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

Percentage Accurate: 77.0% → 88.2%

Time: 8.2s

Alternatives: 7

Speedup: 0.9×

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: 77.0% 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: 88.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(-y, \frac{a - z}{t}, x\right)\\ \mathbf{if}\;t \leq -7.9 \cdot 10^{+102}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 2.65 \cdot 10^{-12}:\\ \;\;\;\;\left(x + y\right) - \frac{z}{a - t} \cdot y\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (- y) (/ (- a z) t) x)))
   (if (<= t -7.9e+102)
     t_1
     (if (<= t 2.65e-12) (- (+ x y) (* (/ z (- a t)) y)) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma(-y, ((a - z) / t), x);
	double tmp;
	if (t <= -7.9e+102) {
		tmp = t_1;
	} else if (t <= 2.65e-12) {
		tmp = (x + y) - ((z / (a - t)) * y);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(-y), Float64(Float64(a - z) / t), x)
	tmp = 0.0
	if (t <= -7.9e+102)
		tmp = t_1;
	elseif (t <= 2.65e-12)
		tmp = Float64(Float64(x + y) - Float64(Float64(z / Float64(a - t)) * y));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[((-y) * N[(N[(a - z), $MachinePrecision] / t), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[t, -7.9e+102], t$95$1, If[LessEqual[t, 2.65e-12], N[(N[(x + y), $MachinePrecision] - N[(N[(z / N[(a - t), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -7.9000000000000003e102 or 2.64999999999999982e-12 < t

   1. Initial program 54.8%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. flip-- N/A

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

      4. associate-*l/ N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. difference-of-squares N/A

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

      8. lift--.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 22.0

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

   4. Applied rewrites 22.0%

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

   5. Taylor expanded in z around inf

      \[\leadsto \color{blue}{-1 \cdot \frac{y \cdot z}{a - t}} \]
   6. 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

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower--.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 24.8

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

   7. Applied rewrites 24.8%

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

   8. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\left(x + -1 \cdot \frac{a \cdot y}{t}\right) - -1 \cdot \frac{y \cdot z}{t}} \]
   9. 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

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

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

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

      6. +-commutative N/A

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

      7. associate-/l* N/A

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

      8. associate-*r* N/A

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

      9. lower-fma.f64 N/A

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

      10. mul-1-neg N/A

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

      11. lower-neg.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower--.f64 87.1

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

   10. Applied rewrites 87.1%

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

   if -7.9000000000000003e102 < t < 2.64999999999999982e-12

   1. Initial program 92.4%

      \[\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \]
   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 94.3

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

   5. Applied rewrites 94.3%

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

Alternative 2: 62.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -4.9 \cdot 10^{-61}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;a \leq 7 \cdot 10^{-305}:\\ \;\;\;\;x\\ \mathbf{elif}\;a \leq 1.35 \cdot 10^{-124}:\\ \;\;\;\;\frac{y}{t} \cdot z\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= a -4.9e-61)
   (+ x y)
   (if (<= a 7e-305) x (if (<= a 1.35e-124) (* (/ y t) z) (+ x y)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (a <= -4.9e-61) {
		tmp = x + y;
	} else if (a <= 7e-305) {
		tmp = x;
	} else if (a <= 1.35e-124) {
		tmp = (y / t) * z;
	} else {
		tmp = x + y;
	}
	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 <= (-4.9d-61)) then
        tmp = x + y
    else if (a <= 7d-305) then
        tmp = x
    else if (a <= 1.35d-124) then
        tmp = (y / t) * z
    else
        tmp = x + y
    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 <= -4.9e-61) {
		tmp = x + y;
	} else if (a <= 7e-305) {
		tmp = x;
	} else if (a <= 1.35e-124) {
		tmp = (y / t) * z;
	} else {
		tmp = x + y;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if a <= -4.9e-61:
		tmp = x + y
	elif a <= 7e-305:
		tmp = x
	elif a <= 1.35e-124:
		tmp = (y / t) * z
	else:
		tmp = x + y
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (a <= -4.9e-61)
		tmp = Float64(x + y);
	elseif (a <= 7e-305)
		tmp = x;
	elseif (a <= 1.35e-124)
		tmp = Float64(Float64(y / t) * z);
	else
		tmp = Float64(x + y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (a <= -4.9e-61)
		tmp = x + y;
	elseif (a <= 7e-305)
		tmp = x;
	elseif (a <= 1.35e-124)
		tmp = (y / t) * z;
	else
		tmp = x + y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[a, -4.9e-61], N[(x + y), $MachinePrecision], If[LessEqual[a, 7e-305], x, If[LessEqual[a, 1.35e-124], N[(N[(y / t), $MachinePrecision] * z), $MachinePrecision], N[(x + y), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if a < -4.90000000000000002e-61 or 1.35000000000000009e-124 < a

