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

Percentage Accurate: 94.6% → 98.3%

Time: 9.6s

Alternatives: 10

Speedup: 0.3×

• 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: 94.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y \cdot \frac{x}{z}\\ t_2 := \frac{y}{z} + \frac{t}{z + -1}\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 4 \cdot 10^{+296}:\\ \;\;\;\;t\_2 \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* y (/ x z))) (t_2 (+ (/ y z) (/ t (+ z -1.0)))))
   (if (<= t_2 (- INFINITY)) t_1 (if (<= t_2 4e+296) (* t_2 x) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = y * (x / z);
	double t_2 = (y / z) + (t / (z + -1.0));
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = t_1;
	} else if (t_2 <= 4e+296) {
		tmp = t_2 * x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = y * (x / z);
	double t_2 = (y / z) + (t / (z + -1.0));
	double tmp;
	if (t_2 <= -Double.POSITIVE_INFINITY) {
		tmp = t_1;
	} else if (t_2 <= 4e+296) {
		tmp = t_2 * x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = y * (x / z)
	t_2 = (y / z) + (t / (z + -1.0))
	tmp = 0
	if t_2 <= -math.inf:
		tmp = t_1
	elif t_2 <= 4e+296:
		tmp = t_2 * x
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(y * Float64(x / z))
	t_2 = Float64(Float64(y / z) + Float64(t / Float64(z + -1.0)))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = t_1;
	elseif (t_2 <= 4e+296)
		tmp = Float64(t_2 * x);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = y * (x / z);
	t_2 = (y / z) + (t / (z + -1.0));
	tmp = 0.0;
	if (t_2 <= -Inf)
		tmp = t_1;
	elseif (t_2 <= 4e+296)
		tmp = t_2 * x;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(y * N[(x / z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(y / z), $MachinePrecision] + N[(t / N[(z + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], t$95$1, If[LessEqual[t$95$2, 4e+296], N[(t$95$2 * x), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (/.f64 y z) (/.f64 t (-.f64 #s(literal 1 binary64) z))) < -inf.0 or 3.99999999999999993e296 < (-.f64 (/.f64 y z) (/.f64 t (-.f64 #s(literal 1 binary64) z)))

   1. Initial program 63.7%

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

   3. Taylor expanded in y around inf

      \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{y}{z}\right)}\right) \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 63.7%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \color{blue}{z}\right)\right) \]

   5. Simplified 63.7%

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

      1. div-inv N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. div-inv N/A

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

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{x}{z}\right), \color{blue}{y}\right) \]

      6. /-lowering-/.f64 99.8%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(x, z\right), y\right) \]

   7. Applied egg-rr 99.8%

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

   if -inf.0 < (-.f64 (/.f64 y z) (/.f64 t (-.f64 #s(literal 1 binary64) z))) < 3.99999999999999993e296

   1. Initial program 98.7%

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

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y}{z} + \frac{t}{z + -1} \leq -\infty:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{elif}\;\frac{y}{z} + \frac{t}{z + -1} \leq 4 \cdot 10^{+296}:\\ \;\;\;\;\left(\frac{y}{z} + \frac{t}{z + -1}\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 75.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x}{\frac{z + -1}{t}}\\ \mathbf{if}\;t \leq -2.4 \cdot 10^{+94}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq -1 \cdot 10^{-246}:\\ \;\;\;\;\frac{y}{z} \cdot x\\ \mathbf{elif}\;t \leq 230:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ x (/ (+ z -1.0) t))))
   (if (<= t -2.4e+94)
     t_1
     (if (<= t -1e-246) (* (/ y z) x) (if (<= t 230.0) (* y (/ x 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 <= -2.4e+94) {
		tmp = t_1;
	} else if (t <= -1e-246) {
		tmp = (y / z) * x;
	} else if (t <= 230.0) {
		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 / ((z + (-1.0d0)) / t)
    if (t <= (-2.4d+94)) then
        tmp = t_1
    else if (t <= (-1d-246)) then
        tmp = (y / z) * x
    else if (t <= 230.0d0) 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 / ((z + -1.0) / t);
	double tmp;
	if (t <= -2.4e+94) {
		tmp = t_1;
	} else if (t <= -1e-246) {
		tmp = (y / z) * x;
	} else if (t <= 230.0) {
		tmp = y * (x / 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 <= -2.4e+94:
		tmp = t_1
	elif t <= -1e-246:
		tmp = (y / z) * x
	elif t <= 230.0:
		tmp = y * (x / z)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(x / Float64(Float64(z + -1.0) / t))
	tmp = 0.0
	if (t <= -2.4e+94)
		tmp = t_1;
	elseif (t <= -1e-246)
		tmp = Float64(Float64(y / z) * x);
	elseif (t <= 230.0)
		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 / ((z + -1.0) / t);
	tmp = 0.0;
	if (t <= -2.4e+94)
		tmp = t_1;
	elseif (t <= -1e-246)
		tmp = (y / z) * x;
	elseif (t <= 230.0)
		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[(z + -1.0), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -2.4e+94], t$95$1, If[LessEqual[t, -1e-246], N[(N[(y / z), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[t, 230.0], N[(y * N[(x / z), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if t < -2.39999999999999983e94 or 230 < t

