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

Percentage Accurate: 76.6% → 89.4%

Time: 7.4s

Alternatives: 11

Speedup: 1.0×

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: 76.6% 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: 89.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -9.2 \cdot 10^{-39} \lor \neg \left(a \leq 1.8 \cdot 10^{-130}\right):\\ \;\;\;\;\mathsf{fma}\left(1 - \frac{z - t}{a - t}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{t}, z - a, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= a -9.2e-39) (not (<= a 1.8e-130)))
   (fma (- 1.0 (/ (- z t) (- a t))) y x)
   (fma (/ y t) (- z a) x)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a <= -9.2e-39) || !(a <= 1.8e-130)) {
		tmp = fma((1.0 - ((z - t) / (a - t))), y, x);
	} else {
		tmp = fma((y / t), (z - a), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((a <= -9.2e-39) || !(a <= 1.8e-130))
		tmp = fma(Float64(1.0 - Float64(Float64(z - t) / Float64(a - t))), y, x);
	else
		tmp = fma(Float64(y / t), Float64(z - a), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[a, -9.2e-39], N[Not[LessEqual[a, 1.8e-130]], $MachinePrecision]], N[(N[(1.0 - N[(N[(z - t), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * y + x), $MachinePrecision], N[(N[(y / t), $MachinePrecision] * N[(z - a), $MachinePrecision] + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -9.20000000000000033e-39 or 1.8000000000000001e-130 < a

   1. Initial program 83.8%

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

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

      4. associate-/l* N/A

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

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

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

      6. unsub-neg N/A

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

      7. mul-1-neg N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. mul-1-neg N/A

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

      11. unsub-neg N/A

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

      12. lower--.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower--.f64 N/A

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

      15. lower--.f64 95.9

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

   5. Applied rewrites 95.9%

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

   if -9.20000000000000033e-39 < a < 1.8000000000000001e-130

   1. Initial program 71.2%

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

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

      5. associate-*r/ N/A

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

      6. mul-1-neg N/A

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

      7. sub-neg N/A

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

      8. +-commutative N/A

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

      9. distribute-neg-in N/A

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

      10. remove-double-neg N/A

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

      11. sub-neg N/A

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

      12. div-sub N/A

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

      13. *-commutative N/A

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

      14. associate-/l* N/A

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

      15. associate-/l* N/A

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

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

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

      17. lower-fma.f64 N/A

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

   5. Applied rewrites 86.3%

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

4. Final simplification 92.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -9.2 \cdot 10^{-39} \lor \neg \left(a \leq 1.8 \cdot 10^{-130}\right):\\ \;\;\;\;\mathsf{fma}\left(1 - \frac{z - t}{a - t}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{t}, z - a, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 86.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -6 \cdot 10^{-38} \lor \neg \left(a \leq 4.5 \cdot 10^{-129}\right):\\ \;\;\;\;\left(x + y\right) - \frac{z}{a - t} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{t}, z - a, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= a -6e-38) (not (<= a 4.5e-129)))
   (- (+ x y) (* (/ z (- a t)) y))
   (fma (/ y t) (- z a) x)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a <= -6e-38) || !(a <= 4.5e-129)) {
		tmp = (x + y) - ((z / (a - t)) * y);
	} else {
		tmp = fma((y / t), (z - a), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((a <= -6e-38) || !(a <= 4.5e-129))
		tmp = Float64(Float64(x + y) - Float64(Float64(z / Float64(a - t)) * y));
	else
		tmp = fma(Float64(y / t), Float64(z - a), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[a, -6e-38], N[Not[LessEqual[a, 4.5e-129]], $MachinePrecision]], N[(N[(x + y), $MachinePrecision] - N[(N[(z / N[(a - t), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision], N[(N[(y / t), $MachinePrecision] * N[(z - a), $MachinePrecision] + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -5.99999999999999977e-38 or 4.50000000000000031e-129 < a

