Development.Shake.Progress:decay from shake-0.15.5

Percentage Accurate: 66.7% → 90.9%

Time: 13.9s

Alternatives: 13

Speedup: 0.9×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 66.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 90.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(z, b - y, y\right)\\ t_2 := \frac{t - a}{b - y}\\ \mathbf{if}\;z \leq -6 \cdot 10^{+43}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;z \leq 4.8 \cdot 10^{+45}:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{y}{t\_1}, \frac{z \cdot \left(t - a\right)}{t\_1}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (fma z (- b y) y)) (t_2 (/ (- t a) (- b y))))
   (if (<= z -6e+43)
     t_2
     (if (<= z 4.8e+45) (fma x (/ y t_1) (/ (* z (- t a)) t_1)) t_2))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = fma(z, (b - y), y);
	double t_2 = (t - a) / (b - y);
	double tmp;
	if (z <= -6e+43) {
		tmp = t_2;
	} else if (z <= 4.8e+45) {
		tmp = fma(x, (y / t_1), ((z * (t - a)) / t_1));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = fma(z, Float64(b - y), y)
	t_2 = Float64(Float64(t - a) / Float64(b - y))
	tmp = 0.0
	if (z <= -6e+43)
		tmp = t_2;
	elseif (z <= 4.8e+45)
		tmp = fma(x, Float64(y / t_1), Float64(Float64(z * Float64(t - a)) / t_1));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(z * N[(b - y), $MachinePrecision] + y), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t - a), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -6e+43], t$95$2, If[LessEqual[z, 4.8e+45], N[(x * N[(y / t$95$1), $MachinePrecision] + N[(N[(z * N[(t - a), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if z < -6.00000000000000033e43 or 4.79999999999999979e45 < z

   1. Initial program 45.1%

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

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\frac{t - a}{b - y}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{t - a}{b - y}} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{\color{blue}{t - a}}{b - y} \]

      3. lower--.f64 87.4

         \[\leadsto \frac{t - a}{\color{blue}{b - y}} \]

   5. Simplified 87.4%

      \[\leadsto \color{blue}{\frac{t - a}{b - y}} \]

   if -6.00000000000000033e43 < z < 4.79999999999999979e45

   1. Initial program 89.3%

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

   3. Taylor expanded in x around 0

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

      1. associate-/l* N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower--.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. lower--.f64 97.8

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

   5. Simplified 97.8%

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

Alternative 2: 84.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t - a}{b - y}\\ \mathbf{if}\;z \leq -1.6 \cdot 10^{+42}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 4.6 \cdot 10^{+45}:\\ \;\;\;\;\frac{z \cdot \left(t - a\right) + y \cdot x}{y + z \cdot \left(b - y\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (- t a) (- b y))))
   (if (<= z -1.6e+42)
     t_1
     (if (<= z 4.6e+45)
       (/ (+ (* z (- t a)) (* y x)) (+ y (* z (- b y))))
       t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (t - a) / (b - y);
	double tmp;
	if (z <= -1.6e+42) {
		tmp = t_1;
	} else if (z <= 4.6e+45) {
		tmp = ((z * (t - a)) + (y * x)) / (y + (z * (b - y)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (t - a) / (b - y)
    if (z <= (-1.6d+42)) then
        tmp = t_1
    else if (z <= 4.6d+45) then
        tmp = ((z * (t - a)) + (y * x)) / (y + (z * (b - y)))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (t - a) / (b - y);
	double tmp;
	if (z <= -1.6e+42) {
		tmp = t_1;
	} else if (z <= 4.6e+45) {
		tmp = ((z * (t - a)) + (y * x)) / (y + (z * (b - y)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = (t - a) / (b - y)
	tmp = 0
	if z <= -1.6e+42:
		tmp = t_1
	elif z <= 4.6e+45:
		tmp = ((z * (t - a)) + (y * x)) / (y + (z * (b - y)))
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(t - a) / Float64(b - y))
	tmp = 0.0
	if (z <= -1.6e+42)
		tmp = t_1;
	elseif (z <= 4.6e+45)
		tmp = Float64(Float64(Float64(z * Float64(t - a)) + Float64(y * x)) / Float64(y + Float64(z * Float64(b - y))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = (t - a) / (b - y);
	tmp = 0.0;
	if (z <= -1.6e+42)
		tmp = t_1;
	elseif (z <= 4.6e+45)
		tmp = ((z * (t - a)) + (y * x)) / (y + (z * (b - y)));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(t - a), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -1.6e+42], t$95$1, If[LessEqual[z, 4.6e+45], N[(N[(N[(z * N[(t - a), $MachinePrecision]), $MachinePrecision] + N[(y * x), $MachinePrecision]), $MachinePrecision] / N[(y + N[(z * N[(b - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.60000000000000001e42 or 4.60000000000000025e45 < z