   1. Initial program 93.7%

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

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(-1 \cdot \frac{t}{1 - z}\right)}\right) \]

      7. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\mathsf{neg}\left(\frac{t}{1 - z}\right)\right)\right) \]

      8. distribute-neg-frac2 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{t}{\color{blue}{\mathsf{neg}\left(\left(1 - z\right)\right)}}\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(t, \color{blue}{\left(\mathsf{neg}\left(\left(1 - z\right)\right)\right)}\right)\right) \]

      10. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(t, \left(\mathsf{neg}\left(\left(1 + \left(\mathsf{neg}\left(z\right)\right)\right)\right)\right)\right)\right) \]

      11. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(t, \left(\mathsf{neg}\left(\left(1 + -1 \cdot z\right)\right)\right)\right)\right) \]

      12. distribute-neg-in N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(t, \left(\left(\mathsf{neg}\left(1\right)\right) + \color{blue}{\left(\mathsf{neg}\left(-1 \cdot z\right)\right)}\right)\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(t, \left(-1 + \left(\mathsf{neg}\left(\color{blue}{-1 \cdot z}\right)\right)\right)\right)\right) \]

      14. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(t, \left(-1 + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(z\right)\right)\right)\right)\right)\right)\right) \]

      15. remove-double-neg N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(t, \left(-1 + z\right)\right)\right) \]

      16. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(t, \left(z + \color{blue}{-1}\right)\right)\right) \]

      17. +-lowering-+.f64 73.0%

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(t, \mathsf{+.f64}\left(z, \color{blue}{-1}\right)\right)\right) \]

   5. Simplified 73.0%

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

      1. clear-num N/A

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

      2. un-div-inv N/A

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

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{\left(\frac{z + -1}{t}\right)}\right) \]

   7. Applied egg-rr 73.0%

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

   if -2.39999999999999983e94 < t < -9.99999999999999956e-247

   1. Initial program 98.4%

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

   3. Taylor expanded in y around inf

      \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{y}{z}\right)}\right) \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 90.0%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \color{blue}{z}\right)\right) \]

   5. Simplified 90.0%

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

   if -9.99999999999999956e-247 < t < 230

   1. Initial program 87.2%

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

   3. Taylor expanded in y around inf

      \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{y}{z}\right)}\right) \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 77.9%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \color{blue}{z}\right)\right) \]

   5. Simplified 77.9%

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

      1. div-inv N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. div-inv N/A

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

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{x}{z}\right), \color{blue}{y}\right) \]

      6. /-lowering-/.f64 84.1%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(x, z\right), y\right) \]

   7. Applied egg-rr 84.1%

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

4. Final simplification 80.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2.4 \cdot 10^{+94}:\\ \;\;\;\;\frac{x}{\frac{z + -1}{t}}\\ \mathbf{elif}\;t \leq -1 \cdot 10^{-246}:\\ \;\;\;\;\frac{y}{z} \cdot x\\ \mathbf{elif}\;t \leq 230:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z + -1}{t}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 75.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \frac{t}{z + -1}\\ \mathbf{if}\;t \leq -6.6 \cdot 10^{+93}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq -1.1 \cdot 10^{-247}:\\ \;\;\;\;\frac{y}{z} \cdot x\\ \mathbf{elif}\;t \leq 10.6:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* x (/ t (+ z -1.0)))))
   (if (<= t -6.6e+93)
     t_1
     (if (<= t -1.1e-247) (* (/ y z) x) (if (<= t 10.6) (* y (/ x 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 <= -6.6e+93) {
		tmp = t_1;
	} else if (t <= -1.1e-247) {
		tmp = (y / z) * x;
	} else if (t <= 10.6) {
		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 * (t / (z + (-1.0d0)))
    if (t <= (-6.6d+93)) then
        tmp = t_1
    else if (t <= (-1.1d-247)) then
        tmp = (y / z) * x
    else if (t <= 10.6d0) 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 * (t / (z + -1.0));
	double tmp;
	if (t <= -6.6e+93) {
		tmp = t_1;
	} else if (t <= -1.1e-247) {
		tmp = (y / z) * x;
	} else if (t <= 10.6) {
		tmp = y * (x / 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 <= -6.6e+93:
		tmp = t_1
	elif t <= -1.1e-247:
		tmp = (y / z) * x
	elif t <= 10.6:
		tmp = y * (x / z)
	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 <= -6.6e+93)
		tmp = t_1;
	elseif (t <= -1.1e-247)
		tmp = Float64(Float64(y / z) * x);
	elseif (t <= 10.6)
		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 * (t / (z + -1.0));
	tmp = 0.0;
	if (t <= -6.6e+93)
		tmp = t_1;
	elseif (t <= -1.1e-247)
		tmp = (y / z) * x;
	elseif (t <= 10.6)
		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[(t / N[(z + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -6.6e+93], t$95$1, If[LessEqual[t, -1.1e-247], N[(N[(y / z), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[t, 10.6], N[(y * N[(x / z), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if t < -6.60000000000000017e93 or 10.5999999999999996 < t

   1. Initial program 93.7%

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

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(-1 \cdot \frac{t}{1 - z}\right)}\right) \]

      7. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\mathsf{neg}\left(\frac{t}{1 - z}\right)\right)\right) \]

