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

Percentage Accurate: 96.8% → 99.5%

Time: 9.7s

Alternatives: 14

Speedup: 1.0×

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: 96.8% 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: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

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)) / (z - y)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 95.7%

   \[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \]
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 99.9

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

   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 99.9

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

4. Applied rewrites 99.9%

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

5. Final simplification 99.9%

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

Alternative 2: 63.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(-a\right) \cdot y\\ t_2 := \frac{z - y}{\frac{-1 - \left(t - z\right)}{a}}\\ \mathbf{if}\;t\_2 \leq -2 \cdot 10^{+276}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 10^{+267}:\\ \;\;\;\;x - a\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* (- a) y)) (t_2 (/ (- z y) (/ (- -1.0 (- t z)) a))))
   (if (<= t_2 -2e+276) t_1 (if (<= t_2 1e+267) (- x a) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = -a * y;
	double t_2 = (z - y) / ((-1.0 - (t - z)) / a);
	double tmp;
	if (t_2 <= -2e+276) {
		tmp = t_1;
	} else if (t_2 <= 1e+267) {
		tmp = x - a;
	} else {
		tmp = t_1;
	}
	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) :: t_2
    real(8) :: tmp
    t_1 = -a * y
    t_2 = (z - y) / (((-1.0d0) - (t - z)) / a)
    if (t_2 <= (-2d+276)) then
        tmp = t_1
    else if (t_2 <= 1d+267) then
        tmp = x - a
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = -a * y;
	double t_2 = (z - y) / ((-1.0 - (t - z)) / a);
	double tmp;
	if (t_2 <= -2e+276) {
		tmp = t_1;
	} else if (t_2 <= 1e+267) {
		tmp = x - a;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = -a * y
	t_2 = (z - y) / ((-1.0 - (t - z)) / a)
	tmp = 0
	if t_2 <= -2e+276:
		tmp = t_1
	elif t_2 <= 1e+267:
		tmp = x - a
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(-a) * y)
	t_2 = Float64(Float64(z - y) / Float64(Float64(-1.0 - Float64(t - z)) / a))
	tmp = 0.0
	if (t_2 <= -2e+276)
		tmp = t_1;
	elseif (t_2 <= 1e+267)
		tmp = Float64(x - a);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = -a * y;
	t_2 = (z - y) / ((-1.0 - (t - z)) / a);
	tmp = 0.0;
	if (t_2 <= -2e+276)
		tmp = t_1;
	elseif (t_2 <= 1e+267)
		tmp = x - a;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[((-a) * y), $MachinePrecision]}, Block[{t$95$2 = N[(N[(z - y), $MachinePrecision] / N[(N[(-1.0 - N[(t - z), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -2e+276], t$95$1, If[LessEqual[t$95$2, 1e+267], N[(x - a), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 y z) (/.f64 (+.f64 (-.f64 t z) #s(literal 1 binary64)) a)) < -2.0000000000000001e276 or 9.9999999999999997e266 < (/.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. *-commutative N/A

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

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

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

      6. lower-fma.f64 N/A

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

      7. distribute-neg-frac N/A

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

      8. mul-1-neg N/A

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

      9. lower-/.f64 N/A

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

      10. mul-1-neg N/A

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

      11. lower-neg.f64 N/A

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

      12. lower--.f64 N/A

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

      13. lower--.f64 76.5

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

   5. Applied rewrites 76.5%

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

   6. Taylor expanded in a around inf

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

   8. Applied rewrites 62.1%

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

   9. Taylor expanded in z around 0

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

   11. Applied rewrites 54.7%

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

   if -2.0000000000000001e276 < (/.f64 (-.f64 y z) (/.f64 (+.f64 (-.f64 t z) #s(literal 1 binary64)) a)) < 9.9999999999999997e266

