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

Percentage Accurate: 97.1% → 99.6%

Time: 10.2s

Alternatives: 19

Speedup: 1.3×

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: 97.1% 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: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z, double t, double a) {
	return x - (a / ((1.0 + (t - z)) / (y - 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 - (a / ((1.0d0 + (t - z)) / (y - z)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 96.8%

   \[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \]
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. *-commutative N/A

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

   5. clear-num N/A

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

   6. un-div-inv N/A

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

   7. lower-/.f64 N/A

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

   8. lower-/.f64 99.2

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

   9. lift-+.f64 N/A

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

   10. +-commutative N/A

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

   11. lower-+.f64 99.2

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

4. Applied rewrites 99.2%

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

Alternative 2: 64.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{y - z}{\frac{\left(t - z\right) + 1}{a}}\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+290} \lor \neg \left(t\_1 \leq 10^{+243}\right):\\ \;\;\;\;\left(-y\right) \cdot a\\ \mathbf{else}:\\ \;\;\;\;x - a\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- y z) (/ (+ (- t z) 1.0) a))))
   (if (or (<= t_1 -5e+290) (not (<= t_1 1e+243))) (* (- y) 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 <= -5e+290) || !(t_1 <= 1e+243)) {
		tmp = -y * a;
	} 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 <= (-5d+290)) .or. (.not. (t_1 <= 1d+243))) then
        tmp = -y * a
    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 <= -5e+290) || !(t_1 <= 1e+243)) {
		tmp = -y * a;
	} 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 <= -5e+290) or not (t_1 <= 1e+243):
		tmp = -y * 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 <= -5e+290) || !(t_1 <= 1e+243))
		tmp = Float64(Float64(-y) * a);
	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 <= -5e+290) || ~((t_1 <= 1e+243)))
		tmp = -y * a;
	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, -5e+290], N[Not[LessEqual[t$95$1, 1e+243]], $MachinePrecision]], N[((-y) * a), $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)) < -4.9999999999999998e290 or 1.0000000000000001e243 < (/.f64 (-.f64 y z) (/.f64 (+.f64 (-.f64 t z) #s(literal 1 binary64)) a))

   1. Initial program 99.8%

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

   3. Taylor expanded in t around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

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

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

      5. lower-fma.f64 N/A

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

      6. lower-neg.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower--.f64 85.3

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

   5. Applied rewrites 85.3%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 85.3%

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

   9. Taylor expanded in z around 0

      \[\leadsto \left(-1 \cdot y\right) \cdot a \]
   10. Step-by-step derivation

   11. Applied rewrites 59.8%

       \[\leadsto \left(-y\right) \cdot a \]

   if -4.9999999999999998e290 < (/.f64 (-.f64 y z) (/.f64 (+.f64 (-.f64 t z) #s(literal 1 binary64)) a)) < 1.0000000000000001e243

   1. Initial program 96.5%

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

   3. Taylor expanded in z around inf

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

      1. lower--.f64 66.0

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

   5. Applied rewrites 66.0%

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

4. Final simplification 65.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \leq -5 \cdot 10^{+290} \lor \neg \left(\frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \leq 10^{+243}\right):\\ \;\;\;\;\left(-y\right) \cdot a\\ \mathbf{else}:\\ \;\;\;\;x - a\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 92.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.7 \cdot 10^{+163} \lor \neg \left(z \leq 1.3 \cdot 10^{+36}\right):\\ \;\;\;\;\mathsf{fma}\left(-a, \frac{1 - y}{z} + 1, x\right)\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\left(y - z\right) \cdot a}{1 + \left(t - z\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= z -2.7e+163) (not (<= z 1.3e+36)))
   (fma (- a) (+ (/ (- 1.0 y) z) 1.0) x)
   (- x (/ (* (- y z) a) (+ 1.0 (- t z))))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z <= -2.7e+163) || !(z <= 1.3e+36)) {
		tmp = fma(-a, (((1.0 - y) / z) + 1.0), x);
	} else {
		tmp = x - (((y - z) * a) / (1.0 + (t - z)));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((z <= -2.7e+163) || !(z <= 1.3e+36))
		tmp = fma(Float64(-a), Float64(Float64(Float64(1.0 - y) / z) + 1.0), x);
	else
		tmp = Float64(x - Float64(Float64(Float64(y - z) * a) / Float64(1.0 + Float64(t - z))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[z, -2.7e+163], N[Not[LessEqual[z, 1.3e+36]], $MachinePrecision]], N[((-a) * N[(N[(N[(1.0 - y), $MachinePrecision] / z), $MachinePrecision] + 1.0), $MachinePrecision] + x), $MachinePrecision], N[(x - N[(N[(N[(y - z), $MachinePrecision] * a), $MachinePrecision] / N[(1.0 + N[(t - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -2.69999999999999999e163 or 1.3000000000000001e36 < z