      8. distribute-neg-frac2 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{t}{\color{blue}{\mathsf{neg}\left(\left(1 - z\right)\right)}}\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(t, \color{blue}{\left(\mathsf{neg}\left(\left(1 - z\right)\right)\right)}\right)\right) \]

      10. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(t, \left(\mathsf{neg}\left(\left(1 + \left(\mathsf{neg}\left(z\right)\right)\right)\right)\right)\right)\right) \]

      11. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(t, \left(\mathsf{neg}\left(\left(1 + -1 \cdot z\right)\right)\right)\right)\right) \]

      12. distribute-neg-in N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(t, \left(\left(\mathsf{neg}\left(1\right)\right) + \color{blue}{\left(\mathsf{neg}\left(-1 \cdot z\right)\right)}\right)\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(t, \left(-1 + \left(\mathsf{neg}\left(\color{blue}{-1 \cdot z}\right)\right)\right)\right)\right) \]

      14. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(t, \left(-1 + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(z\right)\right)\right)\right)\right)\right)\right) \]

      15. remove-double-neg N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(t, \left(-1 + z\right)\right)\right) \]

      16. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(t, \left(z + \color{blue}{-1}\right)\right)\right) \]

      17. +-lowering-+.f64 73.0%

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(t, \mathsf{+.f64}\left(z, \color{blue}{-1}\right)\right)\right) \]

   5. Simplified 73.0%

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

   if -6.60000000000000017e93 < t < -1.09999999999999996e-247

   1. Initial program 98.4%

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

   3. Taylor expanded in y around inf

      \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{y}{z}\right)}\right) \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 90.0%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \color{blue}{z}\right)\right) \]

   5. Simplified 90.0%

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

   if -1.09999999999999996e-247 < t < 10.5999999999999996

   1. Initial program 87.2%

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

   3. Taylor expanded in y around inf

      \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{y}{z}\right)}\right) \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 77.9%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \color{blue}{z}\right)\right) \]

   5. Simplified 77.9%

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

      1. div-inv N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. div-inv N/A

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

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{x}{z}\right), \color{blue}{y}\right) \]

      6. /-lowering-/.f64 84.1%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(x, z\right), y\right) \]

   7. Applied egg-rr 84.1%

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

4. Final simplification 80.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -6.6 \cdot 10^{+93}:\\ \;\;\;\;x \cdot \frac{t}{z + -1}\\ \mathbf{elif}\;t \leq -1.1 \cdot 10^{-247}:\\ \;\;\;\;\frac{y}{z} \cdot x\\ \mathbf{elif}\;t \leq 10.6:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{t}{z + -1}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 68.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x}{\frac{z}{t}}\\ \mathbf{if}\;t \leq -3.2 \cdot 10^{+115}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq -2.3 \cdot 10^{-248}:\\ \;\;\;\;\frac{y}{z} \cdot x\\ \mathbf{elif}\;t \leq 7.8 \cdot 10^{+70}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ x (/ z t))))
   (if (<= t -3.2e+115)
     t_1
     (if (<= t -2.3e-248)
       (* (/ y z) x)
       (if (<= t 7.8e+70) (* y (/ x z)) t_1)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x / (z / t);
	double tmp;
	if (t <= -3.2e+115) {
		tmp = t_1;
	} else if (t <= -2.3e-248) {
		tmp = (y / z) * x;
	} else if (t <= 7.8e+70) {
		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 / (z / t)
    if (t <= (-3.2d+115)) then
        tmp = t_1
    else if (t <= (-2.3d-248)) then
        tmp = (y / z) * x
    else if (t <= 7.8d+70) 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 / (z / t);
	double tmp;
	if (t <= -3.2e+115) {
		tmp = t_1;
	} else if (t <= -2.3e-248) {
		tmp = (y / z) * x;
	} else if (t <= 7.8e+70) {
		tmp = y * (x / z);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = x / (z / t)
	tmp = 0
	if t <= -3.2e+115:
		tmp = t_1
	elif t <= -2.3e-248:
		tmp = (y / z) * x
	elif t <= 7.8e+70:
		tmp = y * (x / z)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(x / Float64(z / t))
	tmp = 0.0
	if (t <= -3.2e+115)
		tmp = t_1;
	elseif (t <= -2.3e-248)
		tmp = Float64(Float64(y / z) * x);
	elseif (t <= 7.8e+70)
		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 / (z / t);
	tmp = 0.0;
	if (t <= -3.2e+115)
		tmp = t_1;
	elseif (t <= -2.3e-248)
		tmp = (y / z) * x;
	elseif (t <= 7.8e+70)
		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[(z / t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -3.2e+115], t$95$1, If[LessEqual[t, -2.3e-248], N[(N[(y / z), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[t, 7.8e+70], N[(y * N[(x / z), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if t < -3.2e115 or 7.79999999999999949e70 < t

   1. Initial program 94.7%

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

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(-1 \cdot \frac{t}{1 - z}\right)}\right) \]

      7. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\mathsf{neg}\left(\frac{t}{1 - z}\right)\right)\right) \]

      8. distribute-neg-frac2 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{t}{\color{blue}{\mathsf{neg}\left(\left(1 - z\right)\right)}}\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(t, \color{blue}{\left(\mathsf{neg}\left(\left(1 - z\right)\right)\right)}\right)\right) \]

