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

Percentage Accurate: 76.1% → 88.8%

Time: 7.8s

Alternatives: 13

Speedup: 1.1×

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 76.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 88.8% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 80.1%

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

3. Taylor expanded in x around 0

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

   1. associate--l+ N/A

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

   2. +-commutative N/A

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

   3. *-lft-identity N/A

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

   4. associate-/l* N/A

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

   5. *-commutative N/A

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

   6. fp-cancel-sub-sign N/A

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

   7. mul-1-neg N/A

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

   8. distribute-rgt-in N/A

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

   9. *-commutative N/A

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

   10. lower-fma.f64 N/A

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

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

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

   12. metadata-eval N/A

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

   13. *-lft-identity N/A

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

   14. lower--.f64 N/A

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

   15. lower-/.f64 N/A

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

   16. lower--.f64 N/A

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

   17. lower--.f64 90.5

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

5. Applied rewrites 90.5%

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

6. Final simplification 90.5%

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

Alternative 2: 62.0% accurate, 0.6× speedup

Math

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

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= (- (+ x y) (/ (* (- z t) y) (- a t))) (- INFINITY))
   (* y (/ z t))
   (+ x y)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (((x + y) - (((z - t) * y) / (a - t))) <= -((double) INFINITY)) {
		tmp = y * (z / t);
	} else {
		tmp = x + y;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (((x + y) - (((z - t) * y) / (a - t))) <= -Double.POSITIVE_INFINITY) {
		tmp = y * (z / t);
	} else {
		tmp = x + y;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if ((x + y) - (((z - t) * y) / (a - t))) <= -math.inf:
		tmp = y * (z / t)
	else:
		tmp = x + y
	return tmp

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (-.f64 (+.f64 x y) (/.f64 (*.f64 (-.f64 z t) y) (-.f64 a t))) < -inf.0

   1. Initial program 55.3%

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

   3. Taylor expanded in t around inf

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

      1. associate--l+ N/A

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

      2. distribute-lft-out-- N/A

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

      3. div-sub N/A

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

      4. *-commutative N/A

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

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

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

      6. mul-1-neg N/A

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

      7. associate-*r* N/A

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

      8. +-commutative N/A

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

      9. *-lft-identity N/A

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

      10. metadata-eval N/A

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

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

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

      12. distribute-lft-out-- N/A

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

      13. mul-1-neg N/A

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

      14. distribute-neg-frac N/A

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

      15. fp-cancel-sub-sign N/A

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

   5. Applied rewrites 50.3%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 38.6%

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

   if -inf.0 < (-.f64 (+.f64 x y) (/.f64 (*.f64 (-.f64 z t) y) (-.f64 a t)))

   1. Initial program 85.1%

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

   3. Taylor expanded in x around inf

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. distribute-rgt-in N/A

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

      4. *-lft-identity N/A

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

      5. lower-fma.f64 N/A

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

   5. Applied rewrites 78.8%

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

   7. Applied rewrites 81.5%

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

   8. Taylor expanded in a around inf

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

   10. Applied rewrites 67.6%

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

4. Final simplification 62.8%

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

Alternative 3: 60.8% accurate, 0.6× speedup

Math

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

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= (- (+ x y) (/ (* (- z t) y) (- a t))) (- INFINITY))
   (/ (* z y) t)
   (+ x y)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (((x + y) - (((z - t) * y) / (a - t))) <= -((double) INFINITY)) {
		tmp = (z * y) / t;
	} else {
		tmp = x + y;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (((x + y) - (((z - t) * y) / (a - t))) <= -Double.POSITIVE_INFINITY) {
		tmp = (z * y) / t;
	} else {
		tmp = x + y;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if ((x + y) - (((z - t) * y) / (a - t))) <= -math.inf:
		tmp = (z * y) / t
	else:
		tmp = x + y
	return tmp

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (-.f64 (+.f64 x y) (/.f64 (*.f64 (-.f64 z t) y) (-.f64 a t))) < -inf.0

   1. Initial program 55.3%

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

   3. Taylor expanded in z around inf

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

      1. associate-*l/ N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower--.f64 68.6

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

   5. Applied rewrites 68.6%

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

   6. Taylor expanded in t around inf

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

   8. Applied rewrites 36.2%

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

   if -inf.0 < (-.f64 (+.f64 x y) (/.f64 (*.f64 (-.f64 z t) y) (-.f64 a t)))