   1. Initial program 91.6%

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

   3. Taylor expanded in t around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

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

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

      5. lower-fma.f64 N/A

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

      6. lower-neg.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower--.f64 94.6

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

   5. Applied rewrites 94.6%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 94.7%

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

   if -2.69999999999999999e163 < z < 1.3000000000000001e36

   1. Initial program 99.3%

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

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

      6. lower-*.f64 91.4

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 91.4

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

   4. Applied rewrites 91.4%

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

4. Final simplification 92.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.7 \cdot 10^{+163} \lor \neg \left(z \leq 1.3 \cdot 10^{+36}\right):\\ \;\;\;\;\mathsf{fma}\left(-a, \frac{1 - y}{z} + 1, x\right)\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\left(y - z\right) \cdot a}{1 + \left(t - z\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 89.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -9.5 \cdot 10^{+29} \lor \neg \left(z \leq 2.7 \cdot 10^{+30}\right):\\ \;\;\;\;\mathsf{fma}\left(-a, \frac{1 - y}{z} + 1, x\right)\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{1 + t} \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= z -9.5e+29) (not (<= z 2.7e+30)))
   (fma (- a) (+ (/ (- 1.0 y) z) 1.0) x)
   (- x (* (/ y (+ 1.0 t)) a))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z <= -9.5e+29) || !(z <= 2.7e+30)) {
		tmp = fma(-a, (((1.0 - y) / z) + 1.0), x);
	} else {
		tmp = x - ((y / (1.0 + t)) * a);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((z <= -9.5e+29) || !(z <= 2.7e+30))
		tmp = fma(Float64(-a), Float64(Float64(Float64(1.0 - y) / z) + 1.0), 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, -9.5e+29], N[Not[LessEqual[z, 2.7e+30]], $MachinePrecision]], N[((-a) * N[(N[(N[(1.0 - y), $MachinePrecision] / z), $MachinePrecision] + 1.0), $MachinePrecision] + 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 < -9.5000000000000003e29 or 2.6999999999999999e30 < z

   1. Initial program 94.1%

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

   3. Taylor expanded in t around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

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

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

      5. lower-fma.f64 N/A

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

      6. lower-neg.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower--.f64 90.5

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

   5. Applied rewrites 90.5%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 90.5%

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

   if -9.5000000000000003e29 < z < 2.6999999999999999e30

   1. Initial program 99.1%

      \[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \]
   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 89.9

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

   5. Applied rewrites 89.9%

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

4. Final simplification 90.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -9.5 \cdot 10^{+29} \lor \neg \left(z \leq 2.7 \cdot 10^{+30}\right):\\ \;\;\;\;\mathsf{fma}\left(-a, \frac{1 - y}{z} + 1, x\right)\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{1 + t} \cdot a\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 72.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x - \frac{y}{t} \cdot a\\ \mathbf{if}\;t \leq -2 \cdot 10^{+61}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq -1.12 \cdot 10^{-228}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{1 - z}, a, x\right)\\ \mathbf{elif}\;t \leq 0.75:\\ \;\;\;\;x - \mathsf{fma}\left(-y, t, y\right) \cdot a\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- x (* (/ y t) a))))
   (if (<= t -2e+61)
     t_1
     (if (<= t -1.12e-228)
       (fma (/ z (- 1.0 z)) a x)
       (if (<= t 0.75) (- x (* (fma (- y) t y) a)) t_1)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = x - ((y / t) * a);
	double tmp;
	if (t <= -2e+61) {
		tmp = t_1;
	} else if (t <= -1.12e-228) {
		tmp = fma((z / (1.0 - z)), a, x);
	} else if (t <= 0.75) {
		tmp = x - (fma(-y, t, y) * a);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(x - Float64(Float64(y / t) * a))
	tmp = 0.0
	if (t <= -2e+61)
		tmp = t_1;
	elseif (t <= -1.12e-228)
		tmp = fma(Float64(z / Float64(1.0 - z)), a, x);
	elseif (t <= 0.75)
		tmp = Float64(x - Float64(fma(Float64(-y), t, y) * a));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(x - N[(N[(y / t), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -2e+61], t$95$1, If[LessEqual[t, -1.12e-228], N[(N[(z / N[(1.0 - z), $MachinePrecision]), $MachinePrecision] * a + x), $MachinePrecision], If[LessEqual[t, 0.75], N[(x - N[(N[((-y) * t + y), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if t < -1.9999999999999999e61 or 0.75 < t