      10. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(t, \left(\mathsf{neg}\left(\left(1 + \left(\mathsf{neg}\left(z\right)\right)\right)\right)\right)\right)\right) \]

      11. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(t, \left(\mathsf{neg}\left(\left(1 + -1 \cdot z\right)\right)\right)\right)\right) \]

      12. distribute-neg-in N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(t, \left(\left(\mathsf{neg}\left(1\right)\right) + \color{blue}{\left(\mathsf{neg}\left(-1 \cdot z\right)\right)}\right)\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(t, \left(-1 + \left(\mathsf{neg}\left(\color{blue}{-1 \cdot z}\right)\right)\right)\right)\right) \]

      14. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(t, \left(-1 + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(z\right)\right)\right)\right)\right)\right)\right) \]

      15. remove-double-neg N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(t, \left(-1 + z\right)\right)\right) \]

      16. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(t, \left(z + \color{blue}{-1}\right)\right)\right) \]

      17. +-lowering-+.f64 74.1%

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(t, \mathsf{+.f64}\left(z, \color{blue}{-1}\right)\right)\right) \]

   5. Simplified 74.1%

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

   6. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{t}{z}\right)}\right) \]

      4. /-lowering-/.f64 55.6%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(t, \color{blue}{z}\right)\right) \]

   8. Simplified 55.6%

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

      1. clear-num N/A

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

      2. un-div-inv N/A

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

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{\left(\frac{z}{t}\right)}\right) \]

      4. /-lowering-/.f64 55.7%

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{/.f64}\left(z, \color{blue}{t}\right)\right) \]

   10. Applied egg-rr 55.7%

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

   if -3.2e115 < t < -2.3e-248

   1. Initial program 98.5%

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

   3. Taylor expanded in y around inf

      \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{y}{z}\right)}\right) \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 87.7%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \color{blue}{z}\right)\right) \]

   5. Simplified 87.7%

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

   if -2.3e-248 < t < 7.79999999999999949e70

   1. Initial program 86.6%

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

   3. Taylor expanded in y around inf

      \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{y}{z}\right)}\right) \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 73.0%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \color{blue}{z}\right)\right) \]

   5. Simplified 73.0%

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

      1. div-inv N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. div-inv N/A

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

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{x}{z}\right), \color{blue}{y}\right) \]

      6. /-lowering-/.f64 80.6%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(x, z\right), y\right) \]

   7. Applied egg-rr 80.6%

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

4. Final simplification 73.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -3.2 \cdot 10^{+115}:\\ \;\;\;\;\frac{x}{\frac{z}{t}}\\ \mathbf{elif}\;t \leq -2.3 \cdot 10^{-248}:\\ \;\;\;\;\frac{y}{z} \cdot x\\ \mathbf{elif}\;t \leq 7.8 \cdot 10^{+70}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{t}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 68.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \frac{t}{z}\\ \mathbf{if}\;t \leq -6.6 \cdot 10^{+117}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq -7.5 \cdot 10^{-249}:\\ \;\;\;\;\frac{y}{z} \cdot x\\ \mathbf{elif}\;t \leq 7.2 \cdot 10^{+68}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* x (/ t z))))
   (if (<= t -6.6e+117)
     t_1
     (if (<= t -7.5e-249)
       (* (/ y z) x)
       (if (<= t 7.2e+68) (* y (/ x z)) t_1)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x * (t / z);
	double tmp;
	if (t <= -6.6e+117) {
		tmp = t_1;
	} else if (t <= -7.5e-249) {
		tmp = (y / z) * x;
	} else if (t <= 7.2e+68) {
		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 * (t / z)
    if (t <= (-6.6d+117)) then
        tmp = t_1
    else if (t <= (-7.5d-249)) then
        tmp = (y / z) * x
    else if (t <= 7.2d+68) 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 * (t / z);
	double tmp;
	if (t <= -6.6e+117) {
		tmp = t_1;
	} else if (t <= -7.5e-249) {
		tmp = (y / z) * x;
	} else if (t <= 7.2e+68) {
		tmp = y * (x / z);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = x * (t / z)
	tmp = 0
	if t <= -6.6e+117:
		tmp = t_1
	elif t <= -7.5e-249:
		tmp = (y / z) * x
	elif t <= 7.2e+68:
		tmp = y * (x / 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 (t <= -6.6e+117)
		tmp = t_1;
	elseif (t <= -7.5e-249)
		tmp = Float64(Float64(y / z) * x);
	elseif (t <= 7.2e+68)
		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 * (t / z);
	tmp = 0.0;
	if (t <= -6.6e+117)
		tmp = t_1;
	elseif (t <= -7.5e-249)
		tmp = (y / z) * x;
	elseif (t <= 7.2e+68)
		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[(t / z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -6.6e+117], t$95$1, If[LessEqual[t, -7.5e-249], N[(N[(y / z), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[t, 7.2e+68], N[(y * N[(x / z), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if t < -6.5999999999999996e117 or 7.1999999999999998e68 < t

   1. Initial program 94.7%

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

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(-1 \cdot \frac{t}{1 - z}\right)}\right) \]

      7. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\mathsf{neg}\left(\frac{t}{1 - z}\right)\right)\right) \]