   1. Initial program 85.1%

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

   3. Taylor expanded in x around inf

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. distribute-rgt-in N/A

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

      4. *-lft-identity N/A

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

      5. lower-fma.f64 N/A

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

   5. Applied rewrites 78.8%

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

   7. Applied rewrites 81.5%

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

   8. Taylor expanded in a around inf

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

   10. Applied rewrites 67.6%

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

4. Final simplification 62.4%

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

Alternative 4: 88.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -4 \cdot 10^{+150} \lor \neg \left(t \leq 6.4 \cdot 10^{+96}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{z - a}{t}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x + y\right) - \frac{z}{a - t} \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= t -4e+150) (not (<= t 6.4e+96)))
   (fma (/ (- z a) t) y x)
   (- (+ x y) (* (/ z (- a t)) y))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((t <= -4e+150) || !(t <= 6.4e+96)) {
		tmp = fma(((z - a) / t), y, x);
	} else {
		tmp = (x + y) - ((z / (a - t)) * y);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((t <= -4e+150) || !(t <= 6.4e+96))
		tmp = fma(Float64(Float64(z - a) / t), y, x);
	else
		tmp = Float64(Float64(x + y) - Float64(Float64(z / Float64(a - t)) * y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[t, -4e+150], N[Not[LessEqual[t, 6.4e+96]], $MachinePrecision]], N[(N[(N[(z - a), $MachinePrecision] / t), $MachinePrecision] * y + x), $MachinePrecision], N[(N[(x + y), $MachinePrecision] - N[(N[(z / N[(a - t), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -3.99999999999999992e150 or 6.40000000000000013e96 < t

   1. Initial program 60.8%

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

   3. Taylor expanded in x around 0

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. *-lft-identity N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. fp-cancel-sub-sign N/A

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

      7. mul-1-neg N/A

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

      8. distribute-rgt-in N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

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

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

      12. metadata-eval N/A

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

      13. *-lft-identity N/A

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

      14. lower--.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower--.f64 N/A

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

      17. lower--.f64 82.8

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

   5. Applied rewrites 82.8%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 79.0%

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

   9. Taylor expanded in t around inf

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

       1. associate--l+ N/A

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

       2. distribute-lft-out-- N/A

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

       3. div-sub N/A

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

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

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

       5. metadata-eval N/A

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

       6. *-lft-identity N/A

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

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

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

       8. mul-1-neg N/A

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

       9. associate-*r* N/A

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

       10. +-commutative N/A

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

       11. div-add N/A

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

       12. *-lft-identity N/A

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

       13. metadata-eval N/A

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

       14. div-add N/A

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

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

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

       16. distribute-lft-out-- N/A

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

       17. mul-1-neg N/A

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

   11. Applied rewrites 89.1%

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

   if -3.99999999999999992e150 < t < 6.40000000000000013e96

   1. Initial program 88.2%

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

   3. Taylor expanded in z around inf

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 90.9

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

   5. Applied rewrites 90.9%

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

4. Final simplification 90.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -4 \cdot 10^{+150} \lor \neg \left(t \leq 6.4 \cdot 10^{+96}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{z - a}{t}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x + y\right) - \frac{z}{a - t} \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 83.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -5.2 \cdot 10^{+71} \lor \neg \left(a \leq 2.5 \cdot 10^{+136}\right):\\ \;\;\;\;\left(x + y\right) - \frac{z \cdot y}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-z}{a - t}, y, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= a -5.2e+71) (not (<= a 2.5e+136)))
   (- (+ x y) (/ (* z y) a))
   (fma (/ (- z) (- a t)) y x)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a <= -5.2e+71) || !(a <= 2.5e+136)) {
		tmp = (x + y) - ((z * y) / a);
	} else {
		tmp = fma((-z / (a - t)), y, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((a <= -5.2e+71) || !(a <= 2.5e+136))
		tmp = Float64(Float64(x + y) - Float64(Float64(z * y) / a));
	else
		tmp = fma(Float64(Float64(-z) / Float64(a - t)), y, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[a, -5.2e+71], N[Not[LessEqual[a, 2.5e+136]], $MachinePrecision]], N[(N[(x + y), $MachinePrecision] - N[(N[(z * y), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[(N[((-z) / N[(a - t), $MachinePrecision]), $MachinePrecision] * y + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -5.19999999999999983e71 or 2.5000000000000001e136 < a

