Graphics.Rendering.Chart.SparkLine:renderSparkLine from Chart-1.5.3

Percentage Accurate: 96.9% → 97.2%

Time: 9.1s

Alternatives: 14

Speedup: 1.2×

Specification

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z, t, a)
    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
    code = x - ((y - z) / (((t - z) + 1.0d0) / a))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 96.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z, t, a)
    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
    code = x - ((y - z) / (((t - z) + 1.0d0) / a))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.2% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z, t, a)
    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
    code = x - ((y - z) * (a / (1.0d0 + (t - z))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.1%

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

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-/r/ N/A

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

   4. associate-*l/ N/A

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

   5. associate-/l* N/A

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

   6. lower-*.f64 N/A

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

   7. lower-/.f64 98.4

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

   8. lift-+.f64 N/A

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

   9. +-commutative N/A

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

   10. lower-+.f64 98.4

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

4. Applied rewrites 98.4%

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

Alternative 2: 64.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{y - z}{\frac{\left(t - z\right) + 1}{a}}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+293} \lor \neg \left(t\_1 \leq 10^{+289}\right):\\ \;\;\;\;\left(-a\right) \cdot \frac{y}{1}\\ \mathbf{else}:\\ \;\;\;\;x - a\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- y z) (/ (+ (- t z) 1.0) a))))
   (if (or (<= t_1 -1e+293) (not (<= t_1 1e+289)))
     (* (- a) (/ y 1.0))
     (- x a))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (y - z) / (((t - z) + 1.0) / a);
	double tmp;
	if ((t_1 <= -1e+293) || !(t_1 <= 1e+289)) {
		tmp = -a * (y / 1.0);
	} else {
		tmp = x - a;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    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) :: t_1
    real(8) :: tmp
    t_1 = (y - z) / (((t - z) + 1.0d0) / a)
    if ((t_1 <= (-1d+293)) .or. (.not. (t_1 <= 1d+289))) then
        tmp = -a * (y / 1.0d0)
    else
        tmp = x - a
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = (y - z) / (((t - z) + 1.0) / a);
	double tmp;
	if ((t_1 <= -1e+293) || !(t_1 <= 1e+289)) {
		tmp = -a * (y / 1.0);
	} else {
		tmp = x - a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = (y - z) / (((t - z) + 1.0) / a)
	tmp = 0
	if (t_1 <= -1e+293) or not (t_1 <= 1e+289):
		tmp = -a * (y / 1.0)
	else:
		tmp = x - a
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(y - z) / Float64(Float64(Float64(t - z) + 1.0) / a))
	tmp = 0.0
	if ((t_1 <= -1e+293) || !(t_1 <= 1e+289))
		tmp = Float64(Float64(-a) * Float64(y / 1.0));
	else
		tmp = Float64(x - a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = (y - z) / (((t - z) + 1.0) / a);
	tmp = 0.0;
	if ((t_1 <= -1e+293) || ~((t_1 <= 1e+289)))
		tmp = -a * (y / 1.0);
	else
		tmp = x - a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(y - z), $MachinePrecision] / N[(N[(N[(t - z), $MachinePrecision] + 1.0), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$1, -1e+293], N[Not[LessEqual[t$95$1, 1e+289]], $MachinePrecision]], N[((-a) * N[(y / 1.0), $MachinePrecision]), $MachinePrecision], N[(x - a), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 y z) (/.f64 (+.f64 (-.f64 t z) #s(literal 1 binary64)) a)) < -9.9999999999999992e292 or 1.0000000000000001e289 < (/.f64 (-.f64 y z) (/.f64 (+.f64 (-.f64 t z) #s(literal 1 binary64)) a))

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

      3. associate-*r/ N/A

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

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

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

      5. lower-*.f64 N/A

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

      6. lower-neg.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-+.f64 100.0

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

   5. Applied rewrites 100.0%

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

   7. Applied rewrites 94.4%

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

   8. Taylor expanded in z around 0

      \[\leadsto -1 \cdot \color{blue}{\frac{a \cdot y}{1 + t}} \]
   9. Step-by-step derivation