   1. Initial program 96.9%

      \[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \]
   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 76.4

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

   5. Applied rewrites 76.4%

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

   6. Taylor expanded in t around inf

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

   8. Applied rewrites 76.3%

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

   if -1.9999999999999999e61 < t < -1.11999999999999996e-228

   1. Initial program 94.4%

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

   3. Taylor expanded in t around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

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

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

      5. lower-fma.f64 N/A

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

      6. lower-neg.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower--.f64 91.3

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

   5. Applied rewrites 91.3%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 68.7%

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

   if -1.11999999999999996e-228 < t < 0.75

   1. Initial program 98.5%

      \[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \]
   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 75.9

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

   5. Applied rewrites 75.9%

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

   6. Taylor expanded in t around 0

      \[\leadsto x - \left(y + -1 \cdot \left(t \cdot y\right)\right) \cdot a \]
   7. Step-by-step derivation

   8. Applied rewrites 75.9%

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

Alternative 6: 72.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x - \frac{y}{t} \cdot a\\ \mathbf{if}\;t \leq -2 \cdot 10^{+61}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq -1.15 \cdot 10^{-228}:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{a}{1 - z}, x\right)\\ \mathbf{elif}\;t \leq 0.75:\\ \;\;\;\;x - \mathsf{fma}\left(-y, t, y\right) \cdot a\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- x (* (/ y t) a))))
   (if (<= t -2e+61)
     t_1
     (if (<= t -1.15e-228)
       (fma z (/ a (- 1.0 z)) x)
       (if (<= t 0.75) (- x (* (fma (- y) t y) a)) t_1)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = x - ((y / t) * a);
	double tmp;
	if (t <= -2e+61) {
		tmp = t_1;
	} else if (t <= -1.15e-228) {
		tmp = fma(z, (a / (1.0 - z)), x);
	} else if (t <= 0.75) {
		tmp = x - (fma(-y, t, y) * a);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(x - Float64(Float64(y / t) * a))
	tmp = 0.0
	if (t <= -2e+61)
		tmp = t_1;
	elseif (t <= -1.15e-228)
		tmp = fma(z, Float64(a / Float64(1.0 - z)), x);
	elseif (t <= 0.75)
		tmp = Float64(x - Float64(fma(Float64(-y), t, y) * a));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(x - N[(N[(y / t), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -2e+61], t$95$1, If[LessEqual[t, -1.15e-228], N[(z * N[(a / N[(1.0 - z), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t, 0.75], N[(x - N[(N[((-y) * t + y), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if t < -1.9999999999999999e61 or 0.75 < t

   1. Initial program 96.9%

      \[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \]
   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 76.4

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

   5. Applied rewrites 76.4%

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

   6. Taylor expanded in t around inf

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

   8. Applied rewrites 76.3%

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

   if -1.9999999999999999e61 < t < -1.1499999999999999e-228

   1. Initial program 94.4%

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

   3. Taylor expanded in t around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

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

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

      5. lower-fma.f64 N/A

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

      6. lower-neg.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower--.f64 91.3

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

   5. Applied rewrites 91.3%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 68.7%

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

   10. Applied rewrites 66.9%

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

   if -1.1499999999999999e-228 < t < 0.75

   1. Initial program 98.5%

      \[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \]
   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 75.9

