Graphics.Rendering.Plot.Render.Plot.Axis:renderAxisTicks from plot-0.2.3.4, B

Percentage Accurate: 85.3% → 98.1%

Time: 6.9s

Alternatives: 9

Speedup: 1.1×

Specification

Math

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

FPCore

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

C

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

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)) / (a - t))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 85.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

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)) / (a - t))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.1% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 84.8%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-/.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. associate-/l* N/A

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

   6. *-commutative N/A

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

   7. lower-fma.f64 N/A

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

   8. lower-/.f64 97.6

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

4. Applied rewrites 97.6%

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

Alternative 2: 79.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -8.6 \cdot 10^{+166}:\\ \;\;\;\;y + x\\ \mathbf{elif}\;t \leq -2.6 \cdot 10^{+19}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-z}{t}, y, x\right)\\ \mathbf{elif}\;t \leq 7.2 \cdot 10^{-10}:\\ \;\;\;\;\mathsf{fma}\left(z - t, \frac{y}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;y + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t -8.6e+166)
   (+ y x)
   (if (<= t -2.6e+19)
     (fma (/ (- z) t) y x)
     (if (<= t 7.2e-10) (fma (- z t) (/ y a) x) (+ y x)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= -8.6e+166) {
		tmp = y + x;
	} else if (t <= -2.6e+19) {
		tmp = fma((-z / t), y, x);
	} else if (t <= 7.2e-10) {
		tmp = fma((z - t), (y / a), x);
	} else {
		tmp = y + x;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= -8.6e+166)
		tmp = Float64(y + x);
	elseif (t <= -2.6e+19)
		tmp = fma(Float64(Float64(-z) / t), y, x);
	elseif (t <= 7.2e-10)
		tmp = fma(Float64(z - t), Float64(y / a), x);
	else
		tmp = Float64(y + x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, -8.6e+166], N[(y + x), $MachinePrecision], If[LessEqual[t, -2.6e+19], N[(N[((-z) / t), $MachinePrecision] * y + x), $MachinePrecision], If[LessEqual[t, 7.2e-10], N[(N[(z - t), $MachinePrecision] * N[(y / a), $MachinePrecision] + x), $MachinePrecision], N[(y + x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if t < -8.5999999999999999e166 or 7.2e-10 < t

   1. Initial program 69.2%

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

   3. Taylor expanded in t around inf

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{y + x} \]

      2. lower-+.f64 85.8

         \[\leadsto \color{blue}{y + x} \]

   5. Applied rewrites 85.8%

      \[\leadsto \color{blue}{y + x} \]

   if -8.5999999999999999e166 < t < -2.6e19

   1. Initial program 80.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 86.1

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

   7. Applied rewrites 86.1%

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

   8. Taylor expanded in t around inf

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

   10. Applied rewrites 75.7%

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

   if -2.6e19 < t < 7.2e-10

   1. Initial program 96.2%

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

   3. Taylor expanded in a around inf

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 81.1

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

   5. Applied rewrites 81.1%

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

Alternative 3: 77.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -8.6 \cdot 10^{+166}:\\ \;\;\;\;y + x\\ \mathbf{elif}\;t \leq -5.9 \cdot 10^{-40}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-z}{t}, y, x\right)\\ \mathbf{elif}\;t \leq 3.4 \cdot 10^{-11}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{a}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;y + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t -8.6e+166)
   (+ y x)
   (if (<= t -5.9e-40)
     (fma (/ (- z) t) y x)
     (if (<= t 3.4e-11) (fma (/ z a) y x) (+ y x)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= -8.6e+166) {
		tmp = y + x;
	} else if (t <= -5.9e-40) {
		tmp = fma((-z / t), y, x);
	} else if (t <= 3.4e-11) {
		tmp = fma((z / a), y, x);
	} else {
		tmp = y + x;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= -8.6e+166)
		tmp = Float64(y + x);
	elseif (t <= -5.9e-40)
		tmp = fma(Float64(Float64(-z) / t), y, x);
	elseif (t <= 3.4e-11)
		tmp = fma(Float64(z / a), y, x);
	else
		tmp = Float64(y + x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, -8.6e+166], N[(y + x), $MachinePrecision], If[LessEqual[t, -5.9e-40], N[(N[((-z) / t), $MachinePrecision] * y + x), $MachinePrecision], If[LessEqual[t, 3.4e-11], N[(N[(z / a), $MachinePrecision] * y + x), $MachinePrecision], N[(y + x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if t < -8.5999999999999999e166 or 3.3999999999999999e-11 < t

   1. Initial program 69.2%

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

   3. Taylor expanded in t around inf

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{y + x} \]

      2. lower-+.f64 85.8

         \[\leadsto \color{blue}{y + x} \]

