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

Percentage Accurate: 85.5% → 98.2%

Time: 5.4s

Alternatives: 8

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 85.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 86.8%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. associate-/l* N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. lower-/.f64 98.4

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

4. Applied rewrites 98.4%

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

Alternative 2: 79.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -4.7 \cdot 10^{-169} \lor \neg \left(z \leq 2.35 \cdot 10^{-154}\right):\\ \;\;\;\;\mathsf{fma}\left(1 - \frac{y}{z}, t, x\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{\left(y - z\right) \cdot t}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= z -4.7e-169) (not (<= z 2.35e-154)))
   (fma (- 1.0 (/ y z)) t x)
   (+ x (/ (* (- y z) t) a))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z <= -4.7e-169) || !(z <= 2.35e-154)) {
		tmp = fma((1.0 - (y / z)), t, x);
	} else {
		tmp = x + (((y - z) * t) / a);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((z <= -4.7e-169) || !(z <= 2.35e-154))
		tmp = fma(Float64(1.0 - Float64(y / z)), t, x);
	else
		tmp = Float64(x + Float64(Float64(Float64(y - z) * t) / a));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[z, -4.7e-169], N[Not[LessEqual[z, 2.35e-154]], $MachinePrecision]], N[(N[(1.0 - N[(y / z), $MachinePrecision]), $MachinePrecision] * t + x), $MachinePrecision], N[(x + N[(N[(N[(y - z), $MachinePrecision] * t), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -4.6999999999999999e-169 or 2.3500000000000001e-154 < z

   1. Initial program 84.1%

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

   3. Taylor expanded in a around 0

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

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

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. associate-/l* N/A

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

      5. div-sub N/A

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

      6. *-inverses N/A

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

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

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

      8. *-commutative N/A

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

      9. associate-/l* N/A

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

      10. *-lft-identity N/A

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

      11. associate-+l- N/A

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

      12. *-lft-identity N/A

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

      13. metadata-eval N/A

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

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

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

      15. lower-+.f64 N/A

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

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

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

      17. metadata-eval N/A

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

      18. *-lft-identity N/A

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

      19. lower--.f64 N/A

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

      20. lower-/.f64 N/A

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

      21. lower-*.f64 85.3

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

   5. Applied rewrites 85.3%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 86.7%

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

   if -4.6999999999999999e-169 < z < 2.3500000000000001e-154

   1. Initial program 98.0%

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

   3. Taylor expanded in a around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 92.1

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

   5. Applied rewrites 92.1%

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

4. Final simplification 87.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4.7 \cdot 10^{-169} \lor \neg \left(z \leq 2.35 \cdot 10^{-154}\right):\\ \;\;\;\;\mathsf{fma}\left(1 - \frac{y}{z}, t, x\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{\left(y - z\right) \cdot t}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 78.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.32 \cdot 10^{-21} \lor \neg \left(a \leq 1.6 \cdot 10^{-129}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{a - z}, -t, x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x - \frac{t \cdot y}{z}\right) + t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= a -1.32e-21) (not (<= a 1.6e-129)))
   (fma (/ z (- a z)) (- t) x)
   (+ (- x (/ (* t y) z)) t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a <= -1.32e-21) || !(a <= 1.6e-129)) {
		tmp = fma((z / (a - z)), -t, x);
	} else {
		tmp = (x - ((t * y) / z)) + t;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((a <= -1.32e-21) || !(a <= 1.6e-129))
		tmp = fma(Float64(z / Float64(a - z)), Float64(-t), x);
	else
		tmp = Float64(Float64(x - Float64(Float64(t * y) / z)) + t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[a, -1.32e-21], N[Not[LessEqual[a, 1.6e-129]], $MachinePrecision]], N[(N[(z / N[(a - z), $MachinePrecision]), $MachinePrecision] * (-t) + x), $MachinePrecision], N[(N[(x - N[(N[(t * y), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -1.32e-21 or 1.6000000000000001e-129 < a