   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 60.9

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

   5. Applied rewrites 60.9%

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

4. Final simplification 60.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z - y}{\frac{-1 - \left(t - z\right)}{a}} \leq -2 \cdot 10^{+276}:\\ \;\;\;\;\left(-a\right) \cdot y\\ \mathbf{elif}\;\frac{z - y}{\frac{-1 - \left(t - z\right)}{a}} \leq 10^{+267}:\\ \;\;\;\;x - a\\ \mathbf{else}:\\ \;\;\;\;\left(-a\right) \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 74.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1:\\ \;\;\;\;x - a\\ \mathbf{elif}\;z \leq 3.2 \cdot 10^{-195}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(1 - y, z, -y\right), a, x\right)\\ \mathbf{elif}\;z \leq 3.3 \cdot 10^{-80}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{t}, -a, x\right)\\ \mathbf{elif}\;z \leq 8.5 \cdot 10^{+68}:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{a}{t - -1}, x\right)\\ \mathbf{else}:\\ \;\;\;\;x - a\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -1.0)
   (- x a)
   (if (<= z 3.2e-195)
     (fma (fma (- 1.0 y) z (- y)) a x)
     (if (<= z 3.3e-80)
       (fma (/ y t) (- a) x)
       (if (<= z 8.5e+68) (fma z (/ a (- t -1.0)) x) (- x a))))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1.0) {
		tmp = x - a;
	} else if (z <= 3.2e-195) {
		tmp = fma(fma((1.0 - y), z, -y), a, x);
	} else if (z <= 3.3e-80) {
		tmp = fma((y / t), -a, x);
	} else if (z <= 8.5e+68) {
		tmp = fma(z, (a / (t - -1.0)), x);
	} else {
		tmp = x - a;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -1.0)
		tmp = Float64(x - a);
	elseif (z <= 3.2e-195)
		tmp = fma(fma(Float64(1.0 - y), z, Float64(-y)), a, x);
	elseif (z <= 3.3e-80)
		tmp = fma(Float64(y / t), Float64(-a), x);
	elseif (z <= 8.5e+68)
		tmp = fma(z, Float64(a / Float64(t - -1.0)), x);
	else
		tmp = Float64(x - a);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -1.0], N[(x - a), $MachinePrecision], If[LessEqual[z, 3.2e-195], N[(N[(N[(1.0 - y), $MachinePrecision] * z + (-y)), $MachinePrecision] * a + x), $MachinePrecision], If[LessEqual[z, 3.3e-80], N[(N[(y / t), $MachinePrecision] * (-a) + x), $MachinePrecision], If[LessEqual[z, 8.5e+68], N[(z * N[(a / N[(t - -1.0), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], N[(x - a), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if z < -1 or 8.49999999999999966e68 < z

   1. Initial program 91.2%

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

   3. Taylor expanded in z around inf

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

      1. lower--.f64 83.8

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

   5. Applied rewrites 83.8%

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

   if -1 < z < 3.2000000000000001e-195

   1. Initial program 98.7%

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

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

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

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

      6. lower-fma.f64 N/A

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

      7. distribute-neg-frac N/A

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

      8. mul-1-neg N/A

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

      9. lower-/.f64 N/A

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

      10. mul-1-neg N/A

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

      11. lower-neg.f64 N/A

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

      12. lower--.f64 N/A

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

      13. lower--.f64 78.7

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

   5. Applied rewrites 78.7%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 78.2%

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

   if 3.2000000000000001e-195 < z < 3.3e-80

   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 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. *-commutative N/A

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

      5. distribute-rgt-neg-in N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-neg.f64 81.6

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

   5. Applied rewrites 81.6%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 77.3%

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

   if 3.3e-80 < z < 8.49999999999999966e68

   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. 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 78.9

