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

Percentage Accurate: 98.2% → 98.2%

Time: 6.2s

Alternatives: 9

Speedup: 1.1×

Specification

Math

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

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(y * Float64(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[(y * N[(N[(z - t), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 98.2% accurate, 1.0× speedup

Math

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

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(y * Float64(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[(y * N[(N[(z - t), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 98.2% 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 99.1%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 99.1

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

4. Applied rewrites 99.1%

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

Alternative 2: 97.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - t}{a - t}\\ t_2 := \mathsf{fma}\left(\frac{z}{a - t}, y, x\right)\\ \mathbf{if}\;t\_1 \leq -1000000:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 0.0002:\\ \;\;\;\;\mathsf{fma}\left(\frac{z - t}{a}, y, x\right)\\ \mathbf{elif}\;t\_1 \leq 2:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- z t) (- a t))) (t_2 (fma (/ z (- a t)) y x)))
   (if (<= t_1 -1000000.0)
     t_2
     (if (<= t_1 0.0002)
       (fma (/ (- z t) a) y x)
       (if (<= t_1 2.0) (+ y x) t_2)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (z - t) / (a - t);
	double t_2 = fma((z / (a - t)), y, x);
	double tmp;
	if (t_1 <= -1000000.0) {
		tmp = t_2;
	} else if (t_1 <= 0.0002) {
		tmp = fma(((z - t) / a), y, x);
	} else if (t_1 <= 2.0) {
		tmp = y + x;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(z - t) / Float64(a - t))
	t_2 = fma(Float64(z / Float64(a - t)), y, x)
	tmp = 0.0
	if (t_1 <= -1000000.0)
		tmp = t_2;
	elseif (t_1 <= 0.0002)
		tmp = fma(Float64(Float64(z - t) / a), y, x);
	elseif (t_1 <= 2.0)
		tmp = Float64(y + x);
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(z - t), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(z / N[(a - t), $MachinePrecision]), $MachinePrecision] * y + x), $MachinePrecision]}, If[LessEqual[t$95$1, -1000000.0], t$95$2, If[LessEqual[t$95$1, 0.0002], N[(N[(N[(z - t), $MachinePrecision] / a), $MachinePrecision] * y + x), $MachinePrecision], If[LessEqual[t$95$1, 2.0], N[(y + x), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 z t) (-.f64 a t)) < -1e6 or 2 < (/.f64 (-.f64 z t) (-.f64 a t))

   1. Initial program 97.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 97.5

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

   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 96.6

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

   7. Applied rewrites 96.6%

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

   if -1e6 < (/.f64 (-.f64 z t) (-.f64 a t)) < 2.0000000000000001e-4

   1. Initial program 99.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in a around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 98.8

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

   7. Applied rewrites 98.8%

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

   if 2.0000000000000001e-4 < (/.f64 (-.f64 z t) (-.f64 a t)) < 2

   1. Initial program 100.0%

      \[x + y \cdot \frac{z - t}{a - t} \]
   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 99.1

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

   5. Applied rewrites 99.1%

      \[\leadsto \color{blue}{y + x} \]
3. Recombined 3 regimes into one program.

4. Final simplification 98.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z - t}{a - t} \leq -1000000:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{a - t}, y, x\right)\\ \mathbf{elif}\;\frac{z - t}{a - t} \leq 0.0002:\\ \;\;\;\;\mathsf{fma}\left(\frac{z - t}{a}, y, x\right)\\ \mathbf{elif}\;\frac{z - t}{a - t} \leq 2:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{a - t}, y, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 96.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - t}{a - t}\\ t_2 := \mathsf{fma}\left(\frac{z}{a - t}, y, x\right)\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{-34}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 0.0002:\\ \;\;\;\;\mathsf{fma}\left(z - t, \frac{y}{a}, x\right)\\ \mathbf{elif}\;t\_1 \leq 2:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- z t) (- a t))) (t_2 (fma (/ z (- a t)) y x)))
   (if (<= t_1 -2e-34)
     t_2
     (if (<= t_1 0.0002)
       (fma (- z t) (/ y a) x)
       (if (<= t_1 2.0) (+ y x) t_2)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (z - t) / (a - t);
	double t_2 = fma((z / (a - t)), y, x);
	double tmp;
	if (t_1 <= -2e-34) {
		tmp = t_2;
	} else if (t_1 <= 0.0002) {
		tmp = fma((z - t), (y / a), x);
	} else if (t_1 <= 2.0) {
		tmp = y + x;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(z - t) / Float64(a - t))
	t_2 = fma(Float64(z / Float64(a - t)), y, x)
	tmp = 0.0
	if (t_1 <= -2e-34)
		tmp = t_2;
	elseif (t_1 <= 0.0002)
		tmp = fma(Float64(z - t), Float64(y / a), x);
	elseif (t_1 <= 2.0)
		tmp = Float64(y + x);
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(z - t), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(z / N[(a - t), $MachinePrecision]), $MachinePrecision] * y + x), $MachinePrecision]}, If[LessEqual[t$95$1, -2e-34], t$95$2, If[LessEqual[t$95$1, 0.0002], N[(N[(z - t), $MachinePrecision] * N[(y / a), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t$95$1, 2.0], N[(y + x), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 z t) (-.f64 a t)) < -1.99999999999999986e-34 or 2 < (/.f64 (-.f64 z t) (-.f64 a t))

