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

Percentage Accurate: 95.1% → 95.1%

Time: 9.9s

Alternatives: 10

Speedup: 1.0×

• 20.2% 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: 95.1% 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: 95.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

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 / (z + (-1.0d0))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 96.8%

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

3. Final simplification 96.8%

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

Alternative 2: 94.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -0.74:\\ \;\;\;\;\frac{x}{\frac{z}{y + t}}\\ \mathbf{elif}\;z \leq 9.6 \cdot 10^{-5}:\\ \;\;\;\;x \cdot \frac{\mathsf{fma}\left(-z, \mathsf{fma}\left(z, t, t\right), y\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y + t}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -0.74)
   (/ x (/ z (+ y t)))
   (if (<= z 9.6e-5)
     (* x (/ (fma (- z) (fma z t t) y) z))
     (* x (/ (+ y t) z)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -0.74) {
		tmp = x / (z / (y + t));
	} else if (z <= 9.6e-5) {
		tmp = x * (fma(-z, fma(z, t, t), y) / z);
	} else {
		tmp = x * ((y + t) / z);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -0.74)
		tmp = Float64(x / Float64(z / Float64(y + t)));
	elseif (z <= 9.6e-5)
		tmp = Float64(x * Float64(fma(Float64(-z), fma(z, t, t), y) / z));
	else
		tmp = Float64(x * Float64(Float64(y + t) / z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -0.74], N[(x / N[(z / N[(y + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 9.6e-5], N[(x * N[(N[((-z) * N[(z * t + t), $MachinePrecision] + y), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], N[(x * N[(N[(y + t), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -0.73999999999999999

   1. Initial program 97.9%

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

   3. Taylor expanded in z around inf

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

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

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

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

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

      3. metadata-eval N/A

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

      4. *-lft-identity N/A

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

      5. +-lowering-+.f64 97.3

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

   5. Simplified 97.3%

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

      1. clear-num N/A

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

      2. un-div-inv N/A

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

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

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

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

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

      5. +-commutative N/A

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

      6. +-lowering-+.f64 97.6

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

   7. Applied egg-rr 97.6%

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

   if -0.73999999999999999 < z < 9.6000000000000002e-5

   1. Initial program 96.0%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. distribute-neg-frac2 N/A

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

      4. div-inv N/A

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

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

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

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

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

      7. sub-neg N/A

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

      8. distribute-neg-in N/A

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

      9. metadata-eval N/A

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

      10. remove-double-neg N/A

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

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

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

      12. /-lowering-/.f64 96.0

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

   4. Applied egg-rr 96.0%

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

   5. Taylor expanded in z around 0

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

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

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. *-commutative N/A

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

      5. distribute-lft-out N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. accelerator-lowering-fma.f64 N/A

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

      9. mul-1-neg N/A

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

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

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. accelerator-lowering-fma.f64 95.1

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

   7. Simplified 95.1%

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

   if 9.6000000000000002e-5 < z

   1. Initial program 97.5%

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

   3. Taylor expanded in z around inf

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

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

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

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

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

      3. metadata-eval N/A

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

      4. *-lft-identity N/A

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

      5. +-lowering-+.f64 97.5

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

   5. Simplified 97.5%

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

4. Final simplification 96.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -0.74:\\ \;\;\;\;\frac{x}{\frac{z}{y + t}}\\ \mathbf{elif}\;z \leq 9.6 \cdot 10^{-5}:\\ \;\;\;\;x \cdot \frac{\mathsf{fma}\left(-z, \mathsf{fma}\left(z, t, t\right), y\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y + t}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 94.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -0.78:\\ \;\;\;\;\frac{x}{\frac{z}{y + t}}\\ \mathbf{elif}\;z \leq 9.6 \cdot 10^{-5}:\\ \;\;\;\;x \cdot \frac{\mathsf{fma}\left(-t, \mathsf{fma}\left(z, z, z\right), y\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y + t}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -0.78)
   (/ x (/ z (+ y t)))
   (if (<= z 9.6e-5)
     (* x (/ (fma (- t) (fma z z z) y) z))
     (* x (/ (+ y t) z)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -0.78) {
		tmp = x / (z / (y + t));
	} else if (z <= 9.6e-5) {
		tmp = x * (fma(-t, fma(z, z, z), y) / z);
	} else {
		tmp = x * ((y + t) / z);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -0.78)
		tmp = Float64(x / Float64(z / Float64(y + t)));
	elseif (z <= 9.6e-5)
		tmp = Float64(x * Float64(fma(Float64(-t), fma(z, z, z), y) / z));
	else
		tmp = Float64(x * Float64(Float64(y + t) / z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -0.78], N[(x / N[(z / N[(y + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 9.6e-5], N[(x * N[(N[((-t) * N[(z * z + z), $MachinePrecision] + y), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], N[(x * N[(N[(y + t), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -0.78000000000000003

