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

Percentage Accurate: 94.7% → 97.9%

Time: 9.8s

Alternatives: 9

Speedup: 0.8×

• 20.9% 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: 94.7% 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: 97.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{y}{z} + \frac{t}{z + -1}\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+293}:\\ \;\;\;\;t\_1 \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (+ (/ y z) (/ t (+ z -1.0)))))
   (if (<= t_1 (- INFINITY))
     (/ y (/ z x))
     (if (<= t_1 5e+293) (* t_1 x) (/ (* y x) z)))))

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 / (z / x);
	} else if (t_1 <= 5e+293) {
		tmp = t_1 * x;
	} else {
		tmp = (y * x) / z;
	}
	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 / (z / x);
	} else if (t_1 <= 5e+293) {
		tmp = t_1 * x;
	} else {
		tmp = (y * x) / z;
	}
	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 / (z / x)
	elif t_1 <= 5e+293:
		tmp = t_1 * x
	else:
		tmp = (y * x) / z
	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(z / x));
	elseif (t_1 <= 5e+293)
		tmp = Float64(t_1 * x);
	else
		tmp = Float64(Float64(y * x) / z);
	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 / (z / x);
	elseif (t_1 <= 5e+293)
		tmp = t_1 * x;
	else
		tmp = (y * x) / z;
	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[(z / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e+293], N[(t$95$1 * x), $MachinePrecision], N[(N[(y * x), $MachinePrecision] / z), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 69.1%

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

   3. Taylor expanded in y around inf

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

      1. /-lowering-/.f64 69.1

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

   5. Simplified 69.1%

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

      1. *-commutative N/A

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

      2. div-inv N/A

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

      3. associate-*l* N/A

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

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

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

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

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

      6. /-lowering-/.f64 99.8

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

   7. Applied egg-rr 99.8%

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

      1. *-commutative N/A

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

      2. div-inv N/A

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

      3. clear-num N/A

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

      4. un-div-inv N/A

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

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

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

      6. /-lowering-/.f64 100.0

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

   9. Applied egg-rr 100.0%

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

   if -inf.0 < (-.f64 (/.f64 y z) (/.f64 t (-.f64 #s(literal 1 binary64) z))) < 5.00000000000000033e293

   1. Initial program 97.7%

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

   if 5.00000000000000033e293 < (-.f64 (/.f64 y z) (/.f64 t (-.f64 #s(literal 1 binary64) z)))

   1. Initial program 64.3%

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

   3. Taylor expanded in y around inf

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

      1. /-lowering-/.f64 N/A

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

      2. *-lowering-*.f64 100.0

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

   5. Simplified 100.0%

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

4. Final simplification 98.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y}{z} + \frac{t}{z + -1} \leq -\infty:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \mathbf{elif}\;\frac{y}{z} + \frac{t}{z + -1} \leq 5 \cdot 10^{+293}:\\ \;\;\;\;\left(\frac{y}{z} + \frac{t}{z + -1}\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 64.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y \cdot \frac{x}{z}\\ t_2 := \frac{y}{z} + \frac{t}{z + -1}\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq -2 \cdot 10^{+250}:\\ \;\;\;\;t \cdot \left(-x\right)\\ \mathbf{elif}\;t\_2 \leq 10^{+144}:\\ \;\;\;\;\frac{y}{z} \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