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

   5. Applied rewrites 78.9%

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

   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 73.2%

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

   10. Applied rewrites 73.1%

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

4. Final simplification 79.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1:\\ \;\;\;\;x - a\\ \mathbf{elif}\;z \leq 3.2 \cdot 10^{-195}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(1 - y, z, -y\right), a, x\right)\\ \mathbf{elif}\;z \leq 3.3 \cdot 10^{-80}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{t}, -a, x\right)\\ \mathbf{elif}\;z \leq 8.5 \cdot 10^{+68}:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{a}{t - -1}, x\right)\\ \mathbf{else}:\\ \;\;\;\;x - a\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 92.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.5 \cdot 10^{+161}:\\ \;\;\;\;x - \frac{a}{\frac{-z}{y - z}}\\ \mathbf{elif}\;z \leq 1.45 \cdot 10^{+73}:\\ \;\;\;\;x - \frac{\left(y - z\right) \cdot a}{\left(t - z\right) - -1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z - y}{1 - z}, a, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -2.5e+161)
   (- x (/ a (/ (- z) (- y z))))
   (if (<= z 1.45e+73)
     (- x (/ (* (- y z) a) (- (- t z) -1.0)))
     (fma (/ (- z y) (- 1.0 z)) a x))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -2.5e+161) {
		tmp = x - (a / (-z / (y - z)));
	} else if (z <= 1.45e+73) {
		tmp = x - (((y - z) * a) / ((t - z) - -1.0));
	} else {
		tmp = fma(((z - y) / (1.0 - z)), a, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -2.5e+161)
		tmp = Float64(x - Float64(a / Float64(Float64(-z) / Float64(y - z))));
	elseif (z <= 1.45e+73)
		tmp = Float64(x - Float64(Float64(Float64(y - z) * a) / Float64(Float64(t - z) - -1.0)));
	else
		tmp = fma(Float64(Float64(z - y) / Float64(1.0 - z)), a, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -2.5e+161], N[(x - N[(a / N[((-z) / N[(y - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.45e+73], N[(x - N[(N[(N[(y - z), $MachinePrecision] * a), $MachinePrecision] / N[(N[(t - z), $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(z - y), $MachinePrecision] / N[(1.0 - z), $MachinePrecision]), $MachinePrecision] * a + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -2.4999999999999998e161

   1. Initial program 87.3%

      \[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \]
   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. *-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 100.0

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

      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 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in z around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 98.6

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

   7. Applied rewrites 98.6%

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

   if -2.4999999999999998e161 < z < 1.4500000000000001e73

   1. Initial program 97.6%

      \[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \]
   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 95.0

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

      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 95.0

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

   4. Applied rewrites 95.0%

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

   if 1.4500000000000001e73 < z

   1. Initial program 95.0%

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

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

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

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

      6. lower-fma.f64 N/A

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

      7. distribute-neg-frac N/A

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

      8. mul-1-neg N/A

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

      9. lower-/.f64 N/A

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

      10. mul-1-neg N/A

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

      11. lower-neg.f64 N/A

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

      12. lower--.f64 N/A

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

      13. lower--.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 100.0%

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

4. Final simplification 96.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.5 \cdot 10^{+161}:\\ \;\;\;\;x - \frac{a}{\frac{-z}{y - z}}\\ \mathbf{elif}\;z \leq 1.45 \cdot 10^{+73}:\\ \;\;\;\;x - \frac{\left(y - z\right) \cdot a}{\left(t - z\right) - -1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z - y}{1 - z}, a, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 92.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{z - y}{1 - z}, a, x\right)\\ \mathbf{if}\;z \leq -2.5 \cdot 10^{+161}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1.45 \cdot 10^{+73}:\\ \;\;\;\;x - \frac{\left(y - z\right) \cdot a}{\left(t - z\right) - -1}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (/ (- z y) (- 1.0 z)) a x)))
   (if (<= z -2.5e+161)
     t_1
     (if (<= z 1.45e+73) (- x (/ (* (- y z) a) (- (- t z) -1.0))) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma(((z - y) / (1.0 - z)), a, x);
	double tmp;
	if (z <= -2.5e+161) {
		tmp = t_1;
	} else if (z <= 1.45e+73) {
		tmp = x - (((y - z) * a) / ((t - z) - -1.0));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(Float64(z - y) / Float64(1.0 - z)), a, x)
	tmp = 0.0
	if (z <= -2.5e+161)
		tmp = t_1;
	elseif (z <= 1.45e+73)
		tmp = Float64(x - Float64(Float64(Float64(y - z) * a) / Float64(Float64(t - z) - -1.0)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(N[(z - y), $MachinePrecision] / N[(1.0 - z), $MachinePrecision]), $MachinePrecision] * a + x), $MachinePrecision]}, If[LessEqual[z, -2.5e+161], t$95$1, If[LessEqual[z, 1.45e+73], N[(x - N[(N[(N[(y - z), $MachinePrecision] * a), $MachinePrecision] / N[(N[(t - z), $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -2.4999999999999998e161 or 1.4500000000000001e73 < z