   1. Initial program 44.7%

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

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\frac{t - a}{b - y}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{t - a}{b - y}} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{\color{blue}{t - a}}{b - y} \]

      3. lower--.f64 86.6

         \[\leadsto \frac{t - a}{\color{blue}{b - y}} \]

   5. Simplified 86.6%

      \[\leadsto \color{blue}{\frac{t - a}{b - y}} \]

   if -1.60000000000000001e42 < z < 4.60000000000000025e45

   1. Initial program 89.9%

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

4. Final simplification 88.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.6 \cdot 10^{+42}:\\ \;\;\;\;\frac{t - a}{b - y}\\ \mathbf{elif}\;z \leq 4.6 \cdot 10^{+45}:\\ \;\;\;\;\frac{z \cdot \left(t - a\right) + y \cdot x}{y + z \cdot \left(b - y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t - a}{b - y}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 64.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(z, b - y, y\right)\\ t_2 := \frac{t - a}{b - y}\\ \mathbf{if}\;z \leq -330:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;z \leq -1.48 \cdot 10^{-164}:\\ \;\;\;\;\frac{z \cdot \left(t - a\right)}{t\_1}\\ \mathbf{elif}\;z \leq 5.8 \cdot 10^{-40}:\\ \;\;\;\;\frac{y \cdot x}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (fma z (- b y) y)) (t_2 (/ (- t a) (- b y))))
   (if (<= z -330.0)
     t_2
     (if (<= z -1.48e-164)
       (/ (* z (- t a)) t_1)
       (if (<= z 5.8e-40) (/ (* y x) t_1) t_2)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = fma(z, (b - y), y);
	double t_2 = (t - a) / (b - y);
	double tmp;
	if (z <= -330.0) {
		tmp = t_2;
	} else if (z <= -1.48e-164) {
		tmp = (z * (t - a)) / t_1;
	} else if (z <= 5.8e-40) {
		tmp = (y * x) / t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = fma(z, Float64(b - y), y)
	t_2 = Float64(Float64(t - a) / Float64(b - y))
	tmp = 0.0
	if (z <= -330.0)
		tmp = t_2;
	elseif (z <= -1.48e-164)
		tmp = Float64(Float64(z * Float64(t - a)) / t_1);
	elseif (z <= 5.8e-40)
		tmp = Float64(Float64(y * x) / t_1);
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(z * N[(b - y), $MachinePrecision] + y), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t - a), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -330.0], t$95$2, If[LessEqual[z, -1.48e-164], N[(N[(z * N[(t - a), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[z, 5.8e-40], N[(N[(y * x), $MachinePrecision] / t$95$1), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 3 regimes

2. if z < -330 or 5.7999999999999998e-40 < z

   1. Initial program 53.0%

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

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\frac{t - a}{b - y}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{t - a}{b - y}} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{\color{blue}{t - a}}{b - y} \]

      3. lower--.f64 83.0

         \[\leadsto \frac{t - a}{\color{blue}{b - y}} \]