   10. Applied rewrites 93.8%

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

   11. Taylor expanded in t around 0

       \[\leadsto \left(-a\right) \cdot \frac{y}{1} \]
   12. Step-by-step derivation

   13. Applied rewrites 94.4%

       \[\leadsto \left(-a\right) \cdot \frac{y}{1} \]

   if -9.9999999999999992e292 < (/.f64 (-.f64 y z) (/.f64 (+.f64 (-.f64 t z) #s(literal 1 binary64)) a)) < 1.0000000000000001e289

   1. Initial program 98.0%

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

   3. Taylor expanded in z around inf

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

      1. lower--.f64 68.3

         \[\leadsto \color{blue}{x - a} \]

   5. Applied rewrites 68.3%

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

4. Final simplification 69.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \leq -1 \cdot 10^{+293} \lor \neg \left(\frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \leq 10^{+289}\right):\\ \;\;\;\;\left(-a\right) \cdot \frac{y}{1}\\ \mathbf{else}:\\ \;\;\;\;x - a\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 63.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{y - z}{\frac{\left(t - z\right) + 1}{a}}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+293} \lor \neg \left(t\_1 \leq 10^{+289}\right):\\ \;\;\;\;y \cdot \mathsf{fma}\left(t, a, -a\right)\\ \mathbf{else}:\\ \;\;\;\;x - a\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- y z) (/ (+ (- t z) 1.0) a))))
   (if (or (<= t_1 -1e+293) (not (<= t_1 1e+289)))
     (* y (fma t a (- a)))
     (- x a))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (y - z) / (((t - z) + 1.0) / a);
	double tmp;
	if ((t_1 <= -1e+293) || !(t_1 <= 1e+289)) {
		tmp = y * fma(t, a, -a);
	} else {
		tmp = x - a;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(y - z) / Float64(Float64(Float64(t - z) + 1.0) / a))
	tmp = 0.0
	if ((t_1 <= -1e+293) || !(t_1 <= 1e+289))
		tmp = Float64(y * fma(t, a, Float64(-a)));
	else
		tmp = Float64(x - a);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(y - z), $MachinePrecision] / N[(N[(N[(t - z), $MachinePrecision] + 1.0), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$1, -1e+293], N[Not[LessEqual[t$95$1, 1e+289]], $MachinePrecision]], N[(y * N[(t * a + (-a)), $MachinePrecision]), $MachinePrecision], N[(x - a), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 y z) (/.f64 (+.f64 (-.f64 t z) #s(literal 1 binary64)) a)) < -9.9999999999999992e292 or 1.0000000000000001e289 < (/.f64 (-.f64 y z) (/.f64 (+.f64 (-.f64 t z) #s(literal 1 binary64)) a))

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

      3. associate-*r/ N/A

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

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

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

      5. lower-*.f64 N/A

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

      6. lower-neg.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-+.f64 100.0

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

   5. Applied rewrites 100.0%

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

   7. Applied rewrites 94.4%

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

   8. Taylor expanded in z around 0

      \[\leadsto -1 \cdot \color{blue}{\frac{a \cdot y}{1 + t}} \]
   9. Step-by-step derivation

   10. Applied rewrites 93.8%

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

   11. Taylor expanded in t around 0

       \[\leadsto -1 \cdot \left(a \cdot y\right) + a \cdot \color{blue}{\left(t \cdot y\right)} \]
   12. Step-by-step derivation

   13. Applied rewrites 75.7%

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

   if -9.9999999999999992e292 < (/.f64 (-.f64 y z) (/.f64 (+.f64 (-.f64 t z) #s(literal 1 binary64)) a)) < 1.0000000000000001e289

   1. Initial program 98.0%

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

   3. Taylor expanded in z around inf

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

      1. lower--.f64 68.3

         \[\leadsto \color{blue}{x - a} \]

