
	else
		tmp = (x * y) / z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(y / z), $MachinePrecision] - N[(t / N[(1.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(x / z), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[t$95$1, 1e+299], N[(x * t$95$1), $MachinePrecision], N[(N[(x * y), $MachinePrecision] / z), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 68.6%

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

   3. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 99.8

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

   5. Applied rewrites 99.8%

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

   7. Applied rewrites 99.9%

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

   if -inf.0 < (-.f64 (/.f64 y z) (/.f64 t (-.f64 #s(literal 1 binary64) z))) < 1.0000000000000001e299

   1. Initial program 98.8%

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

   if 1.0000000000000001e299 < (-.f64 (/.f64 y z) (/.f64 t (-.f64 #s(literal 1 binary64) z)))

   1. Initial program 77.0%

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

   3. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 99.9

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

   5. Applied rewrites 99.9%

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

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y}{z} - \frac{t}{1 - z} \leq -\infty:\\ \;\;\;\;\frac{x}{z} \cdot y\\ \mathbf{elif}\;\frac{y}{z} - \frac{t}{1 - z} \leq 10^{+299}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 93.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t + y}{z} \cdot x\\ \mathbf{if}\;z \leq -63000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1:\\ \;\;\;\;\frac{\mathsf{fma}\left(x \cdot t, -z, x \cdot y\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (/ (+ t y) z) x)))
   (if (<= z -63000.0)
     t_1
     (if (<= z 1.0) (/ (fma (* x t) (- z) (* x y)) z) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = ((t + y) / z) * x;
	double tmp;
	if (z <= -63000.0) {
		tmp = t_1;
	} else if (z <= 1.0) {
		tmp = fma((x * t), -z, (x * y)) / z;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(Float64(t + y) / z) * x)
	tmp = 0.0
	if (z <= -63000.0)
		tmp = t_1;
	elseif (z <= 1.0)
		tmp = Float64(fma(Float64(x * t), Float64(-z), Float64(x * y)) / z);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(N[(t + y), $MachinePrecision] / z), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[z, -63000.0], t$95$1, If[LessEqual[z, 1.0], N[(N[(N[(x * t), $MachinePrecision] * (-z) + N[(x * y), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -63000 or 1 < z

   1. Initial program 98.2%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

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

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

      3. metadata-eval N/A

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

      4. *-lft-identity N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 97.7

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

   5. Applied rewrites 97.7%

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

   if -63000 < z < 1

   1. Initial program 91.5%

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

   3. Taylor expanded in z around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. distribute-lft-out N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. distribute-rgt-out N/A

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

      11. lower-*.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. mul-1-neg N/A

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

      15. lower-neg.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 93.5

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

   5. Applied rewrites 93.5%

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

   6. Taylor expanded in z around 0

      \[\leadsto \frac{\mathsf{fma}\left(t \cdot x, \mathsf{neg}\left(z\right), y \cdot x\right)}{z} \]
   7. Step-by-step derivation

   8. Applied rewrites 93.2%

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

4. Final simplification 95.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -63000:\\ \;\;\;\;\frac{t + y}{z} \cdot x\\ \mathbf{elif}\;z \leq 1:\\ \;\;\;\;\frac{\mathsf{fma}\left(x \cdot t, -z, x \cdot y\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{t + y}{z} \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 81.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t + y}{z} \cdot x\\ \mathbf{if}\;z \leq -3.7 \cdot 10^{-6}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq -1.5 \cdot 10^{-52}:\\ \;\;\;\;\frac{x \cdot t}{z - 1}\\ \mathbf{elif}\;z \leq 1:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (/ (+ t y) z) x)))
   (if (<= z -3.7e-6)
     t_1
     (if (<= z -1.5e-52)
       (/ (* x t) (- z 1.0))
       (if (<= z 1.0) (/ (* x y) z) t_1)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = ((t + y) / z) * x;
	double tmp;
	if (z <= -3.7e-6) {
		tmp = t_1;
	} else if (z <= -1.5e-52) {
		tmp = (x * t) / (z - 1.0);
	} else if (z <= 1.0) {
		tmp = (x * y) / z;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = ((t + y) / z) * x
    if (z <= (-3.7d-6)) then
        tmp = t_1
    else if (z <= (-1.5d-52)) then
        tmp = (x * t) / (z - 1.0d0)
    else if (z <= 1.0d0) then
        tmp = (x * y) / z
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = ((t + y) / z) * x;
	double tmp;
	if (z <= -3.7e-6) {
		tmp = t_1;
	} else if (z <= -1.5e-52) {
		tmp = (x * t) / (z - 1.0);
	} else if (z <= 1.0) {
		tmp = (x * y) / z;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = ((t + y) / z) * x
	tmp = 0
	if z <= -3.7e-6:
		tmp = t_1
	elif z <= -1.5e-52:
		tmp = (x * t) / (z - 1.0)
	elif z <= 1.0:
		tmp = (x * y) / z
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(Float64(t + y) / z) * x)
	tmp = 0.0
	if (z <= -3.7e-6)
		tmp = t_1;
	elseif (z <= -1.5e-52)
		tmp = Float64(Float64(x * t) / Float64(z - 1.0));
	elseif (z <= 1.0)
		tmp = Float64(Float64(x * y) / z);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = ((t + y) / z) * x;
	tmp = 0.0;
	if (z <= -3.7e-6)
		tmp = t_1;
	elseif (z <= -1.5e-52)
		tmp = (x * t) / (z - 1.0);
	elseif (z <= 1.0)
		tmp = (x * y) / z;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(N[(t + y), $MachinePrecision] / z), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[z, -3.7e-6], t$95$1, If[LessEqual[z, -1.5e-52], N[(N[(x * t), $MachinePrecision] / N[(z - 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.0], N[(N[(x * y), $MachinePrecision] / z), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if z < -3.7000000000000002e-6 or 1 < z