   1. Initial program 97.9%

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

   3. Taylor expanded in z around inf

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

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

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

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

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

      3. metadata-eval N/A

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

      4. *-lft-identity N/A

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

      5. +-lowering-+.f64 97.3

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

   5. Simplified 97.3%

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

      1. clear-num N/A

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

      2. un-div-inv N/A

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

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

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

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

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

      5. +-commutative N/A

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

      6. +-lowering-+.f64 97.6

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

   7. Applied egg-rr 97.6%

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

   if -0.78000000000000003 < z < 9.6000000000000002e-5

   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 0

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

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

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

      2. +-commutative N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. distribute-rgt-in N/A

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

      6. associate-*r* N/A

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

      7. associate-*l* N/A

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

      8. unpow2 N/A

         \[\leadsto x \cdot \frac{\left(\left(-1 \cdot t\right) \cdot \color{blue}{{z}^{2}} + \left(-1 \cdot t\right) \cdot z\right) + y}{z} \]

      9. distribute-lft-out N/A

         \[\leadsto x \cdot \frac{\color{blue}{\left(-1 \cdot t\right) \cdot \left({z}^{2} + z\right)} + y}{z} \]

      10. accelerator-lowering-fma.f64 N/A

          \[\leadsto x \cdot \frac{\color{blue}{\mathsf{fma}\left(-1 \cdot t, {z}^{2} + z, y\right)}}{z} \]

      11. mul-1-neg N/A

          \[\leadsto x \cdot \frac{\mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(t\right)}, {z}^{2} + z, y\right)}{z} \]

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

          \[\leadsto x \cdot \frac{\mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(t\right)}, {z}^{2} + z, y\right)}{z} \]

      13. unpow2 N/A

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

      14. accelerator-lowering-fma.f64 95.1

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

   5. Simplified 95.1%

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

   if 9.6000000000000002e-5 < z

   1. Initial program 97.5%

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

   3. Taylor expanded in z around inf

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

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

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

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

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

      3. metadata-eval N/A

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

      4. *-lft-identity N/A

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

      5. +-lowering-+.f64 97.5

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

   5. Simplified 97.5%

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

4. Final simplification 96.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -0.78:\\ \;\;\;\;\frac{x}{\frac{z}{y + t}}\\ \mathbf{elif}\;z \leq 9.6 \cdot 10^{-5}:\\ \;\;\;\;x \cdot \frac{\mathsf{fma}\left(-t, \mathsf{fma}\left(z, z, z\right), y\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y + t}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 94.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1:\\ \;\;\;\;\frac{x}{\frac{z}{y + t}}\\ \mathbf{elif}\;z \leq 9.6 \cdot 10^{-5}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - t\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y + t}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -1.0)
   (/ x (/ z (+ y t)))
   (if (<= z 9.6e-5) (* x (- (/ y z) t)) (* x (/ (+ y t) z)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -1.0) {
		tmp = x / (z / (y + t));
	} else if (z <= 9.6e-5) {
		tmp = x * ((y / z) - t);
	} else {
		tmp = x * ((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 <= (-1.0d0)) then
        tmp = x / (z / (y + t))
    else if (z <= 9.6d-5) then
        tmp = x * ((y / z) - t)
    else
        tmp = x * ((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 <= -1.0) {
		tmp = x / (z / (y + t));
	} else if (z <= 9.6e-5) {
		tmp = x * ((y / z) - t);
	} else {
		tmp = x * ((y + t) / z);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= -1.0:
		tmp = x / (z / (y + t))
	elif z <= 9.6e-5:
		tmp = x * ((y / z) - t)
	else:
		tmp = x * ((y + t) / z)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -1.0)
		tmp = Float64(x / Float64(z / Float64(y + t)));
	elseif (z <= 9.6e-5)
		tmp = Float64(x * Float64(Float64(y / z) - t));
	else
		tmp = Float64(x * Float64(Float64(y + t) / z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= -1.0)
		tmp = x / (z / (y + t));
	elseif (z <= 9.6e-5)
		tmp = x * ((y / z) - t);
	else
		tmp = x * ((y + t) / z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -1.0], N[(x / N[(z / N[(y + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 9.6e-5], N[(x * N[(N[(y / z), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision], N[(x * N[(N[(y + t), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1