   5. Applied rewrites 68.3%

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

4. Final simplification 68.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \leq -1 \cdot 10^{+293} \lor \neg \left(\frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \leq 10^{+289}\right):\\ \;\;\;\;y \cdot \mathsf{fma}\left(t, a, -a\right)\\ \mathbf{else}:\\ \;\;\;\;x - a\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 70.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.55 \cdot 10^{+30}:\\ \;\;\;\;x - a\\ \mathbf{elif}\;z \leq 6.2 \cdot 10^{-94}:\\ \;\;\;\;x - \frac{a \cdot y}{t}\\ \mathbf{elif}\;z \leq 44000:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t + 1}, a, x\right)\\ \mathbf{else}:\\ \;\;\;\;x - a\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -2.55e+30)
   (- x a)
   (if (<= z 6.2e-94)
     (- x (/ (* a y) t))
     (if (<= z 44000.0) (fma (/ z (+ t 1.0)) a x) (- x a)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -2.55e+30) {
		tmp = x - a;
	} else if (z <= 6.2e-94) {
		tmp = x - ((a * y) / t);
	} else if (z <= 44000.0) {
		tmp = fma((z / (t + 1.0)), a, x);
	} else {
		tmp = x - a;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -2.55e+30)
		tmp = Float64(x - a);
	elseif (z <= 6.2e-94)
		tmp = Float64(x - Float64(Float64(a * y) / t));
	elseif (z <= 44000.0)
		tmp = fma(Float64(z / Float64(t + 1.0)), a, x);
	else
		tmp = Float64(x - a);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -2.55e+30], N[(x - a), $MachinePrecision], If[LessEqual[z, 6.2e-94], N[(x - N[(N[(a * y), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 44000.0], N[(N[(z / N[(t + 1.0), $MachinePrecision]), $MachinePrecision] * a + x), $MachinePrecision], N[(x - a), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if z < -2.55000000000000018e30 or 44000 < z

   1. Initial program 96.8%

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

   3. Taylor expanded in z around inf

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

      1. lower--.f64 86.3

         \[\leadsto \color{blue}{x - a} \]

   5. Applied rewrites 86.3%

      \[\leadsto \color{blue}{x - a} \]

   if -2.55000000000000018e30 < z < 6.1999999999999996e-94

   1. Initial program 99.2%

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

   3. Taylor expanded in t around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 75.0

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

   5. Applied rewrites 75.0%

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

   6. Taylor expanded in y around inf

      \[\leadsto x - \frac{a \cdot y}{t} \]
   7. Step-by-step derivation

   8. Applied rewrites 73.2%

      \[\leadsto x - \frac{a \cdot y}{t} \]

   if 6.1999999999999996e-94 < z < 44000

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

      1. fp-cancel-sub-sign-inv N/A

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

      2. +-commutative N/A

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

      3. metadata-eval N/A

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

      4. *-lft-identity N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-+.f64 64.1

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

   5. Applied rewrites 64.1%

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

   6. Taylor expanded in z around 0

      \[\leadsto \mathsf{fma}\left(\frac{z}{1 + t}, a, x\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 64.1%