   5. Simplified 83.0%

      \[\leadsto \color{blue}{\frac{t - a}{b - y}} \]

   if -330 < z < -1.48000000000000001e-164

   1. Initial program 94.9%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower--.f64 65.0

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

   5. Simplified 65.0%

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

   if -1.48000000000000001e-164 < z < 5.7999999999999998e-40

   1. Initial program 86.1%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower--.f64 54.5

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

   5. Simplified 54.5%

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

4. Final simplification 71.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -330:\\ \;\;\;\;\frac{t - a}{b - y}\\ \mathbf{elif}\;z \leq -1.48 \cdot 10^{-164}:\\ \;\;\;\;\frac{z \cdot \left(t - a\right)}{\mathsf{fma}\left(z, b - y, y\right)}\\ \mathbf{elif}\;z \leq 5.8 \cdot 10^{-40}:\\ \;\;\;\;\frac{y \cdot x}{\mathsf{fma}\left(z, b - y, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t - a}{b - y}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 64.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(z, b - y, y\right)\\ t_2 := \frac{t - a}{b - y}\\ \mathbf{if}\;z \leq -1.3 \cdot 10^{+47}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;z \leq -1.48 \cdot 10^{-164}:\\ \;\;\;\;\left(t - a\right) \cdot \frac{z}{t\_1}\\ \mathbf{elif}\;z \leq 5.8 \cdot 10^{-40}:\\ \;\;\;\;\frac{y \cdot x}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (fma z (- b y) y)) (t_2 (/ (- t a) (- b y))))
   (if (<= z -1.3e+47)
     t_2
     (if (<= z -1.48e-164)
       (* (- t a) (/ z t_1))
       (if (<= z 5.8e-40) (/ (* y x) t_1) t_2)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = fma(z, (b - y), y);
	double t_2 = (t - a) / (b - y);
	double tmp;
	if (z <= -1.3e+47) {
		tmp = t_2;
	} else if (z <= -1.48e-164) {
		tmp = (t - a) * (z / t_1);
	} else if (z <= 5.8e-40) {
		tmp = (y * x) / t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = fma(z, Float64(b - y), y)
	t_2 = Float64(Float64(t - a) / Float64(b - y))
	tmp = 0.0
	if (z <= -1.3e+47)
		tmp = t_2;
	elseif (z <= -1.48e-164)
		tmp = Float64(Float64(t - a) * Float64(z / t_1));
	elseif (z <= 5.8e-40)
		tmp = Float64(Float64(y * x) / t_1);
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(z * N[(b - y), $MachinePrecision] + y), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t - a), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -1.3e+47], t$95$2, If[LessEqual[z, -1.48e-164], N[(N[(t - a), $MachinePrecision] * N[(z / t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 5.8e-40], N[(N[(y * x), $MachinePrecision] / t$95$1), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.30000000000000002e47 or 5.7999999999999998e-40 < z

   1. Initial program 51.2%

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

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\frac{t - a}{b - y}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{t - a}{b - y}} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{\color{blue}{t - a}}{b - y} \]

      3. lower--.f64 84.8

         \[\leadsto \frac{t - a}{\color{blue}{b - y}} \]

   5. Simplified 84.8%

      \[\leadsto \color{blue}{\frac{t - a}{b - y}} \]