   1. Initial program 97.9%

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

   3. Taylor expanded in z around inf

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

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

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

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

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

      3. metadata-eval N/A

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

      4. *-lft-identity N/A

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

      5. +-lowering-+.f64 97.3

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

   5. Simplified 97.3%

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

      1. clear-num N/A

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

      2. un-div-inv N/A

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

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

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

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

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

      5. +-commutative N/A

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

      6. +-lowering-+.f64 97.6

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

   7. Applied egg-rr 97.6%

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

   if -1 < z < 9.6000000000000002e-5

   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 0

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

   5. Simplified 94.6%

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

   if 9.6000000000000002e-5 < z

   1. Initial program 97.5%

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

   3. Taylor expanded in z around inf

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

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

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

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

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

      3. metadata-eval N/A

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

      4. *-lft-identity N/A

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

      5. +-lowering-+.f64 97.5

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

   5. Simplified 97.5%

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

4. Final simplification 96.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1:\\ \;\;\;\;\frac{x}{\frac{z}{y + t}}\\ \mathbf{elif}\;z \leq 9.6 \cdot 10^{-5}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - t\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y + t}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 94.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \frac{y + t}{z}\\ \mathbf{if}\;z \leq -1:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 9.6 \cdot 10^{-5}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* x (/ (+ y t) z))))
   (if (<= z -1.0) t_1 (if (<= z 9.6e-5) (* 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 <= -1.0) {
		tmp = t_1;
	} else if (z <= 9.6e-5) {
		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 <= (-1.0d0)) then
        tmp = t_1
    else if (z <= 9.6d-5) 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 <= -1.0) {
		tmp = t_1;
	} else if (z <= 9.6e-5) {
		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 <= -1.0:
		tmp = t_1
	elif z <= 9.6e-5:
		tmp = x * ((y / z) - t)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(x * Float64(Float64(y + t) / z))
	tmp = 0.0
	if (z <= -1.0)
		tmp = t_1;
	elseif (z <= 9.6e-5)
		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 <= -1.0)
		tmp = t_1;
	elseif (z <= 9.6e-5)
		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[(x * N[(N[(y + t), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -1.0], t$95$1, If[LessEqual[z, 9.6e-5], N[(x * N[(N[(y / z), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -1 or 9.6000000000000002e-5 < 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

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

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

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

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

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

      3. metadata-eval N/A

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

      4. *-lft-identity N/A

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

      5. +-lowering-+.f64 97.4

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

   5. Simplified 97.4%

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

   if -1 < z < 9.6000000000000002e-5

   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 0

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

   5. Simplified 94.6%

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

Alternative 6: 76.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \frac{t}{z + -1}\\ \mathbf{if}\;t \leq -1.95 \cdot 10^{+124}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 2.75 \cdot 10^{+101}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* x (/ t (+ z -1.0)))))
   (if (<= t -1.95e+124) t_1 (if (<= t 2.75e+101) (* x (/ y z)) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x * (t / (z + -1.0));
	double tmp;
	if (t <= -1.95e+124) {
		tmp = t_1;
	} else if (t <= 2.75e+101) {
		tmp = x * (y / z);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = x * (t / (z + (-1.0d0)))
    if (t <= (-1.95d+124)) then
        tmp = t_1
    else if (t <= 2.75d+101) then
        tmp = x * (y / z)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = x * (t / (z + -1.0));
	double tmp;
	if (t <= -1.95e+124) {
		tmp = t_1;
	} else if (t <= 2.75e+101) {
		tmp = x * (y / z);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = x * (t / (z + -1.0))
	tmp = 0
	if t <= -1.95e+124:
		tmp = t_1
	elif t <= 2.75e+101:
		tmp = x * (y / 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 <= -1.95e+124)
		tmp = t_1;
	elseif (t <= 2.75e+101)
		tmp = Float64(x * Float64(y / z));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = x * (t / (z + -1.0));
	tmp = 0.0;
	if (t <= -1.95e+124)
		tmp = t_1;
	elseif (t <= 2.75e+101)
		tmp = x * (y / 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, -1.95e+124], t$95$1, If[LessEqual[t, 2.75e+101], N[(x * N[(y / z), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -1.95e124 or 2.75000000000000009e101 < t

   1. Initial program 97.4%

      \[x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \]
   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 82.2