   1. Initial program 83.8%

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

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

   5. Applied rewrites 91.1%

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

   if -5.99999999999999977e-38 < a < 4.50000000000000031e-129

   1. Initial program 71.2%

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

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

      5. associate-*r/ N/A

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

      6. mul-1-neg N/A

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

      7. sub-neg N/A

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

      8. +-commutative N/A

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

      9. distribute-neg-in N/A

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

      10. remove-double-neg N/A

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

      11. sub-neg N/A

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

      12. div-sub N/A

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

      13. *-commutative N/A

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

      14. associate-/l* N/A

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

      15. associate-/l* N/A

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

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

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

      17. lower-fma.f64 N/A

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

   5. Applied rewrites 86.3%

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

4. Final simplification 89.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -6 \cdot 10^{-38} \lor \neg \left(a \leq 4.5 \cdot 10^{-129}\right):\\ \;\;\;\;\left(x + y\right) - \frac{z}{a - t} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{t}, z - a, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 76.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.9 \cdot 10^{+52}:\\ \;\;\;\;\mathsf{fma}\left(y, 1, x\right)\\ \mathbf{elif}\;a \leq 3 \cdot 10^{-92}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, y, x\right)\\ \mathbf{elif}\;a \leq 6.8 \cdot 10^{+62}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{-z}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, 1, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= a -1.9e+52)
   (fma y 1.0 x)
   (if (<= a 3e-92)
     (fma (/ z t) y x)
     (if (<= a 6.8e+62) (fma y (/ (- z) a) x) (fma y 1.0 x)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (a <= -1.9e+52) {
		tmp = fma(y, 1.0, x);
	} else if (a <= 3e-92) {
		tmp = fma((z / t), y, x);
	} else if (a <= 6.8e+62) {
		tmp = fma(y, (-z / a), x);
	} else {
		tmp = fma(y, 1.0, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (a <= -1.9e+52)
		tmp = fma(y, 1.0, x);
	elseif (a <= 3e-92)
		tmp = fma(Float64(z / t), y, x);
	elseif (a <= 6.8e+62)
		tmp = fma(y, Float64(Float64(-z) / a), x);
	else
		tmp = fma(y, 1.0, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[a, -1.9e+52], N[(y * 1.0 + x), $MachinePrecision], If[LessEqual[a, 3e-92], N[(N[(z / t), $MachinePrecision] * y + x), $MachinePrecision], If[LessEqual[a, 6.8e+62], N[(y * N[((-z) / a), $MachinePrecision] + x), $MachinePrecision], N[(y * 1.0 + x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if a < -1.9e52 or 6.80000000000000028e62 < a

   1. Initial program 82.4%

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

   3. Taylor expanded in t around 0

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. *-rgt-identity N/A

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

      4. associate-/l* N/A

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

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

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

      6. lower-fma.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-/.f64 91.1

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

   5. Applied rewrites 91.1%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 82.5%

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

   if -1.9e52 < a < 3.00000000000000013e-92

   1. Initial program 73.0%

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

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

      4. associate-/l* N/A

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

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

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

      6. unsub-neg N/A

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

      7. mul-1-neg N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. mul-1-neg N/A

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

      11. unsub-neg N/A

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

      12. lower--.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower--.f64 N/A

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

      15. lower--.f64 77.5

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

   5. Applied rewrites 77.5%

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

   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 73.4%

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

   if 3.00000000000000013e-92 < a < 6.80000000000000028e62

   1. Initial program 92.9%

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

   3. Taylor expanded in t around 0

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. *-rgt-identity N/A

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

      4. associate-/l* N/A

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

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

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

      6. lower-fma.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-/.f64 74.5

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

   5. Applied rewrites 74.5%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 76.7%

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

Alternative 4: 82.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.45 \cdot 10^{-29} \lor \neg \left(a \leq 5 \cdot 10^{-129}\right):\\ \;\;\;\;\mathsf{fma}\left(y, 1 - \frac{z}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{t}, z - a, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= a -1.45e-29) (not (<= a 5e-129)))
   (fma y (- 1.0 (/ z a)) x)
   (fma (/ y t) (- z a) x)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a <= -1.45e-29) || !(a <= 5e-129)) {
		tmp = fma(y, (1.0 - (z / a)), x);
	} else {
		tmp = fma((y / t), (z - a), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((a <= -1.45e-29) || !(a <= 5e-129))
		tmp = fma(y, Float64(1.0 - Float64(z / a)), x);
	else
		tmp = fma(Float64(y / t), Float64(z - a), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[a, -1.45e-29], N[Not[LessEqual[a, 5e-129]], $MachinePrecision]], N[(y * N[(1.0 - N[(z / a), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], N[(N[(y / t), $MachinePrecision] * N[(z - a), $MachinePrecision] + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -1.45000000000000012e-29 or 5.00000000000000027e-129 < a

