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

Percentage Accurate: 94.3% → 94.9%

Time: 8.9s

Alternatives: 10

Speedup: 1.0×

• 20.7% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 94.3% 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: 94.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \left(\frac{y}{z} - \frac{t}{-z}\right)\\ \mathbf{if}\;z \leq -33000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 0.0014:\\ \;\;\;\;\frac{x \cdot \left(y - z \cdot t\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* x (- (/ y z) (/ t (- z))))))
   (if (<= z -33000.0) t_1 (if (<= z 0.0014) (/ (* 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 <= -33000.0) {
		tmp = t_1;
	} else if (z <= 0.0014) {
		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 <= (-33000.0d0)) then
        tmp = t_1
    else if (z <= 0.0014d0) 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 <= -33000.0) {
		tmp = t_1;
	} else if (z <= 0.0014) {
		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 <= -33000.0:
		tmp = t_1
	elif z <= 0.0014:
		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 / Float64(-z))))
	tmp = 0.0
	if (z <= -33000.0)
		tmp = t_1;
	elseif (z <= 0.0014)
		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 <= -33000.0)
		tmp = t_1;
	elseif (z <= 0.0014)
		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, -33000.0], t$95$1, If[LessEqual[z, 0.0014], 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 < -33000 or 0.00139999999999999999 < z

   1. Initial program 97.6%

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

   3. Taylor expanded in z around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 97.2

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

   5. Applied rewrites 97.2%

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

   if -33000 < z < 0.00139999999999999999

   1. Initial program 93.7%

      \[x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \]
   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. lower-/.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. lower-*.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. lower--.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. lower-*.f64 98.3

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

   5. Applied rewrites 98.3%

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

Alternative 2: 77.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -6 \cdot 10^{-155}:\\ \;\;\;\;\frac{x \cdot \left(y + t\right)}{z}\\ \mathbf{elif}\;z \leq 3.2 \cdot 10^{-28}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{elif}\;z \leq 1.32 \cdot 10^{-8}:\\ \;\;\;\;x \cdot \left(-\mathsf{fma}\left(z, t, t\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y + t\right) \cdot \frac{x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -6e-155)
   (/ (* x (+ y t)) z)
   (if (<= z 3.2e-28)
     (/ (* x y) z)
     (if (<= z 1.32e-8) (* x (- (fma z t t))) (* (+ y t) (/ x z))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -6e-155) {
		tmp = (x * (y + t)) / z;
	} else if (z <= 3.2e-28) {
		tmp = (x * y) / z;
	} else if (z <= 1.32e-8) {
		tmp = x * -fma(z, t, t);
	} else {
		tmp = (y + t) * (x / z);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -6e-155)
		tmp = Float64(Float64(x * Float64(y + t)) / z);
	elseif (z <= 3.2e-28)
		tmp = Float64(Float64(x * y) / z);
	elseif (z <= 1.32e-8)
		tmp = Float64(x * Float64(-fma(z, t, t)));
	else
		tmp = Float64(Float64(y + t) * Float64(x / z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -6e-155], N[(N[(x * N[(y + t), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[z, 3.2e-28], N[(N[(x * y), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[z, 1.32e-8], N[(x * (-N[(z * t + t), $MachinePrecision])), $MachinePrecision], N[(N[(y + t), $MachinePrecision] * N[(x / z), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if z < -5.99999999999999967e-155

   1. Initial program 98.4%

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

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

   5. Applied rewrites 80.6%

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

   if -5.99999999999999967e-155 < z < 3.19999999999999982e-28

   1. Initial program 91.9%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 82.6

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

   5. Applied rewrites 82.6%

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

   if 3.19999999999999982e-28 < z < 1.32000000000000007e-8

   1. Initial program 99.7%

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

   3. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. lower-/.f64 N/A

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

      4. sub-neg N/A

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

      5. mul-1-neg N/A

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

      6. +-commutative N/A

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

      7. distribute-neg-in N/A

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

      8. mul-1-neg N/A

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

      9. remove-double-neg N/A

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

      10. metadata-eval N/A

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

      11. lower-+.f64 99.7

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

   5. Applied rewrites 99.7%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 99.7%

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

   if 1.32000000000000007e-8 < z

   1. Initial program 96.1%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift--.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. frac-sub N/A

