tan-example (used to crash)

Percentage Accurate: 78.8% → 99.7%

Time: 29.3s

Alternatives: 12

Speedup: 1.0×

Specification

\[\left(\left(\left(x = 0 \lor 0.5884142 \leq x \land x \leq 505.5909\right) \land \left(-1.796658 \cdot 10^{+308} \leq y \land y \leq -9.425585 \cdot 10^{-310} \lor 1.284938 \cdot 10^{-309} \leq y \land y \leq 1.751224 \cdot 10^{+308}\right)\right) \land \left(-1.776707 \cdot 10^{+308} \leq z \land z \leq -8.599796 \cdot 10^{-310} \lor 3.293145 \cdot 10^{-311} \leq z \land z \leq 1.725154 \cdot 10^{+308}\right)\right) \land \left(-1.796658 \cdot 10^{+308} \leq a \land a \leq -9.425585 \cdot 10^{-310} \lor 1.284938 \cdot 10^{-309} \leq a \land a \leq 1.751224 \cdot 10^{+308}\right)\]

Math

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

FPCore

(FPCore (x y z a) :precision binary64 (+ x (- (tan (+ y z)) (tan a))))

C

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

Fortran

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

Java

public static double code(double x, double y, double z, double a) {
	return x + (Math.tan((y + z)) - Math.tan(a));
}

Python

def code(x, y, z, a):
	return x + (math.tan((y + z)) - math.tan(a))

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 78.8% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x y z a) :precision binary64 (+ x (- (tan (+ y z)) (tan a))))

C

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

Fortran

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

Java

public static double code(double x, double y, double z, double a) {
	return x + (Math.tan((y + z)) - Math.tan(a));
}

Python

def code(x, y, z, a):
	return x + (math.tan((y + z)) - math.tan(a))

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 0.4× speedup

Math

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

FPCore

(FPCore (x y z a)
 :precision binary64
 (+
  x
  (fma (/ -1.0 (+ -1.0 (* (tan y) (tan z)))) (+ (tan y) (tan z)) (- (tan a)))))

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 79.3%

   \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift--.f64 N/A

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

   2. sub-neg N/A

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

   3. lift-tan.f64 N/A

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

   4. lift-+.f64 N/A

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

   5. tan-sum N/A

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

   6. clear-num N/A

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

   7. associate-/r/ N/A

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

   8. lower-fma.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lower--.f64 N/A

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

   11. lower-*.f64 N/A

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

   12. lower-tan.f64 N/A

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

   13. lower-tan.f64 N/A

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

   14. lower-+.f64 N/A

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

   15. lower-tan.f64 N/A

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

   16. lower-tan.f64 N/A

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

   17. lower-neg.f64 99.6

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

4. Applied rewrites 99.6%

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

5. Final simplification 99.6%

   \[\leadsto x + \mathsf{fma}\left(\frac{-1}{-1 + \tan y \cdot \tan z}, \tan y + \tan z, -\tan a\right) \]
6. Add Preprocessing

Alternative 2: 88.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan y + \tan z\\ t_1 := x + \mathsf{fma}\left(1, t\_0, -\tan a\right)\\ \mathbf{if}\;\tan a \leq -0.005:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\tan a \leq 5 \cdot 10^{-7}:\\ \;\;\;\;x + \left(\frac{t\_0}{1 - \tan y \cdot \tan z} - \mathsf{fma}\left(a \cdot a, a \cdot 0.3333333333333333, a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (let* ((t_0 (+ (tan y) (tan z))) (t_1 (+ x (fma 1.0 t_0 (- (tan a))))))
   (if (<= (tan a) -0.005)
     t_1
     (if (<= (tan a) 5e-7)
       (+
        x
        (-
         (/ t_0 (- 1.0 (* (tan y) (tan z))))
         (fma (* a a) (* a 0.3333333333333333) a)))
       t_1))))

C

double code(double x, double y, double z, double a) {
	double t_0 = tan(y) + tan(z);
	double t_1 = x + fma(1.0, t_0, -tan(a));
	double tmp;
	if (tan(a) <= -0.005) {
		tmp = t_1;
	} else if (tan(a) <= 5e-7) {
		tmp = x + ((t_0 / (1.0 - (tan(y) * tan(z)))) - fma((a * a), (a * 0.3333333333333333), a));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, a)
	t_0 = Float64(tan(y) + tan(z))
	t_1 = Float64(x + fma(1.0, t_0, Float64(-tan(a))))
	tmp = 0.0
	if (tan(a) <= -0.005)
		tmp = t_1;
	elseif (tan(a) <= 5e-7)
		tmp = Float64(x + Float64(Float64(t_0 / Float64(1.0 - Float64(tan(y) * tan(z)))) - fma(Float64(a * a), Float64(a * 0.3333333333333333), a)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, a_] := Block[{t$95$0 = N[(N[Tan[y], $MachinePrecision] + N[Tan[z], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x + N[(1.0 * t$95$0 + (-N[Tan[a], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Tan[a], $MachinePrecision], -0.005], t$95$1, If[LessEqual[N[Tan[a], $MachinePrecision], 5e-7], N[(x + N[(N[(t$95$0 / N[(1.0 - N[(N[Tan[y], $MachinePrecision] * N[Tan[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(a * a), $MachinePrecision] * N[(a * 0.3333333333333333), $MachinePrecision] + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (tan.f64 a) < -0.0050000000000000001 or 4.99999999999999977e-7 < (tan.f64 a)

   1. Initial program 77.3%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. lift-tan.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. tan-sum N/A

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

      6. clear-num N/A

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

      7. associate-/r/ N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-tan.f64 N/A

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

      13. lower-tan.f64 N/A

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

      14. lower-+.f64 N/A

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

      15. lower-tan.f64 N/A

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

      16. lower-tan.f64 N/A

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

      17. lower-neg.f64 99.4

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

   4. Applied rewrites 99.4%

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

   5. Taylor expanded in y around 0

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

   7. Applied rewrites 77.9%

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

   if -0.0050000000000000001 < (tan.f64 a) < 4.99999999999999977e-7

   1. Initial program 81.3%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0

      \[\leadsto x + \left(\tan \left(y + z\right) - \color{blue}{a \cdot \left(1 + \frac{1}{3} \cdot {a}^{2}\right)}\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x + \left(\tan \left(y + z\right) - a \cdot \color{blue}{\left(\frac{1}{3} \cdot {a}^{2} + 1\right)}\right) \]