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 -2e+250)
       (* t (- x))
       (if (<= t_2 1e+144) (* (/ y z) 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 <= -2e+250) {
		tmp = t * -x;
	} else if (t_2 <= 1e+144) {
		tmp = (y / z) * 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 <= -2e+250) {
		tmp = t * -x;
	} else if (t_2 <= 1e+144) {
		tmp = (y / z) * 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 <= -2e+250:
		tmp = t * -x
	elif t_2 <= 1e+144:
		tmp = (y / z) * 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 <= -2e+250)
		tmp = Float64(t * Float64(-x));
	elseif (t_2 <= 1e+144)
		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 = y * (x / z);
	t_2 = (y / z) + (t / (z + -1.0));
	tmp = 0.0;
	if (t_2 <= -Inf)
		tmp = t_1;
	elseif (t_2 <= -2e+250)
		tmp = t * -x;
	elseif (t_2 <= 1e+144)
		tmp = (y / z) * 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, -2e+250], N[(t * (-x)), $MachinePrecision], If[LessEqual[t$95$2, 1e+144], N[(N[(y / z), $MachinePrecision] * x), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (/.f64 y z) (/.f64 t (-.f64 #s(literal 1 binary64) z))) < -inf.0 or 1.00000000000000002e144 < (-.f64 (/.f64 y z) (/.f64 t (-.f64 #s(literal 1 binary64) z)))

   1. Initial program 83.7%

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

   3. Taylor expanded in y around inf

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

      1. /-lowering-/.f64 65.9

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

   5. Simplified 65.9%

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

      1. *-commutative N/A

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

      2. div-inv N/A

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

      3. associate-*l* N/A

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

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

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

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

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

      6. /-lowering-/.f64 84.4

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

   7. Applied egg-rr 84.4%

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

      1. *-commutative N/A

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

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

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

      3. *-commutative N/A

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

      4. div-inv N/A

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

      5. /-lowering-/.f64 84.5

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

   9. Applied egg-rr 84.5%

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

   if -inf.0 < (-.f64 (/.f64 y z) (/.f64 t (-.f64 #s(literal 1 binary64) z))) < -1.9999999999999998e250

   1. Initial program 100.0%

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

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

      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. *-lowering-*.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. /-lowering-/.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. +-lowering-+.f64 100.0

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

   5. Simplified 100.0%

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

   6. Taylor expanded in z around 0

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

      1. associate-*r* N/A

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

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

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

      3. mul-1-neg N/A

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

      4. neg-lowering-neg.f64 100.0

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

   8. Simplified 100.0%

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

   if -1.9999999999999998e250 < (-.f64 (/.f64 y z) (/.f64 t (-.f64 #s(literal 1 binary64) z))) < 1.00000000000000002e144

   1. Initial program 97.1%

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

   3. Taylor expanded in y around inf

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

      1. /-lowering-/.f64 64.7

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

   5. Simplified 64.7%

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

4. Final simplification 71.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y}{z} + \frac{t}{z + -1} \leq -\infty:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{elif}\;\frac{y}{z} + \frac{t}{z + -1} \leq -2 \cdot 10^{+250}:\\ \;\;\;\;t \cdot \left(-x\right)\\ \mathbf{elif}\;\frac{y}{z} + \frac{t}{z + -1} \leq 10^{+144}:\\ \;\;\;\;\frac{y}{z} \cdot x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 94.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \left(\frac{y}{z} + \frac{t}{z}\right)\\ \mathbf{if}\;z \leq -0.245:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 3.5 \cdot 10^{-10}:\\ \;\;\;\;\frac{x \cdot \left(y - z \cdot t\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* x (+ (/ y z) (/ t z)))))
   (if (<= z -0.245) t_1 (if (<= z 3.5e-10) (/ (* x (- y (* z t))) z) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x * ((y / z) + (t / z));
	double tmp;
	if (z <= -0.245) {
		tmp = t_1;
	} else if (z <= 3.5e-10) {
		tmp = (x * (y - (z * t))) / 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 / z) + (t / z))
    if (z <= (-0.245d0)) then
        tmp = t_1
    else if (z <= 3.5d-10) then
        tmp = (x * (y - (z * t))) / 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 / z));
	double tmp;
	if (z <= -0.245) {
		tmp = t_1;
	} else if (z <= 3.5e-10) {
		tmp = (x * (y - (z * t))) / z;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = x * ((y / z) + (t / z))
	tmp = 0
	if z <= -0.245:
		tmp = t_1
	elif z <= 3.5e-10:
		tmp = (x * (y - (z * t))) / z
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(x * Float64(Float64(y / z) + Float64(t / z)))
	tmp = 0.0
	if (z <= -0.245)
		tmp = t_1;
	elseif (z <= 3.5e-10)
		tmp = Float64(Float64(x * Float64(y - Float64(z * t))) / z);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = x * ((y / z) + (t / z));
	tmp = 0.0;
	if (z <= -0.245)
		tmp = t_1;
	elseif (z <= 3.5e-10)
		tmp = (x * (y - (z * t))) / 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 / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -0.245], t$95$1, If[LessEqual[z, 3.5e-10], N[(N[(x * N[(y - N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -0.245 or 3.4999999999999998e-10 < z