   1. Initial program 83.8%

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

   3. Taylor expanded in t around 0

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. *-rgt-identity N/A

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

      4. associate-/l* N/A

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

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

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

      6. lower-fma.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-/.f64 86.1

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

   5. Applied rewrites 86.1%

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

   if -1.45000000000000012e-29 < a < 5.00000000000000027e-129

   1. Initial program 71.2%

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

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

      5. associate-*r/ N/A

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

      6. mul-1-neg N/A

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

      7. sub-neg N/A

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

      8. +-commutative N/A

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

      9. distribute-neg-in N/A

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

      10. remove-double-neg N/A

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

      11. sub-neg N/A

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

      12. div-sub N/A

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

      13. *-commutative N/A

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

      14. associate-/l* N/A

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

      15. associate-/l* N/A

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

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

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

      17. lower-fma.f64 N/A

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

   5. Applied rewrites 86.3%

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

4. Final simplification 86.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.45 \cdot 10^{-29} \lor \neg \left(a \leq 5 \cdot 10^{-129}\right):\\ \;\;\;\;\mathsf{fma}\left(y, 1 - \frac{z}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{t}, z - a, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 81.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.45 \cdot 10^{-29} \lor \neg \left(a \leq 1.05 \cdot 10^{-111}\right):\\ \;\;\;\;\mathsf{fma}\left(y, 1 - \frac{z}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, y, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= a -1.45e-29) (not (<= a 1.05e-111)))
   (fma y (- 1.0 (/ z a)) x)
   (fma (/ z t) y x)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a <= -1.45e-29) || !(a <= 1.05e-111)) {
		tmp = fma(y, (1.0 - (z / a)), x);
	} else {
		tmp = fma((z / t), y, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((a <= -1.45e-29) || !(a <= 1.05e-111))
		tmp = fma(y, Float64(1.0 - Float64(z / a)), x);
	else
		tmp = fma(Float64(z / t), y, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[a, -1.45e-29], N[Not[LessEqual[a, 1.05e-111]], $MachinePrecision]], N[(y * N[(1.0 - N[(z / a), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], N[(N[(z / t), $MachinePrecision] * y + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -1.45000000000000012e-29 or 1.0499999999999999e-111 < a

   1. Initial program 84.0%

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

   3. Taylor expanded in t around 0

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. *-rgt-identity N/A

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

      4. associate-/l* N/A

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

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

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

      6. lower-fma.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-/.f64 86.4

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

   5. Applied rewrites 86.4%

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

   if -1.45000000000000012e-29 < a < 1.0499999999999999e-111

   1. Initial program 71.4%

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

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

      4. associate-/l* N/A

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

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

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

      6. unsub-neg N/A

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

      7. mul-1-neg N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. mul-1-neg N/A

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

      11. unsub-neg N/A

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

      12. lower--.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower--.f64 N/A

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

      15. lower--.f64 73.8

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

   5. Applied rewrites 73.8%

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

   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 76.0%

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

4. Final simplification 82.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.45 \cdot 10^{-29} \lor \neg \left(a \leq 1.05 \cdot 10^{-111}\right):\\ \;\;\;\;\mathsf{fma}\left(y, 1 - \frac{z}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, y, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 82.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.45 \cdot 10^{-29}:\\ \;\;\;\;\left(x + y\right) - y \cdot \frac{z}{a}\\ \mathbf{elif}\;a \leq 5 \cdot 10^{-129}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{t}, z - a, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, 1 - \frac{z}{a}, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= a -1.45e-29)
   (- (+ x y) (* y (/ z a)))
   (if (<= a 5e-129) (fma (/ y t) (- z a) x) (fma y (- 1.0 (/ z a)) x))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (a <= -1.45e-29) {
		tmp = (x + y) - (y * (z / a));
	} else if (a <= 5e-129) {
		tmp = fma((y / t), (z - a), x);
	} else {
		tmp = fma(y, (1.0 - (z / a)), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (a <= -1.45e-29)
		tmp = Float64(Float64(x + y) - Float64(y * Float64(z / a)));
	elseif (a <= 5e-129)
		tmp = fma(Float64(y / t), Float64(z - a), x);
	else
		tmp = fma(y, Float64(1.0 - Float64(z / a)), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[a, -1.45e-29], N[(N[(x + y), $MachinePrecision] - N[(y * N[(z / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 5e-129], N[(N[(y / t), $MachinePrecision] * N[(z - a), $MachinePrecision] + x), $MachinePrecision], N[(y * N[(1.0 - N[(z / a), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -1.45000000000000012e-29