      2. distribute-lft-in N/A

         \[\leadsto x + \left(\tan \left(y + z\right) - \color{blue}{\left(a \cdot \left(\frac{1}{3} \cdot {a}^{2}\right) + a \cdot 1\right)}\right) \]

      3. associate-*r* N/A

         \[\leadsto x + \left(\tan \left(y + z\right) - \left(\color{blue}{\left(a \cdot \frac{1}{3}\right) \cdot {a}^{2}} + a \cdot 1\right)\right) \]

      4. *-commutative N/A

         \[\leadsto x + \left(\tan \left(y + z\right) - \left(\color{blue}{{a}^{2} \cdot \left(a \cdot \frac{1}{3}\right)} + a \cdot 1\right)\right) \]

      5. *-rgt-identity N/A

         \[\leadsto x + \left(\tan \left(y + z\right) - \left({a}^{2} \cdot \left(a \cdot \frac{1}{3}\right) + \color{blue}{a}\right)\right) \]

      6. lower-fma.f64 N/A

         \[\leadsto x + \left(\tan \left(y + z\right) - \color{blue}{\mathsf{fma}\left({a}^{2}, a \cdot \frac{1}{3}, a\right)}\right) \]

      7. unpow2 N/A

         \[\leadsto x + \left(\tan \left(y + z\right) - \mathsf{fma}\left(\color{blue}{a \cdot a}, a \cdot \frac{1}{3}, a\right)\right) \]

      8. lower-*.f64 N/A

         \[\leadsto x + \left(\tan \left(y + z\right) - \mathsf{fma}\left(\color{blue}{a \cdot a}, a \cdot \frac{1}{3}, a\right)\right) \]

      9. lower-*.f64 81.3

         \[\leadsto x + \left(\tan \left(y + z\right) - \mathsf{fma}\left(a \cdot a, \color{blue}{a \cdot 0.3333333333333333}, a\right)\right) \]

   5. Applied rewrites 81.3%

      \[\leadsto x + \left(\tan \left(y + z\right) - \color{blue}{\mathsf{fma}\left(a \cdot a, a \cdot 0.3333333333333333, a\right)}\right) \]
   6. Step-by-step derivation

      1. lift-tan.f64 N/A

         \[\leadsto x + \left(\color{blue}{\tan \left(y + z\right)} - \mathsf{fma}\left(a \cdot a, a \cdot \frac{1}{3}, a\right)\right) \]

      2. lift-+.f64 N/A

         \[\leadsto x + \left(\tan \color{blue}{\left(y + z\right)} - \mathsf{fma}\left(a \cdot a, a \cdot \frac{1}{3}, a\right)\right) \]

      3. tan-sum N/A

         \[\leadsto x + \left(\color{blue}{\frac{\tan y + \tan z}{1 - \tan y \cdot \tan z}} - \mathsf{fma}\left(a \cdot a, a \cdot \frac{1}{3}, a\right)\right) \]

      4. lift-tan.f64 N/A

         \[\leadsto x + \left(\frac{\color{blue}{\tan y} + \tan z}{1 - \tan y \cdot \tan z} - \mathsf{fma}\left(a \cdot a, a \cdot \frac{1}{3}, a\right)\right) \]

      5. lift-tan.f64 N/A

         \[\leadsto x + \left(\frac{\tan y + \color{blue}{\tan z}}{1 - \tan y \cdot \tan z} - \mathsf{fma}\left(a \cdot a, a \cdot \frac{1}{3}, a\right)\right) \]

      6. lift-+.f64 N/A

         \[\leadsto x + \left(\frac{\color{blue}{\tan y + \tan z}}{1 - \tan y \cdot \tan z} - \mathsf{fma}\left(a \cdot a, a \cdot \frac{1}{3}, a\right)\right) \]

      7. lift-tan.f64 N/A

         \[\leadsto x + \left(\frac{\tan y + \tan z}{1 - \color{blue}{\tan y} \cdot \tan z} - \mathsf{fma}\left(a \cdot a, a \cdot \frac{1}{3}, a\right)\right) \]

      8. lift-tan.f64 N/A

         \[\leadsto x + \left(\frac{\tan y + \tan z}{1 - \tan y \cdot \color{blue}{\tan z}} - \mathsf{fma}\left(a \cdot a, a \cdot \frac{1}{3}, a\right)\right) \]

      9. lift-*.f64 N/A

         \[\leadsto x + \left(\frac{\tan y + \tan z}{1 - \color{blue}{\tan y \cdot \tan z}} - \mathsf{fma}\left(a \cdot a, a \cdot \frac{1}{3}, a\right)\right) \]

      10. lift--.f64 N/A

          \[\leadsto x + \left(\frac{\tan y + \tan z}{\color{blue}{1 - \tan y \cdot \tan z}} - \mathsf{fma}\left(a \cdot a, a \cdot \frac{1}{3}, a\right)\right) \]

      11. lift-/.f64 99.6

          \[\leadsto x + \left(\color{blue}{\frac{\tan y + \tan z}{1 - \tan y \cdot \tan z}} - \mathsf{fma}\left(a \cdot a, a \cdot 0.3333333333333333, a\right)\right) \]

   7. Applied rewrites 99.6%

      \[\leadsto x + \left(\color{blue}{\frac{\tan y + \tan z}{1 - \tan y \cdot \tan z}} - \mathsf{fma}\left(a \cdot a, a \cdot 0.3333333333333333, a\right)\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 99.7% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 79.3%

   \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-tan.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. tan-sum N/A

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

   4. lower-/.f64 N/A

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

   5. lower-+.f64 N/A

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

   6. lower-tan.f64 N/A

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

   7. lower-tan.f64 N/A

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

   8. lower--.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. lower-tan.f64 N/A