      \[\leadsto \mathsf{fma}\left(\frac{z}{t + 1}, a, x\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 5: 63.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3.6 \cdot 10^{+42}:\\ \;\;\;\;x - a\\ \mathbf{elif}\;z \leq 6.4 \cdot 10^{-249}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, a, x\right)\\ \mathbf{elif}\;z \leq 8.2 \cdot 10^{-166}:\\ \;\;\;\;\frac{a \cdot y}{-1 - t}\\ \mathbf{else}:\\ \;\;\;\;x - a\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -3.6e+42)
   (- x a)
   (if (<= z 6.4e-249)
     (fma (/ z t) a x)
     (if (<= z 8.2e-166) (/ (* a y) (- -1.0 t)) (- x a)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -3.6e+42) {
		tmp = x - a;
	} else if (z <= 6.4e-249) {
		tmp = fma((z / t), a, x);
	} else if (z <= 8.2e-166) {
		tmp = (a * y) / (-1.0 - t);
	} else {
		tmp = x - a;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -3.6e+42)
		tmp = Float64(x - a);
	elseif (z <= 6.4e-249)
		tmp = fma(Float64(z / t), a, x);
	elseif (z <= 8.2e-166)
		tmp = Float64(Float64(a * y) / Float64(-1.0 - t));
	else
		tmp = Float64(x - a);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -3.6e+42], N[(x - a), $MachinePrecision], If[LessEqual[z, 6.4e-249], N[(N[(z / t), $MachinePrecision] * a + x), $MachinePrecision], If[LessEqual[z, 8.2e-166], N[(N[(a * y), $MachinePrecision] / N[(-1.0 - t), $MachinePrecision]), $MachinePrecision], N[(x - a), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if z < -3.6000000000000001e42 or 8.1999999999999995e-166 < z

   1. Initial program 97.4%

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

   3. Taylor expanded in z around inf

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

      1. lower--.f64 78.0

         \[\leadsto \color{blue}{x - a} \]

   5. Applied rewrites 78.0%

      \[\leadsto \color{blue}{x - a} \]

   if -3.6000000000000001e42 < z < 6.4000000000000003e-249

   1. Initial program 99.0%

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

   3. Taylor expanded in y around 0

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

      1. fp-cancel-sub-sign-inv N/A

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

      2. +-commutative N/A

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

      3. metadata-eval N/A

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

      4. *-lft-identity N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-+.f64 63.8

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

   5. Applied rewrites 63.8%

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

   6. Taylor expanded in t around inf

      \[\leadsto \mathsf{fma}\left(\frac{z}{t}, a, x\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 62.0%

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

   if 6.4000000000000003e-249 < z < 8.1999999999999995e-166

   1. Initial program 99.9%

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

   3. Taylor expanded in y around inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

      3. associate-*r/ N/A

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

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

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

      5. lower-*.f64 N/A

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

      6. lower-neg.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-+.f64 66.2

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

   5. Applied rewrites 66.2%

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

   7. Applied rewrites 66.2%

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

   8. Taylor expanded in z around 0

      \[\leadsto -1 \cdot \color{blue}{\frac{a \cdot y}{1 + t}} \]
   9. Step-by-step derivation

   10. Applied rewrites 66.2%

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

   11. Taylor expanded in y around 0

       \[\leadsto -1 \cdot \frac{a \cdot y}{\color{blue}{1 + t}} \]
   12. Step-by-step derivation

   13. Applied rewrites 66.2%

       \[\leadsto \frac{a \cdot y}{-1 - \color{blue}{t}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 6: 88.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -4.6 \cdot 10^{+40} \lor \neg \left(z \leq 21000\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{\left(1 + t\right) - z}, a, x\right)\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{1 + t} \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= z -4.6e+40) (not (<= z 21000.0)))
   (fma (/ z (- (+ 1.0 t) z)) a x)
   (- x (* (/ y (+ 1.0 t)) a))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z <= -4.6e+40) || !(z <= 21000.0)) {
		tmp = fma((z / ((1.0 + t) - z)), a, x);
	} else {
		tmp = x - ((y / (1.0 + t)) * a);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((z <= -4.6e+40) || !(z <= 21000.0))
		tmp = fma(Float64(z / Float64(Float64(1.0 + t) - z)), a, x);
	else
		tmp = Float64(x - Float64(Float64(y / Float64(1.0 + t)) * a));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[z, -4.6e+40], N[Not[LessEqual[z, 21000.0]], $MachinePrecision]], N[(N[(z / N[(N[(1.0 + t), $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision] * a + x), $MachinePrecision], N[(x - N[(N[(y / N[(1.0 + t), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -4.59999999999999987e40 or 21000 < z

   1. Initial program 96.6%

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

   3. Taylor expanded in y around 0

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

      1. fp-cancel-sub-sign-inv N/A

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

      2. +-commutative N/A

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

      3. metadata-eval N/A

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

      4. *-lft-identity N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-+.f64 94.0

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

   5. Applied rewrites 94.0%

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

   if -4.59999999999999987e40 < z < 21000

   1. Initial program 99.4%

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

   3. Taylor expanded in z around 0

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-+.f64 92.7

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

   5. Applied rewrites 92.7%

      \[\leadsto x - \color{blue}{\frac{y}{1 + t} \cdot a} \]
3. Recombined 2 regimes into one program.