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

      7. associate-*l/ N/A

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

      8. *-commutative N/A

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

      9. times-frac N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 68.5

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

   4. Applied rewrites 68.5%

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

   5. Taylor expanded in z around inf

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

      1. mul-1-neg N/A

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

      2. sub-neg N/A

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

      3. mul-1-neg N/A

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

      4. distribute-neg-out N/A

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

      5. +-commutative N/A

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

      6. remove-double-neg N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 91.2

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

   7. Applied rewrites 91.2%

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

Alternative 3: 77.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x \cdot \left(y + t\right)}{z}\\ \mathbf{if}\;z \leq -6 \cdot 10^{-155}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 3.2 \cdot 10^{-28}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{elif}\;z \leq 1.32 \cdot 10^{-8}:\\ \;\;\;\;x \cdot \left(-\mathsf{fma}\left(z, t, t\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (* x (+ y t)) z)))
   (if (<= z -6e-155)
     t_1
     (if (<= z 3.2e-28)
       (/ (* x y) z)
       (if (<= z 1.32e-8) (* x (- (fma z t t))) t_1)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x * (y + t)) / z;
	double tmp;
	if (z <= -6e-155) {
		tmp = t_1;
	} else if (z <= 3.2e-28) {
		tmp = (x * y) / z;
	} else if (z <= 1.32e-8) {
		tmp = x * -fma(z, t, 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 <= -6e-155)
		tmp = t_1;
	elseif (z <= 3.2e-28)
		tmp = Float64(Float64(x * y) / z);
	elseif (z <= 1.32e-8)
		tmp = Float64(x * Float64(-fma(z, t, t)));
	else
		tmp = t_1;
	end
	return 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, -6e-155], t$95$1, If[LessEqual[z, 3.2e-28], N[(N[(x * y), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[z, 1.32e-8], N[(x * (-N[(z * t + t), $MachinePrecision])), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if z < -5.99999999999999967e-155 or 1.32000000000000007e-8 < z

   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

      \[\leadsto \color{blue}{\frac{x \cdot \left(y - -1 \cdot t\right)}{z}} \]
   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. lower-/.f64 N/A

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

   5. Applied rewrites 84.4%

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

   if -5.99999999999999967e-155 < z < 3.19999999999999982e-28

   1. Initial program 91.9%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 82.6

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

   5. Applied rewrites 82.6%

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

   if 3.19999999999999982e-28 < z < 1.32000000000000007e-8

   1. Initial program 99.7%

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

   3. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. lower-/.f64 N/A

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

      4. sub-neg N/A

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

      5. mul-1-neg N/A

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

      6. +-commutative N/A

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

      7. distribute-neg-in N/A

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

      8. mul-1-neg N/A

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

      9. remove-double-neg N/A

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

      10. metadata-eval N/A

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

      11. lower-+.f64 99.7

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

   5. Applied rewrites 99.7%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 99.7%

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

Alternative 4: 89.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -33000:\\ \;\;\;\;\frac{x \cdot \left(y + t\right)}{z}\\ \mathbf{elif}\;z \leq 0.0014:\\ \;\;\;\;\frac{x \cdot \left(y - z \cdot t\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\left(y + t\right) \cdot \frac{x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -33000.0)
   (/ (* x (+ y t)) z)
   (if (<= z 0.0014) (/ (* x (- y (* z t))) z) (* (+ y t) (/ x z)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -33000.0) {
		tmp = (x * (y + t)) / z;
	} else if (z <= 0.0014) {
		tmp = (x * (y - (z * t))) / z;
	} else {
		tmp = (y + t) * (x / 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 <= (-33000.0d0)) then
        tmp = (x * (y + t)) / z
    else if (z <= 0.0014d0) then
        tmp = (x * (y - (z * t))) / z
    else
        tmp = (y + t) * (x / z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -33000.0) {
		tmp = (x * (y + t)) / z;
	} else if (z <= 0.0014) {
		tmp = (x * (y - (z * t))) / z;
	} else {
		tmp = (y + t) * (x / z);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= -33000.0:
		tmp = (x * (y + t)) / z
	elif z <= 0.0014:
		tmp = (x * (y - (z * t))) / z
	else:
		tmp = (y + t) * (x / z)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -33000.0)
		tmp = Float64(Float64(x * Float64(y + t)) / z);
	elseif (z <= 0.0014)
		tmp = Float64(Float64(x * Float64(y - Float64(z * t))) / z);
	else
		tmp = Float64(Float64(y + t) * Float64(x / z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= -33000.0)
		tmp = (x * (y + t)) / z;
	elseif (z <= 0.0014)
		tmp = (x * (y - (z * t))) / z;
	else
		tmp = (y + t) * (x / z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -33000.0], N[(N[(x * N[(y + t), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[z, 0.0014], N[(N[(x * N[(y - N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], N[(N[(y + t), $MachinePrecision] * N[(x / z), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -33000