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

   11. lower-tan.f64 99.5

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

4. Applied rewrites 99.5%

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

   1. lift--.f64 N/A

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

   2. sub-neg N/A

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

   3. +-commutative N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

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

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

   7. lower-fma.f64 N/A

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

   8. lower-neg.f64 99.6

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

6. Applied rewrites 99.6%

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

Alternative 4: 99.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ x - \left(\tan a + \frac{\tan y + \tan z}{-1 + \tan y \cdot \tan z}\right) \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (- x (+ (tan a) (/ (+ (tan y) (tan z)) (+ -1.0 (* (tan y) (tan z)))))))

C

double code(double x, double y, double z, double a) {
	return x - (tan(a) + ((tan(y) + tan(z)) / (-1.0 + (tan(y) * tan(z)))));
}

Fortran

real(8) function code(x, y, z, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: a
    code = x - (tan(a) + ((tan(y) + tan(z)) / ((-1.0d0) + (tan(y) * tan(z)))))
end function

Java

public static double code(double x, double y, double z, double a) {
	return x - (Math.tan(a) + ((Math.tan(y) + Math.tan(z)) / (-1.0 + (Math.tan(y) * Math.tan(z)))));
}

Python

def code(x, y, z, a):
	return x - (math.tan(a) + ((math.tan(y) + math.tan(z)) / (-1.0 + (math.tan(y) * math.tan(z)))))

Julia

function code(x, y, z, a)
	return Float64(x - Float64(tan(a) + Float64(Float64(tan(y) + tan(z)) / Float64(-1.0 + Float64(tan(y) * tan(z))))))
end

MATLAB

function tmp = code(x, y, z, a)
	tmp = x - (tan(a) + ((tan(y) + tan(z)) / (-1.0 + (tan(y) * tan(z)))));
end

Wolfram

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

Derivation

1. Initial program 79.3%

   \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-tan.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. tan-sum N/A

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

   4. lower-/.f64 N/A

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

   5. lower-+.f64 N/A

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

   6. lower-tan.f64 N/A

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

   7. lower-tan.f64 N/A

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

   8. lower--.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. lower-tan.f64 N/A

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

   11. lower-tan.f64 99.5

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

4. Applied rewrites 99.5%

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

5. Final simplification 99.5%

   \[\leadsto x - \left(\tan a + \frac{\tan y + \tan z}{-1 + \tan y \cdot \tan z}\right) \]
6. Add Preprocessing

Alternative 5: 89.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan y + \tan z\\ t_1 := x + \mathsf{fma}\left(1, t\_0, -\tan a\right)\\ \mathbf{if}\;a \leq -0.0098:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 0.0066:\\ \;\;\;\;\mathsf{fma}\left(\frac{-1}{-1 + \tan y \cdot \tan z}, t\_0, x - \mathsf{fma}\left(a, \left(a \cdot a\right) \cdot 0.3333333333333333, a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (let* ((t_0 (+ (tan y) (tan z))) (t_1 (+ x (fma 1.0 t_0 (- (tan a))))))
   (if (<= a -0.0098)
     t_1
     (if (<= a 0.0066)
       (fma
        (/ -1.0 (+ -1.0 (* (tan y) (tan z))))
        t_0
        (- x (fma a (* (* a a) 0.3333333333333333) a)))
       t_1))))

C

double code(double x, double y, double z, double a) {
	double t_0 = tan(y) + tan(z);
	double t_1 = x + fma(1.0, t_0, -tan(a));
	double tmp;
	if (a <= -0.0098) {
		tmp = t_1;
	} else if (a <= 0.0066) {
		tmp = fma((-1.0 / (-1.0 + (tan(y) * tan(z)))), t_0, (x - fma(a, ((a * a) * 0.3333333333333333), a)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, a)
	t_0 = Float64(tan(y) + tan(z))
	t_1 = Float64(x + fma(1.0, t_0, Float64(-tan(a))))
	tmp = 0.0
	if (a <= -0.0098)
		tmp = t_1;
	elseif (a <= 0.0066)
		tmp = fma(Float64(-1.0 / Float64(-1.0 + Float64(tan(y) * tan(z)))), t_0, Float64(x - fma(a, Float64(Float64(a * a) * 0.3333333333333333), a)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, a_] := Block[{t$95$0 = N[(N[Tan[y], $MachinePrecision] + N[Tan[z], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x + N[(1.0 * t$95$0 + (-N[Tan[a], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -0.0098], t$95$1, If[LessEqual[a, 0.0066], N[(N[(-1.0 / N[(-1.0 + N[(N[Tan[y], $MachinePrecision] * N[Tan[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0 + N[(x - N[(a * N[(N[(a * a), $MachinePrecision] * 0.3333333333333333), $MachinePrecision] + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if a < -0.0097999999999999997 or 0.0066 < a

   1. Initial program 77.3%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. lift-tan.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. tan-sum N/A

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

      6. clear-num N/A

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

      7. associate-/r/ N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-tan.f64 N/A

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

      13. lower-tan.f64 N/A

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

      14. lower-+.f64 N/A

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

      15. lower-tan.f64 N/A

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

      16. lower-tan.f64 N/A

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

      17. lower-neg.f64 99.4

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

   4. Applied rewrites 99.4%

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

   5. Taylor expanded in y around 0

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

   7. Applied rewrites 77.9%

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

   if -0.0097999999999999997 < a < 0.0066

   1. Initial program 81.3%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0

      \[\leadsto x + \left(\tan \left(y + z\right) - \color{blue}{a \cdot \left(1 + \frac{1}{3} \cdot {a}^{2}\right)}\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x + \left(\tan \left(y + z\right) - a \cdot \color{blue}{\left(\frac{1}{3} \cdot {a}^{2} + 1\right)}\right) \]

      2. distribute-lft-in N/A

         \[\leadsto x + \left(\tan \left(y + z\right) - \color{blue}{\left(a \cdot \left(\frac{1}{3} \cdot {a}^{2}\right) + a \cdot 1\right)}\right) \]

      3. associate-*r* N/A

         \[\leadsto x + \left(\tan \left(y + z\right) - \left(\color{blue}{\left(a \cdot \frac{1}{3}\right) \cdot {a}^{2}} + a \cdot 1\right)\right) \]