4. Final simplification 93.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4.6 \cdot 10^{+40} \lor \neg \left(z \leq 21000\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{\left(1 + t\right) - z}, a, x\right)\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{1 + t} \cdot a\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 88.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3.2 \cdot 10^{+40}:\\ \;\;\;\;x - \left(y - z\right) \cdot \frac{a}{1 - z}\\ \mathbf{elif}\;z \leq 21000:\\ \;\;\;\;x - \frac{y}{1 + t} \cdot a\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{\left(1 + t\right) - z}, a, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -3.2e+40)
   (- x (* (- y z) (/ a (- 1.0 z))))
   (if (<= z 21000.0)
     (- x (* (/ y (+ 1.0 t)) a))
     (fma (/ z (- (+ 1.0 t) z)) a x))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -3.2e+40) {
		tmp = x - ((y - z) * (a / (1.0 - z)));
	} else if (z <= 21000.0) {
		tmp = x - ((y / (1.0 + t)) * a);
	} else {
		tmp = fma((z / ((1.0 + t) - z)), a, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -3.2e+40)
		tmp = Float64(x - Float64(Float64(y - z) * Float64(a / Float64(1.0 - z))));
	elseif (z <= 21000.0)
		tmp = Float64(x - Float64(Float64(y / Float64(1.0 + t)) * a));
	else
		tmp = fma(Float64(z / Float64(Float64(1.0 + t) - z)), a, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -3.2e+40], N[(x - N[(N[(y - z), $MachinePrecision] * N[(a / N[(1.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 21000.0], N[(x - N[(N[(y / N[(1.0 + t), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision], N[(N[(z / N[(N[(1.0 + t), $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision] * a + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -3.19999999999999981e40

   1. Initial program 95.2%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 94.2

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

   5. Applied rewrites 94.2%

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

   if -3.19999999999999981e40 < z < 21000

   1. Initial program 99.4%

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

   3. Taylor expanded in z around 0

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-+.f64 92.7

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

   5. Applied rewrites 92.7%

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

   if 21000 < z

   1. Initial program 98.1%

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

   3. Taylor expanded in y around 0

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

      1. fp-cancel-sub-sign-inv N/A

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

      2. +-commutative N/A

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

      3. metadata-eval N/A

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

      4. *-lft-identity N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-+.f64 94.7

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

   5. Applied rewrites 94.7%

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

Alternative 8: 86.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.3 \cdot 10^{+43}:\\ \;\;\;\;x - a\\ \mathbf{elif}\;z \leq 21000:\\ \;\;\;\;x - \frac{y}{1 + t} \cdot a\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{a}{\left(t + 1\right) - z}, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -1.3e+43)
   (- x a)
   (if (<= z 21000.0)
     (- x (* (/ y (+ 1.0 t)) a))
     (fma z (/ a (- (+ t 1.0) z)) x))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1.3e+43) {
		tmp = x - a;
	} else if (z <= 21000.0) {
		tmp = x - ((y / (1.0 + t)) * a);
	} else {
		tmp = fma(z, (a / ((t + 1.0) - z)), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -1.3e+43)
		tmp = Float64(x - a);
	elseif (z <= 21000.0)
		tmp = Float64(x - Float64(Float64(y / Float64(1.0 + t)) * a));
	else
		tmp = fma(z, Float64(a / Float64(Float64(t + 1.0) - z)), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -1.3e+43], N[(x - a), $MachinePrecision], If[LessEqual[z, 21000.0], N[(x - N[(N[(y / N[(1.0 + t), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision], N[(z * N[(a / N[(N[(t + 1.0), $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.3000000000000001e43