   1. Initial program 98.2%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

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

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

      3. metadata-eval N/A

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

      4. *-lft-identity N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 97.8

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

   5. Applied rewrites 97.8%

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

   if -3.7000000000000002e-6 < z < -1.5e-52

   1. Initial program 89.1%

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

   3. Taylor expanded in t around inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. sub-neg N/A

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

      6. mul-1-neg N/A

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

      7. +-commutative N/A

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

      8. distribute-neg-in N/A

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

      9. mul-1-neg N/A

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

      10. remove-double-neg N/A

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

      11. sub-neg N/A

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

      12. lower--.f64 85.4

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

   5. Applied rewrites 85.4%

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

   if -1.5e-52 < z < 1

   1. Initial program 91.5%

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

   3. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 74.3

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

   5. Applied rewrites 74.3%

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

4. Final simplification 86.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.7 \cdot 10^{-6}:\\ \;\;\;\;\frac{t + y}{z} \cdot x\\ \mathbf{elif}\;z \leq -1.5 \cdot 10^{-52}:\\ \;\;\;\;\frac{x \cdot t}{z - 1}\\ \mathbf{elif}\;z \leq 1:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{t + y}{z} \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 72.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x}{z - 1} \cdot t\\ \mathbf{if}\;t \leq -4.5 \cdot 10^{+61}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 1.6 \cdot 10^{-248}:\\ \;\;\;\;\frac{y}{z} \cdot x\\ \mathbf{elif}\;t \leq 7.2 \cdot 10^{+77}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (/ x (- z 1.0)) t)))
   (if (<= t -4.5e+61)
     t_1
     (if (<= t 1.6e-248)
       (* (/ y z) x)
       (if (<= t 7.2e+77) (/ (* x y) z) t_1)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x / (z - 1.0)) * t;
	double tmp;
	if (t <= -4.5e+61) {
		tmp = t_1;
	} else if (t <= 1.6e-248) {
		tmp = (y / z) * x;
	} else if (t <= 7.2e+77) {
		tmp = (x * y) / z;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (x / (z - 1.0d0)) * t
    if (t <= (-4.5d+61)) then
        tmp = t_1
    else if (t <= 1.6d-248) then
        tmp = (y / z) * x
    else if (t <= 7.2d+77) then
        tmp = (x * y) / 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 / (z - 1.0)) * t;
	double tmp;
	if (t <= -4.5e+61) {
		tmp = t_1;
	} else if (t <= 1.6e-248) {
		tmp = (y / z) * x;
	} else if (t <= 7.2e+77) {
		tmp = (x * y) / z;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (x / (z - 1.0)) * t
	tmp = 0
	if t <= -4.5e+61:
		tmp = t_1
	elif t <= 1.6e-248:
		tmp = (y / z) * x
	elif t <= 7.2e+77:
		tmp = (x * y) / z
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x / Float64(z - 1.0)) * t)
	tmp = 0.0
	if (t <= -4.5e+61)
		tmp = t_1;
	elseif (t <= 1.6e-248)
		tmp = Float64(Float64(y / z) * x);
	elseif (t <= 7.2e+77)
		tmp = Float64(Float64(x * y) / z);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (x / (z - 1.0)) * t;
	tmp = 0.0;
	if (t <= -4.5e+61)
		tmp = t_1;
	elseif (t <= 1.6e-248)
		tmp = (y / z) * x;
	elseif (t <= 7.2e+77)
		tmp = (x * y) / z;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x / N[(z - 1.0), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]}, If[LessEqual[t, -4.5e+61], t$95$1, If[LessEqual[t, 1.6e-248], N[(N[(y / z), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[t, 7.2e+77], N[(N[(x * y), $MachinePrecision] / z), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if t < -4.5e61 or 7.1999999999999996e77 < t

   1. Initial program 97.1%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift--.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. frac-sub N/A

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

      7. associate-*l/ N/A

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

      8. div-inv N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

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

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-neg.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 66.2