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

   5. Applied rewrites 75.9%

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

   6. Taylor expanded in t around 0

      \[\leadsto x - \left(y + -1 \cdot \left(t \cdot y\right)\right) \cdot a \]
   7. Step-by-step derivation

   8. Applied rewrites 75.9%

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

Alternative 7: 71.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x - \frac{y}{t} \cdot a\\ \mathbf{if}\;t \leq -4.15 \cdot 10^{+52}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq -1.12 \cdot 10^{-228}:\\ \;\;\;\;x - a\\ \mathbf{elif}\;t \leq 0.75:\\ \;\;\;\;x - \mathsf{fma}\left(-y, t, y\right) \cdot a\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- x (* (/ y t) a))))
   (if (<= t -4.15e+52)
     t_1
     (if (<= t -1.12e-228)
       (- x a)
       (if (<= t 0.75) (- x (* (fma (- y) t y) a)) t_1)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = x - ((y / t) * a);
	double tmp;
	if (t <= -4.15e+52) {
		tmp = t_1;
	} else if (t <= -1.12e-228) {
		tmp = x - a;
	} else if (t <= 0.75) {
		tmp = x - (fma(-y, t, y) * a);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(x - Float64(Float64(y / t) * a))
	tmp = 0.0
	if (t <= -4.15e+52)
		tmp = t_1;
	elseif (t <= -1.12e-228)
		tmp = Float64(x - a);
	elseif (t <= 0.75)
		tmp = Float64(x - Float64(fma(Float64(-y), t, y) * a));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(x - N[(N[(y / t), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -4.15e+52], t$95$1, If[LessEqual[t, -1.12e-228], N[(x - a), $MachinePrecision], If[LessEqual[t, 0.75], N[(x - N[(N[((-y) * t + y), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if t < -4.14999999999999997e52 or 0.75 < t

   1. Initial program 96.3%

      \[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \]
   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 76.4

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

   5. Applied rewrites 76.4%

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

   6. Taylor expanded in t around inf

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

   8. Applied rewrites 76.3%

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

   if -4.14999999999999997e52 < t < -1.11999999999999996e-228

   1. Initial program 95.9%

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

   3. Taylor expanded in z around inf

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

      1. lower--.f64 64.5

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

   5. Applied rewrites 64.5%

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

   if -1.11999999999999996e-228 < t < 0.75

   1. Initial program 98.5%

      \[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \]
   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 75.9

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

   5. Applied rewrites 75.9%

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

   6. Taylor expanded in t around 0

      \[\leadsto x - \left(y + -1 \cdot \left(t \cdot y\right)\right) \cdot a \]
   7. Step-by-step derivation

   8. Applied rewrites 75.9%

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

Alternative 8: 91.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -8.6 \cdot 10^{+60}:\\ \;\;\;\;\mathsf{fma}\left(-a, \frac{y - z}{t}, x\right)\\ \mathbf{elif}\;t \leq 5.5 \cdot 10^{+62}:\\ \;\;\;\;\mathsf{fma}\left(-a, \frac{y - z}{1 - z}, x\right)\\ \mathbf{else}:\\ \;\;\;\;x - \frac{a}{t} \cdot \left(y - z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t -8.6e+60)
   (fma (- a) (/ (- y z) t) x)
   (if (<= t 5.5e+62)
     (fma (- a) (/ (- y z) (- 1.0 z)) x)
     (- x (* (/ a t) (- y z))))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= -8.6e+60) {
		tmp = fma(-a, ((y - z) / t), x);
	} else if (t <= 5.5e+62) {
		tmp = fma(-a, ((y - z) / (1.0 - z)), x);
	} else {
		tmp = x - ((a / t) * (y - z));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= -8.6e+60)
		tmp = fma(Float64(-a), Float64(Float64(y - z) / t), x);
	elseif (t <= 5.5e+62)
		tmp = fma(Float64(-a), Float64(Float64(y - z) / Float64(1.0 - z)), x);
	else
		tmp = Float64(x - Float64(Float64(a / t) * Float64(y - z)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, -8.6e+60], N[((-a) * N[(N[(y - z), $MachinePrecision] / t), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t, 5.5e+62], N[((-a) * N[(N[(y - z), $MachinePrecision] / N[(1.0 - z), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], N[(x - N[(N[(a / t), $MachinePrecision] * N[(y - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -8.59999999999999942e60