      4. *-commutative N/A

         \[\leadsto x + \left(\tan \left(y + z\right) - \left(\color{blue}{{a}^{2} \cdot \left(a \cdot \frac{1}{3}\right)} + a \cdot 1\right)\right) \]

      5. *-rgt-identity N/A

         \[\leadsto x + \left(\tan \left(y + z\right) - \left({a}^{2} \cdot \left(a \cdot \frac{1}{3}\right) + \color{blue}{a}\right)\right) \]

      6. lower-fma.f64 N/A

         \[\leadsto x + \left(\tan \left(y + z\right) - \color{blue}{\mathsf{fma}\left({a}^{2}, a \cdot \frac{1}{3}, a\right)}\right) \]

      7. unpow2 N/A

         \[\leadsto x + \left(\tan \left(y + z\right) - \mathsf{fma}\left(\color{blue}{a \cdot a}, a \cdot \frac{1}{3}, a\right)\right) \]

      8. lower-*.f64 N/A

         \[\leadsto x + \left(\tan \left(y + z\right) - \mathsf{fma}\left(\color{blue}{a \cdot a}, a \cdot \frac{1}{3}, a\right)\right) \]

      9. lower-*.f64 81.3

         \[\leadsto x + \left(\tan \left(y + z\right) - \mathsf{fma}\left(a \cdot a, \color{blue}{a \cdot 0.3333333333333333}, a\right)\right) \]

   5. Applied rewrites 81.3%

      \[\leadsto x + \left(\tan \left(y + z\right) - \color{blue}{\mathsf{fma}\left(a \cdot a, a \cdot 0.3333333333333333, a\right)}\right) \]
   6. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x + \left(\tan \left(y + z\right) - \mathsf{fma}\left(a \cdot a, a \cdot \frac{1}{3}, a\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(\tan \left(y + z\right) - \mathsf{fma}\left(a \cdot a, a \cdot \frac{1}{3}, a\right)\right) + x} \]

      3. lift--.f64 N/A

         \[\leadsto \color{blue}{\left(\tan \left(y + z\right) - \mathsf{fma}\left(a \cdot a, a \cdot \frac{1}{3}, a\right)\right)} + x \]

      4. sub-neg N/A

         \[\leadsto \color{blue}{\left(\tan \left(y + z\right) + \left(\mathsf{neg}\left(\mathsf{fma}\left(a \cdot a, a \cdot \frac{1}{3}, a\right)\right)\right)\right)} + x \]

      5. associate-+l+ N/A

         \[\leadsto \color{blue}{\tan \left(y + z\right) + \left(\left(\mathsf{neg}\left(\mathsf{fma}\left(a \cdot a, a \cdot \frac{1}{3}, a\right)\right)\right) + x\right)} \]

   7. Applied rewrites 99.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{1 - \tan y \cdot \tan z}, \tan y + \tan z, \left(-\mathsf{fma}\left(a, \left(a \cdot a\right) \cdot 0.3333333333333333, a\right)\right) + x\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 88.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -0.0098:\\ \;\;\;\;x + \mathsf{fma}\left(1, \tan y + \tan z, -\tan a\right)\\ \mathbf{elif}\;a \leq 0.0066:\\ \;\;\;\;\mathsf{fma}\left(\frac{-1}{-1 + \tan y \cdot \tan z}, \tan y + \tan z, x - \mathsf{fma}\left(a, \left(a \cdot a\right) \cdot 0.3333333333333333, a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x + \mathsf{fma}\left(1, \tan y + \tan z, -\tan a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 68.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\tan a \leq -0.005:\\ \;\;\;\;x + \left(\tan y - \tan a\right)\\ \mathbf{elif}\;\tan a \leq 2 \cdot 10^{-14}:\\ \;\;\;\;x + \left(\tan \left(y + z\right) - \mathsf{fma}\left(a \cdot a, a \cdot 0.3333333333333333, a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\tan y + \left(x - \tan a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (if (<= (tan a) -0.005)
   (+ x (- (tan y) (tan a)))
   (if (<= (tan a) 2e-14)
     (+ x (- (tan (+ y z)) (fma (* a a) (* a 0.3333333333333333) a)))
     (+ (tan y) (- x (tan a))))))

C

double code(double x, double y, double z, double a) {
	double tmp;
	if (tan(a) <= -0.005) {
		tmp = x + (tan(y) - tan(a));
	} else if (tan(a) <= 2e-14) {
		tmp = x + (tan((y + z)) - fma((a * a), (a * 0.3333333333333333), a));
	} else {
		tmp = tan(y) + (x - tan(a));
	}
	return tmp;
}

Julia

function code(x, y, z, a)
	tmp = 0.0
	if (tan(a) <= -0.005)
		tmp = Float64(x + Float64(tan(y) - tan(a)));
	elseif (tan(a) <= 2e-14)
		tmp = Float64(x + Float64(tan(Float64(y + z)) - fma(Float64(a * a), Float64(a * 0.3333333333333333), a)));
	else
		tmp = Float64(tan(y) + Float64(x - tan(a)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, a_] := If[LessEqual[N[Tan[a], $MachinePrecision], -0.005], N[(x + N[(N[Tan[y], $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Tan[a], $MachinePrecision], 2e-14], N[(x + N[(N[Tan[N[(y + z), $MachinePrecision]], $MachinePrecision] - N[(N[(a * a), $MachinePrecision] * N[(a * 0.3333333333333333), $MachinePrecision] + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Tan[y], $MachinePrecision] + N[(x - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (tan.f64 a) < -0.0050000000000000001

   1. Initial program 74.9%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

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

      1. lower-/.f64 N/A

         \[\leadsto x + \left(\color{blue}{\frac{\sin y}{\cos y}} - \tan a\right) \]

      2. lower-sin.f64 N/A

         \[\leadsto x + \left(\frac{\color{blue}{\sin y}}{\cos y} - \tan a\right) \]

      3. lower-cos.f64 54.2

         \[\leadsto x + \left(\frac{\sin y}{\color{blue}{\cos y}} - \tan a\right) \]