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

   4. Applied rewrites 66.2%

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

   5. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. remove-double-neg N/A

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

      4. mul-1-neg N/A

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

      5. distribute-frac-neg N/A

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

      6. associate-*r/ N/A

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

      7. mul-1-neg N/A

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

      8. distribute-neg-in N/A

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

      9. lower-*.f64 N/A

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

   7. Applied rewrites 90.6%

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

   8. Taylor expanded in t around inf

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

      1. mul-1-neg N/A

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

      4. distribute-neg-in N/A

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

      5. +-commutative N/A

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

      6. metadata-eval N/A

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

      7. sub-neg N/A

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

      8. distribute-neg-frac2 N/A

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

      9. remove-double-neg N/A

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

      10. *-commutative N/A

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

      11. associate-*l/ N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower--.f64 73.3

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

   10. Applied rewrites 73.3%

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

   if -4.5e61 < t < 1.60000000000000009e-248

   1. Initial program 93.9%

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

   3. Taylor expanded in t around 0

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

      1. lower-/.f64 84.8

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

   5. Applied rewrites 84.8%

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

   if 1.60000000000000009e-248 < t < 7.1999999999999996e77

   1. Initial program 92.8%

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

   3. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 76.2

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

   5. Applied rewrites 76.2%

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

4. Final simplification 77.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -4.5 \cdot 10^{+61}:\\ \;\;\;\;\frac{x}{z - 1} \cdot t\\ \mathbf{elif}\;t \leq 1.6 \cdot 10^{-248}:\\ \;\;\;\;\frac{y}{z} \cdot x\\ \mathbf{elif}\;t \leq 7.2 \cdot 10^{+77}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z - 1} \cdot t\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 72.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x \cdot t}{z - 1}\\ \mathbf{if}\;t \leq -4.5 \cdot 10^{+61}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 1.6 \cdot 10^{-248}:\\ \;\;\;\;\frac{y}{z} \cdot x\\ \mathbf{elif}\;t \leq 7.8 \cdot 10^{+77}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (* x t) (- z 1.0))))
   (if (<= t -4.5e+61)
     t_1
     (if (<= t 1.6e-248)
       (* (/ y z) x)
       (if (<= t 7.8e+77) (/ (* x y) z) t_1)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x * t) / (z - 1.0);
	double tmp;
	if (t <= -4.5e+61) {
		tmp = t_1;
	} else if (t <= 1.6e-248) {
		tmp = (y / z) * x;
	} else if (t <= 7.8e+77) {
		tmp = (x * y) / z;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (x * t) / (z - 1.0d0)
    if (t <= (-4.5d+61)) then
        tmp = t_1
    else if (t <= 1.6d-248) then
        tmp = (y / z) * x
    else if (t <= 7.8d+77) then
        tmp = (x * y) / 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 * t) / (z - 1.0);
	double tmp;
	if (t <= -4.5e+61) {
		tmp = t_1;
	} else if (t <= 1.6e-248) {
		tmp = (y / z) * x;
	} else if (t <= 7.8e+77) {
		tmp = (x * y) / z;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (x * t) / (z - 1.0)
	tmp = 0
	if t <= -4.5e+61:
		tmp = t_1
	elif t <= 1.6e-248:
		tmp = (y / z) * x
	elif t <= 7.8e+77:
		tmp = (x * y) / z
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x * t) / Float64(z - 1.0))
	tmp = 0.0
	if (t <= -4.5e+61)
		tmp = t_1;
	elseif (t <= 1.6e-248)
		tmp = Float64(Float64(y / z) * x);
	elseif (t <= 7.8e+77)
		tmp = Float64(Float64(x * y) / z);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (x * t) / (z - 1.0);
	tmp = 0.0;
	if (t <= -4.5e+61)
		tmp = t_1;
	elseif (t <= 1.6e-248)
		tmp = (y / z) * x;
	elseif (t <= 7.8e+77)
		tmp = (x * y) / z;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x * t), $MachinePrecision] / N[(z - 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -4.5e+61], t$95$1, If[LessEqual[t, 1.6e-248], N[(N[(y / z), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[t, 7.8e+77], N[(N[(x * y), $MachinePrecision] / z), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if t < -4.5e61 or 7.7999999999999995e77 < t