   1. Initial program 82.7%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. distribute-rgt-neg-in N/A

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

      6. mul-1-neg N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower--.f64 N/A

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

      10. mul-1-neg N/A

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

      11. lower-neg.f64 83.3

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

   5. Applied rewrites 83.3%

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

   if -1.32e-21 < a < 1.6000000000000001e-129

   1. Initial program 92.2%

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

   3. Taylor expanded in a around 0

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

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

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. associate-/l* N/A

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

      5. div-sub N/A

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

      6. *-inverses N/A

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

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

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

      8. *-commutative N/A

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

      9. associate-/l* N/A

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

      10. *-lft-identity N/A

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

      11. associate-+l- N/A

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

      12. *-lft-identity N/A

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

      13. metadata-eval N/A

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

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

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

      15. lower-+.f64 N/A

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

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

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

      17. metadata-eval N/A

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

      18. *-lft-identity N/A

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

      19. lower--.f64 N/A

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

      20. lower-/.f64 N/A

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

      21. lower-*.f64 92.3

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

   5. Applied rewrites 92.3%

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

4. Final simplification 87.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.32 \cdot 10^{-21} \lor \neg \left(a \leq 1.6 \cdot 10^{-129}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{a - z}, -t, x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x - \frac{t \cdot y}{z}\right) + t\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 79.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -4.7 \cdot 10^{-169} \lor \neg \left(z \leq 2.35 \cdot 10^{-154}\right):\\ \;\;\;\;\mathsf{fma}\left(1 - \frac{y}{z}, t, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y - z, \frac{t}{a}, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= z -4.7e-169) (not (<= z 2.35e-154)))
   (fma (- 1.0 (/ y z)) t x)
   (fma (- y z) (/ t a) x)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z <= -4.7e-169) || !(z <= 2.35e-154)) {
		tmp = fma((1.0 - (y / z)), t, x);
	} else {
		tmp = fma((y - z), (t / a), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((z <= -4.7e-169) || !(z <= 2.35e-154))
		tmp = fma(Float64(1.0 - Float64(y / z)), t, x);
	else
		tmp = fma(Float64(y - z), Float64(t / a), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[z, -4.7e-169], N[Not[LessEqual[z, 2.35e-154]], $MachinePrecision]], N[(N[(1.0 - N[(y / z), $MachinePrecision]), $MachinePrecision] * t + x), $MachinePrecision], N[(N[(y - z), $MachinePrecision] * N[(t / a), $MachinePrecision] + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -4.6999999999999999e-169 or 2.3500000000000001e-154 < z