   1. Initial program 79.5%

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

   3. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 82.6

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

   5. Applied rewrites 82.6%

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

   if -5.19999999999999983e71 < a < 2.5000000000000001e136

   1. Initial program 80.3%

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

   3. Taylor expanded in x around 0

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. *-lft-identity N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. fp-cancel-sub-sign N/A

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

      7. mul-1-neg N/A

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

      8. distribute-rgt-in N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

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

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

      12. metadata-eval N/A

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

      13. *-lft-identity N/A

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

      14. lower--.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower--.f64 N/A

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

      17. lower--.f64 89.6

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

   5. Applied rewrites 89.6%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 86.5%

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

4. Final simplification 85.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -5.2 \cdot 10^{+71} \lor \neg \left(a \leq 2.5 \cdot 10^{+136}\right):\\ \;\;\;\;\left(x + y\right) - \frac{z \cdot y}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-z}{a - t}, y, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 83.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.55 \cdot 10^{+159} \lor \neg \left(a \leq 2.8 \cdot 10^{+138}\right):\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-z}{a - t}, y, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= a -1.55e+159) (not (<= a 2.8e+138)))
   (+ x y)
   (fma (/ (- z) (- a t)) y x)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a <= -1.55e+159) || !(a <= 2.8e+138)) {
		tmp = x + y;
	} else {
		tmp = fma((-z / (a - t)), y, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((a <= -1.55e+159) || !(a <= 2.8e+138))
		tmp = Float64(x + y);
	else
		tmp = fma(Float64(Float64(-z) / Float64(a - t)), y, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[a, -1.55e+159], N[Not[LessEqual[a, 2.8e+138]], $MachinePrecision]], N[(x + y), $MachinePrecision], N[(N[((-z) / N[(a - t), $MachinePrecision]), $MachinePrecision] * y + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -1.5499999999999999e159 or 2.8000000000000001e138 < a

   1. Initial program 80.8%

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

   3. Taylor expanded in x around inf

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. distribute-rgt-in N/A

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

      4. *-lft-identity N/A

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

      5. lower-fma.f64 N/A

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

   5. Applied rewrites 88.9%

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

   7. Applied rewrites 87.3%

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

   8. Taylor expanded in a around inf

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

   10. Applied rewrites 83.6%

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

   if -1.5499999999999999e159 < a < 2.8000000000000001e138

   1. Initial program 79.8%

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

   3. Taylor expanded in x around 0

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. *-lft-identity N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. fp-cancel-sub-sign N/A

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

      7. mul-1-neg N/A

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

      8. distribute-rgt-in N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

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

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

      12. metadata-eval N/A

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

      13. *-lft-identity N/A

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

      14. lower--.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower--.f64 N/A

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

      17. lower--.f64 89.0

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

   5. Applied rewrites 89.0%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 85.0%

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

4. Final simplification 84.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.55 \cdot 10^{+159} \lor \neg \left(a \leq 2.8 \cdot 10^{+138}\right):\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-z}{a - t}, y, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 77.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -5.6 \cdot 10^{+77} \lor \neg \left(a \leq 7.8 \cdot 10^{+34}\right):\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y \cdot \left(a - z\right)}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= a -5.6e+77) (not (<= a 7.8e+34)))
   (+ x y)
   (- x (/ (* y (- a z)) t))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a <= -5.6e+77) || !(a <= 7.8e+34)) {
		tmp = x + y;
	} else {
		tmp = x - ((y * (a - z)) / t);
	}
	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 <= (-5.6d+77)) .or. (.not. (a <= 7.8d+34))) then
        tmp = x + y
    else
        tmp = x - ((y * (a - z)) / t)
    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 <= -5.6e+77) || !(a <= 7.8e+34)) {
		tmp = x + y;
	} else {
		tmp = x - ((y * (a - z)) / t);
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (a <= -5.6e+77) or not (a <= 7.8e+34):
		tmp = x + y
	else:
		tmp = x - ((y * (a - z)) / t)
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((a <= -5.6e+77) || !(a <= 7.8e+34))
		tmp = Float64(x + y);
	else
		tmp = Float64(x - Float64(Float64(y * Float64(a - z)) / t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((a <= -5.6e+77) || ~((a <= 7.8e+34)))
		tmp = x + y;
	else
		tmp = x - ((y * (a - z)) / t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[a, -5.6e+77], N[Not[LessEqual[a, 7.8e+34]], $MachinePrecision]], N[(x + y), $MachinePrecision], N[(x - N[(N[(y * N[(a - z), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -5.60000000000000001e77 or 7.80000000000000038e34 < a