   1. Initial program 97.1%

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

   3. Taylor expanded in t around inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. sub-neg N/A

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

      6. mul-1-neg N/A

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

      7. +-commutative N/A

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

      8. distribute-neg-in N/A

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

      9. mul-1-neg N/A

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

      10. remove-double-neg N/A

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

      11. sub-neg N/A

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

      12. lower--.f64 71.4

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

   5. Applied rewrites 71.4%

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

   if -4.5e61 < t < 1.60000000000000009e-248

   1. Initial program 93.9%

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

   3. Taylor expanded in t around 0

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

      1. lower-/.f64 84.8

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

   5. Applied rewrites 84.8%

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

   if 1.60000000000000009e-248 < t < 7.7999999999999995e77

   1. Initial program 92.8%

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

   3. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 76.2

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

   5. Applied rewrites 76.2%

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

4. Final simplification 77.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -4.5 \cdot 10^{+61}:\\ \;\;\;\;\frac{x \cdot t}{z - 1}\\ \mathbf{elif}\;t \leq 1.6 \cdot 10^{-248}:\\ \;\;\;\;\frac{y}{z} \cdot x\\ \mathbf{elif}\;t \leq 7.8 \cdot 10^{+77}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot t}{z - 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 93.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t + y}{z} \cdot x\\ \mathbf{if}\;z \leq -63000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1:\\ \;\;\;\;\left(\frac{y}{z} - \mathsf{fma}\left(t, z, t\right)\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (/ (+ t y) z) x)))
   (if (<= z -63000.0) t_1 (if (<= z 1.0) (* (- (/ y z) (fma t z t)) x) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = ((t + y) / z) * x;
	double tmp;
	if (z <= -63000.0) {
		tmp = t_1;
	} else if (z <= 1.0) {
		tmp = ((y / z) - fma(t, z, t)) * x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(Float64(t + y) / z) * x)
	tmp = 0.0
	if (z <= -63000.0)
		tmp = t_1;
	elseif (z <= 1.0)
		tmp = Float64(Float64(Float64(y / z) - fma(t, z, t)) * x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(N[(t + y), $MachinePrecision] / z), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[z, -63000.0], t$95$1, If[LessEqual[z, 1.0], N[(N[(N[(y / z), $MachinePrecision] - N[(t * z + t), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -63000 or 1 < z

   1. Initial program 98.2%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

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

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

      3. metadata-eval N/A

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

      4. *-lft-identity N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 97.7

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

   5. Applied rewrites 97.7%

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

   if -63000 < z < 1

   1. Initial program 91.5%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 90.8

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

   5. Applied rewrites 90.8%

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

4. Final simplification 94.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -63000:\\ \;\;\;\;\frac{t + y}{z} \cdot x\\ \mathbf{elif}\;z \leq 1:\\ \;\;\;\;\left(\frac{y}{z} - \mathsf{fma}\left(t, z, t\right)\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{t + y}{z} \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 93.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t + y}{z} \cdot x\\ \mathbf{if}\;z \leq -63000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1:\\ \;\;\;\;\frac{\mathsf{fma}\left(-z, t, y\right)}{z} \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (/ (+ t y) z) x)))
   (if (<= z -63000.0) t_1 (if (<= z 1.0) (* (/ (fma (- z) t y) z) x) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = ((t + y) / z) * x;
	double tmp;
	if (z <= -63000.0) {
		tmp = t_1;
	} else if (z <= 1.0) {
		tmp = (fma(-z, t, y) / z) * x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(Float64(t + y) / z) * x)
	tmp = 0.0
	if (z <= -63000.0)
		tmp = t_1;
	elseif (z <= 1.0)
		tmp = Float64(Float64(fma(Float64(-z), t, y) / z) * x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(N[(t + y), $MachinePrecision] / z), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[z, -63000.0], t$95$1, If[LessEqual[z, 1.0], N[(N[(N[((-z) * t + y), $MachinePrecision] / z), $MachinePrecision] * x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -63000 or 1 < z

   1. Initial program 98.2%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

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

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

      3. metadata-eval N/A

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

      4. *-lft-identity N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 97.7

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

   5. Applied rewrites 97.7%

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

   if -63000 < z < 1

   1. Initial program 91.5%

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

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

      3. metadata-eval N/A

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

      4. *-lft-identity N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 52.2

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

   5. Applied rewrites 52.2%

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

   6. Taylor expanded in z around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-fma.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 90.5

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

   8. Applied rewrites 90.5%

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

4. Final simplification 94.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -63000:\\ \;\;\;\;\frac{t + y}{z} \cdot x\\ \mathbf{elif}\;z \leq 1:\\ \;\;\;\;\frac{\mathsf{fma}\left(-z, t, y\right)}{z} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{t + y}{z} \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 93.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t + y}{z} \cdot x\\ \mathbf{if}\;z \leq -63000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1:\\ \;\;\;\;\frac{y - t \cdot z}{z} \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (/ (+ t y) z) x)))
   (if (<= z -63000.0) t_1 (if (<= z 1.0) (* (/ (- y (* t z)) z) x) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = ((t + y) / z) * x;
	double tmp;
	if (z <= -63000.0) {
		tmp = t_1;
	} else if (z <= 1.0) {
		tmp = ((y - (t * z)) / z) * x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = ((t + y) / z) * x
    if (z <= (-63000.0d0)) then
        tmp = t_1
    else if (z <= 1.0d0) then
        tmp = ((y - (t * z)) / z) * x
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = ((t + y) / z) * x;
	double tmp;
	if (z <= -63000.0) {
		tmp = t_1;
	} else if (z <= 1.0) {
		tmp = ((y - (t * z)) / z) * x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = ((t + y) / z) * x
	tmp = 0
	if z <= -63000.0:
		tmp = t_1
	elif z <= 1.0:
		tmp = ((y - (t * z)) / z) * x
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(Float64(t + y) / z) * x)
	tmp = 0.0
	if (z <= -63000.0)
		tmp = t_1;
	elseif (z <= 1.0)
		tmp = Float64(Float64(Float64(y - Float64(t * z)) / z) * x);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = ((t + y) / z) * x;
	tmp = 0.0;
	if (z <= -63000.0)
		tmp = t_1;
	elseif (z <= 1.0)
		tmp = ((y - (t * z)) / z) * x;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(N[(t + y), $MachinePrecision] / z), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[z, -63000.0], t$95$1, If[LessEqual[z, 1.0], N[(N[(N[(y - N[(t * z), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision] * x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -63000 or 1 < z