   1. Initial program 77.2%

      \[\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \]
   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 88.4

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

   5. Applied rewrites 88.4%

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

   6. Taylor expanded in t around 0

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 83.7

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

   8. Applied rewrites 83.7%

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

   if -1.45000000000000012e-29 < a < 5.00000000000000027e-129

   1. Initial program 71.2%

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

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

      5. associate-*r/ N/A

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

      6. mul-1-neg N/A

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

      7. sub-neg N/A

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

      8. +-commutative N/A

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

      9. distribute-neg-in N/A

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

      10. remove-double-neg N/A

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

      11. sub-neg N/A

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

      12. div-sub N/A

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

      13. *-commutative N/A

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

      14. associate-/l* N/A

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

      15. associate-/l* N/A

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

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

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

      17. lower-fma.f64 N/A

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

   5. Applied rewrites 86.3%

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

   if 5.00000000000000027e-129 < a

   1. Initial program 88.8%

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

   3. Taylor expanded in t around 0

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. *-rgt-identity N/A

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

      4. associate-/l* N/A

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

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

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

      6. lower-fma.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-/.f64 88.0

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

   5. Applied rewrites 88.0%

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

Alternative 7: 77.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.9 \cdot 10^{+52} \lor \neg \left(a \leq 102000000000\right):\\ \;\;\;\;\mathsf{fma}\left(y, 1, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, y, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= a -1.9e+52) (not (<= a 102000000000.0)))
   (fma y 1.0 x)
   (fma (/ z t) y x)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a <= -1.9e+52) || !(a <= 102000000000.0)) {
		tmp = fma(y, 1.0, x);
	} else {
		tmp = fma((z / t), y, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((a <= -1.9e+52) || !(a <= 102000000000.0))
		tmp = fma(y, 1.0, x);
	else
		tmp = fma(Float64(z / t), y, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[a, -1.9e+52], N[Not[LessEqual[a, 102000000000.0]], $MachinePrecision]], N[(y * 1.0 + x), $MachinePrecision], N[(N[(z / t), $MachinePrecision] * y + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -1.9e52 or 1.02e11 < a

   1. Initial program 83.4%

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

   3. Taylor expanded in t around 0

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. *-rgt-identity N/A

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

      4. associate-/l* N/A

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

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

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

      6. lower-fma.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-/.f64 91.7

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

   5. Applied rewrites 91.7%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 81.0%

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

   if -1.9e52 < a < 1.02e11

   1. Initial program 75.6%

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

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

      4. associate-/l* N/A

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

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

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

      6. unsub-neg N/A

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

      7. mul-1-neg N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. mul-1-neg N/A

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

      11. unsub-neg N/A

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

      12. lower--.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower--.f64 N/A

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

      15. lower--.f64 79.4

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

   5. Applied rewrites 79.4%

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

   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 71.7%

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

4. Final simplification 76.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.9 \cdot 10^{+52} \lor \neg \left(a \leq 102000000000\right):\\ \;\;\;\;\mathsf{fma}\left(y, 1, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, y, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 60.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 9.8 \cdot 10^{+215}:\\ \;\;\;\;\mathsf{fma}\left(y, 1, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{t} \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= y 9.8e+215) (fma y 1.0 x) (* (/ z t) y)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (y <= 9.8e+215) {
		tmp = fma(y, 1.0, x);
	} else {
		tmp = (z / t) * y;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (y <= 9.8e+215)
		tmp = fma(y, 1.0, x);
	else
		tmp = Float64(Float64(z / t) * y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[y, 9.8e+215], N[(y * 1.0 + x), $MachinePrecision], N[(N[(z / t), $MachinePrecision] * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 9.8000000000000003e215