   1. Initial program 75.8%

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

   3. Taylor expanded in x around inf

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. distribute-rgt-in N/A

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

      4. *-lft-identity N/A

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

      5. lower-fma.f64 N/A

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

   5. Applied rewrites 81.9%

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

   7. Applied rewrites 81.9%

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

   8. Taylor expanded in a around inf

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

   10. Applied rewrites 75.3%

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

   if -5.60000000000000001e77 < a < 7.80000000000000038e34

   1. Initial program 82.6%

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

   3. Taylor expanded in t around inf

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

      1. associate--l+ N/A

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

      2. distribute-lft-out-- N/A

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

      3. div-sub N/A

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

      4. *-commutative N/A

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

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

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

      6. mul-1-neg N/A

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

      7. associate-*r* N/A

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

      8. +-commutative N/A

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

      9. *-lft-identity N/A

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

      10. metadata-eval N/A

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

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

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

      12. distribute-lft-out-- N/A

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

      13. mul-1-neg N/A

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

      14. distribute-neg-frac N/A

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

      15. fp-cancel-sub-sign N/A

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

   5. Applied rewrites 76.8%

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

4. Final simplification 76.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -5.6 \cdot 10^{+77} \lor \neg \left(a \leq 7.8 \cdot 10^{+34}\right):\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y \cdot \left(a - z\right)}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 83.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.55 \cdot 10^{+159}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{t}{a - t}, y + x\right)\\ \mathbf{elif}\;a \leq 2.5 \cdot 10^{+136}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-z}{a - t}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x + y\right) - \frac{z \cdot y}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= a -1.55e+159)
   (fma y (/ t (- a t)) (+ y x))
   (if (<= a 2.5e+136) (fma (/ (- z) (- a t)) y x) (- (+ x y) (/ (* z y) a)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (a <= -1.55e+159) {
		tmp = fma(y, (t / (a - t)), (y + x));
	} else if (a <= 2.5e+136) {
		tmp = fma((-z / (a - t)), y, x);
	} else {
		tmp = (x + y) - ((z * y) / a);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (a <= -1.55e+159)
		tmp = fma(y, Float64(t / Float64(a - t)), Float64(y + x));
	elseif (a <= 2.5e+136)
		tmp = fma(Float64(Float64(-z) / Float64(a - t)), y, x);
	else
		tmp = Float64(Float64(x + y) - Float64(Float64(z * y) / a));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[a, -1.55e+159], N[(y * N[(t / N[(a - t), $MachinePrecision]), $MachinePrecision] + N[(y + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 2.5e+136], N[(N[((-z) / N[(a - t), $MachinePrecision]), $MachinePrecision] * y + x), $MachinePrecision], N[(N[(x + y), $MachinePrecision] - N[(N[(z * y), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -1.5499999999999999e159

   1. Initial program 74.3%

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

   3. Taylor expanded in z around 0

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

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

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

      2. +-commutative N/A

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

      3. metadata-eval N/A

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

      4. *-lft-identity N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower--.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 81.7

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

   5. Applied rewrites 81.7%

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

   if -1.5499999999999999e159 < a < 2.5000000000000001e136

   1. Initial program 79.8%

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

   3. Taylor expanded in x around 0

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. *-lft-identity N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. fp-cancel-sub-sign N/A