   1. Initial program 78.0%

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

   3. Taylor expanded in a around inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 74.1

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

   5. Applied rewrites 74.1%

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

   if -4.90000000000000002e-61 < a < 6.9999999999999996e-305

   1. Initial program 79.2%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. flip-- N/A

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

      4. associate-*l/ N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. difference-of-squares N/A

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

      8. lift--.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 55.3

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

   4. Applied rewrites 55.3%

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

   5. Taylor expanded in z around 0

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

      1. cancel-sign-sub-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. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. +-commutative N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower--.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 52.2

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

   7. Applied rewrites 52.2%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 60.0%

       \[\leadsto 0 + \color{blue}{x} \]
   11. Step-by-step derivation

   12. Applied rewrites 60.0%

       \[\leadsto x \]

   if 6.9999999999999996e-305 < a < 1.35000000000000009e-124

   1. Initial program 63.2%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. flip-- N/A

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

      4. associate-*l/ N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. difference-of-squares N/A

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

      8. lift--.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 47.4

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

   4. Applied rewrites 47.4%

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

   5. Taylor expanded in z around inf

      \[\leadsto \color{blue}{-1 \cdot \frac{y \cdot z}{a - t}} \]
   6. 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

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower--.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 59.8

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

   7. Applied rewrites 59.8%

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

   9. Applied rewrites 58.4%

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

   10. Taylor expanded in a around 0

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

   12. Applied rewrites 54.6%

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

4. Final simplification 67.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -4.9 \cdot 10^{-61}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;a \leq 7 \cdot 10^{-305}:\\ \;\;\;\;x\\ \mathbf{elif}\;a \leq 1.35 \cdot 10^{-124}:\\ \;\;\;\;\frac{y}{t} \cdot z\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 82.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(y, 1 - \frac{z}{a}, x\right)\\ \mathbf{if}\;a \leq -5.1 \cdot 10^{+64}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 2.2 \cdot 10^{-54}:\\ \;\;\;\;\mathsf{fma}\left(-y, \frac{a - z}{t}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma y (- 1.0 (/ z a)) x)))
   (if (<= a -5.1e+64)
     t_1
     (if (<= a 2.2e-54) (fma (- y) (/ (- a z) t) x) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma(y, (1.0 - (z / a)), x);
	double tmp;
	if (a <= -5.1e+64) {
		tmp = t_1;
	} else if (a <= 2.2e-54) {
		tmp = fma(-y, ((a - z) / t), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(y, Float64(1.0 - Float64(z / a)), x)
	tmp = 0.0
	if (a <= -5.1e+64)
		tmp = t_1;
	elseif (a <= 2.2e-54)
		tmp = fma(Float64(-y), Float64(Float64(a - z) / t), x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(y * N[(1.0 - N[(z / a), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[a, -5.1e+64], t$95$1, If[LessEqual[a, 2.2e-54], N[((-y) * N[(N[(a - z), $MachinePrecision] / t), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if a < -5.10000000000000024e64 or 2.2e-54 < a

   1. Initial program 78.6%

      \[\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \]
   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 90.1

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

   5. Applied rewrites 90.1%

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

   if -5.10000000000000024e64 < a < 2.2e-54

   1. Initial program 74.0%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. flip-- N/A

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

      4. associate-*l/ N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. difference-of-squares N/A