   1. Initial program 84.1%

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

   3. Taylor expanded in a around 0

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

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

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. associate-/l* N/A

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

      5. div-sub N/A

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

      6. *-inverses N/A

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

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

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

      8. *-commutative N/A

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

      9. associate-/l* N/A

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

      10. *-lft-identity N/A

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

      11. associate-+l- N/A

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

      12. *-lft-identity N/A

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

      13. metadata-eval N/A

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

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

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

      15. lower-+.f64 N/A

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

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

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

      17. metadata-eval N/A

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

      18. *-lft-identity N/A

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

      19. lower--.f64 N/A

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

      20. lower-/.f64 N/A

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

      21. lower-*.f64 85.3

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

   5. Applied rewrites 85.3%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 86.7%

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

   if -4.6999999999999999e-169 < z < 2.3500000000000001e-154

   1. Initial program 98.0%

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

   3. Taylor expanded in a around inf

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 88.3

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

   5. Applied rewrites 88.3%

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

4. Final simplification 87.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4.7 \cdot 10^{-169} \lor \neg \left(z \leq 2.35 \cdot 10^{-154}\right):\\ \;\;\;\;\mathsf{fma}\left(1 - \frac{y}{z}, t, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y - z, \frac{t}{a}, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 78.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -4.7 \cdot 10^{-169} \lor \neg \left(z \leq 2.35 \cdot 10^{-154}\right):\\ \;\;\;\;\mathsf{fma}\left(1 - \frac{y}{z}, t, x\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{t \cdot y}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= z -4.7e-169) (not (<= z 2.35e-154)))
   (fma (- 1.0 (/ y z)) t x)
   (+ x (/ (* t y) a))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z <= -4.7e-169) || !(z <= 2.35e-154)) {
		tmp = fma((1.0 - (y / z)), t, x);
	} else {
		tmp = x + ((t * y) / a);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((z <= -4.7e-169) || !(z <= 2.35e-154))
		tmp = fma(Float64(1.0 - Float64(y / z)), t, x);
	else
		tmp = Float64(x + Float64(Float64(t * y) / a));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[z, -4.7e-169], N[Not[LessEqual[z, 2.35e-154]], $MachinePrecision]], N[(N[(1.0 - N[(y / z), $MachinePrecision]), $MachinePrecision] * t + x), $MachinePrecision], N[(x + N[(N[(t * y), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -4.6999999999999999e-169 or 2.3500000000000001e-154 < z

   1. Initial program 84.1%

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

   3. Taylor expanded in a around 0

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

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

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. associate-/l* N/A

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

      5. div-sub N/A

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

      6. *-inverses N/A

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

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

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

      8. *-commutative N/A

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

      9. associate-/l* N/A

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

      10. *-lft-identity N/A

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

      11. associate-+l- N/A

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

      12. *-lft-identity N/A

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

      13. metadata-eval N/A

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

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

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

      15. lower-+.f64 N/A

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

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

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

      17. metadata-eval N/A

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

      18. *-lft-identity N/A

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

      19. lower--.f64 N/A

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

      20. lower-/.f64 N/A

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

      21. lower-*.f64 85.3

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

   5. Applied rewrites 85.3%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 86.7%

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

   if -4.6999999999999999e-169 < z < 2.3500000000000001e-154

   1. Initial program 98.0%

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

   3. Taylor expanded in z around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 84.0

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

   5. Applied rewrites 84.0%

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

4. Final simplification 86.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4.7 \cdot 10^{-169} \lor \neg \left(z \leq 2.35 \cdot 10^{-154}\right):\\ \;\;\;\;\mathsf{fma}\left(1 - \frac{y}{z}, t, x\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{t \cdot y}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 77.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3.4 \cdot 10^{+20} \lor \neg \left(z \leq 6.5 \cdot 10^{+25}\right):\\ \;\;\;\;t + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= z -3.4e+20) (not (<= z 6.5e+25))) (+ t x) (fma (/ y a) t x)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z <= -3.4e+20) || !(z <= 6.5e+25)) {
		tmp = t + x;
	} else {
		tmp = fma((y / a), t, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((z <= -3.4e+20) || !(z <= 6.5e+25))
		tmp = Float64(t + x);
	else
		tmp = fma(Float64(y / a), t, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[z, -3.4e+20], N[Not[LessEqual[z, 6.5e+25]], $MachinePrecision]], N[(t + x), $MachinePrecision], N[(N[(y / a), $MachinePrecision] * t + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -3.4e20 or 6.50000000000000005e25 < z

   1. Initial program 75.5%

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

   3. Taylor expanded in z around inf

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

      1. lower-+.f64 88.6

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

   5. Applied rewrites 88.6%

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

   if -3.4e20 < z < 6.50000000000000005e25

   1. Initial program 97.6%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 68.6

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

   5. Applied rewrites 68.6%

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

4. Final simplification 78.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.4 \cdot 10^{+20} \lor \neg \left(z \leq 6.5 \cdot 10^{+25}\right):\\ \;\;\;\;t + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 60.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -5.5 \cdot 10^{+117}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{x}, x, x\right)\\ \mathbf{else}:\\ \;\;\;\;t + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= y -5.5e+117) (fma (/ t x) x x) (+ t x)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (y <= -5.5e+117) {
		tmp = fma((t / x), x, x);
	} else {
		tmp = t + x;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (y <= -5.5e+117)
		tmp = fma(Float64(t / x), x, x);
	else
		tmp = Float64(t + x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[y, -5.5e+117], N[(N[(t / x), $MachinePrecision] * x + x), $MachinePrecision], N[(t + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -5.49999999999999965e117