   1. Initial program 95.2%

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

   3. Taylor expanded in z around inf

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

      1. lower--.f64 91.6

         \[\leadsto \color{blue}{x - a} \]

   5. Applied rewrites 91.6%

      \[\leadsto \color{blue}{x - a} \]

   if -1.3000000000000001e43 < z < 21000

   1. Initial program 99.4%

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

   3. Taylor expanded in z around 0

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-+.f64 92.7

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

   5. Applied rewrites 92.7%

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

   if 21000 < z

   1. Initial program 98.1%

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

   3. Taylor expanded in y around 0

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

      1. fp-cancel-sub-sign-inv N/A

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

      2. +-commutative N/A

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

      3. metadata-eval N/A

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

      4. *-lft-identity N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-+.f64 94.7

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

   5. Applied rewrites 94.7%

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

   7. Applied rewrites 93.0%

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

Alternative 9: 85.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.3 \cdot 10^{+43}:\\ \;\;\;\;x - a\\ \mathbf{elif}\;z \leq 46000:\\ \;\;\;\;x - \frac{y}{1 + t} \cdot a\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{1 - z}, a, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -1.3e+43)
   (- x a)
   (if (<= z 46000.0) (- x (* (/ y (+ 1.0 t)) a)) (fma (/ z (- 1.0 z)) a x))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1.3e+43) {
		tmp = x - a;
	} else if (z <= 46000.0) {
		tmp = x - ((y / (1.0 + t)) * a);
	} else {
		tmp = fma((z / (1.0 - z)), a, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -1.3e+43)
		tmp = Float64(x - a);
	elseif (z <= 46000.0)
		tmp = Float64(x - Float64(Float64(y / Float64(1.0 + t)) * a));
	else
		tmp = fma(Float64(z / Float64(1.0 - z)), a, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -1.3e+43], N[(x - a), $MachinePrecision], If[LessEqual[z, 46000.0], N[(x - N[(N[(y / N[(1.0 + t), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision], N[(N[(z / N[(1.0 - z), $MachinePrecision]), $MachinePrecision] * a + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.3000000000000001e43

   1. Initial program 95.2%

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

   3. Taylor expanded in z around inf

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

      1. lower--.f64 91.6

         \[\leadsto \color{blue}{x - a} \]

   5. Applied rewrites 91.6%

      \[\leadsto \color{blue}{x - a} \]

   if -1.3000000000000001e43 < z < 46000

   1. Initial program 99.4%

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

   3. Taylor expanded in z around 0

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-+.f64 92.7

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

   5. Applied rewrites 92.7%

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

   if 46000 < z

   1. Initial program 98.1%

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

   3. Taylor expanded in y around 0

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

      1. fp-cancel-sub-sign-inv N/A

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

      2. +-commutative N/A

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

      3. metadata-eval N/A

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

      4. *-lft-identity N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-+.f64 94.7

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

   5. Applied rewrites 94.7%

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

   6. Taylor expanded in t around 0

      \[\leadsto \mathsf{fma}\left(\frac{z}{1 - z}, a, x\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 86.5%

      \[\leadsto \mathsf{fma}\left(\frac{z}{1 - z}, a, x\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 10: 70.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.55 \cdot 10^{+30}:\\ \;\;\;\;x - a\\ \mathbf{elif}\;z \leq 3.5 \cdot 10^{-92}:\\ \;\;\;\;x - \frac{a \cdot y}{t}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{1 - z}, a, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -2.55e+30)
   (- x a)
   (if (<= z 3.5e-92) (- x (/ (* a y) t)) (fma (/ z (- 1.0 z)) a x))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -2.55e+30) {
		tmp = x - a;
	} else if (z <= 3.5e-92) {
		tmp = x - ((a * y) / t);
	} else {
		tmp = fma((z / (1.0 - z)), a, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -2.55e+30)
		tmp = Float64(x - a);
	elseif (z <= 3.5e-92)
		tmp = Float64(x - Float64(Float64(a * y) / t));
	else
		tmp = fma(Float64(z / Float64(1.0 - z)), a, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -2.55e+30], N[(x - a), $MachinePrecision], If[LessEqual[z, 3.5e-92], N[(x - N[(N[(a * y), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], N[(N[(z / N[(1.0 - z), $MachinePrecision]), $MachinePrecision] * a + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -2.55000000000000018e30