   if -1.30000000000000002e47 < z < -1.48000000000000001e-164

   1. Initial program 91.8%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower--.f64 61.9

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

   5. Simplified 61.9%

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

      1. lift--.f64 N/A

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

      2. *-commutative N/A

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

      3. lift--.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 61.8

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

   7. Applied egg-rr 61.8%

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

   if -1.48000000000000001e-164 < z < 5.7999999999999998e-40

   1. Initial program 86.1%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower--.f64 54.5

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

   5. Simplified 54.5%

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

4. Final simplification 70.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.3 \cdot 10^{+47}:\\ \;\;\;\;\frac{t - a}{b - y}\\ \mathbf{elif}\;z \leq -1.48 \cdot 10^{-164}:\\ \;\;\;\;\left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(z, b - y, y\right)}\\ \mathbf{elif}\;z \leq 5.8 \cdot 10^{-40}:\\ \;\;\;\;\frac{y \cdot x}{\mathsf{fma}\left(z, b - y, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t - a}{b - y}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 72.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t - a}{b - y}\\ \mathbf{if}\;z \leq -2.65 \cdot 10^{-17}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 6.4 \cdot 10^{-33}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, y, z \cdot t\right)}{\mathsf{fma}\left(z, b - y, y\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (- t a) (- b y))))
   (if (<= z -2.65e-17)
     t_1
     (if (<= z 6.4e-33) (/ (fma x y (* z t)) (fma z (- b y) y)) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (t - a) / (b - y);
	double tmp;
	if (z <= -2.65e-17) {
		tmp = t_1;
	} else if (z <= 6.4e-33) {
		tmp = fma(x, y, (z * t)) / fma(z, (b - y), y);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(t - a) / Float64(b - y))
	tmp = 0.0
	if (z <= -2.65e-17)
		tmp = t_1;
	elseif (z <= 6.4e-33)
		tmp = Float64(fma(x, y, Float64(z * t)) / fma(z, Float64(b - y), y));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(t - a), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -2.65e-17], t$95$1, If[LessEqual[z, 6.4e-33], N[(N[(x * y + N[(z * t), $MachinePrecision]), $MachinePrecision] / N[(z * N[(b - y), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -2.6499999999999999e-17 or 6.39999999999999954e-33 < z

   1. Initial program 54.1%

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

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\frac{t - a}{b - y}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{t - a}{b - y}} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{\color{blue}{t - a}}{b - y} \]

      3. lower--.f64 83.4

         \[\leadsto \frac{t - a}{\color{blue}{b - y}} \]

   5. Simplified 83.4%

      \[\leadsto \color{blue}{\frac{t - a}{b - y}} \]

   if -2.6499999999999999e-17 < z < 6.39999999999999954e-33

   1. Initial program 88.7%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. div-inv N/A

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

      9. lift-+.f64 N/A

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

      10. flip-+ N/A

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

      11. clear-num N/A

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

      12. lower-*.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lift-*.f64 N/A

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

      16. lower-fma.f64 N/A

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

   4. Applied egg-rr 88.5%

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

   5. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower--.f64 70.0

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

   7. Simplified 70.0%

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

Alternative 6: 69.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t - a}{b - y}\\ \mathbf{if}\;z \leq -3.15 \cdot 10^{-19}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 5.3 \cdot 10^{-45}:\\ \;\;\;\;\mathsf{fma}\left(z, t - a, y \cdot x\right) \cdot \frac{1}{y}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (- t a) (- b y))))
   (if (<= z -3.15e-19)
     t_1
     (if (<= z 5.3e-45) (* (fma z (- t a) (* y x)) (/ 1.0 y)) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (t - a) / (b - y);
	double tmp;
	if (z <= -3.15e-19) {
		tmp = t_1;
	} else if (z <= 5.3e-45) {
		tmp = fma(z, (t - a), (y * x)) * (1.0 / y);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(t - a) / Float64(b - y))
	tmp = 0.0
	if (z <= -3.15e-19)
		tmp = t_1;
	elseif (z <= 5.3e-45)
		tmp = Float64(fma(z, Float64(t - a), Float64(y * x)) * Float64(1.0 / y));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(t - a), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -3.15e-19], t$95$1, If[LessEqual[z, 5.3e-45], N[(N[(z * N[(t - a), $MachinePrecision] + N[(y * x), $MachinePrecision]), $MachinePrecision] * N[(1.0 / y), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -3.15000000000000009e-19 or 5.2999999999999997e-45 < z

   1. Initial program 55.7%

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

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\frac{t - a}{b - y}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{t - a}{b - y}} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{\color{blue}{t - a}}{b - y} \]

      3. lower--.f64 82.0

         \[\leadsto \frac{t - a}{\color{blue}{b - y}} \]