   1. Initial program 91.8%

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

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

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

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

      6. lower-fma.f64 N/A

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

      7. distribute-neg-frac N/A

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

      8. mul-1-neg N/A

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

      9. lower-/.f64 N/A

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

      10. mul-1-neg N/A

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

      11. lower-neg.f64 N/A

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

      12. lower--.f64 N/A

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

      13. lower--.f64 99.4

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

   5. Applied rewrites 99.4%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 99.4%

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

   if -2.4999999999999998e161 < z < 1.4500000000000001e73

   1. Initial program 97.6%

      \[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \]
   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 95.0

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

      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 95.0

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

   4. Applied rewrites 95.0%

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

4. Final simplification 96.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.5 \cdot 10^{+161}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z - y}{1 - z}, a, x\right)\\ \mathbf{elif}\;z \leq 1.45 \cdot 10^{+73}:\\ \;\;\;\;x - \frac{\left(y - z\right) \cdot a}{\left(t - z\right) - -1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z - y}{1 - z}, a, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 74.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1:\\ \;\;\;\;x - a\\ \mathbf{elif}\;z \leq 3.2 \cdot 10^{-195}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(1 - y, z, -y\right), a, x\right)\\ \mathbf{elif}\;z \leq 1700000000000:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{t}, -a, x\right)\\ \mathbf{else}:\\ \;\;\;\;x - a\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -1.0)
   (- x a)
   (if (<= z 3.2e-195)
     (fma (fma (- 1.0 y) z (- y)) a x)
     (if (<= z 1700000000000.0) (fma (/ y t) (- a) x) (- x a)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1.0) {
		tmp = x - a;
	} else if (z <= 3.2e-195) {
		tmp = fma(fma((1.0 - y), z, -y), a, x);
	} else if (z <= 1700000000000.0) {
		tmp = fma((y / t), -a, x);
	} else {
		tmp = x - a;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -1.0)
		tmp = Float64(x - a);
	elseif (z <= 3.2e-195)
		tmp = fma(fma(Float64(1.0 - y), z, Float64(-y)), a, x);
	elseif (z <= 1700000000000.0)
		tmp = fma(Float64(y / t), Float64(-a), x);
	else
		tmp = Float64(x - a);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -1.0], N[(x - a), $MachinePrecision], If[LessEqual[z, 3.2e-195], N[(N[(N[(1.0 - y), $MachinePrecision] * z + (-y)), $MachinePrecision] * a + x), $MachinePrecision], If[LessEqual[z, 1700000000000.0], N[(N[(y / t), $MachinePrecision] * (-a) + x), $MachinePrecision], N[(x - a), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if z < -1 or 1.7e12 < z

   1. Initial program 92.0%

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

   3. Taylor expanded in z around inf

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

      1. lower--.f64 81.1

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

   5. Applied rewrites 81.1%

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

   if -1 < z < 3.2000000000000001e-195

   1. Initial program 98.7%

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

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

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

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

      6. lower-fma.f64 N/A

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

      7. distribute-neg-frac N/A

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

      8. mul-1-neg N/A

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

      9. lower-/.f64 N/A

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

      10. mul-1-neg N/A

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

      11. lower-neg.f64 N/A

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

      12. lower--.f64 N/A

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

      13. lower--.f64 78.7

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

   5. Applied rewrites 78.7%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 78.2%

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

   if 3.2000000000000001e-195 < z < 1.7e12

   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

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

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

      5. distribute-rgt-neg-in N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-neg.f64 69.3