   1. Initial program 99.2%

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

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

   5. Applied rewrites 85.9%

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

   if -33000 < z < 0.00139999999999999999

   1. Initial program 93.7%

      \[x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \]
   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. lower-/.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. lower-*.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. lower--.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. lower-*.f64 98.3

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

   5. Applied rewrites 98.3%

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

   if 0.00139999999999999999 < z

   1. Initial program 96.1%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift--.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. frac-sub N/A

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

      7. associate-*l/ N/A

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

      8. *-commutative N/A

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

      9. times-frac N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 68.0

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

   4. Applied rewrites 68.0%

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

   5. Taylor expanded in z around inf

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

      1. mul-1-neg N/A

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

      2. sub-neg N/A

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

      3. mul-1-neg N/A

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

      4. distribute-neg-out N/A

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

      5. +-commutative N/A

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

      6. remove-double-neg N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 91.1

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

   7. Applied rewrites 91.1%

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

Alternative 5: 88.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -36000:\\ \;\;\;\;\frac{x \cdot \left(y + t\right)}{z}\\ \mathbf{elif}\;z \leq 0.00011:\\ \;\;\;\;x \cdot \mathsf{fma}\left(t, -1, \frac{y}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y + t\right) \cdot \frac{x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -36000.0)
   (/ (* x (+ y t)) z)
   (if (<= z 0.00011) (* x (fma t -1.0 (/ y z))) (* (+ y t) (/ x z)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -36000.0) {
		tmp = (x * (y + t)) / z;
	} else if (z <= 0.00011) {
		tmp = x * fma(t, -1.0, (y / z));
	} else {
		tmp = (y + t) * (x / z);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -36000.0)
		tmp = Float64(Float64(x * Float64(y + t)) / z);
	elseif (z <= 0.00011)
		tmp = Float64(x * fma(t, -1.0, Float64(y / z)));
	else
		tmp = Float64(Float64(y + t) * Float64(x / z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -36000.0], N[(N[(x * N[(y + t), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[z, 0.00011], N[(x * N[(t * -1.0 + N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y + t), $MachinePrecision] * N[(x / z), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -36000

   1. Initial program 99.2%

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

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

   5. Applied rewrites 85.9%

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

   if -36000 < z < 1.10000000000000004e-4

   1. Initial program 93.6%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. distribute-neg-frac2 N/A

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

      6. div-inv N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift--.f64 N/A

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

      10. sub-neg N/A

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

      11. distribute-neg-in N/A

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

      12. metadata-eval N/A

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

      13. remove-double-neg N/A

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

      14. lower-+.f64 93.6

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

   4. Applied rewrites 93.6%

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

   5. Taylor expanded in z around 0

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

   7. Applied rewrites 93.1%

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

   if 1.10000000000000004e-4 < z

   1. Initial program 96.1%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift--.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. frac-sub N/A

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

      7. associate-*l/ N/A

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

      8. *-commutative N/A

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

      9. times-frac N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 68.5

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

   4. Applied rewrites 68.5%

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

   5. Taylor expanded in z around inf

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

      1. mul-1-neg N/A

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

      2. sub-neg N/A

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

      3. mul-1-neg N/A

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

      4. distribute-neg-out N/A

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

      5. +-commutative N/A

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

      6. remove-double-neg N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 91.2

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

   7. Applied rewrites 91.2%

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

Alternative 6: 94.3% 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 95.7%

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

3. Final simplification 95.7%

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

Alternative 7: 62.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \frac{t}{z}\\ \mathbf{if}\;z \leq -1.1 \cdot 10^{+52}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 3.5 \cdot 10^{+71}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* x (/ t z))))
   (if (<= z -1.1e+52) t_1 (if (<= z 3.5e+71) (/ (* 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 (z <= -1.1e+52) {
		tmp = t_1;
	} else if (z <= 3.5e+71) {
		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 (z <= (-1.1d+52)) then
        tmp = t_1
    else if (z <= 3.5d+71) 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 (z <= -1.1e+52) {
		tmp = t_1;
	} else if (z <= 3.5e+71) {
		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 z <= -1.1e+52:
		tmp = t_1
	elif z <= 3.5e+71:
		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 (z <= -1.1e+52)
		tmp = t_1;
	elseif (z <= 3.5e+71)
		tmp = Float64(Float64(x * y) / z);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = x * (t / z);
	tmp = 0.0;
	if (z <= -1.1e+52)
		tmp = t_1;
	elseif (z <= 3.5e+71)
		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[z, -1.1e+52], t$95$1, If[LessEqual[z, 3.5e+71], N[(N[(x * y), $MachinePrecision] / z), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.1e52 or 3.4999999999999999e71 < z

