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

Percentage Accurate: 94.0% → 96.4%

Time: 10.0s

Alternatives: 11

Speedup: 0.5×

• 20.7% 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: 94.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 96.4% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 64.8%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 64.8

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

   5. Simplified 64.8%

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

      1. lift-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. div-inv N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 99.7

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

   7. Applied egg-rr 99.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 99.7

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-/.f64 N/A

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

      8. div-inv N/A

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

      9. lower-/.f64 99.9

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

   9. Applied egg-rr 99.9%

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

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

   1. Initial program 97.5%

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

4. Final simplification 97.6%

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

Alternative 2: 93.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -0.75:\\ \;\;\;\;x \cdot \frac{y + t}{z}\\ \mathbf{elif}\;z \leq 1:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - \mathsf{fma}\left(z, t, t\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{y + t}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -0.75)
   (* x (/ (+ y t) z))
   (if (<= z 1.0) (* x (- (/ y z) (fma z t t))) (/ x (/ z (+ y t))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -0.75) {
		tmp = x * ((y + t) / z);
	} else if (z <= 1.0) {
		tmp = x * ((y / z) - fma(z, t, t));
	} else {
		tmp = x / (z / (y + t));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -0.75)
		tmp = Float64(x * Float64(Float64(y + t) / z));
	elseif (z <= 1.0)
		tmp = Float64(x * Float64(Float64(y / z) - fma(z, t, t)));
	else
		tmp = Float64(x / Float64(z / Float64(y + t)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -0.75], N[(x * N[(N[(y + t), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.0], N[(x * N[(N[(y / z), $MachinePrecision] - N[(z * t + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x / N[(z / N[(y + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -0.75

   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. lower-+.f64 97.9

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

   5. Simplified 97.9%

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

   if -0.75 < z < 1

   1. Initial program 93.2%

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

   3. Taylor expanded in z around 0

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

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

      3. lower-fma.f64 92.5

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

   5. Simplified 92.5%

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

   if 1 < z

   1. Initial program 97.9%

      \[x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \]
   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. lower-+.f64 97.1

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

   5. Simplified 97.1%

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

      1. lift-+.f64 N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 97.2

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 97.2

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

   7. Applied egg-rr 97.2%

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

4. Final simplification 94.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -0.75:\\ \;\;\;\;x \cdot \frac{y + t}{z}\\ \mathbf{elif}\;z \leq 1:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - \mathsf{fma}\left(z, t, t\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{y + t}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 93.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \frac{y + t}{z}\\ \mathbf{if}\;z \leq -0.75:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - \mathsf{fma}\left(z, t, t\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* x (/ (+ y t) z))))
   (if (<= z -0.75) t_1 (if (<= z 1.0) (* x (- (/ y z) (fma z t t))) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x * ((y + t) / z);
	double tmp;
	if (z <= -0.75) {
		tmp = t_1;
	} else if (z <= 1.0) {
		tmp = x * ((y / z) - fma(z, t, t));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(x * Float64(Float64(y + t) / z))
	tmp = 0.0
	if (z <= -0.75)
		tmp = t_1;
	elseif (z <= 1.0)
		tmp = Float64(x * Float64(Float64(y / z) - fma(z, t, t)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x * N[(N[(y + t), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -0.75], t$95$1, If[LessEqual[z, 1.0], N[(x * N[(N[(y / z), $MachinePrecision] - N[(z * t + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -0.75 or 1 < z

   1. Initial program 98.1%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

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

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

      3. metadata-eval N/A

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

      4. *-lft-identity N/A

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

      5. lower-+.f64 97.5

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

   5. Simplified 97.5%

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

   if -0.75 < z < 1

   1. Initial program 93.2%

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

   3. Taylor expanded in z around 0

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

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

      3. lower-fma.f64 92.5

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

   5. Simplified 92.5%

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

Alternative 4: 93.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if z < -1 or 1 < z