   1. Initial program 95.6%

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

   3. Taylor expanded in z around inf

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

      1. lower--.f64 87.7

         \[\leadsto \color{blue}{x - a} \]

   5. Applied rewrites 87.7%

      \[\leadsto \color{blue}{x - a} \]

   if -2.55000000000000018e30 < z < 3.5e-92

   1. Initial program 99.2%

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

   3. Taylor expanded in t around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 75.0

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

   5. Applied rewrites 75.0%

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

   6. Taylor expanded in y around inf

      \[\leadsto x - \frac{a \cdot y}{t} \]
   7. Step-by-step derivation

   8. Applied rewrites 73.2%

      \[\leadsto x - \frac{a \cdot y}{t} \]

   if 3.5e-92 < z

   1. Initial program 98.7%

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

   3. Taylor expanded in y around 0

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

      1. fp-cancel-sub-sign-inv N/A

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

      2. +-commutative N/A

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

      3. metadata-eval N/A

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

      4. *-lft-identity N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-+.f64 85.6

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

   5. Applied rewrites 85.6%

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

   6. Taylor expanded in t around 0

      \[\leadsto \mathsf{fma}\left(\frac{z}{1 - z}, a, x\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 78.6%

      \[\leadsto \mathsf{fma}\left(\frac{z}{1 - z}, a, x\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 11: 70.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.55 \cdot 10^{+30} \lor \neg \left(z \leq 46000\right):\\ \;\;\;\;x - a\\ \mathbf{else}:\\ \;\;\;\;x - \frac{a \cdot y}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= z -2.55e+30) (not (<= z 46000.0))) (- x a) (- x (/ (* a y) t))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z <= -2.55e+30) || !(z <= 46000.0)) {
		tmp = x - a;
	} else {
		tmp = x - ((a * y) / t);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    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) :: tmp
    if ((z <= (-2.55d+30)) .or. (.not. (z <= 46000.0d0))) then
        tmp = x - a
    else
        tmp = x - ((a * y) / t)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z <= -2.55e+30) || !(z <= 46000.0)) {
		tmp = x - a;
	} else {
		tmp = x - ((a * y) / t);
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (z <= -2.55e+30) or not (z <= 46000.0):
		tmp = x - a
	else:
		tmp = x - ((a * y) / t)
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((z <= -2.55e+30) || !(z <= 46000.0))
		tmp = Float64(x - a);
	else
		tmp = Float64(x - Float64(Float64(a * y) / t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((z <= -2.55e+30) || ~((z <= 46000.0)))
		tmp = x - a;
	else
		tmp = x - ((a * y) / t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[z, -2.55e+30], N[Not[LessEqual[z, 46000.0]], $MachinePrecision]], N[(x - a), $MachinePrecision], N[(x - N[(N[(a * y), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -2.55000000000000018e30 or 46000 < z

   1. Initial program 96.8%

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

   3. Taylor expanded in z around inf

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

      1. lower--.f64 86.3

         \[\leadsto \color{blue}{x - a} \]

   5. Applied rewrites 86.3%

      \[\leadsto \color{blue}{x - a} \]

   if -2.55000000000000018e30 < z < 46000

   1. Initial program 99.3%

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

   3. Taylor expanded in t around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 68.3

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

   5. Applied rewrites 68.3%

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

   6. Taylor expanded in y around inf

      \[\leadsto x - \frac{a \cdot y}{t} \]
   7. Step-by-step derivation

   8. Applied rewrites 68.3%

      \[\leadsto x - \frac{a \cdot y}{t} \]
3. Recombined 2 regimes into one program.