      8. distribute-neg-frac2 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{t}{\color{blue}{\mathsf{neg}\left(\left(1 - z\right)\right)}}\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(t, \color{blue}{\left(\mathsf{neg}\left(\left(1 - z\right)\right)\right)}\right)\right) \]

      10. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(t, \left(\mathsf{neg}\left(\left(1 + \left(\mathsf{neg}\left(z\right)\right)\right)\right)\right)\right)\right) \]

      11. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(t, \left(\mathsf{neg}\left(\left(1 + -1 \cdot z\right)\right)\right)\right)\right) \]

      12. distribute-neg-in N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(t, \left(\left(\mathsf{neg}\left(1\right)\right) + \color{blue}{\left(\mathsf{neg}\left(-1 \cdot z\right)\right)}\right)\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(t, \left(-1 + \left(\mathsf{neg}\left(\color{blue}{-1 \cdot z}\right)\right)\right)\right)\right) \]

      14. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(t, \left(-1 + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(z\right)\right)\right)\right)\right)\right)\right) \]

      15. remove-double-neg N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(t, \left(-1 + z\right)\right)\right) \]

      16. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(t, \left(z + \color{blue}{-1}\right)\right)\right) \]

      17. +-lowering-+.f64 74.1%

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(t, \mathsf{+.f64}\left(z, \color{blue}{-1}\right)\right)\right) \]

   5. Simplified 74.1%

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

   6. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{t}{z}\right)}\right) \]

      4. /-lowering-/.f64 55.6%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(t, \color{blue}{z}\right)\right) \]

   8. Simplified 55.6%

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

   if -6.5999999999999996e117 < t < -7.50000000000000034e-249

   1. Initial program 98.5%

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

   3. Taylor expanded in y around inf

      \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{y}{z}\right)}\right) \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 87.7%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \color{blue}{z}\right)\right) \]

   5. Simplified 87.7%

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

   if -7.50000000000000034e-249 < t < 7.1999999999999998e68

   1. Initial program 86.6%

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

   3. Taylor expanded in y around inf

      \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{y}{z}\right)}\right) \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 73.0%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \color{blue}{z}\right)\right) \]

   5. Simplified 73.0%

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

      1. div-inv N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. div-inv N/A

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

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{x}{z}\right), \color{blue}{y}\right) \]

      6. /-lowering-/.f64 80.6%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(x, z\right), y\right) \]

   7. Applied egg-rr 80.6%

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

4. Final simplification 73.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -6.6 \cdot 10^{+117}:\\ \;\;\;\;x \cdot \frac{t}{z}\\ \mathbf{elif}\;t \leq -7.5 \cdot 10^{-249}:\\ \;\;\;\;\frac{y}{z} \cdot x\\ \mathbf{elif}\;t \leq 7.2 \cdot 10^{+68}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{t}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 88.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -165000:\\ \;\;\;\;\frac{x \cdot \left(y + t\right)}{z}\\ \mathbf{elif}\;z \leq 1:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - t\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{\frac{1}{y + t}}}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -165000.0)
   (/ (* x (+ y t)) z)
   (if (<= z 1.0) (* x (- (/ y z) t)) (/ (/ x (/ 1.0 (+ y t))) z))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -165000.0) {
		tmp = (x * (y + t)) / z;
	} else if (z <= 1.0) {
		tmp = x * ((y / z) - t);
	} else {
		tmp = (x / (1.0 / (y + t))) / z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (z <= (-165000.0d0)) then
        tmp = (x * (y + t)) / z
    else if (z <= 1.0d0) then
        tmp = x * ((y / z) - t)
    else
        tmp = (x / (1.0d0 / (y + t))) / z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -165000.0) {
		tmp = (x * (y + t)) / z;
	} else if (z <= 1.0) {
		tmp = x * ((y / z) - t);
	} else {
		tmp = (x / (1.0 / (y + t))) / z;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= -165000.0:
		tmp = (x * (y + t)) / z
	elif z <= 1.0:
		tmp = x * ((y / z) - t)
	else:
		tmp = (x / (1.0 / (y + t))) / z
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -165000.0)
		tmp = Float64(Float64(x * Float64(y + t)) / z);
	elseif (z <= 1.0)
		tmp = Float64(x * Float64(Float64(y / z) - t));
	else
		tmp = Float64(Float64(x / Float64(1.0 / Float64(y + t))) / z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= -165000.0)
		tmp = (x * (y + t)) / z;
	elseif (z <= 1.0)
		tmp = x * ((y / z) - t);
	else
		tmp = (x / (1.0 / (y + t))) / z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -165000.0], N[(N[(x * N[(y + t), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[z, 1.0], N[(x * N[(N[(y / z), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(1.0 / N[(y + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -165000

   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

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

      1. *-commutative N/A

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

      2. remove-double-neg N/A

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

      3. neg-mul-1 N/A

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

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

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

      5. neg-mul-1 N/A

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

      6. mul-1-neg N/A

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

      7. remove-double-neg N/A

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

      8. distribute-lft-neg-in N/A

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

      9. *-commutative N/A

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

      10. mul-1-neg N/A

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

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(-1 \cdot \left(x \cdot \left(-1 \cdot y - t\right)\right)\right), \color{blue}{z}\right) \]