   1. Initial program 98.1%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

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

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

      3. metadata-eval N/A

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

      4. *-lft-identity N/A

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

      5. lower-+.f64 97.5

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

   5. Simplified 97.5%

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

   if -1 < z < 1

   1. Initial program 93.2%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. sub-neg N/A

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

      5. +-commutative N/A

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

      6. lift-/.f64 N/A

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

      7. distribute-neg-frac2 N/A

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

      8. div-inv N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift--.f64 N/A

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

      12. sub-neg N/A

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

      13. distribute-neg-in N/A

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

      14. metadata-eval N/A

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

      15. remove-double-neg N/A

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

      16. lower-+.f64 93.3

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

   4. Applied egg-rr 93.3%

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

   5. Taylor expanded in z around 0

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

   7. Simplified 92.0%

      \[\leadsto x \cdot \mathsf{fma}\left(t, \color{blue}{-1}, \frac{y}{z}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 82.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \frac{y + t}{z}\\ \mathbf{if}\;z \leq -1.1 \cdot 10^{-99}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 2.5 \cdot 10^{-61}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* x (/ (+ y t) z))))
   (if (<= z -1.1e-99) t_1 (if (<= z 2.5e-61) (* y (/ x z)) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x * ((y + t) / z);
	double tmp;
	if (z <= -1.1e-99) {
		tmp = t_1;
	} else if (z <= 2.5e-61) {
		tmp = y * (x / z);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = x * ((y + t) / z)
    if (z <= (-1.1d-99)) then
        tmp = t_1
    else if (z <= 2.5d-61) then
        tmp = y * (x / z)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = x * ((y + t) / z);
	double tmp;
	if (z <= -1.1e-99) {
		tmp = t_1;
	} else if (z <= 2.5e-61) {
		tmp = y * (x / z);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = x * ((y + t) / z)
	tmp = 0
	if z <= -1.1e-99:
		tmp = t_1
	elif z <= 2.5e-61:
		tmp = y * (x / z)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(x * Float64(Float64(y + t) / z))
	tmp = 0.0
	if (z <= -1.1e-99)
		tmp = t_1;
	elseif (z <= 2.5e-61)
		tmp = Float64(y * Float64(x / z));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = x * ((y + t) / z);
	tmp = 0.0;
	if (z <= -1.1e-99)
		tmp = t_1;
	elseif (z <= 2.5e-61)
		tmp = y * (x / 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 + t), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -1.1e-99], t$95$1, If[LessEqual[z, 2.5e-61], N[(y * N[(x / z), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.10000000000000002e-99 or 2.4999999999999999e-61 < z

   1. Initial program 98.5%

      \[x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \]
   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. lower-+.f64 90.2

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

   5. Simplified 90.2%

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

   if -1.10000000000000002e-99 < z < 2.4999999999999999e-61

   1. Initial program 90.8%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 70.7

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

   5. Simplified 70.7%

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

      1. lift-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. div-inv N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 77.1

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

   7. Applied egg-rr 77.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 77.1

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-/.f64 N/A

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

      8. div-inv N/A

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

      9. lower-/.f64 77.2

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

   9. Applied egg-rr 77.2%

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

4. Final simplification 85.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.1 \cdot 10^{-99}:\\ \;\;\;\;x \cdot \frac{y + t}{z}\\ \mathbf{elif}\;z \leq 2.5 \cdot 10^{-61}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y + t}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 74.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \frac{t}{z + -1}\\ \mathbf{if}\;t \leq -1.02 \cdot 10^{+21}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 1.8 \cdot 10^{+165}:\\ \;\;\;\;\frac{y}{z} \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* x (/ t (+ z -1.0)))))
   (if (<= t -1.02e+21) t_1 (if (<= t 1.8e+165) (* (/ y z) x) 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 <= -1.02e+21) {
		tmp = t_1;
	} else if (t <= 1.8e+165) {
		tmp = (y / z) * x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = x * (t / (z + (-1.0d0)))
    if (t <= (-1.02d+21)) then
        tmp = t_1
    else if (t <= 1.8d+165) then
        tmp = (y / z) * x
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = x * (t / (z + -1.0));
	double tmp;
	if (t <= -1.02e+21) {
		tmp = t_1;
	} else if (t <= 1.8e+165) {
		tmp = (y / z) * x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = x * (t / (z + -1.0))
	tmp = 0
	if t <= -1.02e+21:
		tmp = t_1
	elif t <= 1.8e+165:
		tmp = (y / z) * x
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(x * Float64(t / Float64(z + -1.0)))
	tmp = 0.0
	if (t <= -1.02e+21)
		tmp = t_1;
	elseif (t <= 1.8e+165)
		tmp = Float64(Float64(y / z) * x);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = x * (t / (z + -1.0));
	tmp = 0.0;
	if (t <= -1.02e+21)
		tmp = t_1;
	elseif (t <= 1.8e+165)
		tmp = (y / z) * x;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x * N[(t / N[(z + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -1.02e+21], t$95$1, If[LessEqual[t, 1.8e+165], N[(N[(y / z), $MachinePrecision] * x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -1.02e21 or 1.7999999999999999e165 < t

   1. Initial program 95.1%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. mul-1-neg N/A

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

      8. distribute-neg-frac2 N/A

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

      9. lower-/.f64 N/A

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

      10. sub-neg N/A

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

      11. mul-1-neg N/A

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

      12. distribute-neg-in N/A

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

      13. metadata-eval N/A

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

      14. mul-1-neg N/A

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

      15. remove-double-neg N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 78.9