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

      8. lift--.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 50.9

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

   4. Applied rewrites 50.9%

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

   5. Taylor expanded in z around inf

      \[\leadsto \color{blue}{-1 \cdot \frac{y \cdot z}{a - t}} \]
   6. 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

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower--.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 43.1

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

   7. Applied rewrites 43.1%

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

   8. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\left(x + -1 \cdot \frac{a \cdot y}{t}\right) - -1 \cdot \frac{y \cdot z}{t}} \]
   9. 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

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

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

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

      6. +-commutative N/A

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

      7. associate-/l* N/A

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

      8. associate-*r* N/A

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

      9. lower-fma.f64 N/A

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

      10. mul-1-neg N/A

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

      11. lower-neg.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower--.f64 86.9

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

   10. Applied rewrites 86.9%

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

Alternative 4: 81.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(y, 1 - \frac{z}{a}, x\right)\\ \mathbf{if}\;a \leq -5.1 \cdot 10^{+64}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 8.2 \cdot 10^{-32}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z}{t}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma y (- 1.0 (/ z a)) x)))
   (if (<= a -5.1e+64) t_1 (if (<= a 8.2e-32) (fma y (/ z t) x) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma(y, (1.0 - (z / a)), x);
	double tmp;
	if (a <= -5.1e+64) {
		tmp = t_1;
	} else if (a <= 8.2e-32) {
		tmp = fma(y, (z / t), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(y, Float64(1.0 - Float64(z / a)), x)
	tmp = 0.0
	if (a <= -5.1e+64)
		tmp = t_1;
	elseif (a <= 8.2e-32)
		tmp = fma(y, Float64(z / t), x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(y * N[(1.0 - N[(z / a), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[a, -5.1e+64], t$95$1, If[LessEqual[a, 8.2e-32], N[(y * N[(z / t), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if a < -5.10000000000000024e64 or 8.1999999999999995e-32 < a

   1. Initial program 78.0%

      \[\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \]
   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 90.7

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

   5. Applied rewrites 90.7%

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

   if -5.10000000000000024e64 < a < 8.1999999999999995e-32

   1. Initial program 74.6%

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

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\left(x + \left(-1 \cdot \frac{a \cdot y}{t} + -1 \cdot \frac{a \cdot \left(-1 \cdot \left(y \cdot z\right) - -1 \cdot \left(a \cdot y\right)\right)}{{t}^{2}}\right)\right) - -1 \cdot \frac{y \cdot z}{t}} \]

   5. Applied rewrites 80.3%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 79.7%

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

   10. Applied rewrites 85.7%

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

Alternative 5: 76.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -6.5 \cdot 10^{+65}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;a \leq 5.5 \cdot 10^{-30}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z}{t}, x\right)\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= a -6.5e+65) (+ x y) (if (<= a 5.5e-30) (fma y (/ z t) x) (+ x y))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (a <= -6.5e+65) {
		tmp = x + y;
	} else if (a <= 5.5e-30) {
		tmp = fma(y, (z / t), x);
	} else {
		tmp = x + y;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (a <= -6.5e+65)
		tmp = Float64(x + y);
	elseif (a <= 5.5e-30)
		tmp = fma(y, Float64(z / t), x);
	else
		tmp = Float64(x + y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[a, -6.5e+65], N[(x + y), $MachinePrecision], If[LessEqual[a, 5.5e-30], N[(y * N[(z / t), $MachinePrecision] + x), $MachinePrecision], N[(x + y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if a < -6.5000000000000003e65 or 5.49999999999999976e-30 < a

   1. Initial program 78.0%

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

   3. Taylor expanded in a around inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 83.2

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

   5. Applied rewrites 83.2%

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

   if -6.5000000000000003e65 < a < 5.49999999999999976e-30

   1. Initial program 74.6%

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

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\left(x + \left(-1 \cdot \frac{a \cdot y}{t} + -1 \cdot \frac{a \cdot \left(-1 \cdot \left(y \cdot z\right) - -1 \cdot \left(a \cdot y\right)\right)}{{t}^{2}}\right)\right) - -1 \cdot \frac{y \cdot z}{t}} \]