   1. Initial program 80.8%

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

   3. Taylor expanded in z around inf

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

      1. lower-+.f64 44.4

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

   5. Applied rewrites 44.4%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 56.0%

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

   if -5.49999999999999965e117 < y

   1. Initial program 87.9%

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

   3. Taylor expanded in z around inf

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

      1. lower-+.f64 71.1

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

   5. Applied rewrites 71.1%

      \[\leadsto \color{blue}{t + x} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 60.3% accurate, 6.5× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z, double t, double a) {
	return t + 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 = t + x
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 86.8%

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

3. Taylor expanded in z around inf

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

   1. lower-+.f64 67.0

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

5. Applied rewrites 67.0%

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

Developer Target 1: 99.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x + \frac{y - z}{a - z} \cdot t\\ \mathbf{if}\;t < -1.0682974490174067 \cdot 10^{-39}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t < 3.9110949887586375 \cdot 10^{-141}:\\ \;\;\;\;x + \frac{\left(y - z\right) \cdot t}{a - z}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ x (* (/ (- y z) (- a z)) t))))
   (if (< t -1.0682974490174067e-39)
     t_1
     (if (< t 3.9110949887586375e-141) (+ x (/ (* (- y z) t) (- a z))) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = x + (((y - z) / (a - z)) * t);
	double tmp;
	if (t < -1.0682974490174067e-39) {
		tmp = t_1;
	} else if (t < 3.9110949887586375e-141) {
		tmp = x + (((y - z) * t) / (a - z));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: tmp
    t_1 = x + (((y - z) / (a - z)) * t)
    if (t < (-1.0682974490174067d-39)) then
        tmp = t_1
    else if (t < 3.9110949887586375d-141) then
        tmp = x + (((y - z) * t) / (a - z))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = x + (((y - z) / (a - z)) * t);
	double tmp;
	if (t < -1.0682974490174067e-39) {
		tmp = t_1;
	} else if (t < 3.9110949887586375e-141) {
		tmp = x + (((y - z) * t) / (a - z));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = x + (((y - z) / (a - z)) * t)
	tmp = 0
	if t < -1.0682974490174067e-39:
		tmp = t_1
	elif t < 3.9110949887586375e-141:
		tmp = x + (((y - z) * t) / (a - z))
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(x + Float64(Float64(Float64(y - z) / Float64(a - z)) * t))
	tmp = 0.0
	if (t < -1.0682974490174067e-39)
		tmp = t_1;
	elseif (t < 3.9110949887586375e-141)
		tmp = Float64(x + Float64(Float64(Float64(y - z) * t) / Float64(a - z)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = x + (((y - z) / (a - z)) * t);
	tmp = 0.0;
	if (t < -1.0682974490174067e-39)
		tmp = t_1;
	elseif (t < 3.9110949887586375e-141)
		tmp = x + (((y - z) * t) / (a - z));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(x + N[(N[(N[(y - z), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]}, If[Less[t, -1.0682974490174067e-39], t$95$1, If[Less[t, 3.9110949887586375e-141], N[(x + N[(N[(N[(y - z), $MachinePrecision] * t), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Reproduce

herbie shell --seed 2024320 
(FPCore (x y z t a)
  :name "Graphics.Rendering.Plot.Render.Plot.Axis:renderAxisTick from plot-0.2.3.4, A"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< t -10682974490174067/10000000000000000000000000000000000000000000000000000000) (+ x (* (/ (- y z) (- a z)) t)) (if (< t 312887599100691/80000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (+ x (/ (* (- y z) t) (- a z))) (+ x (* (/ (- y z) (- a z)) t)))))

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