   1. Initial program 96.8%

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

   3. Taylor expanded in t around inf

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

      1. mul-1-neg N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

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

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

      5. lower-fma.f64 N/A

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

      6. lower-neg.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 86.3

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

   5. Applied rewrites 86.3%

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

   if -8.59999999999999942e60 < t < 5.4999999999999997e62

   1. Initial program 95.6%

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

   3. Taylor expanded in t around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

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

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

      5. lower-fma.f64 N/A

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

      6. lower-neg.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower--.f64 94.3

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

   5. Applied rewrites 94.3%

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

   if 5.4999999999999997e62 < t

   1. Initial program 99.9%

      \[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \]
   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 74.0

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

   5. Applied rewrites 74.0%

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

   7. Applied rewrites 88.3%

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

Alternative 9: 88.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -4 \cdot 10^{-25} \lor \neg \left(z \leq 3.5 \cdot 10^{-37}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{\left(1 + t\right) - z}, a, x\right)\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{1 + t} \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= z -4e-25) (not (<= z 3.5e-37)))
   (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 <= -4e-25) || !(z <= 3.5e-37)) {
		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 <= -4e-25) || !(z <= 3.5e-37))
		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, -4e-25], N[Not[LessEqual[z, 3.5e-37]], $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 < -4.00000000000000015e-25 or 3.5000000000000001e-37 < z

   1. Initial program 95.0%

      \[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \]
   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. cancel-sign-sub-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. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. +-commutative 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 86.3

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

   5. Applied rewrites 86.3%

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

   if -4.00000000000000015e-25 < z < 3.5000000000000001e-37

   1. Initial program 99.0%

      \[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \]
   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 93.2

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

   5. Applied rewrites 93.2%

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

4. Final simplification 89.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4 \cdot 10^{-25} \lor \neg \left(z \leq 3.5 \cdot 10^{-37}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{\left(1 + t\right) - z}, a, x\right)\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{1 + t} \cdot a\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 84.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -5.2 \cdot 10^{+43} \lor \neg \left(z \leq 2.45 \cdot 10^{+42}\right):\\ \;\;\;\;x - a\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{1 + t} \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= z -5.2e+43) (not (<= z 2.45e+42)))
   (- x a)
   (- x (* (/ y (+ 1.0 t)) a))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z <= -5.2e+43) || !(z <= 2.45e+42)) {
		tmp = x - a;
	} else {
		tmp = x - ((y / (1.0 + t)) * 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) :: tmp
    if ((z <= (-5.2d+43)) .or. (.not. (z <= 2.45d+42))) then
        tmp = x - a
    else
        tmp = x - ((y / (1.0d0 + t)) * a)
    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 <= -5.2e+43) || !(z <= 2.45e+42)) {
		tmp = x - a;
	} else {
		tmp = x - ((y / (1.0 + t)) * a);
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (z <= -5.2e+43) or not (z <= 2.45e+42):
		tmp = x - a
	else:
		tmp = x - ((y / (1.0 + t)) * a)
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((z <= -5.2e+43) || !(z <= 2.45e+42))
		tmp = Float64(x - a);
	else
		tmp = Float64(x - Float64(Float64(y / Float64(1.0 + t)) * a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((z <= -5.2e+43) || ~((z <= 2.45e+42)))
		tmp = x - a;
	else
		tmp = x - ((y / (1.0 + t)) * a);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[z, -5.2e+43], N[Not[LessEqual[z, 2.45e+42]], $MachinePrecision]], N[(x - a), $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 < -5.20000000000000042e43 or 2.4500000000000001e42 < z

   1. Initial program 93.8%

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

   3. Taylor expanded in z around inf

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

      1. lower--.f64 78.2

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

   5. Applied rewrites 78.2%

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

   if -5.20000000000000042e43 < z < 2.4500000000000001e42

   1. Initial program 99.2%

      \[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \]
   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 89.8