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

      7. mul-1-neg N/A

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

      8. distribute-rgt-in N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

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

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

      12. metadata-eval N/A

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

      13. *-lft-identity N/A

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

      14. lower--.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower--.f64 N/A

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

      17. lower--.f64 89.0

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

   5. Applied rewrites 89.0%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 85.0%

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

   if 2.5000000000000001e136 < a

   1. Initial program 86.4%

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

   3. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 90.2

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

   5. Applied rewrites 90.2%

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

4. Final simplification 85.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.55 \cdot 10^{+159}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{t}{a - t}, y + x\right)\\ \mathbf{elif}\;a \leq 2.5 \cdot 10^{+136}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-z}{a - t}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x + y\right) - \frac{z \cdot y}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 79.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -8.4 \cdot 10^{+80} \lor \neg \left(a \leq 7.8 \cdot 10^{+34}\right):\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z - a}{t}, y, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= a -8.4e+80) (not (<= a 7.8e+34)))
   (+ x y)
   (fma (/ (- z a) t) y x)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a <= -8.4e+80) || !(a <= 7.8e+34)) {
		tmp = x + y;
	} else {
		tmp = fma(((z - a) / t), y, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((a <= -8.4e+80) || !(a <= 7.8e+34))
		tmp = Float64(x + y);
	else
		tmp = fma(Float64(Float64(z - a) / t), y, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[a, -8.4e+80], N[Not[LessEqual[a, 7.8e+34]], $MachinePrecision]], N[(x + y), $MachinePrecision], N[(N[(N[(z - a), $MachinePrecision] / t), $MachinePrecision] * y + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -8.40000000000000005e80 or 7.80000000000000038e34 < a

   1. Initial program 75.8%

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

   3. Taylor expanded in x around inf

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. distribute-rgt-in N/A

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

      4. *-lft-identity N/A

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

      5. lower-fma.f64 N/A

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

   5. Applied rewrites 81.9%

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

   7. Applied rewrites 81.9%

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

   8. Taylor expanded in a around inf

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

   10. Applied rewrites 75.3%

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

   if -8.40000000000000005e80 < a < 7.80000000000000038e34

   1. Initial program 82.6%

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

   3. Taylor expanded in x around 0

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. *-lft-identity N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. fp-cancel-sub-sign N/A

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

      7. mul-1-neg N/A

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

      8. distribute-rgt-in N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

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

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

      12. metadata-eval N/A

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

      13. *-lft-identity N/A

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

      14. lower--.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower--.f64 N/A

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

      17. lower--.f64 90.4

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

   5. Applied rewrites 90.4%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 88.2%

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

   9. Taylor expanded in t around inf

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

       1. associate--l+ N/A

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

       2. distribute-lft-out-- N/A

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

       3. div-sub N/A

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

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

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

       5. metadata-eval N/A

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

       6. *-lft-identity N/A

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

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

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

       8. mul-1-neg N/A

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

       9. associate-*r* N/A

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

       10. +-commutative N/A

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

       11. div-add N/A

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

       12. *-lft-identity N/A

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

       13. metadata-eval N/A

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

       14. div-add N/A

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

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

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

       16. distribute-lft-out-- N/A

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

       17. mul-1-neg N/A

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

   11. Applied rewrites 76.2%

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

4. Final simplification 75.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -8.4 \cdot 10^{+80} \lor \neg \left(a \leq 7.8 \cdot 10^{+34}\right):\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z - a}{t}, y, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 76.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -4.4 \cdot 10^{-63} \lor \neg \left(a \leq 35000000000\right):\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, y, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= a -4.4e-63) (not (<= a 35000000000.0)))
   (+ x y)
   (fma (/ z t) y x)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a <= -4.4e-63) || !(a <= 35000000000.0)) {
		tmp = x + y;
	} else {
		tmp = fma((z / t), y, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((a <= -4.4e-63) || !(a <= 35000000000.0))
		tmp = Float64(x + y);
	else
		tmp = fma(Float64(z / t), y, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[a, -4.4e-63], N[Not[LessEqual[a, 35000000000.0]], $MachinePrecision]], N[(x + y), $MachinePrecision], N[(N[(z / t), $MachinePrecision] * y + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -4.3999999999999999e-63 or 3.5e10 < a

   1. Initial program 78.4%

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

   3. Taylor expanded in x around inf

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. distribute-rgt-in N/A

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

      4. *-lft-identity N/A

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

      5. lower-fma.f64 N/A

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

   5. Applied rewrites 80.6%

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

   7. Applied rewrites 81.9%

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

   8. Taylor expanded in a around inf

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

   10. Applied rewrites 67.8%

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

   if -4.3999999999999999e-63 < a < 3.5e10

   1. Initial program 81.9%

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

   3. Taylor expanded in x around 0

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. *-lft-identity N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. fp-cancel-sub-sign N/A