   1. Initial program 97.7%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 97.7

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

   4. Applied rewrites 97.7%

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

   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 95.9

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

   7. Applied rewrites 95.9%

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

   if -1.99999999999999986e-34 < (/.f64 (-.f64 z t) (-.f64 a t)) < 2.0000000000000001e-4

   1. Initial program 99.9%

      \[x + y \cdot \frac{z - t}{a - t} \]
   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 97.8

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

   5. Applied rewrites 97.8%

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

   if 2.0000000000000001e-4 < (/.f64 (-.f64 z t) (-.f64 a t)) < 2

   1. Initial program 100.0%

      \[x + y \cdot \frac{z - t}{a - t} \]
   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 99.1

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

   5. Applied rewrites 99.1%

      \[\leadsto \color{blue}{y + x} \]
3. Recombined 3 regimes into one program.

4. Final simplification 97.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z - t}{a - t} \leq -2 \cdot 10^{-34}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{a - t}, y, x\right)\\ \mathbf{elif}\;\frac{z - t}{a - t} \leq 0.0002:\\ \;\;\;\;\mathsf{fma}\left(z - t, \frac{y}{a}, x\right)\\ \mathbf{elif}\;\frac{z - t}{a - t} \leq 2:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{a - t}, y, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 87.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - t}{a - t}\\ t_2 := \frac{z}{a - t} \cdot y\\ \mathbf{if}\;t\_1 \leq -5000000000000:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 0.0002:\\ \;\;\;\;\mathsf{fma}\left(z - t, \frac{y}{a}, x\right)\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+31}:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- z t) (- a t))) (t_2 (* (/ z (- a t)) y)))
   (if (<= t_1 -5000000000000.0)
     t_2
     (if (<= t_1 0.0002)
       (fma (- z t) (/ y a) x)
       (if (<= t_1 2e+31) (+ y x) t_2)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (z - t) / (a - t);
	double t_2 = (z / (a - t)) * y;
	double tmp;
	if (t_1 <= -5000000000000.0) {
		tmp = t_2;
	} else if (t_1 <= 0.0002) {
		tmp = fma((z - t), (y / a), x);
	} else if (t_1 <= 2e+31) {
		tmp = y + x;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(z - t) / Float64(a - t))
	t_2 = Float64(Float64(z / Float64(a - t)) * y)
	tmp = 0.0
	if (t_1 <= -5000000000000.0)
		tmp = t_2;
	elseif (t_1 <= 0.0002)
		tmp = fma(Float64(z - t), Float64(y / a), x);
	elseif (t_1 <= 2e+31)
		tmp = Float64(y + x);
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(z - t), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(z / N[(a - t), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision]}, If[LessEqual[t$95$1, -5000000000000.0], t$95$2, If[LessEqual[t$95$1, 0.0002], N[(N[(z - t), $MachinePrecision] * N[(y / a), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t$95$1, 2e+31], N[(y + x), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 z t) (-.f64 a t)) < -5e12 or 1.9999999999999999e31 < (/.f64 (-.f64 z t) (-.f64 a t))

   1. Initial program 97.4%

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

   3. Taylor expanded in z around inf

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 77.7

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

   5. Applied rewrites 77.7%

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

   if -5e12 < (/.f64 (-.f64 z t) (-.f64 a t)) < 2.0000000000000001e-4

   1. Initial program 99.9%

      \[x + y \cdot \frac{z - t}{a - t} \]
   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 95.5

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

   5. Applied rewrites 95.5%

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

   if 2.0000000000000001e-4 < (/.f64 (-.f64 z t) (-.f64 a t)) < 1.9999999999999999e31

   1. Initial program 100.0%

      \[x + y \cdot \frac{z - t}{a - t} \]
   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 98.1

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

   5. Applied rewrites 98.1%

      \[\leadsto \color{blue}{y + x} \]
3. Recombined 3 regimes into one program.

