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

Percentage Accurate: 97.0% → 99.6%

Time: 9.3s

Alternatives: 12

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: 97.0% 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, 0.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.4%

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

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

   4. associate-/r* N/A

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

   5. frac-2neg N/A

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

   6. distribute-frac-neg2 N/A

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

   7. lower-/.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. lift-+.f64 N/A

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

   10. +-commutative N/A

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

   11. distribute-neg-in N/A

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

   12. metadata-eval N/A

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

   13. unsub-neg N/A

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

   14. lower--.f64 N/A

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

   15. distribute-neg-frac N/A

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

   16. metadata-eval N/A

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

   17. lower-/.f64 99.9

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

4. Applied rewrites 99.9%

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

5. Final simplification 99.9%

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

Alternative 2: 60.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{z - y}{\frac{-1 - \left(t - z\right)}{a}} \leq -5 \cdot 10^{+255}:\\ \;\;\;\;\left(-a\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;x - a\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= (/ (- z y) (/ (- -1.0 (- t z)) a)) -5e+255) (* (- a) y) (- x a)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (((z - y) / ((-1.0 - (t - z)) / a)) <= -5e+255) {
		tmp = -a * y;
	} 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) :: tmp
    if (((z - y) / (((-1.0d0) - (t - z)) / a)) <= (-5d+255)) then
        tmp = -a * y
    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 tmp;
	if (((z - y) / ((-1.0 - (t - z)) / a)) <= -5e+255) {
		tmp = -a * y;
	} else {
		tmp = x - a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if ((z - y) / ((-1.0 - (t - z)) / a)) <= -5e+255:
		tmp = -a * y
	else:
		tmp = x - a
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (Float64(Float64(z - y) / Float64(Float64(-1.0 - Float64(t - z)) / a)) <= -5e+255)
		tmp = Float64(Float64(-a) * y);
	else
		tmp = Float64(x - a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (((z - y) / ((-1.0 - (t - z)) / a)) <= -5e+255)
		tmp = -a * y;
	else
		tmp = x - a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[N[(N[(z - y), $MachinePrecision] / N[(N[(-1.0 - N[(t - z), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], -5e+255], N[((-a) * y), $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)) < -5.0000000000000002e255

   1. Initial program 99.8%

      \[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \]
   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 72.2

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

   5. Applied rewrites 72.2%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 72.0%

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

   9. Taylor expanded in t around 0

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

   11. Applied rewrites 55.0%

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

   if -5.0000000000000002e255 < (/.f64 (-.f64 y z) (/.f64 (+.f64 (-.f64 t z) #s(literal 1 binary64)) a))

   1. Initial program 98.3%

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

   3. Taylor expanded in z around inf

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

      1. lower--.f64 66.7

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

   5. Applied rewrites 66.7%

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

4. Final simplification 65.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z - y}{\frac{-1 - \left(t - z\right)}{a}} \leq -5 \cdot 10^{+255}:\\ \;\;\;\;\left(-a\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;x - a\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 89.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.36 \cdot 10^{+82}:\\ \;\;\;\;\mathsf{fma}\left(y - z, \frac{-a}{t}, x\right)\\ \mathbf{elif}\;t \leq 5.4 \cdot 10^{+53}:\\ \;\;\;\;x - \frac{a}{1 - z} \cdot \left(y - z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{-1 - t}, a, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t -1.36e+82)
   (fma (- y z) (/ (- a) t) x)
   (if (<= t 5.4e+53)
     (- x (* (/ a (- 1.0 z)) (- y z)))
     (fma (/ y (- -1.0 t)) a x))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= -1.36e+82) {
		tmp = fma((y - z), (-a / t), x);
	} else if (t <= 5.4e+53) {
		tmp = x - ((a / (1.0 - z)) * (y - z));
	} else {
		tmp = fma((y / (-1.0 - t)), a, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= -1.36e+82)
		tmp = fma(Float64(y - z), Float64(Float64(-a) / t), x);
	elseif (t <= 5.4e+53)
		tmp = Float64(x - Float64(Float64(a / Float64(1.0 - z)) * Float64(y - z)));
	else
		tmp = fma(Float64(y / Float64(-1.0 - t)), a, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, -1.36e+82], N[(N[(y - z), $MachinePrecision] * N[((-a) / t), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t, 5.4e+53], N[(x - N[(N[(a / N[(1.0 - z), $MachinePrecision]), $MachinePrecision] * N[(y - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y / N[(-1.0 - t), $MachinePrecision]), $MachinePrecision] * a + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -1.36000000000000001e82

   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 93.9

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

   5. Applied rewrites 93.9%

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

   7. Applied rewrites 93.9%

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

   if -1.36000000000000001e82 < t < 5.40000000000000039e53

   1. Initial program 98.1%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 95.7

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

   5. Applied rewrites 95.7%

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

   if 5.40000000000000039e53 < t

   1. Initial program 98.2%

      \[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \]
   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 93.1