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

   5. Simplified 78.9%

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

   if -1.02e21 < t < 1.7999999999999999e165

   1. Initial program 95.6%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 81.4

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

   5. Simplified 81.4%

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

4. Final simplification 80.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.02 \cdot 10^{+21}:\\ \;\;\;\;x \cdot \frac{t}{z + -1}\\ \mathbf{elif}\;t \leq 1.8 \cdot 10^{+165}:\\ \;\;\;\;\frac{y}{z} \cdot x\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{t}{z + -1}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 68.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \frac{t}{z}\\ \mathbf{if}\;t \leq -1.5 \cdot 10^{+148}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 3 \cdot 10^{+165}:\\ \;\;\;\;\frac{y}{z} \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* x (/ t z))))
   (if (<= t -1.5e+148) t_1 (if (<= t 3e+165) (* (/ y z) x) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x * (t / z);
	double tmp;
	if (t <= -1.5e+148) {
		tmp = t_1;
	} else if (t <= 3e+165) {
		tmp = (y / z) * x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = x * (t / z)
    if (t <= (-1.5d+148)) then
        tmp = t_1
    else if (t <= 3d+165) then
        tmp = (y / z) * x
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = x * (t / z);
	double tmp;
	if (t <= -1.5e+148) {
		tmp = t_1;
	} else if (t <= 3e+165) {
		tmp = (y / z) * x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = x * (t / z)
	tmp = 0
	if t <= -1.5e+148:
		tmp = t_1
	elif t <= 3e+165:
		tmp = (y / z) * x
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(x * Float64(t / z))
	tmp = 0.0
	if (t <= -1.5e+148)
		tmp = t_1;
	elseif (t <= 3e+165)
		tmp = Float64(Float64(y / z) * x);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = x * (t / z);
	tmp = 0.0;
	if (t <= -1.5e+148)
		tmp = t_1;
	elseif (t <= 3e+165)
		tmp = (y / z) * x;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x * N[(t / z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -1.5e+148], t$95$1, If[LessEqual[t, 3e+165], N[(N[(y / z), $MachinePrecision] * x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -1.50000000000000007e148 or 2.9999999999999999e165 < t

   1. Initial program 95.3%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

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

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

      3. metadata-eval N/A

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

      4. *-lft-identity N/A

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

      5. lower-+.f64 67.6

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

   5. Simplified 67.6%

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

   6. Taylor expanded in y around 0

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

      1. lower-/.f64 59.6

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

   8. Simplified 59.6%

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

   if -1.50000000000000007e148 < t < 2.9999999999999999e165

   1. Initial program 95.5%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 78.1

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

   5. Simplified 78.1%

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

4. Final simplification 73.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.5 \cdot 10^{+148}:\\ \;\;\;\;x \cdot \frac{t}{z}\\ \mathbf{elif}\;t \leq 3 \cdot 10^{+165}:\\ \;\;\;\;\frac{y}{z} \cdot x\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{t}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 44.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \frac{t}{z}\\ \mathbf{if}\;z \leq -0.75:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1:\\ \;\;\;\;x \cdot \left(t \cdot \left(-1 - z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* x (/ t z))))
   (if (<= z -0.75) t_1 (if (<= z 1.0) (* x (* t (- -1.0 z))) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x * (t / z);
	double tmp;
	if (z <= -0.75) {
		tmp = t_1;
	} else if (z <= 1.0) {
		tmp = x * (t * (-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) :: tmp
    t_1 = x * (t / z)
    if (z <= (-0.75d0)) then
        tmp = t_1
    else if (z <= 1.0d0) then
        tmp = x * (t * ((-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 * (t / z);
	double tmp;
	if (z <= -0.75) {
		tmp = t_1;
	} else if (z <= 1.0) {
		tmp = x * (t * (-1.0 - z));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = x * (t / z)
	tmp = 0
	if z <= -0.75:
		tmp = t_1
	elif z <= 1.0:
		tmp = x * (t * (-1.0 - z))
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(x * Float64(t / z))
	tmp = 0.0
	if (z <= -0.75)
		tmp = t_1;
	elseif (z <= 1.0)
		tmp = Float64(x * Float64(t * Float64(-1.0 - z)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = x * (t / z);
	tmp = 0.0;
	if (z <= -0.75)
		tmp = t_1;
	elseif (z <= 1.0)
		tmp = x * (t * (-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[(t / z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -0.75], t$95$1, If[LessEqual[z, 1.0], N[(x * N[(t * N[(-1.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -0.75 or 1 < z