4. Final simplification 90.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z - t}{a - t} \leq -5000000000000:\\ \;\;\;\;\frac{z}{a - t} \cdot y\\ \mathbf{elif}\;\frac{z - t}{a - t} \leq 0.0002:\\ \;\;\;\;\mathsf{fma}\left(z - t, \frac{y}{a}, x\right)\\ \mathbf{elif}\;\frac{z - t}{a - t} \leq 2 \cdot 10^{+31}:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{a - t} \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 82.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - t}{a - t}\\ t_2 := \frac{z}{a - t} \cdot y\\ \mathbf{if}\;t\_1 \leq -5000000000000:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-16}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{a}, y, x\right)\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+31}:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- z t) (- a t))) (t_2 (* (/ z (- a t)) y)))
   (if (<= t_1 -5000000000000.0)
     t_2
     (if (<= t_1 5e-16) (fma (/ z a) y x) (if (<= t_1 2e+31) (+ y x) t_2)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (z - t) / (a - t);
	double t_2 = (z / (a - t)) * y;
	double tmp;
	if (t_1 <= -5000000000000.0) {
		tmp = t_2;
	} else if (t_1 <= 5e-16) {
		tmp = fma((z / a), y, x);
	} else if (t_1 <= 2e+31) {
		tmp = y + x;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(z - t) / Float64(a - t))
	t_2 = Float64(Float64(z / Float64(a - t)) * y)
	tmp = 0.0
	if (t_1 <= -5000000000000.0)
		tmp = t_2;
	elseif (t_1 <= 5e-16)
		tmp = fma(Float64(z / a), y, x);
	elseif (t_1 <= 2e+31)
		tmp = Float64(y + x);
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(z - t), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(z / N[(a - t), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision]}, If[LessEqual[t$95$1, -5000000000000.0], t$95$2, If[LessEqual[t$95$1, 5e-16], N[(N[(z / a), $MachinePrecision] * y + x), $MachinePrecision], If[LessEqual[t$95$1, 2e+31], N[(y + x), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 z t) (-.f64 a t)) < -5e12 or 1.9999999999999999e31 < (/.f64 (-.f64 z t) (-.f64 a t))

   1. Initial program 97.4%

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

   3. Taylor expanded in z around inf

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 77.7

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

   5. Applied rewrites 77.7%

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

   if -5e12 < (/.f64 (-.f64 z t) (-.f64 a t)) < 5.0000000000000004e-16

   1. Initial program 99.9%

      \[x + y \cdot \frac{z - t}{a - t} \]
   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 85.6

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

   5. Applied rewrites 85.6%

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

   if 5.0000000000000004e-16 < (/.f64 (-.f64 z t) (-.f64 a t)) < 1.9999999999999999e31

   1. Initial program 100.0%

      \[x + y \cdot \frac{z - t}{a - t} \]
   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 97.2

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

   5. Applied rewrites 97.2%

      \[\leadsto \color{blue}{y + x} \]
3. Recombined 3 regimes into one program.

4. Final simplification 87.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z - t}{a - t} \leq -5000000000000:\\ \;\;\;\;\frac{z}{a - t} \cdot y\\ \mathbf{elif}\;\frac{z - t}{a - t} \leq 5 \cdot 10^{-16}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{a}, y, x\right)\\ \mathbf{elif}\;\frac{z - t}{a - t} \leq 2 \cdot 10^{+31}:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{a - t} \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 78.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - t}{a - t}\\ \mathbf{if}\;t\_1 \leq 5 \cdot 10^{-16}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{a}, y, x\right)\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+31}:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;\frac{-z}{t} \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- z t) (- a t))))
   (if (<= t_1 5e-16)
     (fma (/ z a) y x)
     (if (<= t_1 2e+31) (+ y x) (* (/ (- z) t) y)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (z - t) / (a - t);
	double tmp;
	if (t_1 <= 5e-16) {
		tmp = fma((z / a), y, x);
	} else if (t_1 <= 2e+31) {
		tmp = y + x;
	} else {
		tmp = (-z / t) * y;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(z - t) / Float64(a - t))
	tmp = 0.0
	if (t_1 <= 5e-16)
		tmp = fma(Float64(z / a), y, x);
	elseif (t_1 <= 2e+31)
		tmp = Float64(y + x);
	else
		tmp = Float64(Float64(Float64(-z) / t) * y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(z - t), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 5e-16], N[(N[(z / a), $MachinePrecision] * y + x), $MachinePrecision], If[LessEqual[t$95$1, 2e+31], N[(y + x), $MachinePrecision], N[(N[((-z) / t), $MachinePrecision] * y), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 z t) (-.f64 a t)) < 5.0000000000000004e-16