   5. Applied rewrites 54.2%

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

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x + \left(\frac{\sin y}{\cos y} - \tan a\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{\sin y}{\cos y} - \tan a\right) + x} \]

      3. lower-+.f64 54.2

         \[\leadsto \color{blue}{\left(\frac{\sin y}{\cos y} - \tan a\right) + x} \]

   7. Applied rewrites 54.2%

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

   if -0.0050000000000000001 < (tan.f64 a) < 2e-14

   1. Initial program 82.2%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0

      \[\leadsto x + \left(\tan \left(y + z\right) - \color{blue}{a \cdot \left(1 + \frac{1}{3} \cdot {a}^{2}\right)}\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x + \left(\tan \left(y + z\right) - a \cdot \color{blue}{\left(\frac{1}{3} \cdot {a}^{2} + 1\right)}\right) \]

      2. distribute-lft-in N/A

         \[\leadsto x + \left(\tan \left(y + z\right) - \color{blue}{\left(a \cdot \left(\frac{1}{3} \cdot {a}^{2}\right) + a \cdot 1\right)}\right) \]

      3. associate-*r* N/A

         \[\leadsto x + \left(\tan \left(y + z\right) - \left(\color{blue}{\left(a \cdot \frac{1}{3}\right) \cdot {a}^{2}} + a \cdot 1\right)\right) \]

      4. *-commutative N/A

         \[\leadsto x + \left(\tan \left(y + z\right) - \left(\color{blue}{{a}^{2} \cdot \left(a \cdot \frac{1}{3}\right)} + a \cdot 1\right)\right) \]

      5. *-rgt-identity N/A

         \[\leadsto x + \left(\tan \left(y + z\right) - \left({a}^{2} \cdot \left(a \cdot \frac{1}{3}\right) + \color{blue}{a}\right)\right) \]

      6. lower-fma.f64 N/A

         \[\leadsto x + \left(\tan \left(y + z\right) - \color{blue}{\mathsf{fma}\left({a}^{2}, a \cdot \frac{1}{3}, a\right)}\right) \]

      7. unpow2 N/A

         \[\leadsto x + \left(\tan \left(y + z\right) - \mathsf{fma}\left(\color{blue}{a \cdot a}, a \cdot \frac{1}{3}, a\right)\right) \]

      8. lower-*.f64 N/A

         \[\leadsto x + \left(\tan \left(y + z\right) - \mathsf{fma}\left(\color{blue}{a \cdot a}, a \cdot \frac{1}{3}, a\right)\right) \]

      9. lower-*.f64 82.2

         \[\leadsto x + \left(\tan \left(y + z\right) - \mathsf{fma}\left(a \cdot a, \color{blue}{a \cdot 0.3333333333333333}, a\right)\right) \]

   5. Applied rewrites 82.2%

      \[\leadsto x + \left(\tan \left(y + z\right) - \color{blue}{\mathsf{fma}\left(a \cdot a, a \cdot 0.3333333333333333, a\right)}\right) \]

   if 2e-14 < (tan.f64 a)

   1. Initial program 78.2%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

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

      1. lower-/.f64 N/A

         \[\leadsto x + \left(\color{blue}{\frac{\sin y}{\cos y}} - \tan a\right) \]

      2. lower-sin.f64 N/A

         \[\leadsto x + \left(\frac{\color{blue}{\sin y}}{\cos y} - \tan a\right) \]

      3. lower-cos.f64 54.9

         \[\leadsto x + \left(\frac{\sin y}{\color{blue}{\cos y}} - \tan a\right) \]

   5. Applied rewrites 54.9%

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

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x + \left(\frac{\sin y}{\cos y} - \tan a\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{\sin y}{\cos y} - \tan a\right) + x} \]

      3. lift--.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{\sin y}{\cos y} - \tan a\right)} + x \]

      4. associate-+l- N/A

         \[\leadsto \color{blue}{\frac{\sin y}{\cos y} - \left(\tan a - x\right)} \]

      5. lower--.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin y}{\cos y} - \left(\tan a - x\right)} \]

      6. lower--.f64 N/A

         \[\leadsto \mathsf{Rewrite<=}\left(lift-tan.f64, \tan y\right) - \color{blue}{\left(\tan a - x\right)} \]

   7. Applied rewrites 54.9%

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

4. Final simplification 67.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\tan a \leq -0.005:\\ \;\;\;\;x + \left(\tan y - \tan a\right)\\ \mathbf{elif}\;\tan a \leq 2 \cdot 10^{-14}:\\ \;\;\;\;x + \left(\tan \left(y + z\right) - \mathsf{fma}\left(a \cdot a, a \cdot 0.3333333333333333, a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\tan y + \left(x - \tan a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 68.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x + \left(\tan y - \tan a\right)\\ \mathbf{if}\;\tan a \leq -0.005:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\tan a \leq 2 \cdot 10^{-14}:\\ \;\;\;\;x + \left(\tan \left(y + z\right) - \mathsf{fma}\left(a \cdot a, a \cdot 0.3333333333333333, a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (let* ((t_0 (+ x (- (tan y) (tan a)))))
   (if (<= (tan a) -0.005)
     t_0
     (if (<= (tan a) 2e-14)
       (+ x (- (tan (+ y z)) (fma (* a a) (* a 0.3333333333333333) a)))
       t_0))))

C

double code(double x, double y, double z, double a) {
	double t_0 = x + (tan(y) - tan(a));
	double tmp;
	if (tan(a) <= -0.005) {
		tmp = t_0;
	} else if (tan(a) <= 2e-14) {
		tmp = x + (tan((y + z)) - fma((a * a), (a * 0.3333333333333333), a));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z, a)
	t_0 = Float64(x + Float64(tan(y) - tan(a)))
	tmp = 0.0
	if (tan(a) <= -0.005)
		tmp = t_0;
	elseif (tan(a) <= 2e-14)
		tmp = Float64(x + Float64(tan(Float64(y + z)) - fma(Float64(a * a), Float64(a * 0.3333333333333333), a)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, a_] := Block[{t$95$0 = N[(x + N[(N[Tan[y], $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Tan[a], $MachinePrecision], -0.005], t$95$0, If[LessEqual[N[Tan[a], $MachinePrecision], 2e-14], N[(x + N[(N[Tan[N[(y + z), $MachinePrecision]], $MachinePrecision] - N[(N[(a * a), $MachinePrecision] * N[(a * 0.3333333333333333), $MachinePrecision] + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if (tan.f64 a) < -0.0050000000000000001 or 2e-14 < (tan.f64 a)