   1. Initial program 98.2%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

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

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

      3. metadata-eval N/A

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

      4. *-lft-identity N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 97.7

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

   5. Applied rewrites 97.7%

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

   if -63000 < z < 1

   1. Initial program 91.5%

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

   3. Taylor expanded in z around 0

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

      1. lower-/.f64 N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 90.4

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

   5. Applied rewrites 90.4%

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

4. Final simplification 94.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -63000:\\ \;\;\;\;\frac{t + y}{z} \cdot x\\ \mathbf{elif}\;z \leq 1:\\ \;\;\;\;\frac{y - t \cdot z}{z} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{t + y}{z} \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 67.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t}{z} \cdot x\\ \mathbf{if}\;t \leq -4.9 \cdot 10^{+61}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 1.6 \cdot 10^{-248}:\\ \;\;\;\;\frac{y}{z} \cdot x\\ \mathbf{elif}\;t \leq 1.26 \cdot 10^{+102}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (/ t z) x)))
   (if (<= t -4.9e+61)
     t_1
     (if (<= t 1.6e-248)
       (* (/ y z) x)
       (if (<= t 1.26e+102) (/ (* x y) z) t_1)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (t / z) * x;
	double tmp;
	if (t <= -4.9e+61) {
		tmp = t_1;
	} else if (t <= 1.6e-248) {
		tmp = (y / z) * x;
	} else if (t <= 1.26e+102) {
		tmp = (x * y) / z;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (t / z) * x
    if (t <= (-4.9d+61)) then
        tmp = t_1
    else if (t <= 1.6d-248) then
        tmp = (y / z) * x
    else if (t <= 1.26d+102) then
        tmp = (x * y) / z
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (t / z) * x;
	double tmp;
	if (t <= -4.9e+61) {
		tmp = t_1;
	} else if (t <= 1.6e-248) {
		tmp = (y / z) * x;
	} else if (t <= 1.26e+102) {
		tmp = (x * y) / z;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (t / z) * x
	tmp = 0
	if t <= -4.9e+61:
		tmp = t_1
	elif t <= 1.6e-248:
		tmp = (y / z) * x
	elif t <= 1.26e+102:
		tmp = (x * y) / z
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(t / z) * x)
	tmp = 0.0
	if (t <= -4.9e+61)
		tmp = t_1;
	elseif (t <= 1.6e-248)
		tmp = Float64(Float64(y / z) * x);
	elseif (t <= 1.26e+102)
		tmp = Float64(Float64(x * y) / z);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (t / z) * x;
	tmp = 0.0;
	if (t <= -4.9e+61)
		tmp = t_1;
	elseif (t <= 1.6e-248)
		tmp = (y / z) * x;
	elseif (t <= 1.26e+102)
		tmp = (x * y) / z;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(t / z), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[t, -4.9e+61], t$95$1, If[LessEqual[t, 1.6e-248], N[(N[(y / z), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[t, 1.26e+102], N[(N[(x * y), $MachinePrecision] / z), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if t < -4.90000000000000025e61 or 1.26000000000000001e102 < t