   5. Applied rewrites 85.8%

      \[\leadsto \color{blue}{y + x} \]

   if -8.5999999999999999e166 < t < -5.89999999999999966e-40

   1. Initial program 84.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 98.1

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

   4. Applied rewrites 98.1%

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

   5. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 80.5

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

   7. Applied rewrites 80.5%

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

   8. Taylor expanded in t around inf

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

   10. Applied rewrites 67.9%

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

   if -5.89999999999999966e-40 < t < 3.3999999999999999e-11

   1. Initial program 96.6%

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

   3. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 80.3

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

   5. Applied rewrites 80.3%

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

Alternative 4: 87.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -4.5 \cdot 10^{-49} \lor \neg \left(z \leq 1.3 \cdot 10^{-47}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{a - t}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;x - y \cdot \frac{t}{a - t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= z -4.5e-49) (not (<= z 1.3e-47)))
   (fma (/ z (- a t)) y x)
   (- x (* y (/ t (- a t))))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z <= -4.5e-49) || !(z <= 1.3e-47)) {
		tmp = fma((z / (a - t)), y, x);
	} else {
		tmp = x - (y * (t / (a - t)));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((z <= -4.5e-49) || !(z <= 1.3e-47))
		tmp = fma(Float64(z / Float64(a - t)), y, x);
	else
		tmp = Float64(x - Float64(y * Float64(t / Float64(a - t))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[z, -4.5e-49], N[Not[LessEqual[z, 1.3e-47]], $MachinePrecision]], N[(N[(z / N[(a - t), $MachinePrecision]), $MachinePrecision] * y + x), $MachinePrecision], N[(x - N[(y * N[(t / N[(a - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -4.5000000000000002e-49 or 1.3e-47 < z

   1. Initial program 85.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 97.5

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

   4. Applied rewrites 97.5%

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

   5. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 89.1

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

   7. Applied rewrites 89.1%

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

   if -4.5000000000000002e-49 < z < 1.3e-47

   1. Initial program 84.2%

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

   3. Taylor expanded in z around 0

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

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

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. lower--.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower--.f64 91.3

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

   5. Applied rewrites 91.3%

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

4. Final simplification 89.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4.5 \cdot 10^{-49} \lor \neg \left(z \leq 1.3 \cdot 10^{-47}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{a - t}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;x - y \cdot \frac{t}{a - t}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 84.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -8.6 \cdot 10^{+166} \lor \neg \left(t \leq 7.5 \cdot 10^{+41}\right):\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{a - t}, y, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= t -8.6e+166) (not (<= t 7.5e+41)))
   (+ y x)
   (fma (/ z (- a t)) y x)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((t <= -8.6e+166) || !(t <= 7.5e+41)) {
		tmp = y + x;
	} else {
		tmp = fma((z / (a - t)), y, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((t <= -8.6e+166) || !(t <= 7.5e+41))
		tmp = Float64(y + x);
	else
		tmp = fma(Float64(z / Float64(a - t)), y, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[t, -8.6e+166], N[Not[LessEqual[t, 7.5e+41]], $MachinePrecision]], N[(y + x), $MachinePrecision], N[(N[(z / N[(a - t), $MachinePrecision]), $MachinePrecision] * y + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -8.5999999999999999e166 or 7.50000000000000072e41 < t

   1. Initial program 66.1%

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

   3. Taylor expanded in t around inf

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{y + x} \]

      2. lower-+.f64 86.9

         \[\leadsto \color{blue}{y + x} \]

   5. Applied rewrites 86.9%

      \[\leadsto \color{blue}{y + x} \]

   if -8.5999999999999999e166 < t < 7.50000000000000072e41

   1. Initial program 93.0%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 96.6

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

   4. Applied rewrites 96.6%

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

   5. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 87.5

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

   7. Applied rewrites 87.5%

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

4. Final simplification 87.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -8.6 \cdot 10^{+166} \lor \neg \left(t \leq 7.5 \cdot 10^{+41}\right):\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{a - t}, y, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 77.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -5.2 \cdot 10^{+31} \lor \neg \left(t \leq 3.4 \cdot 10^{-11}\right):\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{a} \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= t -5.2e+31) (not (<= t 3.4e-11))) (+ y x) (+ x (* (/ y a) z))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((t <= -5.2e+31) || !(t <= 3.4e-11)) {
		tmp = y + x;
	} else {
		tmp = x + ((y / a) * z);
	}
	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 ((t <= (-5.2d+31)) .or. (.not. (t <= 3.4d-11))) then
        tmp = y + x
    else
        tmp = x + ((y / a) * z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((t <= -5.2e+31) || !(t <= 3.4e-11)) {
		tmp = y + x;
	} else {
		tmp = x + ((y / a) * z);
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (t <= -5.2e+31) or not (t <= 3.4e-11):
		tmp = y + x
	else:
		tmp = x + ((y / a) * z)
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((t <= -5.2e+31) || !(t <= 3.4e-11))
		tmp = Float64(y + x);
	else
		tmp = Float64(x + Float64(Float64(y / a) * z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((t <= -5.2e+31) || ~((t <= 3.4e-11)))
		tmp = y + x;
	else
		tmp = x + ((y / a) * z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[t, -5.2e+31], N[Not[LessEqual[t, 3.4e-11]], $MachinePrecision]], N[(y + x), $MachinePrecision], N[(x + N[(N[(y / a), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -5.2e31 or 3.3999999999999999e-11 < t