   1. Initial program 81.4%

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

   3. Taylor expanded in t around 0

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. *-rgt-identity N/A

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

      4. associate-/l* N/A

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

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

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

      6. lower-fma.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-/.f64 69.4

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

   5. Applied rewrites 69.4%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 67.3%

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

   if 9.8000000000000003e215 < y

   1. Initial program 46.3%

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

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

      5. associate-*r/ N/A

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

      6. mul-1-neg N/A

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

      7. sub-neg N/A

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

      8. +-commutative N/A

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

      9. distribute-neg-in N/A

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

      10. remove-double-neg N/A

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

      11. sub-neg N/A

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

      12. div-sub N/A

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

      13. *-commutative N/A

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

      14. associate-/l* N/A

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

      15. associate-/l* N/A

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

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

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

      17. lower-fma.f64 N/A

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

   5. Applied rewrites 69.6%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 58.0%

      \[\leadsto \frac{z}{t} \cdot \color{blue}{y} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 62.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 1.3 \cdot 10^{+205}:\\ \;\;\;\;\mathsf{fma}\left(y, 1, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, -1 + 1, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t 1.3e+205) (fma y 1.0 x) (fma y (+ -1.0 1.0) x)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 1.3e+205) {
		tmp = fma(y, 1.0, x);
	} else {
		tmp = fma(y, (-1.0 + 1.0), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= 1.3e+205)
		tmp = fma(y, 1.0, x);
	else
		tmp = fma(y, Float64(-1.0 + 1.0), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, 1.3e+205], N[(y * 1.0 + x), $MachinePrecision], N[(y * N[(-1.0 + 1.0), $MachinePrecision] + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 1.2999999999999999e205

   1. Initial program 82.2%

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

   3. Taylor expanded in t around 0

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. *-rgt-identity N/A

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

      4. associate-/l* N/A

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

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

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

      6. lower-fma.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-/.f64 70.5

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

   5. Applied rewrites 70.5%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 65.7%

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

   if 1.2999999999999999e205 < t

   1. Initial program 43.7%

      \[\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \]
   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. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. sub-neg N/A

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

      4. *-rgt-identity N/A

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

      5. mul-1-neg N/A

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

      6. remove-double-neg N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. distribute-lft-in N/A

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

      10. lower-fma.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower--.f64 68.7

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

   5. Applied rewrites 68.7%

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

   6. Taylor expanded in t around inf

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

   8. Applied rewrites 68.7%

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

Alternative 10: 60.8% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(y, 1, x\right) \end{array} \]

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 79.2%

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

3. Taylor expanded in t around 0

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

   1. associate--l+ N/A

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

   2. +-commutative N/A

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

   3. *-rgt-identity N/A

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

   4. associate-/l* N/A

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

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

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

   6. lower-fma.f64 N/A

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

   7. lower--.f64 N/A

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

   8. lower-/.f64 67.8

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

5. Applied rewrites 67.8%

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

6. Taylor expanded in z around 0

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

8. Applied rewrites 64.0%

   \[\leadsto \mathsf{fma}\left(y, 1, x\right) \]
9. Add Preprocessing

Alternative 11: 18.6% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ 1 \cdot y \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_, a_] := N[(1.0 * y), $MachinePrecision]

Derivation

1. Initial program 79.2%

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

3. Taylor expanded in t around 0

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

   1. associate--l+ N/A

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

   2. +-commutative N/A

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

   3. *-rgt-identity N/A

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

   4. associate-/l* N/A

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

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

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

   6. lower-fma.f64 N/A

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

   7. lower--.f64 N/A

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

   8. lower-/.f64 67.8

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

5. Applied rewrites 67.8%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 29.2%

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

9. Taylor expanded in z around 0

   \[\leadsto 1 \cdot y \]
10. Step-by-step derivation

11. Applied rewrites 17.8%

    \[\leadsto 1 \cdot y \]
12. 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 2024324 
(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))))