   1. Initial program 97.0%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

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

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

      3. metadata-eval N/A

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

      4. *-lft-identity N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 68.1

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

   5. Applied rewrites 68.1%

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

   6. Taylor expanded in t around inf

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

   8. Applied rewrites 56.1%

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

   if -4.90000000000000025e61 < t < 1.60000000000000009e-248

   1. Initial program 93.9%

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

   3. Taylor expanded in t around 0

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

      1. lower-/.f64 84.8

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

   5. Applied rewrites 84.8%

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

   if 1.60000000000000009e-248 < t < 1.26000000000000001e102

   1. Initial program 93.1%

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

   3. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 74.5

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

   5. Applied rewrites 74.5%

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

4. Final simplification 70.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -4.9 \cdot 10^{+61}:\\ \;\;\;\;\frac{t}{z} \cdot x\\ \mathbf{elif}\;t \leq 1.6 \cdot 10^{-248}:\\ \;\;\;\;\frac{y}{z} \cdot x\\ \mathbf{elif}\;t \leq 1.26 \cdot 10^{+102}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{t}{z} \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 66.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t}{z} \cdot x\\ \mathbf{if}\;t \leq -2.5 \cdot 10^{+63}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 1.02 \cdot 10^{-252}:\\ \;\;\;\;\frac{x}{z} \cdot y\\ \mathbf{elif}\;t \leq 1.26 \cdot 10^{+102}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (/ t z) x)))
   (if (<= t -2.5e+63)
     t_1
     (if (<= t 1.02e-252)
       (* (/ x z) y)
       (if (<= t 1.26e+102) (/ (* x y) z) t_1)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (t / z) * x;
	double tmp;
	if (t <= -2.5e+63) {
		tmp = t_1;
	} else if (t <= 1.02e-252) {
		tmp = (x / z) * y;
	} else if (t <= 1.26e+102) {
		tmp = (x * y) / z;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (t / z) * x
    if (t <= (-2.5d+63)) then
        tmp = t_1
    else if (t <= 1.02d-252) then
        tmp = (x / z) * y
    else if (t <= 1.26d+102) then
        tmp = (x * y) / z
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (t / z) * x;
	double tmp;
	if (t <= -2.5e+63) {
		tmp = t_1;
	} else if (t <= 1.02e-252) {
		tmp = (x / z) * y;
	} else if (t <= 1.26e+102) {
		tmp = (x * y) / z;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (t / z) * x
	tmp = 0
	if t <= -2.5e+63:
		tmp = t_1
	elif t <= 1.02e-252:
		tmp = (x / z) * y
	elif t <= 1.26e+102:
		tmp = (x * y) / z
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(t / z) * x)
	tmp = 0.0
	if (t <= -2.5e+63)
		tmp = t_1;
	elseif (t <= 1.02e-252)
		tmp = Float64(Float64(x / z) * y);
	elseif (t <= 1.26e+102)
		tmp = Float64(Float64(x * y) / z);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (t / z) * x;
	tmp = 0.0;
	if (t <= -2.5e+63)
		tmp = t_1;
	elseif (t <= 1.02e-252)
		tmp = (x / z) * y;
	elseif (t <= 1.26e+102)
		tmp = (x * y) / z;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(t / z), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[t, -2.5e+63], t$95$1, If[LessEqual[t, 1.02e-252], N[(N[(x / z), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[t, 1.26e+102], N[(N[(x * y), $MachinePrecision] / z), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if t < -2.50000000000000005e63 or 1.26000000000000001e102 < t

   1. Initial program 97.0%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

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

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

      3. metadata-eval N/A

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

      4. *-lft-identity N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 67.8

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

   5. Applied rewrites 67.8%

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

   6. Taylor expanded in t around inf

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

   8. Applied rewrites 55.7%

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

   if -2.50000000000000005e63 < t < 1.02000000000000002e-252

   1. Initial program 93.9%

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

   3. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 77.5

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

   5. Applied rewrites 77.5%

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

   7. Applied rewrites 84.2%

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

   if 1.02000000000000002e-252 < t < 1.26000000000000001e102

   1. Initial program 93.2%

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

   3. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 74.8

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

   5. Applied rewrites 74.8%

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

4. Final simplification 70.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2.5 \cdot 10^{+63}:\\ \;\;\;\;\frac{t}{z} \cdot x\\ \mathbf{elif}\;t \leq 1.02 \cdot 10^{-252}:\\ \;\;\;\;\frac{x}{z} \cdot y\\ \mathbf{elif}\;t \leq 1.26 \cdot 10^{+102}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{t}{z} \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 64.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x}{z} \cdot t\\ \mathbf{if}\;t \leq -2.5 \cdot 10^{+63}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 1.02 \cdot 10^{-252}:\\ \;\;\;\;\frac{x}{z} \cdot y\\ \mathbf{elif}\;t \leq 5.5 \cdot 10^{+153}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (/ x z) t)))
   (if (<= t -2.5e+63)
     t_1
     (if (<= t 1.02e-252)
       (* (/ x z) y)
       (if (<= t 5.5e+153) (/ (* x y) z) t_1)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x / z) * t;
	double tmp;
	if (t <= -2.5e+63) {
		tmp = t_1;
	} else if (t <= 1.02e-252) {
		tmp = (x / z) * y;
	} else if (t <= 5.5e+153) {
		tmp = (x * y) / z;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (x / z) * t
    if (t <= (-2.5d+63)) then
        tmp = t_1
    else if (t <= 1.02d-252) then
        tmp = (x / z) * y
    else if (t <= 5.5d+153) then
        tmp = (x * y) / 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 / z) * t;
	double tmp;
	if (t <= -2.5e+63) {
		tmp = t_1;
	} else if (t <= 1.02e-252) {
		tmp = (x / z) * y;
	} else if (t <= 5.5e+153) {
		tmp = (x * y) / z;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (x / z) * t
	tmp = 0
	if t <= -2.5e+63:
		tmp = t_1
	elif t <= 1.02e-252:
		tmp = (x / z) * y
	elif t <= 5.5e+153:
		tmp = (x * y) / z
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x / z) * t)
	tmp = 0.0
	if (t <= -2.5e+63)
		tmp = t_1;
	elseif (t <= 1.02e-252)
		tmp = Float64(Float64(x / z) * y);
	elseif (t <= 5.5e+153)
		tmp = Float64(Float64(x * y) / z);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (x / z) * t;
	tmp = 0.0;
	if (t <= -2.5e+63)
		tmp = t_1;
	elseif (t <= 1.02e-252)
		tmp = (x / z) * y;
	elseif (t <= 5.5e+153)
		tmp = (x * y) / z;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x / z), $MachinePrecision] * t), $MachinePrecision]}, If[LessEqual[t, -2.5e+63], t$95$1, If[LessEqual[t, 1.02e-252], N[(N[(x / z), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[t, 5.5e+153], N[(N[(x * y), $MachinePrecision] / z), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if t < -2.50000000000000005e63 or 5.5000000000000003e153 < t