   5. Simplified 79.6%

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

   if -165000 < z < 1

   1. Initial program 87.9%

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

   3. Taylor expanded in z around 0

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(y, z\right), \color{blue}{t}\right)\right) \]
   4. Step-by-step derivation

   5. Simplified 86.8%

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

   if 1 < z

   1. Initial program 97.7%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. remove-double-neg N/A

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

      3. neg-mul-1 N/A

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

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

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

      5. neg-mul-1 N/A

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

      6. mul-1-neg N/A

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

      7. remove-double-neg N/A

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

      8. distribute-lft-neg-in N/A

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

      9. *-commutative N/A

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

      10. mul-1-neg N/A

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

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(-1 \cdot \left(x \cdot \left(-1 \cdot y - t\right)\right)\right), \color{blue}{z}\right) \]

   5. Simplified 83.4%

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

      1. flip-+ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x \cdot \frac{y \cdot y - t \cdot t}{y - t}\right), z\right) \]

      2. clear-num N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x \cdot \frac{1}{\frac{y - t}{y \cdot y - t \cdot t}}\right), z\right) \]

      3. un-div-inv N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{x}{\frac{y - t}{y \cdot y - t \cdot t}}\right), z\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \left(\frac{y - t}{y \cdot y - t \cdot t}\right)\right), z\right) \]

      5. clear-num N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \left(\frac{1}{\frac{y \cdot y - t \cdot t}{y - t}}\right)\right), z\right) \]

      6. flip-+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \left(\frac{1}{y + t}\right)\right), z\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \mathsf{/.f64}\left(1, \left(y + t\right)\right)\right), z\right) \]

      8. +-lowering-+.f64 83.5%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(y, t\right)\right)\right), z\right) \]

   7. Applied egg-rr 83.5%

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

Alternative 7: 88.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x \cdot \left(y + t\right)}{z}\\ \mathbf{if}\;z \leq -165000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (* x (+ y t)) z)))
   (if (<= z -165000.0) t_1 (if (<= z 1.0) (* x (- (/ y z) 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 <= -165000.0) {
		tmp = t_1;
	} else if (z <= 1.0) {
		tmp = x * ((y / z) - 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 * (y + t)) / z
    if (z <= (-165000.0d0)) then
        tmp = t_1
    else if (z <= 1.0d0) then
        tmp = x * ((y / z) - 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 * (y + t)) / z;
	double tmp;
	if (z <= -165000.0) {
		tmp = t_1;
	} else if (z <= 1.0) {
		tmp = x * ((y / z) - t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (x * (y + t)) / z
	tmp = 0
	if z <= -165000.0:
		tmp = t_1
	elif z <= 1.0:
		tmp = x * ((y / z) - t)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x * Float64(y + t)) / z)
	tmp = 0.0
	if (z <= -165000.0)
		tmp = t_1;
	elseif (z <= 1.0)
		tmp = Float64(x * Float64(Float64(y / z) - t));
	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 <= -165000.0)
		tmp = t_1;
	elseif (z <= 1.0)
		tmp = x * ((y / z) - t);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x * N[(y + t), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]}, If[LessEqual[z, -165000.0], t$95$1, If[LessEqual[z, 1.0], N[(x * N[(N[(y / z), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -165000 or 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

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

      1. *-commutative N/A

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

      2. remove-double-neg N/A

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

      3. neg-mul-1 N/A

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

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

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

      5. neg-mul-1 N/A

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

      6. mul-1-neg N/A

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

      7. remove-double-neg N/A

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

      8. distribute-lft-neg-in N/A

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

      9. *-commutative N/A

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

      10. mul-1-neg N/A

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

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(-1 \cdot \left(x \cdot \left(-1 \cdot y - t\right)\right)\right), \color{blue}{z}\right) \]

   5. Simplified 81.5%

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

   if -165000 < z < 1

   1. Initial program 87.9%

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

   3. Taylor expanded in z around 0

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(y, z\right), \color{blue}{t}\right)\right) \]
   4. Step-by-step derivation

   5. Simplified 86.8%

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

Alternative 8: 73.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3.1 \cdot 10^{+149}:\\ \;\;\;\;\frac{x}{\frac{z}{t}}\\ \mathbf{elif}\;z \leq 2.7 \cdot 10^{+16}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - t\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{t}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -3.1e+149)
   (/ x (/ z t))
   (if (<= z 2.7e+16) (* x (- (/ y z) t)) (* x (/ t z)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -3.1e+149) {
		tmp = x / (z / t);
	} else if (z <= 2.7e+16) {
		tmp = x * ((y / z) - t);
	} else {
		tmp = x * (t / z);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (z <= (-3.1d+149)) then
        tmp = x / (z / t)
    else if (z <= 2.7d+16) then
        tmp = x * ((y / z) - t)
    else
        tmp = x * (t / z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -3.1e+149) {
		tmp = x / (z / t);
	} else if (z <= 2.7e+16) {
		tmp = x * ((y / z) - t);
	} else {
		tmp = x * (t / z);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= -3.1e+149:
		tmp = x / (z / t)
	elif z <= 2.7e+16:
		tmp = x * ((y / z) - t)
	else:
		tmp = x * (t / z)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -3.1e+149)
		tmp = Float64(x / Float64(z / t));
	elseif (z <= 2.7e+16)
		tmp = Float64(x * Float64(Float64(y / z) - t));
	else
		tmp = Float64(x * Float64(t / z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= -3.1e+149)
		tmp = x / (z / t);
	elseif (z <= 2.7e+16)
		tmp = x * ((y / z) - t);
	else
		tmp = x * (t / z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -3.1e+149], N[(x / N[(z / t), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 2.7e+16], N[(x * N[(N[(y / z), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision], N[(x * N[(t / z), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -3.09999999999999987e149