   5. Applied rewrites 80.3%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 79.7%

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

   10. Applied rewrites 85.7%

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

4. Final simplification 84.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -6.5 \cdot 10^{+65}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;a \leq 5.5 \cdot 10^{-30}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z}{t}, x\right)\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 64.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -4.9 \cdot 10^{-61}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;a \leq 6 \cdot 10^{-63}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= a -4.9e-61) (+ x y) (if (<= a 6e-63) x (+ x y))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (a <= -4.9e-61) {
		tmp = x + y;
	} else if (a <= 6e-63) {
		tmp = x;
	} else {
		tmp = x + y;
	}
	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 <= (-4.9d-61)) then
        tmp = x + y
    else if (a <= 6d-63) then
        tmp = x
    else
        tmp = x + y
    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 <= -4.9e-61) {
		tmp = x + y;
	} else if (a <= 6e-63) {
		tmp = x;
	} else {
		tmp = x + y;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if a <= -4.9e-61:
		tmp = x + y
	elif a <= 6e-63:
		tmp = x
	else:
		tmp = x + y
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (a <= -4.9e-61)
		tmp = Float64(x + y);
	elseif (a <= 6e-63)
		tmp = x;
	else
		tmp = Float64(x + y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (a <= -4.9e-61)
		tmp = x + y;
	elseif (a <= 6e-63)
		tmp = x;
	else
		tmp = x + y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[a, -4.9e-61], N[(x + y), $MachinePrecision], If[LessEqual[a, 6e-63], x, N[(x + y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if a < -4.90000000000000002e-61 or 5.99999999999999959e-63 < a

   1. Initial program 78.5%

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

   3. Taylor expanded in a around inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 78.1

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

   5. Applied rewrites 78.1%

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

   if -4.90000000000000002e-61 < a < 5.99999999999999959e-63

   1. Initial program 73.4%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. flip-- N/A

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

      4. associate-*l/ N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. difference-of-squares N/A

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

      8. lift--.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 51.6

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

   4. Applied rewrites 51.6%

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

   5. Taylor expanded in z around 0

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

      1. cancel-sign-sub-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. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. +-commutative N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower--.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 43.1

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

   7. Applied rewrites 43.1%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 51.7%

       \[\leadsto 0 + \color{blue}{x} \]
   11. Step-by-step derivation

   12. Applied rewrites 51.7%

       \[\leadsto x \]
3. Recombined 2 regimes into one program.

4. Final simplification 66.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -4.9 \cdot 10^{-61}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;a \leq 6 \cdot 10^{-63}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 51.5% accurate, 29.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 76.1%

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

   1. lift-*.f64 N/A

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

   2. lift--.f64 N/A

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

   3. flip-- N/A

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

   4. associate-*l/ N/A

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

   5. lower-/.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. difference-of-squares N/A

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

   8. lift--.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. +-commutative N/A

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

   11. lower-+.f64 N/A

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

   12. +-commutative N/A

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

   13. lower-+.f64 52.2

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

4. Applied rewrites 52.2%

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

5. Taylor expanded in z around 0

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

   1. cancel-sign-sub-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. metadata-eval N/A

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

   3. *-lft-identity N/A

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

   4. +-commutative N/A

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

   5. associate-/l* N/A

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

   6. *-commutative N/A

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

   7. lower-fma.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. lower--.f64 N/A

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

   10. +-commutative N/A

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

   11. lower-+.f64 62.3

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

7. Applied rewrites 62.3%

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

8. Taylor expanded in a around 0

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

10. Applied rewrites 52.0%

    \[\leadsto 0 + \color{blue}{x} \]
11. Step-by-step derivation

12. Applied rewrites 52.0%

    \[\leadsto x \]
13. Add Preprocessing

Developer Target 1: 88.1% 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 2024277 
(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))))