   5. Simplified 82.0%

      \[\leadsto \color{blue}{\frac{t - a}{b - y}} \]

   if -3.15000000000000009e-19 < z < 5.2999999999999997e-45

   1. Initial program 88.2%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. div-inv N/A

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

      9. lift-+.f64 N/A

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

      10. flip-+ N/A

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

      11. clear-num N/A

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

      12. lower-*.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lift-*.f64 N/A

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

      16. lower-fma.f64 N/A

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

   4. Applied egg-rr 88.0%

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

   5. Taylor expanded in z around 0

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

      1. lower-/.f64 63.1

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

   7. Simplified 63.1%

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

4. Final simplification 73.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.15 \cdot 10^{-19}:\\ \;\;\;\;\frac{t - a}{b - y}\\ \mathbf{elif}\;z \leq 5.3 \cdot 10^{-45}:\\ \;\;\;\;\mathsf{fma}\left(z, t - a, y \cdot x\right) \cdot \frac{1}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{t - a}{b - y}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 63.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t - a}{b - y}\\ \mathbf{if}\;z \leq -2.8 \cdot 10^{-100}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 5.8 \cdot 10^{-40}:\\ \;\;\;\;\frac{y \cdot x}{\mathsf{fma}\left(z, b - y, y\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (- t a) (- b y))))
   (if (<= z -2.8e-100)
     t_1
     (if (<= z 5.8e-40) (/ (* y x) (fma z (- b y) y)) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (t - a) / (b - y);
	double tmp;
	if (z <= -2.8e-100) {
		tmp = t_1;
	} else if (z <= 5.8e-40) {
		tmp = (y * x) / fma(z, (b - y), y);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(t - a) / Float64(b - y))
	tmp = 0.0
	if (z <= -2.8e-100)
		tmp = t_1;
	elseif (z <= 5.8e-40)
		tmp = Float64(Float64(y * x) / fma(z, Float64(b - y), y));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(t - a), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -2.8e-100], t$95$1, If[LessEqual[z, 5.8e-40], N[(N[(y * x), $MachinePrecision] / N[(z * N[(b - y), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -2.79999999999999995e-100 or 5.7999999999999998e-40 < z

   1. Initial program 59.2%

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

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\frac{t - a}{b - y}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{t - a}{b - y}} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{\color{blue}{t - a}}{b - y} \]

      3. lower--.f64 76.3

         \[\leadsto \frac{t - a}{\color{blue}{b - y}} \]

   5. Simplified 76.3%

      \[\leadsto \color{blue}{\frac{t - a}{b - y}} \]

   if -2.79999999999999995e-100 < z < 5.7999999999999998e-40

   1. Initial program 88.2%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower--.f64 53.6

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

   5. Simplified 53.6%

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

4. Final simplification 67.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.8 \cdot 10^{-100}:\\ \;\;\;\;\frac{t - a}{b - y}\\ \mathbf{elif}\;z \leq 5.8 \cdot 10^{-40}:\\ \;\;\;\;\frac{y \cdot x}{\mathsf{fma}\left(z, b - y, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t - a}{b - y}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 53.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x}{1 - z}\\ \mathbf{if}\;y \leq -2.1 \cdot 10^{+85}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 2.1 \cdot 10^{-10}:\\ \;\;\;\;\frac{t - a}{b}\\ \mathbf{elif}\;y \leq 4.2 \cdot 10^{+81}:\\ \;\;\;\;\frac{a}{y - b}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ x (- 1.0 z))))
   (if (<= y -2.1e+85)
     t_1
     (if (<= y 2.1e-10) (/ (- t a) b) (if (<= y 4.2e+81) (/ a (- y b)) t_1)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x / (1.0 - z);
	double tmp;
	if (y <= -2.1e+85) {
		tmp = t_1;
	} else if (y <= 2.1e-10) {
		tmp = (t - a) / b;
	} else if (y <= 4.2e+81) {
		tmp = a / (y - b);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: tmp
    t_1 = x / (1.0d0 - z)
    if (y <= (-2.1d+85)) then
        tmp = t_1
    else if (y <= 2.1d-10) then
        tmp = (t - a) / b
    else if (y <= 4.2d+81) then
        tmp = a / (y - b)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x / (1.0 - z);
	double tmp;
	if (y <= -2.1e+85) {
		tmp = t_1;
	} else if (y <= 2.1e-10) {
		tmp = (t - a) / b;
	} else if (y <= 4.2e+81) {
		tmp = a / (y - b);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = x / (1.0 - z)
	tmp = 0
	if y <= -2.1e+85:
		tmp = t_1
	elif y <= 2.1e-10:
		tmp = (t - a) / b
	elif y <= 4.2e+81:
		tmp = a / (y - b)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(x / Float64(1.0 - z))
	tmp = 0.0
	if (y <= -2.1e+85)
		tmp = t_1;
	elseif (y <= 2.1e-10)
		tmp = Float64(Float64(t - a) / b);
	elseif (y <= 4.2e+81)
		tmp = Float64(a / Float64(y - b));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = x / (1.0 - z);
	tmp = 0.0;
	if (y <= -2.1e+85)
		tmp = t_1;
	elseif (y <= 2.1e-10)
		tmp = (t - a) / b;
	elseif (y <= 4.2e+81)
		tmp = a / (y - b);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(x / N[(1.0 - z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -2.1e+85], t$95$1, If[LessEqual[y, 2.1e-10], N[(N[(t - a), $MachinePrecision] / b), $MachinePrecision], If[LessEqual[y, 4.2e+81], N[(a / N[(y - b), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.1000000000000001e85 or 4.1999999999999997e81 < y