   1. Initial program 97.0%

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

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(-1 \cdot \frac{t}{1 - z}\right)}\right) \]

      7. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\mathsf{neg}\left(\frac{t}{1 - z}\right)\right)\right) \]

      8. distribute-neg-frac2 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{t}{\color{blue}{\mathsf{neg}\left(\left(1 - z\right)\right)}}\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(t, \color{blue}{\left(\mathsf{neg}\left(\left(1 - z\right)\right)\right)}\right)\right) \]

      10. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(t, \left(\mathsf{neg}\left(\left(1 + \left(\mathsf{neg}\left(z\right)\right)\right)\right)\right)\right)\right) \]

      11. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(t, \left(\mathsf{neg}\left(\left(1 + -1 \cdot z\right)\right)\right)\right)\right) \]

      12. distribute-neg-in N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(t, \left(\left(\mathsf{neg}\left(1\right)\right) + \color{blue}{\left(\mathsf{neg}\left(-1 \cdot z\right)\right)}\right)\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(t, \left(-1 + \left(\mathsf{neg}\left(\color{blue}{-1 \cdot z}\right)\right)\right)\right)\right) \]

      14. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(t, \left(-1 + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(z\right)\right)\right)\right)\right)\right)\right) \]

      15. remove-double-neg N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(t, \left(-1 + z\right)\right)\right) \]

      16. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(t, \left(z + \color{blue}{-1}\right)\right)\right) \]

      17. +-lowering-+.f64 66.2%

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(t, \mathsf{+.f64}\left(z, \color{blue}{-1}\right)\right)\right) \]

   5. Simplified 66.2%

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

   6. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{t}{z}\right)}\right) \]

      4. /-lowering-/.f64 66.2%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(t, \color{blue}{z}\right)\right) \]

   8. Simplified 66.2%

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

      1. clear-num N/A

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

      2. un-div-inv N/A

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

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{\left(\frac{z}{t}\right)}\right) \]

      4. /-lowering-/.f64 66.3%

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{/.f64}\left(z, \color{blue}{t}\right)\right) \]

   10. Applied egg-rr 66.3%

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

   if -3.09999999999999987e149 < z < 2.7e16

   1. Initial program 90.2%

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

   3. Taylor expanded in z around 0

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(y, z\right), \color{blue}{t}\right)\right) \]
   4. Step-by-step derivation

   5. Simplified 80.9%

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

   if 2.7e16 < z

   1. Initial program 97.6%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(-1 \cdot \frac{t}{1 - z}\right)}\right) \]

      7. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\mathsf{neg}\left(\frac{t}{1 - z}\right)\right)\right) \]

      8. distribute-neg-frac2 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{t}{\color{blue}{\mathsf{neg}\left(\left(1 - z\right)\right)}}\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(t, \color{blue}{\left(\mathsf{neg}\left(\left(1 - z\right)\right)\right)}\right)\right) \]

      10. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(t, \left(\mathsf{neg}\left(\left(1 + \left(\mathsf{neg}\left(z\right)\right)\right)\right)\right)\right)\right) \]

      11. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(t, \left(\mathsf{neg}\left(\left(1 + -1 \cdot z\right)\right)\right)\right)\right) \]

      12. distribute-neg-in N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(t, \left(\left(\mathsf{neg}\left(1\right)\right) + \color{blue}{\left(\mathsf{neg}\left(-1 \cdot z\right)\right)}\right)\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(t, \left(-1 + \left(\mathsf{neg}\left(\color{blue}{-1 \cdot z}\right)\right)\right)\right)\right) \]

      14. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(t, \left(-1 + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(z\right)\right)\right)\right)\right)\right)\right) \]

      15. remove-double-neg N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(t, \left(-1 + z\right)\right)\right) \]

      16. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(t, \left(z + \color{blue}{-1}\right)\right)\right) \]

      17. +-lowering-+.f64 61.8%

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(t, \mathsf{+.f64}\left(z, \color{blue}{-1}\right)\right)\right) \]

   5. Simplified 61.8%

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

   6. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{t}{z}\right)}\right) \]

      4. /-lowering-/.f64 61.8%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(t, \color{blue}{z}\right)\right) \]

   8. Simplified 61.8%

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

Alternative 9: 68.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \frac{t}{z}\\ \mathbf{if}\;t \leq -1.05 \cdot 10^{+119}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 1.02 \cdot 10^{+54}:\\ \;\;\;\;\frac{y}{z} \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* x (/ t z))))
   (if (<= t -1.05e+119) t_1 (if (<= t 1.02e+54) (* (/ 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 <= -1.05e+119) {
		tmp = t_1;
	} else if (t <= 1.02e+54) {
		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 <= (-1.05d+119)) then
        tmp = t_1
    else if (t <= 1.02d+54) 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 <= -1.05e+119) {
		tmp = t_1;
	} else if (t <= 1.02e+54) {
		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 <= -1.05e+119:
		tmp = t_1
	elif t <= 1.02e+54:
		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 <= -1.05e+119)
		tmp = t_1;
	elseif (t <= 1.02e+54)
		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 <= -1.05e+119)
		tmp = t_1;
	elseif (t <= 1.02e+54)
		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, -1.05e+119], t$95$1, If[LessEqual[t, 1.02e+54], N[(N[(y / z), $MachinePrecision] * x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -1.04999999999999991e119 or 1.02e54 < t