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

   5. Applied rewrites 89.8%

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

4. Final simplification 84.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5.2 \cdot 10^{+43} \lor \neg \left(z \leq 2.45 \cdot 10^{+42}\right):\\ \;\;\;\;x - a\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{1 + t} \cdot a\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 78.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.1 \cdot 10^{+33} \lor \neg \left(t \leq 410000000000\right):\\ \;\;\;\;\mathsf{fma}\left(-a, \frac{y - z}{t}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-a, \frac{y}{1 - z}, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= t -1.1e+33) (not (<= t 410000000000.0)))
   (fma (- a) (/ (- y z) t) x)
   (fma (- a) (/ y (- 1.0 z)) x)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((t <= -1.1e+33) || !(t <= 410000000000.0)) {
		tmp = fma(-a, ((y - z) / t), x);
	} else {
		tmp = fma(-a, (y / (1.0 - z)), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((t <= -1.1e+33) || !(t <= 410000000000.0))
		tmp = fma(Float64(-a), Float64(Float64(y - z) / t), x);
	else
		tmp = fma(Float64(-a), Float64(y / Float64(1.0 - z)), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[t, -1.1e+33], N[Not[LessEqual[t, 410000000000.0]], $MachinePrecision]], N[((-a) * N[(N[(y - z), $MachinePrecision] / t), $MachinePrecision] + x), $MachinePrecision], N[((-a) * N[(y / N[(1.0 - z), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -1.09999999999999997e33 or 4.1e11 < t

   1. Initial program 96.4%

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

   3. Taylor expanded in t around inf

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

      1. mul-1-neg N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

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

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

      5. lower-fma.f64 N/A

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

      6. lower-neg.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 82.5

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

   5. Applied rewrites 82.5%

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

   if -1.09999999999999997e33 < t < 4.1e11

   1. Initial program 97.3%

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

   3. Taylor expanded in t around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

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

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

      5. lower-fma.f64 N/A

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

      6. lower-neg.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower--.f64 97.9

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

   5. Applied rewrites 97.9%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 78.3%

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

4. Final simplification 80.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.1 \cdot 10^{+33} \lor \neg \left(t \leq 410000000000\right):\\ \;\;\;\;\mathsf{fma}\left(-a, \frac{y - z}{t}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-a, \frac{y}{1 - z}, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 76.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.1 \cdot 10^{+33} \lor \neg \left(t \leq 4800000000\right):\\ \;\;\;\;x - \frac{y}{t} \cdot a\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-a, \frac{y}{1 - z}, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= t -1.1e+33) (not (<= t 4800000000.0)))
   (- x (* (/ y t) a))
   (fma (- a) (/ y (- 1.0 z)) x)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((t <= -1.1e+33) || !(t <= 4800000000.0)) {
		tmp = x - ((y / t) * a);
	} else {
		tmp = fma(-a, (y / (1.0 - z)), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((t <= -1.1e+33) || !(t <= 4800000000.0))
		tmp = Float64(x - Float64(Float64(y / t) * a));
	else
		tmp = fma(Float64(-a), Float64(y / Float64(1.0 - z)), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[t, -1.1e+33], N[Not[LessEqual[t, 4800000000.0]], $MachinePrecision]], N[(x - N[(N[(y / t), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision], N[((-a) * N[(y / N[(1.0 - z), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -1.09999999999999997e33 or 4.8e9 < t

   1. Initial program 96.4%

      \[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \]
   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 75.2

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

   5. Applied rewrites 75.2%

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

   6. Taylor expanded in t around inf

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

   8. Applied rewrites 75.0%

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

   if -1.09999999999999997e33 < t < 4.8e9

   1. Initial program 97.3%

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

   3. Taylor expanded in t around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

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

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

      5. lower-fma.f64 N/A

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

      6. lower-neg.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower--.f64 97.9