4. Final simplification 76.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.55 \cdot 10^{+30} \lor \neg \left(z \leq 46000\right):\\ \;\;\;\;x - a\\ \mathbf{else}:\\ \;\;\;\;x - \frac{a \cdot y}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 65.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -2.3 \cdot 10^{+133} \lor \neg \left(t \leq 7 \cdot 10^{+68}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, a, x\right)\\ \mathbf{else}:\\ \;\;\;\;x - a\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= t -2.3e+133) (not (<= t 7e+68))) (fma (/ z t) a x) (- x a)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((t <= -2.3e+133) || !(t <= 7e+68)) {
		tmp = fma((z / t), a, x);
	} else {
		tmp = x - a;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((t <= -2.3e+133) || !(t <= 7e+68))
		tmp = fma(Float64(z / t), a, x);
	else
		tmp = Float64(x - a);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[t, -2.3e+133], N[Not[LessEqual[t, 7e+68]], $MachinePrecision]], N[(N[(z / t), $MachinePrecision] * a + x), $MachinePrecision], N[(x - a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -2.2999999999999999e133 or 6.99999999999999955e68 < t

   1. Initial program 99.0%

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

   3. Taylor expanded in y around 0

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

      1. fp-cancel-sub-sign-inv N/A

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

      2. +-commutative N/A

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

      3. metadata-eval N/A

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

      4. *-lft-identity N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-+.f64 80.3

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

   5. Applied rewrites 80.3%

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

   6. Taylor expanded in t around inf

      \[\leadsto \mathsf{fma}\left(\frac{z}{t}, a, x\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 76.5%

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

   if -2.2999999999999999e133 < t < 6.99999999999999955e68

   1. Initial program 97.8%

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

   3. Taylor expanded in z around inf

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

      1. lower--.f64 66.7

         \[\leadsto \color{blue}{x - a} \]

   5. Applied rewrites 66.7%

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

4. Final simplification 69.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2.3 \cdot 10^{+133} \lor \neg \left(t \leq 7 \cdot 10^{+68}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, a, x\right)\\ \mathbf{else}:\\ \;\;\;\;x - a\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 60.7% accurate, 8.8× speedup

Math

\[\begin{array}{l} \\ x - a \end{array} \]

FPCore

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

C

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

Fortran

real(8) function code(x, y, z, t, a)
    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
    code = x - a
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.1%

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

3. Taylor expanded in z around inf

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

   1. lower--.f64 64.2

      \[\leadsto \color{blue}{x - a} \]

5. Applied rewrites 64.2%

   \[\leadsto \color{blue}{x - a} \]
6. Add Preprocessing

Alternative 14: 16.7% accurate, 11.7× speedup

Math

\[\begin{array}{l} \\ -a \end{array} \]

FPCore

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

C

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

Fortran

real(8) function code(x, y, z, t, a)
    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
    code = -a
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_, a_] := (-a)

Derivation

1. Initial program 98.1%

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

3. Taylor expanded in z around inf

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

   1. lower--.f64 64.2

      \[\leadsto \color{blue}{x - a} \]

5. Applied rewrites 64.2%

   \[\leadsto \color{blue}{x - a} \]

6. Taylor expanded in x around 0

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

8. Applied rewrites 18.5%

   \[\leadsto -a \]
9. Add Preprocessing

Developer Target 1: 99.6% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z, t, a)
    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
    code = x - (((y - z) / ((t - z) + 1.0d0)) * a)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024338 
(FPCore (x y z t a)
  :name "Graphics.Rendering.Chart.SparkLine:renderSparkLine from Chart-1.5.3"
  :precision binary64

  :alt
  (! :herbie-platform default (- x (* (/ (- y z) (+ (- t z) 1)) a)))

  (- x (/ (- y z) (/ (+ (- t z) 1.0) a))))