   1. Initial program 94.0%

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

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(-1 \cdot \frac{t}{1 - z}\right)}\right) \]

      7. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\mathsf{neg}\left(\frac{t}{1 - z}\right)\right)\right) \]

      8. distribute-neg-frac2 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{t}{\color{blue}{\mathsf{neg}\left(\left(1 - z\right)\right)}}\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(t, \color{blue}{\left(\mathsf{neg}\left(\left(1 - z\right)\right)\right)}\right)\right) \]

      10. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(t, \left(\mathsf{neg}\left(\left(1 + \left(\mathsf{neg}\left(z\right)\right)\right)\right)\right)\right)\right) \]

      11. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(t, \left(\mathsf{neg}\left(\left(1 + -1 \cdot z\right)\right)\right)\right)\right) \]

      12. distribute-neg-in N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(t, \left(\left(\mathsf{neg}\left(1\right)\right) + \color{blue}{\left(\mathsf{neg}\left(-1 \cdot z\right)\right)}\right)\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(t, \left(-1 + \left(\mathsf{neg}\left(\color{blue}{-1 \cdot z}\right)\right)\right)\right)\right) \]

      14. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(t, \left(-1 + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(z\right)\right)\right)\right)\right)\right)\right) \]

      15. remove-double-neg N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(t, \left(-1 + z\right)\right)\right) \]

      16. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(t, \left(z + \color{blue}{-1}\right)\right)\right) \]

      17. +-lowering-+.f64 73.9%

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(t, \mathsf{+.f64}\left(z, \color{blue}{-1}\right)\right)\right) \]

   5. Simplified 73.9%

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

   6. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{t}{z}\right)}\right) \]

      4. /-lowering-/.f64 54.2%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(t, \color{blue}{z}\right)\right) \]

   8. Simplified 54.2%

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

   if -1.04999999999999991e119 < t < 1.02e54

   1. Initial program 92.3%

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

   3. Taylor expanded in y around inf

      \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{y}{z}\right)}\right) \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 81.0%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \color{blue}{z}\right)\right) \]

   5. Simplified 81.0%

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

4. Final simplification 70.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.05 \cdot 10^{+119}:\\ \;\;\;\;x \cdot \frac{t}{z}\\ \mathbf{elif}\;t \leq 1.02 \cdot 10^{+54}:\\ \;\;\;\;\frac{y}{z} \cdot x\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{t}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 35.7% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ x \cdot \frac{t}{z} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 92.9%

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

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

   2. associate-/l* N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. associate-*r* N/A

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

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(-1 \cdot \frac{t}{1 - z}\right)}\right) \]

   7. mul-1-neg N/A

      \[\leadsto \mathsf{*.f64}\left(x, \left(\mathsf{neg}\left(\frac{t}{1 - z}\right)\right)\right) \]

   8. distribute-neg-frac2 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{t}{\color{blue}{\mathsf{neg}\left(\left(1 - z\right)\right)}}\right)\right) \]

   9. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(t, \color{blue}{\left(\mathsf{neg}\left(\left(1 - z\right)\right)\right)}\right)\right) \]

   10. sub-neg N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(t, \left(\mathsf{neg}\left(\left(1 + \left(\mathsf{neg}\left(z\right)\right)\right)\right)\right)\right)\right) \]

   11. mul-1-neg N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(t, \left(\mathsf{neg}\left(\left(1 + -1 \cdot z\right)\right)\right)\right)\right) \]

   12. distribute-neg-in N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(t, \left(\left(\mathsf{neg}\left(1\right)\right) + \color{blue}{\left(\mathsf{neg}\left(-1 \cdot z\right)\right)}\right)\right)\right) \]

   13. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(t, \left(-1 + \left(\mathsf{neg}\left(\color{blue}{-1 \cdot z}\right)\right)\right)\right)\right) \]

   14. mul-1-neg N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(t, \left(-1 + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(z\right)\right)\right)\right)\right)\right)\right) \]

   15. remove-double-neg N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(t, \left(-1 + z\right)\right)\right) \]

   16. +-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(t, \left(z + \color{blue}{-1}\right)\right)\right) \]

   17. +-lowering-+.f64 43.0%

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(t, \mathsf{+.f64}\left(z, \color{blue}{-1}\right)\right)\right) \]

5. Simplified 43.0%

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

6. Taylor expanded in z around inf

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

   1. *-commutative N/A

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

   2. associate-/l* N/A

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

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{t}{z}\right)}\right) \]

   4. /-lowering-/.f64 34.0%

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(t, \color{blue}{z}\right)\right) \]

8. Simplified 34.0%

   \[\leadsto \color{blue}{x \cdot \frac{t}{z}} \]
9. Add Preprocessing

Developer Target 1: 94.7% accurate, 0.2× 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 2024185 
(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)))))