   1. Initial program 56.4%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. lower--.f64 54.9

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

   5. Simplified 54.9%

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

   if -2.1000000000000001e85 < y < 2.1e-10

   1. Initial program 80.0%

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

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{t - a}{b}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{t - a}{b}} \]

      2. lower--.f64 58.0

         \[\leadsto \frac{\color{blue}{t - a}}{b} \]

   5. Simplified 58.0%

      \[\leadsto \color{blue}{\frac{t - a}{b}} \]

   if 2.1e-10 < y < 4.1999999999999997e81

   1. Initial program 54.5%

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

   3. Taylor expanded in a around inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-neg.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower--.f64 16.0

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

   5. Simplified 16.0%

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

   6. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\frac{a}{y - b}} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{a}{y - b}} \]

      2. lower--.f64 48.1

         \[\leadsto \frac{a}{\color{blue}{y - b}} \]

   8. Simplified 48.1%

      \[\leadsto \color{blue}{\frac{a}{y - b}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 9: 63.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t - a}{b - y}\\ \mathbf{if}\;z \leq -6.5 \cdot 10^{-136}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 7.5 \cdot 10^{-51}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (- t a) (- b y))))
   (if (<= z -6.5e-136) t_1 (if (<= z 7.5e-51) x t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (t - a) / (b - y);
	double tmp;
	if (z <= -6.5e-136) {
		tmp = t_1;
	} else if (z <= 7.5e-51) {
		tmp = x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (t - a) / (b - y)
    if (z <= (-6.5d-136)) then
        tmp = t_1
    else if (z <= 7.5d-51) then
        tmp = x
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (t - a) / (b - y);
	double tmp;
	if (z <= -6.5e-136) {
		tmp = t_1;
	} else if (z <= 7.5e-51) {
		tmp = x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = (t - a) / (b - y)
	tmp = 0
	if z <= -6.5e-136:
		tmp = t_1
	elif z <= 7.5e-51:
		tmp = x
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(t - a) / Float64(b - y))
	tmp = 0.0
	if (z <= -6.5e-136)
		tmp = t_1;
	elseif (z <= 7.5e-51)
		tmp = x;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = (t - a) / (b - y);
	tmp = 0.0;
	if (z <= -6.5e-136)
		tmp = t_1;
	elseif (z <= 7.5e-51)
		tmp = x;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(t - a), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -6.5e-136], t$95$1, If[LessEqual[z, 7.5e-51], x, t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -6.50000000000000011e-136 or 7.49999999999999976e-51 < z

   1. Initial program 61.6%

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

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\frac{t - a}{b - y}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{t - a}{b - y}} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{\color{blue}{t - a}}{b - y} \]

      3. lower--.f64 74.3

         \[\leadsto \frac{t - a}{\color{blue}{b - y}} \]