   1. Initial program 96.1%

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

   3. Taylor expanded in z around inf

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

      1. associate-*r/ N/A

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

      2. /-lowering-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. neg-lowering-neg.f64 95.8

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

   5. Simplified 95.8%

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

      1. div-inv N/A

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

      2. cancel-sign-sub N/A

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

      3. div-inv N/A

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

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

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

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

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

      6. /-lowering-/.f64 95.8

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

   7. Applied egg-rr 95.8%

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

   if -0.245 < z < 3.4999999999999998e-10

   1. Initial program 90.5%

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

   3. Taylor expanded in z around 0

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

      1. /-lowering-/.f64 N/A

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

      2. +-commutative N/A

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

      3. mul-1-neg N/A

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

      4. unsub-neg N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

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

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

      9. unsub-neg N/A

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

      10. mul-1-neg N/A

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

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

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

      12. mul-1-neg N/A

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

      13. unsub-neg N/A

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

      14. --lowering--.f64 N/A

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

      15. *-commutative N/A

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

      16. *-lowering-*.f64 96.0

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

   5. Simplified 96.0%

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

Alternative 4: 89.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x \cdot \left(y + t\right)}{z}\\ \mathbf{if}\;z \leq -3.2 \cdot 10^{+16}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 3.5 \cdot 10^{-10}:\\ \;\;\;\;\frac{x \cdot \left(y - z \cdot t\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (* x (+ y t)) z)))
   (if (<= z -3.2e+16) t_1 (if (<= z 3.5e-10) (/ (* x (- y (* z t))) 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 <= -3.2e+16) {
		tmp = t_1;
	} else if (z <= 3.5e-10) {
		tmp = (x * (y - (z * t))) / 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 <= (-3.2d+16)) then
        tmp = t_1
    else if (z <= 3.5d-10) then
        tmp = (x * (y - (z * t))) / 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 <= -3.2e+16) {
		tmp = t_1;
	} else if (z <= 3.5e-10) {
		tmp = (x * (y - (z * t))) / z;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (x * (y + t)) / z
	tmp = 0
	if z <= -3.2e+16:
		tmp = t_1
	elif z <= 3.5e-10:
		tmp = (x * (y - (z * t))) / z
	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 <= -3.2e+16)
		tmp = t_1;
	elseif (z <= 3.5e-10)
		tmp = Float64(Float64(x * Float64(y - Float64(z * t))) / 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 <= -3.2e+16)
		tmp = t_1;
	elseif (z <= 3.5e-10)
		tmp = (x * (y - (z * t))) / z;
	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, -3.2e+16], t$95$1, If[LessEqual[z, 3.5e-10], N[(N[(x * N[(y - N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -3.2e16 or 3.4999999999999998e-10 < z