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

      7. mul-1-neg N/A

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

      8. distribute-rgt-in N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

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

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

      12. metadata-eval N/A

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

      13. *-lft-identity N/A

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

      14. lower--.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower--.f64 N/A

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

      17. lower--.f64 88.6

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

   5. Applied rewrites 88.6%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 81.2%

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

4. Final simplification 74.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -4.4 \cdot 10^{-63} \lor \neg \left(a \leq 35000000000\right):\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, y, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 61.3% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 3 \cdot 10^{+129}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0, y, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t 3e+129) (+ x y) (fma 0.0 y x)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 3e+129) {
		tmp = x + y;
	} else {
		tmp = fma(0.0, y, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= 3e+129)
		tmp = Float64(x + y);
	else
		tmp = fma(0.0, y, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, 3e+129], N[(x + y), $MachinePrecision], N[(0.0 * y + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 3.0000000000000003e129

   1. Initial program 84.3%

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

   3. Taylor expanded in x around inf

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. distribute-rgt-in N/A

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

      4. *-lft-identity N/A

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

      5. lower-fma.f64 N/A

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

   5. Applied rewrites 79.9%

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

   7. Applied rewrites 81.1%

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

   8. Taylor expanded in a around inf

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

   10. Applied rewrites 59.1%

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

   if 3.0000000000000003e129 < t

   1. Initial program 54.1%

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

   3. Taylor expanded in x around inf

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. distribute-rgt-in N/A

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

      4. *-lft-identity N/A

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

      5. lower-fma.f64 N/A

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

   5. Applied rewrites 63.1%

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

   6. Taylor expanded in t around inf

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

   8. Applied rewrites 70.0%

      \[\leadsto \mathsf{fma}\left(0, \color{blue}{y}, x\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 60.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 3 \cdot 10^{+129}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0, y, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 59.9% accurate, 7.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 80.1%

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

3. Taylor expanded in x around inf

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

   1. associate--l+ N/A

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

   2. +-commutative N/A

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

   3. distribute-rgt-in N/A

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

   4. *-lft-identity N/A

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

   5. lower-fma.f64 N/A

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

5. Applied rewrites 77.5%

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

7. Applied rewrites 79.6%

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

8. Taylor expanded in a around inf

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

10. Applied rewrites 58.3%

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

11. Final simplification 58.3%

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

Alternative 13: 2.7% accurate, 29.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (x y z t a) :precision binary64 0.0)

C

double code(double x, double y, double z, double t, double a) {
	return 0.0;
}

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 = 0.0d0
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	return 0.0;
}

Python

def code(x, y, z, t, a):
	return 0.0

Julia

function code(x, y, z, t, a)
	return 0.0
end

MATLAB

function tmp = code(x, y, z, t, a)
	tmp = 0.0;
end

Wolfram

code[x_, y_, z_, t_, a_] := 0.0

Derivation

1. Initial program 80.1%

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

3. Taylor expanded in x around 0

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

   1. associate-/l* N/A

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

   2. *-commutative N/A

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

   3. fp-cancel-sub-sign N/A

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

   4. mul-1-neg N/A

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

   5. *-lft-identity N/A

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

   6. distribute-rgt-in N/A

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

   7. +-commutative N/A

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

   8. distribute-rgt-in N/A

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

   9. *-lft-identity N/A

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

   10. mul-1-neg N/A

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

   11. distribute-lft-neg-out N/A

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

   12. *-commutative N/A

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

   13. associate-/l* N/A

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

   14. *-commutative N/A

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

   15. associate-/l* N/A

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

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

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

   17. lower-fma.f64 N/A

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

5. Applied rewrites 41.9%

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

6. Taylor expanded in t around inf

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

8. Applied rewrites 2.6%

   \[\leadsto 0 \cdot \color{blue}{y} \]

9. Taylor expanded in y around 0

   \[\leadsto 0 \]
10. Step-by-step derivation

11. Applied rewrites 2.6%

    \[\leadsto 0 \]