   5. Simplified 74.3%

      \[\leadsto \color{blue}{\frac{t - a}{b - y}} \]

   if -6.50000000000000011e-136 < z < 7.49999999999999976e-51

   1. Initial program 86.9%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. lower--.f64 51.4

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

   5. Simplified 51.4%

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

   6. Taylor expanded in z around 0

      \[\leadsto \frac{x}{\color{blue}{1}} \]
   7. Step-by-step derivation

   8. Simplified 51.4%

      \[\leadsto \frac{x}{\color{blue}{1}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 66.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -6.5 \cdot 10^{-136}:\\ \;\;\;\;\frac{t - a}{b - y}\\ \mathbf{elif}\;z \leq 7.5 \cdot 10^{-51}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{t - a}{b - y}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 44.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{a}{y - b}\\ \mathbf{if}\;z \leq -3.3 \cdot 10^{-19}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 5 \cdot 10^{-49}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ a (- y b)))) (if (<= z -3.3e-19) t_1 (if (<= z 5e-49) x t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = a / (y - b);
	double tmp;
	if (z <= -3.3e-19) {
		tmp = t_1;
	} else if (z <= 5e-49) {
		tmp = x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: tmp
    t_1 = a / (y - b)
    if (z <= (-3.3d-19)) then
        tmp = t_1
    else if (z <= 5d-49) then
        tmp = x
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = a / (y - b);
	double tmp;
	if (z <= -3.3e-19) {
		tmp = t_1;
	} else if (z <= 5e-49) {
		tmp = x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = a / (y - b)
	tmp = 0
	if z <= -3.3e-19:
		tmp = t_1
	elif z <= 5e-49:
		tmp = x
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(a / Float64(y - b))
	tmp = 0.0
	if (z <= -3.3e-19)
		tmp = t_1;
	elseif (z <= 5e-49)
		tmp = x;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = a / (y - b);
	tmp = 0.0;
	if (z <= -3.3e-19)
		tmp = t_1;
	elseif (z <= 5e-49)
		tmp = x;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(a / N[(y - b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -3.3e-19], t$95$1, If[LessEqual[z, 5e-49], x, t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -3.2999999999999998e-19 or 4.9999999999999999e-49 < z

   1. Initial program 55.7%

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

   3. Taylor expanded in a around inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-neg.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower--.f64 30.4

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

   5. Simplified 30.4%

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

   6. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\frac{a}{y - b}} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{a}{y - b}} \]

      2. lower--.f64 48.1

         \[\leadsto \frac{a}{\color{blue}{y - b}} \]

   8. Simplified 48.1%

      \[\leadsto \color{blue}{\frac{a}{y - b}} \]

   if -3.2999999999999998e-19 < z < 4.9999999999999999e-49

   1. Initial program 88.2%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. lower--.f64 46.0

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

   5. Simplified 46.0%

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

   6. Taylor expanded in z around 0

      \[\leadsto \frac{x}{\color{blue}{1}} \]
   7. Step-by-step derivation

   8. Simplified 46.0%

      \[\leadsto \frac{x}{\color{blue}{1}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 47.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.3 \cdot 10^{-19}:\\ \;\;\;\;\frac{a}{y - b}\\ \mathbf{elif}\;z \leq 5 \cdot 10^{-49}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{y - b}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 36.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{a}{-b}\\ \mathbf{if}\;z \leq -3.3 \cdot 10^{-19}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 5 \cdot 10^{-49}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ a (- b)))) (if (<= z -3.3e-19) t_1 (if (<= z 5e-49) x t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = a / -b;
	double tmp;
	if (z <= -3.3e-19) {
		tmp = t_1;
	} else if (z <= 5e-49) {
		tmp = x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: tmp
    t_1 = a / -b
    if (z <= (-3.3d-19)) then
        tmp = t_1
    else if (z <= 5d-49) then
        tmp = x
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = a / -b;
	double tmp;
	if (z <= -3.3e-19) {
		tmp = t_1;
	} else if (z <= 5e-49) {
		tmp = x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = a / -b
	tmp = 0
	if z <= -3.3e-19:
		tmp = t_1
	elif z <= 5e-49:
		tmp = x
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(a / Float64(-b))
	tmp = 0.0
	if (z <= -3.3e-19)
		tmp = t_1;
	elseif (z <= 5e-49)
		tmp = x;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = a / -b;
	tmp = 0.0;
	if (z <= -3.3e-19)
		tmp = t_1;
	elseif (z <= 5e-49)
		tmp = x;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(a / (-b)), $MachinePrecision]}, If[LessEqual[z, -3.3e-19], t$95$1, If[LessEqual[z, 5e-49], x, t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -3.2999999999999998e-19 or 4.9999999999999999e-49 < z