   1. Initial program 76.6%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

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

      1. lower-/.f64 N/A

         \[\leadsto x + \left(\color{blue}{\frac{\sin y}{\cos y}} - \tan a\right) \]

      2. lower-sin.f64 N/A

         \[\leadsto x + \left(\frac{\color{blue}{\sin y}}{\cos y} - \tan a\right) \]

      3. lower-cos.f64 54.5

         \[\leadsto x + \left(\frac{\sin y}{\color{blue}{\cos y}} - \tan a\right) \]

   5. Applied rewrites 54.5%

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

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x + \left(\frac{\sin y}{\cos y} - \tan a\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{\sin y}{\cos y} - \tan a\right) + x} \]

      3. lower-+.f64 54.5

         \[\leadsto \color{blue}{\left(\frac{\sin y}{\cos y} - \tan a\right) + x} \]

   7. Applied rewrites 54.6%

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

   if -0.0050000000000000001 < (tan.f64 a) < 2e-14

   1. Initial program 82.2%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0

      \[\leadsto x + \left(\tan \left(y + z\right) - \color{blue}{a \cdot \left(1 + \frac{1}{3} \cdot {a}^{2}\right)}\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x + \left(\tan \left(y + z\right) - a \cdot \color{blue}{\left(\frac{1}{3} \cdot {a}^{2} + 1\right)}\right) \]

      2. distribute-lft-in N/A

         \[\leadsto x + \left(\tan \left(y + z\right) - \color{blue}{\left(a \cdot \left(\frac{1}{3} \cdot {a}^{2}\right) + a \cdot 1\right)}\right) \]

      3. associate-*r* N/A

         \[\leadsto x + \left(\tan \left(y + z\right) - \left(\color{blue}{\left(a \cdot \frac{1}{3}\right) \cdot {a}^{2}} + a \cdot 1\right)\right) \]

      4. *-commutative N/A

         \[\leadsto x + \left(\tan \left(y + z\right) - \left(\color{blue}{{a}^{2} \cdot \left(a \cdot \frac{1}{3}\right)} + a \cdot 1\right)\right) \]

      5. *-rgt-identity N/A

         \[\leadsto x + \left(\tan \left(y + z\right) - \left({a}^{2} \cdot \left(a \cdot \frac{1}{3}\right) + \color{blue}{a}\right)\right) \]

      6. lower-fma.f64 N/A

         \[\leadsto x + \left(\tan \left(y + z\right) - \color{blue}{\mathsf{fma}\left({a}^{2}, a \cdot \frac{1}{3}, a\right)}\right) \]

      7. unpow2 N/A

         \[\leadsto x + \left(\tan \left(y + z\right) - \mathsf{fma}\left(\color{blue}{a \cdot a}, a \cdot \frac{1}{3}, a\right)\right) \]

      8. lower-*.f64 N/A

         \[\leadsto x + \left(\tan \left(y + z\right) - \mathsf{fma}\left(\color{blue}{a \cdot a}, a \cdot \frac{1}{3}, a\right)\right) \]

      9. lower-*.f64 82.2

         \[\leadsto x + \left(\tan \left(y + z\right) - \mathsf{fma}\left(a \cdot a, \color{blue}{a \cdot 0.3333333333333333}, a\right)\right) \]

   5. Applied rewrites 82.2%

      \[\leadsto x + \left(\tan \left(y + z\right) - \color{blue}{\mathsf{fma}\left(a \cdot a, a \cdot 0.3333333333333333, a\right)}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 67.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\tan a \leq -0.005:\\ \;\;\;\;x + \left(\tan y - \tan a\right)\\ \mathbf{elif}\;\tan a \leq 2 \cdot 10^{-14}:\\ \;\;\;\;x + \left(\tan \left(y + z\right) - \mathsf{fma}\left(a \cdot a, a \cdot 0.3333333333333333, a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(\tan y - \tan a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 79.1% accurate, 0.7× speedup

Math

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

FPCore

(FPCore (x y z a)
 :precision binary64
 (+ x (fma 1.0 (+ (tan y) (tan z)) (- (tan a)))))

C

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

Julia

function code(x, y, z, a)
	return Float64(x + fma(1.0, Float64(tan(y) + tan(z)), Float64(-tan(a))))
end

Wolfram

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

Derivation

1. Initial program 79.3%

   \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift--.f64 N/A

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

   2. sub-neg N/A

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

   3. lift-tan.f64 N/A

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

   4. lift-+.f64 N/A

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

   5. tan-sum N/A

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

   6. clear-num N/A

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

   7. associate-/r/ N/A

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

   8. lower-fma.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lower--.f64 N/A

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

   11. lower-*.f64 N/A

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

   12. lower-tan.f64 N/A

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

   13. lower-tan.f64 N/A

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

   14. lower-+.f64 N/A

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

   15. lower-tan.f64 N/A

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

   16. lower-tan.f64 N/A

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

   17. lower-neg.f64 99.6

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

4. Applied rewrites 99.6%

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

5. Taylor expanded in y around 0

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

7. Applied rewrites 79.5%

   \[\leadsto x + \mathsf{fma}\left(\color{blue}{1}, \tan y + \tan z, -\tan a\right) \]
8. Add Preprocessing

Alternative 9: 78.8% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x y z a) :precision binary64 (+ x (- (tan (+ y z)) (tan a))))

C

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

Fortran

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

Java

public static double code(double x, double y, double z, double a) {
	return x + (Math.tan((y + z)) - Math.tan(a));
}

Python

def code(x, y, z, a):
	return x + (math.tan((y + z)) - math.tan(a))

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.3%

   \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
2. Add Preprocessing
3. Add Preprocessing