   1. Initial program 96.0%

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

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

      2. remove-double-neg N/A

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

      3. neg-mul-1 N/A

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

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

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

      5. neg-mul-1 N/A

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

      6. mul-1-neg N/A

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

      7. remove-double-neg N/A

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

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

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

      9. *-commutative N/A

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

      10. mul-1-neg N/A

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

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

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

   5. Simplified 85.7%

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

   if -3.2e16 < z < 3.4999999999999998e-10

   1. Initial program 90.8%

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

   3. Taylor expanded in z around 0

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

      1. /-lowering-/.f64 N/A

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

      2. +-commutative N/A

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

      3. mul-1-neg N/A

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

      4. unsub-neg N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

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

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

      9. unsub-neg N/A

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

      10. mul-1-neg N/A

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

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

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

      12. mul-1-neg N/A

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

      13. unsub-neg N/A

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

      14. --lowering--.f64 N/A

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

      15. *-commutative N/A

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

      16. *-lowering-*.f64 96.2

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

   5. Simplified 96.2%

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

Alternative 5: 74.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -6 \cdot 10^{+133}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{elif}\;z \leq 3.5 \cdot 10^{-10}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - t\right)\\ \mathbf{elif}\;z \leq 5.2 \cdot 10^{+251}:\\ \;\;\;\;\frac{y}{z} \cdot x\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{t}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -6e+133)
   (* y (/ x z))
   (if (<= z 3.5e-10)
     (* x (- (/ y z) t))
     (if (<= z 5.2e+251) (* (/ y z) x) (* x (/ t z))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -6e+133) {
		tmp = y * (x / z);
	} else if (z <= 3.5e-10) {
		tmp = x * ((y / z) - t);
	} else if (z <= 5.2e+251) {
		tmp = (y / z) * x;
	} else {
		tmp = x * (t / z);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (z <= (-6d+133)) then
        tmp = y * (x / z)
    else if (z <= 3.5d-10) then
        tmp = x * ((y / z) - t)
    else if (z <= 5.2d+251) then
        tmp = (y / z) * x
    else
        tmp = x * (t / z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -6e+133) {
		tmp = y * (x / z);
	} else if (z <= 3.5e-10) {
		tmp = x * ((y / z) - t);
	} else if (z <= 5.2e+251) {
		tmp = (y / z) * x;
	} else {
		tmp = x * (t / z);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= -6e+133:
		tmp = y * (x / z)
	elif z <= 3.5e-10:
		tmp = x * ((y / z) - t)
	elif z <= 5.2e+251:
		tmp = (y / z) * x
	else:
		tmp = x * (t / z)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -6e+133)
		tmp = Float64(y * Float64(x / z));
	elseif (z <= 3.5e-10)
		tmp = Float64(x * Float64(Float64(y / z) - t));
	elseif (z <= 5.2e+251)
		tmp = Float64(Float64(y / z) * x);
	else
		tmp = Float64(x * Float64(t / z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= -6e+133)
		tmp = y * (x / z);
	elseif (z <= 3.5e-10)
		tmp = x * ((y / z) - t);
	elseif (z <= 5.2e+251)
		tmp = (y / z) * x;
	else
		tmp = x * (t / z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -6e+133], N[(y * N[(x / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 3.5e-10], N[(x * N[(N[(y / z), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 5.2e+251], N[(N[(y / z), $MachinePrecision] * x), $MachinePrecision], N[(x * N[(t / z), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if z < -6.00000000000000013e133

   1. Initial program 91.4%

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

   3. Taylor expanded in y around inf

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

      1. /-lowering-/.f64 64.9

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

   5. Simplified 64.9%

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

      1. *-commutative N/A

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

      2. div-inv N/A

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

      3. associate-*l* N/A

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

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

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

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

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

      6. /-lowering-/.f64 66.7

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

   7. Applied egg-rr 66.7%

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

      1. *-commutative N/A

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

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

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

      3. *-commutative N/A

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

      4. div-inv N/A

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

      5. /-lowering-/.f64 66.7

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

   9. Applied egg-rr 66.7%

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

   if -6.00000000000000013e133 < z < 3.4999999999999998e-10

   1. Initial program 92.6%

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

   3. Taylor expanded in z around 0

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

   5. Simplified 85.8%

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

   if 3.4999999999999998e-10 < z < 5.2000000000000004e251

   1. Initial program 97.7%

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

   3. Taylor expanded in y around inf

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

      1. /-lowering-/.f64 71.5

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

   5. Simplified 71.5%

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

   if 5.2000000000000004e251 < z

   1. Initial program 88.3%

      \[x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \]
   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. *-lowering-*.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. /-lowering-/.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. +-lowering-+.f64 76.5