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

   5. Applied rewrites 97.9%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 78.3%

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

4. Final simplification 76.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.1 \cdot 10^{+33} \lor \neg \left(t \leq 4800000000\right):\\ \;\;\;\;x - \frac{y}{t} \cdot a\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-a, \frac{y}{1 - z}, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 78.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.1 \cdot 10^{+33}:\\ \;\;\;\;\mathsf{fma}\left(-a, \frac{y - z}{t}, x\right)\\ \mathbf{elif}\;t \leq 410000000000:\\ \;\;\;\;\mathsf{fma}\left(-a, \frac{y}{1 - z}, x\right)\\ \mathbf{else}:\\ \;\;\;\;x - \frac{a}{t} \cdot \left(y - z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t -1.1e+33)
   (fma (- a) (/ (- y z) t) x)
   (if (<= t 410000000000.0)
     (fma (- a) (/ y (- 1.0 z)) x)
     (- x (* (/ a t) (- y z))))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= -1.1e+33) {
		tmp = fma(-a, ((y - z) / t), x);
	} else if (t <= 410000000000.0) {
		tmp = fma(-a, (y / (1.0 - z)), x);
	} else {
		tmp = x - ((a / t) * (y - z));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= -1.1e+33)
		tmp = fma(Float64(-a), Float64(Float64(y - z) / t), x);
	elseif (t <= 410000000000.0)
		tmp = fma(Float64(-a), Float64(y / Float64(1.0 - z)), x);
	else
		tmp = Float64(x - Float64(Float64(a / t) * Float64(y - z)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, -1.1e+33], N[((-a) * N[(N[(y - z), $MachinePrecision] / t), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t, 410000000000.0], N[((-a) * N[(y / N[(1.0 - z), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], N[(x - N[(N[(a / t), $MachinePrecision] * N[(y - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -1.09999999999999997e33

   1. Initial program 95.8%

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

   3. Taylor expanded in t around inf

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

      1. mul-1-neg N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

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

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

      5. lower-fma.f64 N/A

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

      6. lower-neg.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 82.5

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

   5. Applied rewrites 82.5%

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

   if -1.09999999999999997e33 < t < 4.1e11

   1. Initial program 97.3%

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

   3. Taylor expanded in t around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

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

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

      5. lower-fma.f64 N/A

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

      6. lower-neg.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower--.f64 97.9

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

   5. Applied rewrites 97.9%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 78.3%

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

   if 4.1e11 < t

   1. Initial program 97.1%

      \[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \]
   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 73.1

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

   5. Applied rewrites 73.1%

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

   7. Applied rewrites 85.4%

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

Alternative 14: 75.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -350000:\\ \;\;\;\;\left(x - a\right) - \frac{a}{z}\\ \mathbf{elif}\;z \leq 1650000000:\\ \;\;\;\;\mathsf{fma}\left(-a, \mathsf{fma}\left(y - 1, z, y\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;x - a\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -350000.0)
   (- (- x a) (/ a z))
   (if (<= z 1650000000.0) (fma (- a) (fma (- y 1.0) z y) x) (- x a))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -350000.0) {
		tmp = (x - a) - (a / z);
	} else if (z <= 1650000000.0) {
		tmp = fma(-a, fma((y - 1.0), z, y), x);
	} else {
		tmp = x - a;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -350000.0)
		tmp = Float64(Float64(x - a) - Float64(a / z));
	elseif (z <= 1650000000.0)
		tmp = fma(Float64(-a), fma(Float64(y - 1.0), z, y), x);
	else
		tmp = Float64(x - a);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -350000.0], N[(N[(x - a), $MachinePrecision] - N[(a / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1650000000.0], N[((-a) * N[(N[(y - 1.0), $MachinePrecision] * z + y), $MachinePrecision] + x), $MachinePrecision], N[(x - a), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -3.5e5

   1. Initial program 95.7%

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

   3. Taylor expanded in t around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

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

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

      5. lower-fma.f64 N/A

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

      6. lower-neg.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower--.f64 86.5

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

   5. Applied rewrites 86.5%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 75.5%

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

   9. Taylor expanded in z around inf

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

   11. Applied rewrites 75.5%

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

   if -3.5e5 < z < 1.65e9

   1. Initial program 99.1%

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

   3. Taylor expanded in t around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

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

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

      5. lower-fma.f64 N/A

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

      6. lower-neg.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower--.f64 68.9