   1. Initial program 72.0%

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

   3. Taylor expanded in t around inf

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{y + x} \]

      2. lower-+.f64 78.7

         \[\leadsto \color{blue}{y + x} \]

   5. Applied rewrites 78.7%

      \[\leadsto \color{blue}{y + x} \]

   if -5.2e31 < t < 3.3999999999999999e-11

   1. Initial program 95.7%

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

   3. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 73.7

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

   5. Applied rewrites 73.7%

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

   7. Applied rewrites 75.0%

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

4. Final simplification 76.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -5.2 \cdot 10^{+31} \lor \neg \left(t \leq 3.4 \cdot 10^{-11}\right):\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{a} \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 77.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -5.8 \cdot 10^{+31} \lor \neg \left(t \leq 3.4 \cdot 10^{-11}\right):\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{a}, y, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= t -5.8e+31) (not (<= t 3.4e-11))) (+ y x) (fma (/ z a) y x)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((t <= -5.8e+31) || !(t <= 3.4e-11)) {
		tmp = y + x;
	} else {
		tmp = fma((z / a), y, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((t <= -5.8e+31) || !(t <= 3.4e-11))
		tmp = Float64(y + x);
	else
		tmp = fma(Float64(z / a), y, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[t, -5.8e+31], N[Not[LessEqual[t, 3.4e-11]], $MachinePrecision]], N[(y + x), $MachinePrecision], N[(N[(z / a), $MachinePrecision] * y + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -5.8000000000000001e31 or 3.3999999999999999e-11 < t

   1. Initial program 72.0%

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

   3. Taylor expanded in t around inf

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{y + x} \]

      2. lower-+.f64 78.7

         \[\leadsto \color{blue}{y + x} \]

   5. Applied rewrites 78.7%

      \[\leadsto \color{blue}{y + x} \]

   if -5.8000000000000001e31 < t < 3.3999999999999999e-11

   1. Initial program 95.7%

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

   3. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 75.0

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

   5. Applied rewrites 75.0%

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

4. Final simplification 76.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -5.8 \cdot 10^{+31} \lor \neg \left(t \leq 3.4 \cdot 10^{-11}\right):\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{a}, y, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 61.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 1.22 \cdot 10^{+178}:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;\left(-x\right) \cdot -1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= a 1.22e+178) (+ y x) (* (- x) -1.0)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (a <= 1.22e+178) {
		tmp = y + x;
	} else {
		tmp = -x * -1.0;
	}
	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 (a <= 1.22d+178) then
        tmp = y + x
    else
        tmp = -x * (-1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (a <= 1.22e+178) {
		tmp = y + x;
	} else {
		tmp = -x * -1.0;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if a <= 1.22e+178:
		tmp = y + x
	else:
		tmp = -x * -1.0
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (a <= 1.22e+178)
		tmp = Float64(y + x);
	else
		tmp = Float64(Float64(-x) * -1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (a <= 1.22e+178)
		tmp = y + x;
	else
		tmp = -x * -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[a, 1.22e+178], N[(y + x), $MachinePrecision], N[((-x) * -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 1.2199999999999999e178

   1. Initial program 84.6%

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

   3. Taylor expanded in t around inf

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{y + x} \]

      2. lower-+.f64 61.5

         \[\leadsto \color{blue}{y + x} \]

   5. Applied rewrites 61.5%

      \[\leadsto \color{blue}{y + x} \]

   if 1.2199999999999999e178 < a

   1. Initial program 86.6%

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

   3. Taylor expanded in x around -inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower--.f64 N/A

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

   5. Applied rewrites 95.4%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 86.5%

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

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 82.3%

       \[\leadsto \left(-x\right) \cdot -1 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 60.4% accurate, 6.5× speedup

Math

\[\begin{array}{l} \\ y + x \end{array} \]

FPCore

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

C

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

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 = y + x
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 84.8%

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

3. Taylor expanded in t around inf

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

   1. +-commutative N/A

      \[\leadsto \color{blue}{y + x} \]

   2. lower-+.f64 61.7

      \[\leadsto \color{blue}{y + x} \]

5. Applied rewrites 61.7%

   \[\leadsto \color{blue}{y + x} \]
6. Add Preprocessing

Developer Target 1: 98.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ x + \frac{y}{\frac{a - t}{z - t}} \end{array} \]

FPCore

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

C

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

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 / ((a - t) / (z - t)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024329 
(FPCore (x y z t a)
  :name "Graphics.Rendering.Plot.Render.Plot.Axis:renderAxisTicks from plot-0.2.3.4, B"
  :precision binary64

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

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