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

   5. Simplified 82.2%

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

   if -1.95e124 < t < 2.75000000000000009e101

   1. Initial program 96.6%

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

   3. Taylor expanded in y around inf

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

      1. /-lowering-/.f64 78.7

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

   5. Simplified 78.7%

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

Alternative 7: 74.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -5.8 \cdot 10^{+41}:\\ \;\;\;\;x \cdot \frac{t}{z}\\ \mathbf{elif}\;z \leq 9.6 \cdot 10^{-5}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - t\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -5.8e+41)
   (* x (/ t z))
   (if (<= z 9.6e-5) (* x (- (/ y z) t)) (* x (/ y z)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -5.8e+41) {
		tmp = x * (t / z);
	} else if (z <= 9.6e-5) {
		tmp = x * ((y / z) - t);
	} else {
		tmp = x * (y / 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 <= (-5.8d+41)) then
        tmp = x * (t / z)
    else if (z <= 9.6d-5) then
        tmp = x * ((y / z) - t)
    else
        tmp = x * (y / z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -5.8e+41) {
		tmp = x * (t / z);
	} else if (z <= 9.6e-5) {
		tmp = x * ((y / z) - t);
	} else {
		tmp = x * (y / z);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= -5.8e+41:
		tmp = x * (t / z)
	elif z <= 9.6e-5:
		tmp = x * ((y / z) - t)
	else:
		tmp = x * (y / z)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -5.8e+41)
		tmp = Float64(x * Float64(t / z));
	elseif (z <= 9.6e-5)
		tmp = Float64(x * Float64(Float64(y / z) - t));
	else
		tmp = Float64(x * Float64(y / z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= -5.8e+41)
		tmp = x * (t / z);
	elseif (z <= 9.6e-5)
		tmp = x * ((y / z) - t);
	else
		tmp = x * (y / z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -5.8e+41], N[(x * N[(t / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 9.6e-5], N[(x * N[(N[(y / z), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision], N[(x * N[(y / z), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -5.79999999999999977e41

   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

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

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

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

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

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

      3. metadata-eval N/A

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

      4. *-lft-identity N/A

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

      5. +-lowering-+.f64 97.7

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

   5. Simplified 97.7%

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

   6. Taylor expanded in y around 0

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

      1. /-lowering-/.f64 68.8

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

   8. Simplified 68.8%

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

   if -5.79999999999999977e41 < z < 9.6000000000000002e-5

   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 0

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

   5. Simplified 92.9%

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

   if 9.6000000000000002e-5 < z

   1. Initial program 97.5%

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

   3. Taylor expanded in y around inf

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

      1. /-lowering-/.f64 62.7

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

   5. Simplified 62.7%

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

Alternative 8: 68.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \frac{t}{z}\\ \mathbf{if}\;t \leq -3.1 \cdot 10^{+124}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 6.4 \cdot 10^{+104}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* x (/ t z))))
   (if (<= t -3.1e+124) t_1 (if (<= t 6.4e+104) (* x (/ y z)) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x * (t / z);
	double tmp;
	if (t <= -3.1e+124) {
		tmp = t_1;
	} else if (t <= 6.4e+104) {
		tmp = x * (y / z);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = x * (t / z)
    if (t <= (-3.1d+124)) then
        tmp = t_1
    else if (t <= 6.4d+104) then
        tmp = x * (y / z)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = x * (t / z);
	double tmp;
	if (t <= -3.1e+124) {
		tmp = t_1;
	} else if (t <= 6.4e+104) {
		tmp = x * (y / z);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = x * (t / z)
	tmp = 0
	if t <= -3.1e+124:
		tmp = t_1
	elif t <= 6.4e+104:
		tmp = x * (y / 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 <= -3.1e+124)
		tmp = t_1;
	elseif (t <= 6.4e+104)
		tmp = Float64(x * Float64(y / 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 <= -3.1e+124)
		tmp = t_1;
	elseif (t <= 6.4e+104)
		tmp = x * (y / 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, -3.1e+124], t$95$1, If[LessEqual[t, 6.4e+104], N[(x * N[(y / z), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -3.1000000000000002e124 or 6.4e104 < t

   1. Initial program 97.4%

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

   3. Taylor expanded in z around inf

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

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

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

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

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

      3. metadata-eval N/A

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

      4. *-lft-identity N/A

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

      5. +-lowering-+.f64 64.7

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

   5. Simplified 64.7%

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

   6. Taylor expanded in y around 0

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

      1. /-lowering-/.f64 58.9

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

   8. Simplified 58.9%

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

   if -3.1000000000000002e124 < t < 6.4e104

   1. Initial program 96.6%

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

   3. Taylor expanded in y around inf

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

      1. /-lowering-/.f64 78.3

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

   5. Simplified 78.3%

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

Alternative 9: 36.1% accurate, 2.0× 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 96.8%

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

3. Taylor expanded in z around inf

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

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

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

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

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

   3. metadata-eval N/A

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

   4. *-lft-identity N/A

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

   5. +-lowering-+.f64 74.0

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

5. Simplified 74.0%

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

6. Taylor expanded in y around 0

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

   1. /-lowering-/.f64 36.4

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

8. Simplified 36.4%

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

Alternative 10: 34.8% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 96.8%

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

3. Taylor expanded in z around inf

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

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

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

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

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

   3. metadata-eval N/A

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

   4. *-lft-identity N/A

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

   5. +-lowering-+.f64 74.0

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

5. Simplified 74.0%

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

6. Taylor expanded in y around 0

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

   1. /-lowering-/.f64 36.4

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

8. Simplified 36.4%

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

   1. *-commutative N/A

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

   2. div-inv N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. div-inv N/A

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

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

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

   7. /-lowering-/.f64 34.6

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

10. Applied egg-rr 34.6%

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

Developer Target 1: 95.1% 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 2024198 
(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)))))