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

   5. Applied rewrites 69.3%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 67.1%

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

Alternative 7: 73.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1:\\ \;\;\;\;x - a\\ \mathbf{elif}\;z \leq 4.9 \cdot 10^{-195}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(1 - y, z, -y\right), a, x\right)\\ \mathbf{elif}\;z \leq 1700000000000:\\ \;\;\;\;x - \frac{y \cdot a}{t}\\ \mathbf{else}:\\ \;\;\;\;x - a\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -1.0)
   (- x a)
   (if (<= z 4.9e-195)
     (fma (fma (- 1.0 y) z (- y)) a x)
     (if (<= z 1700000000000.0) (- x (/ (* y a) t)) (- x a)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1.0) {
		tmp = x - a;
	} else if (z <= 4.9e-195) {
		tmp = fma(fma((1.0 - y), z, -y), a, x);
	} else if (z <= 1700000000000.0) {
		tmp = x - ((y * a) / t);
	} else {
		tmp = x - a;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -1.0)
		tmp = Float64(x - a);
	elseif (z <= 4.9e-195)
		tmp = fma(fma(Float64(1.0 - y), z, Float64(-y)), a, x);
	elseif (z <= 1700000000000.0)
		tmp = Float64(x - Float64(Float64(y * a) / t));
	else
		tmp = Float64(x - a);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -1.0], N[(x - a), $MachinePrecision], If[LessEqual[z, 4.9e-195], N[(N[(N[(1.0 - y), $MachinePrecision] * z + (-y)), $MachinePrecision] * a + x), $MachinePrecision], If[LessEqual[z, 1700000000000.0], N[(x - N[(N[(y * a), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], N[(x - a), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if z < -1 or 1.7e12 < z

   1. Initial program 92.0%

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

   3. Taylor expanded in z around inf

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

      1. lower--.f64 81.1

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

   5. Applied rewrites 81.1%

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

   if -1 < z < 4.9000000000000003e-195

   1. Initial program 98.7%

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

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

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

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

      6. lower-fma.f64 N/A

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

      7. distribute-neg-frac N/A

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

      8. mul-1-neg N/A

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

      9. lower-/.f64 N/A

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

      10. mul-1-neg N/A

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

      11. lower-neg.f64 N/A

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

      12. lower--.f64 N/A

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

      13. lower--.f64 78.7

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

   5. Applied rewrites 78.7%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 78.2%

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

   if 4.9000000000000003e-195 < z < 1.7e12

   1. Initial program 99.9%

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

   3. Taylor expanded in z around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

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

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

      6. lower-fma.f64 N/A

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

      7. distribute-neg-frac2 N/A

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

      8. lower-/.f64 N/A

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

      9. distribute-neg-in N/A

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

      10. metadata-eval N/A

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

      11. unsub-neg N/A

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

      12. lower--.f64 88.6

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

   5. Applied rewrites 88.6%

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

   6. Taylor expanded in t around inf

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

   8. Applied rewrites 62.6%

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

4. Final simplification 77.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1:\\ \;\;\;\;x - a\\ \mathbf{elif}\;z \leq 4.9 \cdot 10^{-195}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(1 - y, z, -y\right), a, x\right)\\ \mathbf{elif}\;z \leq 1700000000000:\\ \;\;\;\;x - \frac{y \cdot a}{t}\\ \mathbf{else}:\\ \;\;\;\;x - a\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 88.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{z - y}{1 - z}, a, x\right)\\ \mathbf{if}\;z \leq -2.15 \cdot 10^{+51}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 340:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{-1 - t}, a, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (/ (- z y) (- 1.0 z)) a x)))
   (if (<= z -2.15e+51) t_1 (if (<= z 340.0) (fma (/ y (- -1.0 t)) a x) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma(((z - y) / (1.0 - z)), a, x);
	double tmp;
	if (z <= -2.15e+51) {
		tmp = t_1;
	} else if (z <= 340.0) {
		tmp = fma((y / (-1.0 - t)), a, x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(Float64(z - y) / Float64(1.0 - z)), a, x)
	tmp = 0.0
	if (z <= -2.15e+51)
		tmp = t_1;
	elseif (z <= 340.0)
		tmp = fma(Float64(y / Float64(-1.0 - t)), a, x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(N[(z - y), $MachinePrecision] / N[(1.0 - z), $MachinePrecision]), $MachinePrecision] * a + x), $MachinePrecision]}, If[LessEqual[z, -2.15e+51], t$95$1, If[LessEqual[z, 340.0], N[(N[(y / N[(-1.0 - t), $MachinePrecision]), $MachinePrecision] * a + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -2.1499999999999999e51 or 340 < z