   1. Initial program 99.1%

      \[x + y \cdot \frac{z - t}{a - t} \]
   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 72.2

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

   5. Applied rewrites 72.2%

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

   if 5.0000000000000004e-16 < (/.f64 (-.f64 z t) (-.f64 a t)) < 1.9999999999999999e31

   1. Initial program 100.0%

      \[x + y \cdot \frac{z - t}{a - t} \]
   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 97.2

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

   5. Applied rewrites 97.2%

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

   if 1.9999999999999999e31 < (/.f64 (-.f64 z t) (-.f64 a t))

   1. Initial program 96.5%

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

   3. Taylor expanded in a around 0

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

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

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower--.f64 57.8

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

   5. Applied rewrites 57.8%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 56.1%

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

   9. Taylor expanded in z around inf

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

   11. Applied rewrites 56.1%

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

4. Final simplification 79.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z - t}{a - t} \leq 5 \cdot 10^{-16}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{a}, y, x\right)\\ \mathbf{elif}\;\frac{z - t}{a - t} \leq 2 \cdot 10^{+31}:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;\frac{-z}{t} \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 80.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - t}{a - t}\\ \mathbf{if}\;t\_1 \leq 5 \cdot 10^{-16} \lor \neg \left(t\_1 \leq 10^{+56}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{a}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;y + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- z t) (- a t))))
   (if (or (<= t_1 5e-16) (not (<= t_1 1e+56))) (fma (/ z a) y x) (+ y x))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (z - t) / (a - t);
	double tmp;
	if ((t_1 <= 5e-16) || !(t_1 <= 1e+56)) {
		tmp = fma((z / a), y, x);
	} else {
		tmp = y + x;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(z - t) / Float64(a - t))
	tmp = 0.0
	if ((t_1 <= 5e-16) || !(t_1 <= 1e+56))
		tmp = fma(Float64(z / a), y, x);
	else
		tmp = Float64(y + x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(z - t), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$1, 5e-16], N[Not[LessEqual[t$95$1, 1e+56]], $MachinePrecision]], N[(N[(z / a), $MachinePrecision] * y + x), $MachinePrecision], N[(y + x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 z t) (-.f64 a t)) < 5.0000000000000004e-16 or 1.00000000000000009e56 < (/.f64 (-.f64 z t) (-.f64 a t))

   1. Initial program 98.6%

      \[x + y \cdot \frac{z - t}{a - t} \]
   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 69.9

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

   5. Applied rewrites 69.9%

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

   if 5.0000000000000004e-16 < (/.f64 (-.f64 z t) (-.f64 a t)) < 1.00000000000000009e56

   1. Initial program 99.9%

      \[x + y \cdot \frac{z - t}{a - t} \]
   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 93.4

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

   5. Applied rewrites 93.4%

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

4. Final simplification 78.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z - t}{a - t} \leq 5 \cdot 10^{-16} \lor \neg \left(\frac{z - t}{a - t} \leq 10^{+56}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{a}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;y + x\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 80.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - t}{a - t}\\ \mathbf{if}\;t\_1 \leq 5 \cdot 10^{-16}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{a}, y, x\right)\\ \mathbf{elif}\;t\_1 \leq 10^{+56}:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, z, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- z t) (- a t))))
   (if (<= t_1 5e-16)
     (fma (/ z a) y x)
     (if (<= t_1 1e+56) (+ y x) (fma (/ y a) z x)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (z - t) / (a - t);
	double tmp;
	if (t_1 <= 5e-16) {
		tmp = fma((z / a), y, x);
	} else if (t_1 <= 1e+56) {
		tmp = y + x;
	} else {
		tmp = fma((y / a), z, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(z - t) / Float64(a - t))
	tmp = 0.0
	if (t_1 <= 5e-16)
		tmp = fma(Float64(z / a), y, x);
	elseif (t_1 <= 1e+56)
		tmp = Float64(y + x);
	else
		tmp = fma(Float64(y / a), z, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(z - t), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 5e-16], N[(N[(z / a), $MachinePrecision] * y + x), $MachinePrecision], If[LessEqual[t$95$1, 1e+56], N[(y + x), $MachinePrecision], N[(N[(y / a), $MachinePrecision] * z + x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 z t) (-.f64 a t)) < 5.0000000000000004e-16

   1. Initial program 99.1%

      \[x + y \cdot \frac{z - t}{a - t} \]
   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 72.2