Alternative 10: 54.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x + \left(\mathsf{fma}\left(y, y \cdot \left(y \cdot 0.3333333333333333\right), y\right) - \tan a\right)\\ \mathbf{if}\;a \leq -112000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;a \leq 320000000:\\ \;\;\;\;x + \left(\tan \left(y + z\right) - \mathsf{fma}\left(a \cdot a, a \cdot 0.3333333333333333, a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (let* ((t_0 (+ x (- (fma y (* y (* y 0.3333333333333333)) y) (tan a)))))
   (if (<= a -112000000.0)
     t_0
     (if (<= a 320000000.0)
       (+ x (- (tan (+ y z)) (fma (* a a) (* a 0.3333333333333333) a)))
       t_0))))

C

double code(double x, double y, double z, double a) {
	double t_0 = x + (fma(y, (y * (y * 0.3333333333333333)), y) - tan(a));
	double tmp;
	if (a <= -112000000.0) {
		tmp = t_0;
	} else if (a <= 320000000.0) {
		tmp = x + (tan((y + z)) - fma((a * a), (a * 0.3333333333333333), a));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z, a)
	t_0 = Float64(x + Float64(fma(y, Float64(y * Float64(y * 0.3333333333333333)), y) - tan(a)))
	tmp = 0.0
	if (a <= -112000000.0)
		tmp = t_0;
	elseif (a <= 320000000.0)
		tmp = Float64(x + Float64(tan(Float64(y + z)) - fma(Float64(a * a), Float64(a * 0.3333333333333333), a)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, a_] := Block[{t$95$0 = N[(x + N[(N[(y * N[(y * N[(y * 0.3333333333333333), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -112000000.0], t$95$0, If[LessEqual[a, 320000000.0], N[(x + N[(N[Tan[N[(y + z), $MachinePrecision]], $MachinePrecision] - N[(N[(a * a), $MachinePrecision] * N[(a * 0.3333333333333333), $MachinePrecision] + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if a < -1.12e8 or 3.2e8 < a

   1. Initial program 77.7%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

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

      1. lower-/.f64 N/A

         \[\leadsto x + \left(\color{blue}{\frac{\sin y}{\cos y}} - \tan a\right) \]

      2. lower-sin.f64 N/A

         \[\leadsto x + \left(\frac{\color{blue}{\sin y}}{\cos y} - \tan a\right) \]

      3. lower-cos.f64 55.8

         \[\leadsto x + \left(\frac{\sin y}{\color{blue}{\cos y}} - \tan a\right) \]

   5. Applied rewrites 55.8%

      \[\leadsto x + \left(\color{blue}{\frac{\sin y}{\cos y}} - \tan a\right) \]

   6. Taylor expanded in y around 0

      \[\leadsto x + \left(y \cdot \color{blue}{\left(1 + \frac{1}{3} \cdot {y}^{2}\right)} - \tan a\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 32.5%

      \[\leadsto x + \left(\mathsf{fma}\left(y, \color{blue}{y \cdot \left(y \cdot 0.3333333333333333\right)}, y\right) - \tan a\right) \]

   if -1.12e8 < a < 3.2e8

   1. Initial program 80.8%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0

      \[\leadsto x + \left(\tan \left(y + z\right) - \color{blue}{a \cdot \left(1 + \frac{1}{3} \cdot {a}^{2}\right)}\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x + \left(\tan \left(y + z\right) - a \cdot \color{blue}{\left(\frac{1}{3} \cdot {a}^{2} + 1\right)}\right) \]

      2. distribute-lft-in N/A

         \[\leadsto x + \left(\tan \left(y + z\right) - \color{blue}{\left(a \cdot \left(\frac{1}{3} \cdot {a}^{2}\right) + a \cdot 1\right)}\right) \]

      3. associate-*r* N/A

         \[\leadsto x + \left(\tan \left(y + z\right) - \left(\color{blue}{\left(a \cdot \frac{1}{3}\right) \cdot {a}^{2}} + a \cdot 1\right)\right) \]

      4. *-commutative N/A

         \[\leadsto x + \left(\tan \left(y + z\right) - \left(\color{blue}{{a}^{2} \cdot \left(a \cdot \frac{1}{3}\right)} + a \cdot 1\right)\right) \]

      5. *-rgt-identity N/A

         \[\leadsto x + \left(\tan \left(y + z\right) - \left({a}^{2} \cdot \left(a \cdot \frac{1}{3}\right) + \color{blue}{a}\right)\right) \]

      6. lower-fma.f64 N/A

         \[\leadsto x + \left(\tan \left(y + z\right) - \color{blue}{\mathsf{fma}\left({a}^{2}, a \cdot \frac{1}{3}, a\right)}\right) \]

      7. unpow2 N/A

         \[\leadsto x + \left(\tan \left(y + z\right) - \mathsf{fma}\left(\color{blue}{a \cdot a}, a \cdot \frac{1}{3}, a\right)\right) \]

      8. lower-*.f64 N/A

         \[\leadsto x + \left(\tan \left(y + z\right) - \mathsf{fma}\left(\color{blue}{a \cdot a}, a \cdot \frac{1}{3}, a\right)\right) \]

      9. lower-*.f64 78.4

         \[\leadsto x + \left(\tan \left(y + z\right) - \mathsf{fma}\left(a \cdot a, \color{blue}{a \cdot 0.3333333333333333}, a\right)\right) \]

   5. Applied rewrites 78.4%

      \[\leadsto x + \left(\tan \left(y + z\right) - \color{blue}{\mathsf{fma}\left(a \cdot a, a \cdot 0.3333333333333333, a\right)}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 31.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ x + \left(\mathsf{fma}\left(y, y \cdot \left(y \cdot 0.3333333333333333\right), y\right) - \tan a\right) \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (+ x (- (fma y (* y (* y 0.3333333333333333)) y) (tan a))))

C

double code(double x, double y, double z, double a) {
	return x + (fma(y, (y * (y * 0.3333333333333333)), y) - tan(a));
}

Julia

function code(x, y, z, a)
	return Float64(x + Float64(fma(y, Float64(y * Float64(y * 0.3333333333333333)), y) - tan(a)))
end