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

   5. Applied rewrites 93.1%

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

4. Final simplification 94.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.36 \cdot 10^{+82}:\\ \;\;\;\;\mathsf{fma}\left(y - z, \frac{-a}{t}, x\right)\\ \mathbf{elif}\;t \leq 5.4 \cdot 10^{+53}:\\ \;\;\;\;x - \frac{a}{1 - z} \cdot \left(y - z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{-1 - t}, a, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 87.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3.1 \cdot 10^{+49}:\\ \;\;\;\;x - \frac{-a}{z} \cdot \left(y - z\right)\\ \mathbf{elif}\;z \leq 4.5 \cdot 10^{+43}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{-1 - t}, a, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{\left(t - -1\right) - z}, a, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -3.1e+49)
   (- x (* (/ (- a) z) (- y z)))
   (if (<= z 4.5e+43)
     (fma (/ y (- -1.0 t)) a x)
     (fma (/ z (- (- t -1.0) z)) a x))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -3.1e+49) {
		tmp = x - ((-a / z) * (y - z));
	} else if (z <= 4.5e+43) {
		tmp = fma((y / (-1.0 - t)), a, x);
	} else {
		tmp = fma((z / ((t - -1.0) - z)), a, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -3.1e+49)
		tmp = Float64(x - Float64(Float64(Float64(-a) / z) * Float64(y - z)));
	elseif (z <= 4.5e+43)
		tmp = fma(Float64(y / Float64(-1.0 - t)), a, x);
	else
		tmp = fma(Float64(z / Float64(Float64(t - -1.0) - z)), a, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -3.1e+49], N[(x - N[(N[((-a) / z), $MachinePrecision] * N[(y - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 4.5e+43], N[(N[(y / N[(-1.0 - t), $MachinePrecision]), $MachinePrecision] * a + x), $MachinePrecision], N[(N[(z / N[(N[(t - -1.0), $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision] * a + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -3.09999999999999992e49

   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 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 96.3

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

   5. Applied rewrites 96.3%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 96.3%

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

   if -3.09999999999999992e49 < z < 4.5e43

   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

      \[\leadsto \color{blue}{x - \frac{a \cdot y}{1 + t}} \]
   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 95.2

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

   5. Applied rewrites 95.2%

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

   if 4.5e43 < z

   1. Initial program 94.9%

      \[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \]
   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 91.1

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

   5. Applied rewrites 91.1%

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

4. Final simplification 94.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.1 \cdot 10^{+49}:\\ \;\;\;\;x - \frac{-a}{z} \cdot \left(y - z\right)\\ \mathbf{elif}\;z \leq 4.5 \cdot 10^{+43}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{-1 - t}, a, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{\left(t - -1\right) - z}, a, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 87.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{z}{\left(t - -1\right) - z}, a, x\right)\\ \mathbf{if}\;z \leq -1.55 \cdot 10^{+18}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 4.5 \cdot 10^{+43}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{-1 - t}, a, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (/ z (- (- t -1.0) z)) a x)))
   (if (<= z -1.55e+18)
     t_1
     (if (<= z 4.5e+43) (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 <= -1.55e+18) {
		tmp = t_1;
	} else if (z <= 4.5e+43) {
		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 <= -1.55e+18)
		tmp = t_1;
	elseif (z <= 4.5e+43)
		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, -1.55e+18], t$95$1, If[LessEqual[z, 4.5e+43], 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 < -1.55e18 or 4.5e43 < z

   1. Initial program 97.6%

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

   3. Taylor expanded in y around 0

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

      1. 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 91.0

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

   5. Applied rewrites 91.0%

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

   if -1.55e18 < z < 4.5e43

   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 96.4

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

   5. Applied rewrites 96.4%

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

4. Final simplification 93.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.55 \cdot 10^{+18}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{\left(t - -1\right) - z}, a, x\right)\\ \mathbf{elif}\;z \leq 4.5 \cdot 10^{+43}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{-1 - t}, a, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{\left(t - -1\right) - z}, a, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 83.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3.3 \cdot 10^{+18}:\\ \;\;\;\;x - a\\ \mathbf{elif}\;z \leq 1.65 \cdot 10^{+83}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{-1 - t}, a, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{1 - z}, a, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -3.3e+18)
   (- x a)
   (if (<= z 1.65e+83) (fma (/ y (- -1.0 t)) a x) (fma (/ z (- 1.0 z)) a x))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -3.3e+18) {
		tmp = x - a;
	} else if (z <= 1.65e+83) {
		tmp = fma((y / (-1.0 - t)), a, x);
	} else {
		tmp = fma((z / (1.0 - z)), a, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -3.3e+18)
		tmp = Float64(x - a);
	elseif (z <= 1.65e+83)
		tmp = fma(Float64(y / Float64(-1.0 - t)), a, x);
	else
		tmp = fma(Float64(z / Float64(1.0 - z)), a, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -3.3e+18], N[(x - a), $MachinePrecision], If[LessEqual[z, 1.65e+83], N[(N[(y / N[(-1.0 - t), $MachinePrecision]), $MachinePrecision] * a + x), $MachinePrecision], N[(N[(z / N[(1.0 - z), $MachinePrecision]), $MachinePrecision] * a + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -3.3e18