   1. Initial program 98.1%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

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

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

      3. metadata-eval N/A

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

      4. *-lft-identity N/A

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

      5. lower-+.f64 97.5

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

   5. Simplified 97.5%

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

   6. Taylor expanded in y around 0

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

      1. lower-/.f64 58.4

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

   8. Simplified 58.4%

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

   if -0.75 < z < 1

   1. Initial program 93.2%

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

   3. Taylor expanded in z around 0

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

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

      3. lower-fma.f64 92.5

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

   5. Simplified 92.5%

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

   6. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. distribute-rgt-neg-in N/A

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

      3. lower-*.f64 N/A

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

      4. *-rgt-identity N/A

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

      5. distribute-lft-in N/A

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

      6. distribute-rgt-neg-in N/A

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

      7. +-commutative N/A

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

      8. distribute-neg-in N/A

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

      9. mul-1-neg N/A

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

      10. sub-neg N/A

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

      11. lower-*.f64 N/A

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

      12. sub-neg N/A

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

      13. metadata-eval N/A

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

      14. +-commutative N/A

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

      15. mul-1-neg N/A

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

      16. unsub-neg N/A

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

      17. lower--.f64 35.1

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

   8. Simplified 35.1%

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

Alternative 9: 20.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -4 \cdot 10^{+36}:\\ \;\;\;\;z \cdot \left(t \cdot \left(-x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(t \cdot \left(-1 - z\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -4e+36) (* z (* t (- x))) (* x (* t (- -1.0 z)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -4e+36) {
		tmp = z * (t * -x);
	} else {
		tmp = x * (t * (-1.0 - z));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (z <= (-4d+36)) then
        tmp = z * (t * -x)
    else
        tmp = x * (t * ((-1.0d0) - z))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -4e+36) {
		tmp = z * (t * -x);
	} else {
		tmp = x * (t * (-1.0 - z));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= -4e+36:
		tmp = z * (t * -x)
	else:
		tmp = x * (t * (-1.0 - z))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -4e+36)
		tmp = Float64(z * Float64(t * Float64(-x)));
	else
		tmp = Float64(x * Float64(t * Float64(-1.0 - z)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= -4e+36)
		tmp = z * (t * -x);
	else
		tmp = x * (t * (-1.0 - z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -4e+36], N[(z * N[(t * (-x)), $MachinePrecision]), $MachinePrecision], N[(x * N[(t * N[(-1.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -4.00000000000000017e36

   1. Initial program 98.1%

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

   3. Taylor expanded in z around 0

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

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

      3. lower-fma.f64 5.6

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

   5. Simplified 5.6%

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

   6. Taylor expanded in z around inf

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

      1. associate-*r* N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. mul-1-neg N/A

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

      6. distribute-rgt-neg-in N/A

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

      7. mul-1-neg N/A

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

      8. lower-*.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 13.3

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

   8. Simplified 13.3%

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

   if -4.00000000000000017e36 < z

   1. Initial program 94.7%

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

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

      3. lower-fma.f64 68.9

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

   5. Simplified 68.9%

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

   6. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. distribute-rgt-neg-in N/A

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

      3. lower-*.f64 N/A

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

      4. *-rgt-identity N/A

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

      5. distribute-lft-in N/A

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

      6. distribute-rgt-neg-in N/A

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

      7. +-commutative N/A

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

      8. distribute-neg-in N/A

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

      9. mul-1-neg N/A

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

      10. sub-neg N/A

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

      11. lower-*.f64 N/A

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

      12. sub-neg N/A

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

      13. metadata-eval N/A

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

      14. +-commutative N/A

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

      15. mul-1-neg N/A

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

      16. unsub-neg N/A

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

      17. lower--.f64 26.2

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

   8. Simplified 26.2%

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

Alternative 10: 9.7% accurate, 2.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 95.4%

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

3. Taylor expanded in z around 0

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

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

   3. lower-fma.f64 55.8

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

5. Simplified 55.8%

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

6. Taylor expanded in z around inf

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

   1. associate-*r* N/A

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

   2. associate-*r* N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

   5. mul-1-neg N/A

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

   6. distribute-rgt-neg-in N/A

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

   7. mul-1-neg N/A

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

   8. lower-*.f64 N/A

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

   9. mul-1-neg N/A

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

   10. lower-neg.f64 10.6

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

8. Simplified 10.6%

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

Alternative 11: 5.7% accurate, 2.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 95.4%

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

3. Taylor expanded in z around 0

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

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

   3. lower-fma.f64 55.8

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

5. Simplified 55.8%

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

6. Taylor expanded in z around inf

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

   1. associate-*r* N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 N/A

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

   4. mul-1-neg N/A

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

   5. lower-neg.f64 6.6

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

8. Simplified 6.6%

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

Developer Target 1: 94.6% 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 2024207 
(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)))))