   1. Initial program 97.0%

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

   3. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. lower-/.f64 N/A

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

      4. sub-neg N/A

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

      5. mul-1-neg N/A

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

      6. +-commutative N/A

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

      7. distribute-neg-in N/A

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

      8. mul-1-neg N/A

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

      9. remove-double-neg N/A

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

      10. metadata-eval N/A

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

      11. lower-+.f64 71.9

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

   5. Applied rewrites 71.9%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 71.9%

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

   if -1.1e52 < z < 3.4999999999999999e71

   1. Initial program 94.7%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 74.1

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

   5. Applied rewrites 74.1%

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

Alternative 8: 60.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 4.7 \cdot 10^{+266}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(-t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t 4.7e+266) (/ (* x y) z) (* x (- t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= 4.7e+266) {
		tmp = (x * y) / z;
	} else {
		tmp = x * -t;
	}
	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 <= 4.7d+266) then
        tmp = (x * y) / z
    else
        tmp = x * -t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= 4.7e+266) {
		tmp = (x * y) / z;
	} else {
		tmp = x * -t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if t <= 4.7e+266:
		tmp = (x * y) / z
	else:
		tmp = x * -t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= 4.7e+266)
		tmp = Float64(Float64(x * y) / z);
	else
		tmp = Float64(x * Float64(-t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (t <= 4.7e+266)
		tmp = (x * y) / z;
	else
		tmp = x * -t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, 4.7e+266], N[(N[(x * y), $MachinePrecision] / z), $MachinePrecision], N[(x * (-t)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 4.6999999999999998e266

   1. Initial program 95.5%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 65.6

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

   5. Applied rewrites 65.6%

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

   if 4.6999999999999998e266 < t

   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

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. lower-/.f64 N/A

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

      4. sub-neg N/A

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

      5. mul-1-neg N/A

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

      6. +-commutative N/A

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

      7. distribute-neg-in N/A

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

      8. mul-1-neg N/A

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

      9. remove-double-neg N/A

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

      10. metadata-eval N/A

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

      11. lower-+.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 82.0%

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

Alternative 9: 61.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 1.25 \cdot 10^{+267}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(-t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t 1.25e+267) (* y (/ x z)) (* x (- t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= 1.25e+267) {
		tmp = y * (x / z);
	} else {
		tmp = x * -t;
	}
	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 <= 1.25d+267) then
        tmp = y * (x / z)
    else
        tmp = x * -t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= 1.25e+267) {
		tmp = y * (x / z);
	} else {
		tmp = x * -t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if t <= 1.25e+267:
		tmp = y * (x / z)
	else:
		tmp = x * -t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= 1.25e+267)
		tmp = Float64(y * Float64(x / z));
	else
		tmp = Float64(x * Float64(-t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (t <= 1.25e+267)
		tmp = y * (x / z);
	else
		tmp = x * -t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, 1.25e+267], N[(y * N[(x / z), $MachinePrecision]), $MachinePrecision], N[(x * (-t)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 1.25e267

   1. Initial program 95.5%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 65.6

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

   5. Applied rewrites 65.6%

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

   7. Applied rewrites 64.4%

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

   if 1.25e267 < t

   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

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. lower-/.f64 N/A

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

      4. sub-neg N/A

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

      5. mul-1-neg N/A

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

      6. +-commutative N/A

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

      7. distribute-neg-in N/A

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

      8. mul-1-neg N/A

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

      9. remove-double-neg N/A

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

      10. metadata-eval N/A

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

      11. lower-+.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 82.0%

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

Alternative 10: 23.4% accurate, 4.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 95.7%

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

3. Taylor expanded in y around 0

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

   1. mul-1-neg N/A

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

   2. distribute-neg-frac2 N/A

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

   3. lower-/.f64 N/A

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

   4. sub-neg N/A

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

   5. mul-1-neg N/A

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

   6. +-commutative N/A

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

   7. distribute-neg-in N/A

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

   8. mul-1-neg N/A

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

   9. remove-double-neg N/A

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

   10. metadata-eval N/A

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

   11. lower-+.f64 50.1

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

5. Applied rewrites 50.1%

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

6. Taylor expanded in z around 0

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

8. Applied rewrites 23.3%

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

Developer Target 1: 94.8% 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 2024233 
(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)))))