12. Final simplification 2.6%

    \[\leadsto 0 \]
13. Add Preprocessing

Developer Target 1: 87.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(y + x\right) - \left(\left(z - t\right) \cdot \frac{1}{a - t}\right) \cdot y\\ t_2 := \left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t}\\ \mathbf{if}\;t\_2 < -1.3664970889390727 \cdot 10^{-7}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 < 1.4754293444577233 \cdot 10^{-239}:\\ \;\;\;\;\frac{y \cdot \left(a - z\right) - x \cdot t}{a - t}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- (+ y x) (* (* (- z t) (/ 1.0 (- a t))) y)))
        (t_2 (- (+ x y) (/ (* (- z t) y) (- a t)))))
   (if (< t_2 -1.3664970889390727e-7)
     t_1
     (if (< t_2 1.4754293444577233e-239)
       (/ (- (* y (- a z)) (* x t)) (- a t))
       t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (y + x) - (((z - t) * (1.0 / (a - t))) * y);
	double t_2 = (x + y) - (((z - t) * y) / (a - t));
	double tmp;
	if (t_2 < -1.3664970889390727e-7) {
		tmp = t_1;
	} else if (t_2 < 1.4754293444577233e-239) {
		tmp = ((y * (a - z)) - (x * t)) / (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) :: t_2
    real(8) :: tmp
    t_1 = (y + x) - (((z - t) * (1.0d0 / (a - t))) * y)
    t_2 = (x + y) - (((z - t) * y) / (a - t))
    if (t_2 < (-1.3664970889390727d-7)) then
        tmp = t_1
    else if (t_2 < 1.4754293444577233d-239) then
        tmp = ((y * (a - z)) - (x * t)) / (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 = (y + x) - (((z - t) * (1.0 / (a - t))) * y);
	double t_2 = (x + y) - (((z - t) * y) / (a - t));
	double tmp;
	if (t_2 < -1.3664970889390727e-7) {
		tmp = t_1;
	} else if (t_2 < 1.4754293444577233e-239) {
		tmp = ((y * (a - z)) - (x * t)) / (a - t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = (y + x) - (((z - t) * (1.0 / (a - t))) * y)
	t_2 = (x + y) - (((z - t) * y) / (a - t))
	tmp = 0
	if t_2 < -1.3664970889390727e-7:
		tmp = t_1
	elif t_2 < 1.4754293444577233e-239:
		tmp = ((y * (a - z)) - (x * t)) / (a - t)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(y + x) - Float64(Float64(Float64(z - t) * Float64(1.0 / Float64(a - t))) * y))
	t_2 = Float64(Float64(x + y) - Float64(Float64(Float64(z - t) * y) / Float64(a - t)))
	tmp = 0.0
	if (t_2 < -1.3664970889390727e-7)
		tmp = t_1;
	elseif (t_2 < 1.4754293444577233e-239)
		tmp = Float64(Float64(Float64(y * Float64(a - z)) - Float64(x * t)) / Float64(a - t));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = (y + x) - (((z - t) * (1.0 / (a - t))) * y);
	t_2 = (x + y) - (((z - t) * y) / (a - t));
	tmp = 0.0;
	if (t_2 < -1.3664970889390727e-7)
		tmp = t_1;
	elseif (t_2 < 1.4754293444577233e-239)
		tmp = ((y * (a - z)) - (x * t)) / (a - t);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(y + x), $MachinePrecision] - N[(N[(N[(z - t), $MachinePrecision] * N[(1.0 / N[(a - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x + y), $MachinePrecision] - N[(N[(N[(z - t), $MachinePrecision] * y), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Less[t$95$2, -1.3664970889390727e-7], t$95$1, If[Less[t$95$2, 1.4754293444577233e-239], N[(N[(N[(y * N[(a - z), $MachinePrecision]), $MachinePrecision] - N[(x * t), $MachinePrecision]), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Reproduce

herbie shell --seed 2024338 
(FPCore (x y z t a)
  :name "Graphics.Rendering.Plot.Render.Plot.Axis:renderAxisTick from plot-0.2.3.4, B"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< (- (+ x y) (/ (* (- z t) y) (- a t))) -13664970889390727/100000000000000000000000) (- (+ y x) (* (* (- z t) (/ 1 (- a t))) y)) (if (< (- (+ x y) (/ (* (- z t) y) (- a t))) 14754293444577233/1000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (/ (- (* y (- a z)) (* x t)) (- a t)) (- (+ y x) (* (* (- z t) (/ 1 (- a t))) y)))))

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