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

   5. Applied rewrites 72.2%

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

   if 5.0000000000000004e-16 < (/.f64 (-.f64 z t) (-.f64 a t)) < 1.00000000000000009e56

   1. Initial program 99.9%

      \[x + y \cdot \frac{z - t}{a - t} \]
   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 93.4

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

   5. Applied rewrites 93.4%

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

   if 1.00000000000000009e56 < (/.f64 (-.f64 z t) (-.f64 a t))

   1. Initial program 95.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 95.9

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

   4. Applied rewrites 95.9%

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

   5. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. associate-*r/ N/A

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

      5. div-add-rev N/A

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

      6. +-commutative N/A

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

      7. lower-/.f64 N/A

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

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

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

      9. metadata-eval N/A

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

      10. *-lft-identity N/A

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

      11. lower--.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower--.f64 95.8

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

   7. Applied rewrites 95.8%

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

   8. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. associate-*l/ N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 57.7

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

   10. Applied rewrites 57.7%

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

4. Final simplification 78.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z - t}{a - t} \leq 5 \cdot 10^{-16}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{a}, y, x\right)\\ \mathbf{elif}\;\frac{z - t}{a - t} \leq 10^{+56}:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, z, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 60.9% 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 99.1%

   \[x + y \cdot \frac{z - t}{a - t} \]
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 56.1

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

5. Applied rewrites 56.1%

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

6. Final simplification 56.1%

   \[\leadsto y + x \]
7. Add Preprocessing

Developer Target 1: 99.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x + y \cdot \frac{z - t}{a - t}\\ \mathbf{if}\;y < -8.508084860551241 \cdot 10^{-17}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y < 2.894426862792089 \cdot 10^{-49}:\\ \;\;\;\;x + \left(y \cdot \left(z - t\right)\right) \cdot \frac{1}{a - t}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ x (* y (/ (- z t) (- a t))))))
   (if (< y -8.508084860551241e-17)
     t_1
     (if (< y 2.894426862792089e-49)
       (+ x (* (* y (- z t)) (/ 1.0 (- a t))))
       t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = x + (y * ((z - t) / (a - t)));
	double tmp;
	if (y < -8.508084860551241e-17) {
		tmp = t_1;
	} else if (y < 2.894426862792089e-49) {
		tmp = x + ((y * (z - t)) * (1.0 / (a - t)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: tmp
    t_1 = x + (y * ((z - t) / (a - t)))
    if (y < (-8.508084860551241d-17)) then
        tmp = t_1
    else if (y < 2.894426862792089d-49) then
        tmp = x + ((y * (z - t)) * (1.0d0 / (a - t)))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = x + (y * ((z - t) / (a - t)));
	double tmp;
	if (y < -8.508084860551241e-17) {
		tmp = t_1;
	} else if (y < 2.894426862792089e-49) {
		tmp = x + ((y * (z - t)) * (1.0 / (a - t)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = x + (y * ((z - t) / (a - t)))
	tmp = 0
	if y < -8.508084860551241e-17:
		tmp = t_1
	elif y < 2.894426862792089e-49:
		tmp = x + ((y * (z - t)) * (1.0 / (a - t)))
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(x + Float64(y * Float64(Float64(z - t) / Float64(a - t))))
	tmp = 0.0
	if (y < -8.508084860551241e-17)
		tmp = t_1;
	elseif (y < 2.894426862792089e-49)
		tmp = Float64(x + Float64(Float64(y * Float64(z - t)) * Float64(1.0 / Float64(a - t))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = x + (y * ((z - t) / (a - t)));
	tmp = 0.0;
	if (y < -8.508084860551241e-17)
		tmp = t_1;
	elseif (y < 2.894426862792089e-49)
		tmp = x + ((y * (z - t)) * (1.0 / (a - t)));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(x + N[(y * N[(N[(z - t), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Less[y, -8.508084860551241e-17], t$95$1, If[Less[y, 2.894426862792089e-49], N[(x + N[(N[(y * N[(z - t), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(a - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Reproduce

herbie shell --seed 2024337 
(FPCore (x y z t a)
  :name "Graphics.Rendering.Plot.Render.Plot.Axis:renderAxisLine from plot-0.2.3.4, B"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< y -8508084860551241/100000000000000000000000000000000) (+ x (* y (/ (- z t) (- a t)))) (if (< y 2894426862792089/10000000000000000000000000000000000000000000000000000000000000000) (+ x (* (* y (- z t)) (/ 1 (- a t)))) (+ x (* y (/ (- z t) (- a t)))))))

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