   1. Initial program 97.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift--.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. frac-sub N/A

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

      7. associate-*l/ N/A

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

      8. div-inv N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

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

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-neg.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 65.4

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

   4. Applied rewrites 65.4%

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

   5. Taylor expanded in t around inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. sub-neg N/A

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

      7. mul-1-neg N/A

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

      8. distribute-lft-in N/A

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

      9. metadata-eval N/A

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

      10. neg-mul-1 N/A

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

      11. mul-1-neg N/A

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

      12. remove-double-neg N/A

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

      13. lower-+.f64 69.7

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

   7. Applied rewrites 69.7%

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

   8. Taylor expanded in z around inf

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

   10. Applied rewrites 51.1%

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

   if -2.50000000000000005e63 < t < 1.02000000000000002e-252

   1. Initial program 93.9%

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

   3. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 77.5

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

   5. Applied rewrites 77.5%

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

   7. Applied rewrites 84.2%

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

   if 1.02000000000000002e-252 < t < 5.5000000000000003e153

   1. Initial program 92.7%

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

   3. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 70.4

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

   5. Applied rewrites 70.4%

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

4. Final simplification 68.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2.5 \cdot 10^{+63}:\\ \;\;\;\;\frac{x}{z} \cdot t\\ \mathbf{elif}\;t \leq 1.02 \cdot 10^{-252}:\\ \;\;\;\;\frac{x}{z} \cdot y\\ \mathbf{elif}\;t \leq 5.5 \cdot 10^{+153}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot t\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 59.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2 \cdot 10^{-156}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{elif}\;y \leq 3.9 \cdot 10^{-14}:\\ \;\;\;\;\left(-x\right) \cdot t\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y -2e-156)
   (/ (* x y) z)
   (if (<= y 3.9e-14) (* (- x) t) (* (/ x z) y))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -2e-156) {
		tmp = (x * y) / z;
	} else if (y <= 3.9e-14) {
		tmp = -x * t;
	} else {
		tmp = (x / z) * y;
	}
	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 <= (-2d-156)) then
        tmp = (x * y) / z
    else if (y <= 3.9d-14) then
        tmp = -x * t
    else
        tmp = (x / z) * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -2e-156) {
		tmp = (x * y) / z;
	} else if (y <= 3.9e-14) {
		tmp = -x * t;
	} else {
		tmp = (x / z) * y;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if y <= -2e-156:
		tmp = (x * y) / z
	elif y <= 3.9e-14:
		tmp = -x * t
	else:
		tmp = (x / z) * y
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= -2e-156)
		tmp = Float64(Float64(x * y) / z);
	elseif (y <= 3.9e-14)
		tmp = Float64(Float64(-x) * t);
	else
		tmp = Float64(Float64(x / z) * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (y <= -2e-156)
		tmp = (x * y) / z;
	elseif (y <= 3.9e-14)
		tmp = -x * t;
	else
		tmp = (x / z) * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, -2e-156], N[(N[(x * y), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[y, 3.9e-14], N[((-x) * t), $MachinePrecision], N[(N[(x / z), $MachinePrecision] * y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.00000000000000008e-156

   1. Initial program 94.5%

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

   3. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 67.7

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

   5. Applied rewrites 67.7%

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

   if -2.00000000000000008e-156 < y < 3.8999999999999998e-14

   1. Initial program 98.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift--.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. frac-sub N/A

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

      7. associate-*l/ N/A

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

      8. div-inv N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

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

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-neg.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 73.3