   1. Initial program 91.5%

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

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

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

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

      6. lower-fma.f64 N/A

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

      7. distribute-neg-frac N/A

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

      8. mul-1-neg N/A

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

      9. lower-/.f64 N/A

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

      10. mul-1-neg N/A

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

      11. lower-neg.f64 N/A

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

      12. lower--.f64 N/A

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

      13. lower--.f64 95.3

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

   5. Applied rewrites 95.3%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 95.3%

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

   if -2.1499999999999999e51 < z < 340

   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

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

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

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

      6. lower-fma.f64 N/A

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

      7. distribute-neg-frac2 N/A

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

      8. lower-/.f64 N/A

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

      9. distribute-neg-in N/A

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

      10. metadata-eval N/A

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

      11. unsub-neg N/A

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

      12. lower--.f64 89.9

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

   5. Applied rewrites 89.9%

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

Alternative 9: 86.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{z}{\left(t - -1\right) - z}, a, x\right)\\ \mathbf{if}\;z \leq -2.5 \cdot 10^{+161}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 4.7 \cdot 10^{-23}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{-1 - t}, a, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (/ z (- (- t -1.0) z)) a x)))
   (if (<= z -2.5e+161)
     t_1
     (if (<= z 4.7e-23) (fma (/ y (- -1.0 t)) a x) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma((z / ((t - -1.0) - z)), a, x);
	double tmp;
	if (z <= -2.5e+161) {
		tmp = t_1;
	} else if (z <= 4.7e-23) {
		tmp = fma((y / (-1.0 - t)), a, x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(z / Float64(Float64(t - -1.0) - z)), a, x)
	tmp = 0.0
	if (z <= -2.5e+161)
		tmp = t_1;
	elseif (z <= 4.7e-23)
		tmp = fma(Float64(y / Float64(-1.0 - t)), a, x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(z / N[(N[(t - -1.0), $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision] * a + x), $MachinePrecision]}, If[LessEqual[z, -2.5e+161], t$95$1, If[LessEqual[z, 4.7e-23], N[(N[(y / N[(-1.0 - t), $MachinePrecision]), $MachinePrecision] * a + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -2.4999999999999998e161 or 4.7000000000000001e-23 < z

   1. Initial program 93.1%

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

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

   5. Applied rewrites 89.9%

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

   if -2.4999999999999998e161 < z < 4.7000000000000001e-23

   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 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

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

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

      6. lower-fma.f64 N/A

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

      7. distribute-neg-frac2 N/A

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

      8. lower-/.f64 N/A

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

      9. distribute-neg-in N/A

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

      10. metadata-eval N/A

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

      11. unsub-neg N/A

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

      12. lower--.f64 89.2

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

   5. Applied rewrites 89.2%

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

4. Final simplification 89.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.5 \cdot 10^{+161}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{\left(t - -1\right) - z}, a, x\right)\\ \mathbf{elif}\;z \leq 4.7 \cdot 10^{-23}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{-1 - t}, a, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{\left(t - -1\right) - z}, a, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 83.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.5 \cdot 10^{+161}:\\ \;\;\;\;x - a\\ \mathbf{elif}\;z \leq 8.5 \cdot 10^{+69}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{-1 - t}, a, x\right)\\ \mathbf{else}:\\ \;\;\;\;x - a\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -2.5e+161)
   (- x a)
   (if (<= z 8.5e+69) (fma (/ y (- -1.0 t)) a x) (- x a))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -2.5e+161) {
		tmp = x - a;
	} else if (z <= 8.5e+69) {
		tmp = fma((y / (-1.0 - t)), a, x);
	} else {
		tmp = x - a;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -2.5e+161)
		tmp = Float64(x - a);
	elseif (z <= 8.5e+69)
		tmp = fma(Float64(y / Float64(-1.0 - t)), a, x);
	else
		tmp = Float64(x - a);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -2.5e+161], N[(x - a), $MachinePrecision], If[LessEqual[z, 8.5e+69], N[(N[(y / N[(-1.0 - t), $MachinePrecision]), $MachinePrecision] * a + x), $MachinePrecision], N[(x - a), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -2.4999999999999998e161 or 8.5000000000000002e69 < z