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

   5. Applied rewrites 68.9%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 68.8%

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

   if 1.65e9 < z

   1. Initial program 92.6%

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

   3. Taylor expanded in z around inf

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

      1. lower--.f64 75.4

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

   5. Applied rewrites 75.4%

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

Alternative 15: 97.3% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 96.8%

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

   1. lift--.f64 N/A

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

   2. sub-neg N/A

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

   3. +-commutative N/A

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

   4. lift-/.f64 N/A

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

   5. distribute-neg-frac2 N/A

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

   6. div-inv N/A

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

   7. lower-fma.f64 N/A

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

   8. lift-/.f64 N/A

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

   9. clear-num N/A

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

   10. distribute-neg-frac2 N/A

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

   11. remove-double-div N/A

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

   12. lower-neg.f64 N/A

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

   13. lower-/.f64 97.4

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

   14. lift-+.f64 N/A

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

   15. +-commutative N/A

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

   16. lower-+.f64 97.4

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

4. Applied rewrites 97.4%

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

5. Final simplification 97.4%

   \[\leadsto \mathsf{fma}\left(y - z, \frac{a}{-1 - \left(t - z\right)}, x\right) \]
6. Add Preprocessing

Alternative 16: 73.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.35 \cdot 10^{+42} \lor \neg \left(z \leq 2.7 \cdot 10^{+30}\right):\\ \;\;\;\;x - a\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-a, y, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= z -1.35e+42) (not (<= z 2.7e+30))) (- x a) (fma (- a) y x)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z <= -1.35e+42) || !(z <= 2.7e+30)) {
		tmp = x - a;
	} else {
		tmp = fma(-a, y, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((z <= -1.35e+42) || !(z <= 2.7e+30))
		tmp = Float64(x - a);
	else
		tmp = fma(Float64(-a), y, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[z, -1.35e+42], N[Not[LessEqual[z, 2.7e+30]], $MachinePrecision]], N[(x - a), $MachinePrecision], N[((-a) * y + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.35e42 or 2.6999999999999999e30 < z

   1. Initial program 93.9%

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

   3. Taylor expanded in z around inf

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

      1. lower--.f64 77.5

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

   5. Applied rewrites 77.5%

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

   if -1.35e42 < z < 2.6999999999999999e30

   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 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

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

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

      5. lower-fma.f64 N/A

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

      6. lower-neg.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower--.f64 68.4

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

   5. Applied rewrites 68.4%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 66.1%

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

4. Final simplification 71.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.35 \cdot 10^{+42} \lor \neg \left(z \leq 2.7 \cdot 10^{+30}\right):\\ \;\;\;\;x - a\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-a, y, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 67.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -490 \lor \neg \left(z \leq 2.7 \cdot 10^{+30}\right):\\ \;\;\;\;x - a\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, a, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= z -490.0) (not (<= z 2.7e+30))) (- x a) (fma z a x)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z <= -490.0) || !(z <= 2.7e+30)) {
		tmp = x - a;
	} else {
		tmp = fma(z, a, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((z <= -490.0) || !(z <= 2.7e+30))
		tmp = Float64(x - a);
	else
		tmp = fma(z, a, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[z, -490.0], N[Not[LessEqual[z, 2.7e+30]], $MachinePrecision]], N[(x - a), $MachinePrecision], N[(z * a + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -490 or 2.6999999999999999e30 < z

   1. Initial program 94.2%

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

   3. Taylor expanded in z around inf

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

      1. lower--.f64 76.4

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

   5. Applied rewrites 76.4%

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

   if -490 < z < 2.6999999999999999e30

   1. Initial program 99.1%

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

   3. Taylor expanded in t around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

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

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

      5. lower-fma.f64 N/A

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

      6. lower-neg.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower--.f64 68.1

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

   5. Applied rewrites 68.1%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 55.6%

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

   9. Taylor expanded in z around 0

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

   11. Applied rewrites 55.6%

       \[\leadsto \mathsf{fma}\left(z, a, x\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 65.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -490 \lor \neg \left(z \leq 2.7 \cdot 10^{+30}\right):\\ \;\;\;\;x - a\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, a, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 61.6% 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 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 60.3

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

5. Applied rewrites 60.3%

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

Alternative 19: 17.6% 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 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 60.3

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

5. Applied rewrites 60.3%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 15.0%

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

Developer Target 1: 99.7% 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 2024299 
(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))))