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

   5. Simplified 76.5%

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

   6. Taylor expanded in z around inf

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

      1. /-lowering-/.f64 76.5

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

   8. Simplified 76.5%

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

4. Final simplification 80.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -6 \cdot 10^{+133}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{elif}\;z \leq 3.5 \cdot 10^{-10}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - t\right)\\ \mathbf{elif}\;z \leq 5.2 \cdot 10^{+251}:\\ \;\;\;\;\frac{y}{z} \cdot x\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{t}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 88.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x \cdot \left(y + t\right)}{z}\\ \mathbf{if}\;z \leq -3.2 \cdot 10^{+16}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 3.5 \cdot 10^{-10}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (* x (+ y t)) z)))
   (if (<= z -3.2e+16) t_1 (if (<= z 3.5e-10) (* 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 <= -3.2e+16) {
		tmp = t_1;
	} else if (z <= 3.5e-10) {
		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 <= (-3.2d+16)) then
        tmp = t_1
    else if (z <= 3.5d-10) 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 <= -3.2e+16) {
		tmp = t_1;
	} else if (z <= 3.5e-10) {
		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 <= -3.2e+16:
		tmp = t_1
	elif z <= 3.5e-10:
		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 <= -3.2e+16)
		tmp = t_1;
	elseif (z <= 3.5e-10)
		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 <= -3.2e+16)
		tmp = t_1;
	elseif (z <= 3.5e-10)
		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, -3.2e+16], t$95$1, If[LessEqual[z, 3.5e-10], N[(x * N[(N[(y / z), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -3.2e16 or 3.4999999999999998e-10 < z

   1. Initial program 96.0%

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

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

      2. remove-double-neg N/A

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

      3. neg-mul-1 N/A

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

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

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

      5. neg-mul-1 N/A

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

      6. mul-1-neg N/A

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

      7. remove-double-neg N/A

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

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

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

      9. *-commutative N/A

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

      10. mul-1-neg N/A

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

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

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

   5. Simplified 85.7%

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

   if -3.2e16 < z < 3.4999999999999998e-10

   1. Initial program 90.8%

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

   3. Taylor expanded in z around 0

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

   5. Simplified 90.8%

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

Alternative 7: 66.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -5.2 \cdot 10^{+208}:\\ \;\;\;\;x \cdot \frac{t}{z}\\ \mathbf{elif}\;t \leq 1.7 \cdot 10^{+153}:\\ \;\;\;\;\frac{y}{z} \cdot x\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(-x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t -5.2e+208)
   (* x (/ t z))
   (if (<= t 1.7e+153) (* (/ y z) x) (* t (- x)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -5.2e+208) {
		tmp = x * (t / z);
	} else if (t <= 1.7e+153) {
		tmp = (y / z) * x;
	} else {
		tmp = t * -x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-5.2d+208)) then
        tmp = x * (t / z)
    else if (t <= 1.7d+153) then
        tmp = (y / z) * x
    else
        tmp = t * -x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -5.2e+208) {
		tmp = x * (t / z);
	} else if (t <= 1.7e+153) {
		tmp = (y / z) * x;
	} else {
		tmp = t * -x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if t <= -5.2e+208:
		tmp = x * (t / z)
	elif t <= 1.7e+153:
		tmp = (y / z) * x
	else:
		tmp = t * -x
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= -5.2e+208)
		tmp = Float64(x * Float64(t / z));
	elseif (t <= 1.7e+153)
		tmp = Float64(Float64(y / z) * x);
	else
		tmp = Float64(t * Float64(-x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (t <= -5.2e+208)
		tmp = x * (t / z);
	elseif (t <= 1.7e+153)
		tmp = (y / z) * x;
	else
		tmp = t * -x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, -5.2e+208], N[(x * N[(t / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.7e+153], N[(N[(y / z), $MachinePrecision] * x), $MachinePrecision], N[(t * (-x)), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -5.2000000000000001e208