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

   4. Applied rewrites 73.3%

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

   5. Taylor expanded in t around inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. sub-neg N/A

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

      7. mul-1-neg N/A

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

      8. distribute-lft-in N/A

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

      9. metadata-eval N/A

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

      10. neg-mul-1 N/A

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

      11. mul-1-neg N/A

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

      12. remove-double-neg N/A

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

      13. lower-+.f64 76.8

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

   7. Applied rewrites 76.8%

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

   8. Taylor expanded in z around 0

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

   10. Applied rewrites 50.4%

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

   if 3.8999999999999998e-14 < y

   1. Initial program 92.0%

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

   3. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 72.2

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

   5. Applied rewrites 72.2%

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

   7. Applied rewrites 74.7%

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

4. Final simplification 64.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2 \cdot 10^{-156}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{elif}\;y \leq 3.9 \cdot 10^{-14}:\\ \;\;\;\;\left(-x\right) \cdot t\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 59.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x}{z} \cdot y\\ \mathbf{if}\;y \leq -3.8 \cdot 10^{-151}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 3.9 \cdot 10^{-14}:\\ \;\;\;\;\left(-x\right) \cdot t\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (/ x z) y)))
   (if (<= y -3.8e-151) t_1 (if (<= y 3.9e-14) (* (- x) t) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x / z) * y;
	double tmp;
	if (y <= -3.8e-151) {
		tmp = t_1;
	} else if (y <= 3.9e-14) {
		tmp = -x * t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (x / z) * y
    if (y <= (-3.8d-151)) then
        tmp = t_1
    else if (y <= 3.9d-14) then
        tmp = -x * t
    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 / z) * y;
	double tmp;
	if (y <= -3.8e-151) {
		tmp = t_1;
	} else if (y <= 3.9e-14) {
		tmp = -x * t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (x / z) * y
	tmp = 0
	if y <= -3.8e-151:
		tmp = t_1
	elif y <= 3.9e-14:
		tmp = -x * t
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x / z) * y)
	tmp = 0.0
	if (y <= -3.8e-151)
		tmp = t_1;
	elseif (y <= 3.9e-14)
		tmp = Float64(Float64(-x) * t);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (x / z) * y;
	tmp = 0.0;
	if (y <= -3.8e-151)
		tmp = t_1;
	elseif (y <= 3.9e-14)
		tmp = -x * t;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x / z), $MachinePrecision] * y), $MachinePrecision]}, If[LessEqual[y, -3.8e-151], t$95$1, If[LessEqual[y, 3.9e-14], N[((-x) * t), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -3.7999999999999997e-151 or 3.8999999999999998e-14 < y

   1. Initial program 93.5%

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

   3. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 69.5

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

   5. Applied rewrites 69.5%

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

   7. Applied rewrites 69.2%

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

   if -3.7999999999999997e-151 < y < 3.8999999999999998e-14

   1. Initial program 98.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift--.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. frac-sub N/A

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

      7. associate-*l/ N/A

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

      8. div-inv N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

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

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-neg.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 73.3

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

   4. Applied rewrites 73.3%

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

   5. Taylor expanded in t around inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. sub-neg N/A

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

      7. mul-1-neg N/A

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

      8. distribute-lft-in N/A

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

      9. metadata-eval N/A

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

      10. neg-mul-1 N/A

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

      11. mul-1-neg N/A

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

      12. remove-double-neg N/A

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

      13. lower-+.f64 76.8

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

   7. Applied rewrites 76.8%

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

   8. Taylor expanded in z around 0

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

   10. Applied rewrites 50.4%

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

Alternative 15: 23.5% accurate, 4.3× speedup

Math

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

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(Float64(-x) * t)
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.9%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift--.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. frac-sub N/A

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

   7. associate-*l/ N/A

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

   8. div-inv N/A

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

   9. lower-*.f64 N/A

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

   10. lower-*.f64 N/A

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

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

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

   12. *-commutative N/A

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

   13. lower-fma.f64 N/A

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

   14. lower-*.f64 N/A

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

   15. lower-neg.f64 N/A

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

   16. lower-/.f64 N/A

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

   17. *-commutative N/A

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

   18. lower-*.f64 69.1

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

4. Applied rewrites 69.1%

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

5. Taylor expanded in t around inf

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

   1. mul-1-neg N/A

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

   2. distribute-neg-frac2 N/A

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

   3. mul-1-neg N/A

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

   4. lower-/.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. sub-neg N/A

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

   7. mul-1-neg N/A

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

   8. distribute-lft-in N/A

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

   9. metadata-eval N/A

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

   10. neg-mul-1 N/A

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

   11. mul-1-neg N/A

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

   12. remove-double-neg N/A

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

   13. lower-+.f64 46.6

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

7. Applied rewrites 46.6%

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

8. Taylor expanded in z around 0

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

10. Applied rewrites 25.7%

    \[\leadsto \left(-x\right) \cdot \color{blue}{t} \]
11. 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 2024235 
(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)))))