   1. Initial program 55.7%

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

   3. Taylor expanded in a around inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-neg.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower--.f64 30.4

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

   5. Simplified 30.4%

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

   6. Taylor expanded in b around inf

      \[\leadsto \color{blue}{-1 \cdot \frac{a}{b}} \]
   7. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{-1 \cdot a}{b}} \]

      2. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{-1 \cdot a}{b}} \]

      3. mul-1-neg N/A

         \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(a\right)}}{b} \]

      4. lower-neg.f64 31.3

         \[\leadsto \frac{\color{blue}{-a}}{b} \]

   8. Simplified 31.3%

      \[\leadsto \color{blue}{\frac{-a}{b}} \]

   if -3.2999999999999998e-19 < z < 4.9999999999999999e-49

   1. Initial program 88.2%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. lower--.f64 46.0

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

   5. Simplified 46.0%

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

   6. Taylor expanded in z around 0

      \[\leadsto \frac{x}{\color{blue}{1}} \]
   7. Step-by-step derivation

   8. Simplified 46.0%

      \[\leadsto \frac{x}{\color{blue}{1}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 37.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.3 \cdot 10^{-19}:\\ \;\;\;\;\frac{a}{-b}\\ \mathbf{elif}\;z \leq 5 \cdot 10^{-49}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{-b}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 25.7% accurate, 5.6× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(z, x, x\right) \end{array} \]

FPCore

(FPCore (x y z t a b) :precision binary64 (fma z x x))

C

double code(double x, double y, double z, double t, double a, double b) {
	return fma(z, x, x);
}

Julia

function code(x, y, z, t, a, b)
	return fma(z, x, x)
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(z * x + x), $MachinePrecision]

Derivation

1. Initial program 70.3%

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

3. Taylor expanded in y around inf

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

   1. lower-/.f64 N/A

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

   2. mul-1-neg N/A

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

   3. unsub-neg N/A

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

   4. lower--.f64 28.7

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

5. Simplified 28.7%

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

6. Taylor expanded in z around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 23.2

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

8. Simplified 23.2%

   \[\leadsto \color{blue}{\mathsf{fma}\left(z, x, x\right)} \]
9. Add Preprocessing

Alternative 13: 3.8% accurate, 6.5× speedup

Math

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

FPCore

(FPCore (x y z t a b) :precision binary64 (* z x))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 70.3%

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

3. Taylor expanded in y around inf

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

   1. lower-/.f64 N/A

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

   2. mul-1-neg N/A

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

   3. unsub-neg N/A

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

   4. lower--.f64 28.7

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

5. Simplified 28.7%

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

6. Taylor expanded in z around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 23.2

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

8. Simplified 23.2%

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

9. Taylor expanded in z around inf

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

    1. lower-*.f64 4.1

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

11. Simplified 4.1%

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

12. Final simplification 4.1%

    \[\leadsto z \cdot x \]
13. Add Preprocessing

Developer Target 1: 73.4% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024207 
(FPCore (x y z t a b)
  :name "Development.Shake.Progress:decay from shake-0.15.5"
  :precision binary64

  :alt
  (! :herbie-platform default (- (/ (+ (* z t) (* y x)) (+ y (* z (- b y)))) (/ a (+ (- b y) (/ y z)))))

  (/ (+ (* x y) (* z (- t a))) (+ y (* z (- b y)))))