   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 inf

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

      1. lower--.f64 84.7

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

   5. Applied rewrites 84.7%

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

   if -3.3e18 < z < 1.64999999999999992e83

   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

      \[\leadsto \color{blue}{x - \frac{a \cdot y}{1 + t}} \]
   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 94.7

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

   5. Applied rewrites 94.7%

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

   if 1.64999999999999992e83 < z

   1. Initial program 93.4%

      \[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \]
   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 93.3

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

   5. Applied rewrites 93.3%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 86.8%

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

Alternative 7: 83.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3.3 \cdot 10^{+18}:\\ \;\;\;\;x - a\\ \mathbf{elif}\;z \leq 1.65 \cdot 10^{+83}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{-1 - t}, a, x\right)\\ \mathbf{else}:\\ \;\;\;\;x - a\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -3.3e+18)
   (- x a)
   (if (<= z 1.65e+83) (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 <= -3.3e+18) {
		tmp = x - a;
	} else if (z <= 1.65e+83) {
		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 <= -3.3e+18)
		tmp = Float64(x - a);
	elseif (z <= 1.65e+83)
		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, -3.3e+18], N[(x - a), $MachinePrecision], If[LessEqual[z, 1.65e+83], 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 < -3.3e18 or 1.64999999999999992e83 < z

   1. Initial program 97.3%

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

   3. Taylor expanded in z around inf

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

      1. lower--.f64 85.6

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

   5. Applied rewrites 85.6%

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

   if -3.3e18 < z < 1.64999999999999992e83

   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

      \[\leadsto \color{blue}{x - \frac{a \cdot y}{1 + t}} \]
   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 94.7

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

   5. Applied rewrites 94.7%

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

Alternative 8: 71.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{y}{t}, -a, x\right)\\ \mathbf{if}\;t \leq -1:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 4.5 \cdot 10^{+16}:\\ \;\;\;\;\mathsf{fma}\left(y, \mathsf{fma}\left(a, t, -a\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (/ y t) (- a) x)))
   (if (<= t -1.0) t_1 (if (<= t 4.5e+16) (fma y (fma a t (- a)) x) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma((y / t), -a, x);
	double tmp;
	if (t <= -1.0) {
		tmp = t_1;
	} else if (t <= 4.5e+16) {
		tmp = fma(y, fma(a, t, -a), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(y / t), Float64(-a), x)
	tmp = 0.0
	if (t <= -1.0)
		tmp = t_1;
	elseif (t <= 4.5e+16)
		tmp = fma(y, fma(a, t, Float64(-a)), x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(y / t), $MachinePrecision] * (-a) + x), $MachinePrecision]}, If[LessEqual[t, -1.0], t$95$1, If[LessEqual[t, 4.5e+16], N[(y * N[(a * t + (-a)), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -1 or 4.5e16 < t

   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 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 86.6

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

   5. Applied rewrites 86.6%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 85.6%

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

   if -1 < t < 4.5e16

   1. Initial program 97.7%

      \[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \]
   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 75.8

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

   5. Applied rewrites 75.8%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 75.9%

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

Alternative 9: 97.3% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 98.4%

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

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

   6. associate-/r/ N/A

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

   7. lift-/.f64 N/A

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

   8. clear-num N/A

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

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

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

   10. clear-num N/A

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

   11. lift-/.f64 N/A

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

   12. distribute-frac-neg2 N/A

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

   13. lower-fma.f64 N/A

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

4. Applied rewrites 98.5%

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

Alternative 10: 72.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.06 \cdot 10^{+18}:\\ \;\;\;\;x - a\\ \mathbf{elif}\;z \leq 9 \cdot 10^{+29}:\\ \;\;\;\;\mathsf{fma}\left(-y, a, x\right)\\ \mathbf{else}:\\ \;\;\;\;x - a\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -1.06e+18) (- x a) (if (<= z 9e+29) (fma (- y) a x) (- x a))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1.06e+18) {
		tmp = x - a;
	} else if (z <= 9e+29) {
		tmp = fma(-y, a, x);
	} else {
		tmp = x - a;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -1.06e+18)
		tmp = Float64(x - a);
	elseif (z <= 9e+29)
		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, -1.06e+18], N[(x - a), $MachinePrecision], If[LessEqual[z, 9e+29], N[((-y) * a + x), $MachinePrecision], N[(x - a), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.06e18 or 9.0000000000000005e29 < z

   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 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 -1.06e18 < z < 9.0000000000000005e29

   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 96.3

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

   5. Applied rewrites 96.3%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 71.8%

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

Alternative 11: 59.8% accurate, 8.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.4%

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

3. Taylor expanded in z around inf

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

   1. lower--.f64 63.3

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

5. Applied rewrites 63.3%

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

Alternative 12: 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 98.4%

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

3. Taylor expanded in z around inf

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

   1. lower--.f64 63.3

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

5. Applied rewrites 63.3%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 14.3%

   \[\leadsto -a \]
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 2024296 
(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))))