   1. Initial program 91.6%

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

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

      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. *-lowering-*.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. /-lowering-/.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. +-lowering-+.f64 75.2

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

   5. Simplified 75.2%

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

   6. Taylor expanded in z around inf

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

      1. /-lowering-/.f64 53.9

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

   8. Simplified 53.9%

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

   if -5.2000000000000001e208 < t < 1.6999999999999999e153

   1. Initial program 93.9%

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

   3. Taylor expanded in y around inf

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

      1. /-lowering-/.f64 74.0

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

   5. Simplified 74.0%

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

   if 1.6999999999999999e153 < t

   1. Initial program 90.2%

      \[x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \]
   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. *-lowering-*.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. /-lowering-/.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. +-lowering-+.f64 66.2

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

   5. Simplified 66.2%

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

   6. Taylor expanded in z around 0

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

      1. associate-*r* N/A

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

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

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

      3. mul-1-neg N/A

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

      4. neg-lowering-neg.f64 46.3

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

   8. Simplified 46.3%

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

4. Final simplification 69.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -5.2 \cdot 10^{+208}:\\ \;\;\;\;x \cdot \frac{t}{z}\\ \mathbf{elif}\;t \leq 1.7 \cdot 10^{+153}:\\ \;\;\;\;\frac{y}{z} \cdot x\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(-x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 44.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \frac{t}{z}\\ \mathbf{if}\;z \leq -48000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 125000000:\\ \;\;\;\;-t \cdot \mathsf{fma}\left(z, x, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* x (/ t z))))
   (if (<= z -48000000.0)
     t_1
     (if (<= z 125000000.0) (- (* t (fma z x x))) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x * (t / z);
	double tmp;
	if (z <= -48000000.0) {
		tmp = t_1;
	} else if (z <= 125000000.0) {
		tmp = -(t * fma(z, x, 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 (z <= -48000000.0)
		tmp = t_1;
	elseif (z <= 125000000.0)
		tmp = Float64(-Float64(t * fma(z, x, x)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x * N[(t / z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -48000000.0], t$95$1, If[LessEqual[z, 125000000.0], (-N[(t * N[(z * x + x), $MachinePrecision]), $MachinePrecision]), t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -4.8e7 or 1.25e8 < z

   1. Initial program 96.0%

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

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

      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. *-lowering-*.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. /-lowering-/.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. +-lowering-+.f64 53.9

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

   5. Simplified 53.9%

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

   6. Taylor expanded in z around inf

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

      1. /-lowering-/.f64 53.6

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

   8. Simplified 53.6%

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

   if -4.8e7 < z < 1.25e8

   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 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. *-lowering-*.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. /-lowering-/.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. +-lowering-+.f64 28.1

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

   5. Simplified 28.1%

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

   6. Taylor expanded in z around 0

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

      1. distribute-lft-out N/A

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

      2. mul-1-neg N/A

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

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

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

      4. +-commutative N/A

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

      5. distribute-lft-out N/A

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

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

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

      7. *-commutative N/A

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

      8. accelerator-lowering-fma.f64 28.3

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

   8. Simplified 28.3%

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

Alternative 9: 23.1% accurate, 4.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 93.3%

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

3. Taylor expanded in y around 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. *-lowering-*.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. /-lowering-/.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. +-lowering-+.f64 40.4

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

5. Simplified 40.4%

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

6. Taylor expanded in z around 0

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

   1. associate-*r* N/A

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

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

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

   3. mul-1-neg N/A

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

   4. neg-lowering-neg.f64 22.9

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

8. Simplified 22.9%

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

9. Final simplification 22.9%

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

Developer Target 1: 95.3% 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 2024199 
(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)))))