   1. Initial program 91.8%

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

   3. Taylor expanded in z around inf

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

      1. lower--.f64 90.9

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

   5. Applied rewrites 90.9%

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

   if -2.4999999999999998e161 < z < 8.5000000000000002e69

   1. Initial program 97.6%

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

   3. Taylor expanded in z around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

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

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

      6. lower-fma.f64 N/A

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

      7. distribute-neg-frac2 N/A

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

      8. lower-/.f64 N/A

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

      9. distribute-neg-in N/A

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

      10. metadata-eval N/A

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

      11. unsub-neg N/A

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

      12. lower--.f64 86.3

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

   5. Applied rewrites 86.3%

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

Alternative 11: 74.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1:\\ \;\;\;\;x - a\\ \mathbf{elif}\;z \leq 32:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(1 - y, z, -y\right), a, x\right)\\ \mathbf{else}:\\ \;\;\;\;x - a\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -1.0)
   (- x a)
   (if (<= z 32.0) (fma (fma (- 1.0 y) z (- y)) a x) (- x a))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1.0) {
		tmp = x - a;
	} else if (z <= 32.0) {
		tmp = fma(fma((1.0 - y), z, -y), a, x);
	} else {
		tmp = x - a;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -1.0)
		tmp = Float64(x - a);
	elseif (z <= 32.0)
		tmp = fma(fma(Float64(1.0 - y), z, Float64(-y)), a, x);
	else
		tmp = Float64(x - a);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -1.0], N[(x - a), $MachinePrecision], If[LessEqual[z, 32.0], N[(N[(N[(1.0 - y), $MachinePrecision] * z + (-y)), $MachinePrecision] * a + x), $MachinePrecision], N[(x - a), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -1 or 32 < z

   1. Initial program 92.1%

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

   3. Taylor expanded in z around inf

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

      1. lower--.f64 80.6

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

   5. Applied rewrites 80.6%

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

   if -1 < z < 32

   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. *-commutative N/A

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

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

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

      6. lower-fma.f64 N/A

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

      7. distribute-neg-frac N/A

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

      8. mul-1-neg N/A

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

      9. lower-/.f64 N/A

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

      10. mul-1-neg N/A

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

      11. lower-neg.f64 N/A

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

      12. lower--.f64 N/A

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

      13. lower--.f64 70.3

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

   5. Applied rewrites 70.3%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 70.0%

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

Alternative 12: 72.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -6 \cdot 10^{+20}:\\ \;\;\;\;x - a\\ \mathbf{elif}\;z \leq 2.4 \cdot 10^{+48}:\\ \;\;\;\;\mathsf{fma}\left(-y, a, x\right)\\ \mathbf{else}:\\ \;\;\;\;x - a\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -6e+20) (- x a) (if (<= z 2.4e+48) (fma (- y) a x) (- x a))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -6e+20) {
		tmp = x - a;
	} else if (z <= 2.4e+48) {
		tmp = fma(-y, a, x);
	} else {
		tmp = x - a;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -6e+20)
		tmp = Float64(x - a);
	elseif (z <= 2.4e+48)
		tmp = fma(Float64(-y), a, x);
	else
		tmp = Float64(x - a);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -6e+20], N[(x - a), $MachinePrecision], If[LessEqual[z, 2.4e+48], N[((-y) * a + x), $MachinePrecision], N[(x - a), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -6e20 or 2.4000000000000001e48 < z

   1. Initial program 91.1%

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

   3. Taylor expanded in z around inf

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

      1. lower--.f64 83.3

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

   5. Applied rewrites 83.3%

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

   if -6e20 < z < 2.4000000000000001e48

   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. *-commutative N/A

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

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

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

      6. lower-fma.f64 N/A

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

      7. distribute-neg-frac N/A

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

      8. mul-1-neg N/A

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

      9. lower-/.f64 N/A

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

      10. mul-1-neg N/A

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

      11. lower-neg.f64 N/A

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

      12. lower--.f64 N/A

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

      13. lower--.f64 71.8

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

   5. Applied rewrites 71.8%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 67.2%

      \[\leadsto \mathsf{fma}\left(-y, a, x\right) \]
3. Recombined 2 regimes into one program.
4. 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 95.7%

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

3. Taylor expanded in z around inf

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

   1. lower--.f64 56.1

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

5. Applied rewrites 56.1%

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

Alternative 14: 16.8% 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 95.7%

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

3. Taylor expanded in z around inf

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

   1. lower--.f64 56.1

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

5. Applied rewrites 56.1%

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

6. Taylor expanded in a around inf

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

8. Applied rewrites 16.4%

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

Developer Target 1: 99.5% 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 2024276 
(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))))