Wolfram

code[x_, y_, z_, a_] := N[(x + N[(N[(y * N[(y * N[(y * 0.3333333333333333), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 79.3%

   \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
2. Add Preprocessing

3. Taylor expanded in z around 0

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

   1. lower-/.f64 N/A

      \[\leadsto x + \left(\color{blue}{\frac{\sin y}{\cos y}} - \tan a\right) \]

   2. lower-sin.f64 N/A

      \[\leadsto x + \left(\frac{\color{blue}{\sin y}}{\cos y} - \tan a\right) \]

   3. lower-cos.f64 55.7

      \[\leadsto x + \left(\frac{\sin y}{\color{blue}{\cos y}} - \tan a\right) \]

5. Applied rewrites 55.7%

   \[\leadsto x + \left(\color{blue}{\frac{\sin y}{\cos y}} - \tan a\right) \]

6. Taylor expanded in y around 0

   \[\leadsto x + \left(y \cdot \color{blue}{\left(1 + \frac{1}{3} \cdot {y}^{2}\right)} - \tan a\right) \]
7. Step-by-step derivation

8. Applied rewrites 29.3%

   \[\leadsto x + \left(\mathsf{fma}\left(y, \color{blue}{y \cdot \left(y \cdot 0.3333333333333333\right)}, y\right) - \tan a\right) \]
9. Add Preprocessing

Alternative 12: 30.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ 0.3333333333333333 \cdot \left(y \cdot \left(y \cdot y\right)\right) + \left(x - \tan a\right) \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (+ (* 0.3333333333333333 (* y (* y y))) (- x (tan a))))

C

double code(double x, double y, double z, double a) {
	return (0.3333333333333333 * (y * (y * y))) + (x - tan(a));
}

Fortran

real(8) function code(x, y, z, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: a
    code = (0.3333333333333333d0 * (y * (y * y))) + (x - tan(a))
end function

Java

public static double code(double x, double y, double z, double a) {
	return (0.3333333333333333 * (y * (y * y))) + (x - Math.tan(a));
}

Python

def code(x, y, z, a):
	return (0.3333333333333333 * (y * (y * y))) + (x - math.tan(a))

Julia

function code(x, y, z, a)
	return Float64(Float64(0.3333333333333333 * Float64(y * Float64(y * y))) + Float64(x - tan(a)))
end

MATLAB

function tmp = code(x, y, z, a)
	tmp = (0.3333333333333333 * (y * (y * y))) + (x - tan(a));
end

Wolfram

code[x_, y_, z_, a_] := N[(N[(0.3333333333333333 * N[(y * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 79.3%

   \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
2. Add Preprocessing

3. Taylor expanded in z around 0

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

   1. lower-/.f64 N/A

      \[\leadsto x + \left(\color{blue}{\frac{\sin y}{\cos y}} - \tan a\right) \]

   2. lower-sin.f64 N/A

      \[\leadsto x + \left(\frac{\color{blue}{\sin y}}{\cos y} - \tan a\right) \]

   3. lower-cos.f64 55.7

      \[\leadsto x + \left(\frac{\sin y}{\color{blue}{\cos y}} - \tan a\right) \]

5. Applied rewrites 55.7%

   \[\leadsto x + \left(\color{blue}{\frac{\sin y}{\cos y}} - \tan a\right) \]

6. Taylor expanded in y around 0

   \[\leadsto x + \left(y \cdot \color{blue}{\left(1 + \frac{1}{3} \cdot {y}^{2}\right)} - \tan a\right) \]
7. Step-by-step derivation

8. Applied rewrites 29.3%

   \[\leadsto x + \left(\mathsf{fma}\left(y, \color{blue}{y \cdot \left(y \cdot 0.3333333333333333\right)}, y\right) - \tan a\right) \]
9. Step-by-step derivation

   1. lift-+.f64 N/A

      \[\leadsto \color{blue}{x + \left(\mathsf{fma}\left(y, y \cdot \left(y \cdot \frac{1}{3}\right), y\right) - \tan a\right)} \]

   2. +-commutative N/A

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(y, y \cdot \left(y \cdot \frac{1}{3}\right), y\right) - \tan a\right) + x} \]

   3. lift--.f64 N/A

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(y, y \cdot \left(y \cdot \frac{1}{3}\right), y\right) - \tan a\right)} + x \]

   4. associate-+l- N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, y \cdot \left(y \cdot \frac{1}{3}\right), y\right) - \left(\tan a - x\right)} \]

   5. lower--.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, y \cdot \left(y \cdot \frac{1}{3}\right), y\right) - \left(\tan a - x\right)} \]

   6. lower--.f64 N/A

      \[\leadsto \mathsf{fma}\left(y, \frac{1}{3} \cdot \mathsf{Rewrite=>}\left(lower-*.f64, \left(y \cdot y\right)\right), y\right) - \color{blue}{\left(\tan a - x\right)} \]

10. Applied rewrites 29.3%

    \[\leadsto \color{blue}{\mathsf{fma}\left(y, 0.3333333333333333 \cdot \left(y \cdot y\right), y\right) - \left(\tan a - x\right)} \]

11. Taylor expanded in y around inf

    \[\leadsto \frac{1}{3} \cdot {y}^{\color{blue}{3}} - \left(\tan a - x\right) \]
12. Step-by-step derivation

13. Applied rewrites 28.7%

    \[\leadsto 0.3333333333333333 \cdot \left(y \cdot \color{blue}{\left(y \cdot y\right)}\right) - \left(\tan a - x\right) \]

14. Final simplification 28.7%

    \[\leadsto 0.3333333333333333 \cdot \left(y \cdot \left(y \cdot y\right)\right) + \left(x - \tan a\right) \]
15. Add Preprocessing

Reproduce

herbie shell --seed 2024216 
(FPCore (x y z a)
  :name "tan-example (used to crash)"
  :precision binary64
  :pre (and (and (and (or (== x 0.0) (and (<= 0.5884142 x) (<= x 505.5909))) (or (and (<= -1.796658e+308 y) (<= y -9.425585e-310)) (and (<= 1.284938e-309 y) (<= y 1.751224e+308)))) (or (and (<= -1.776707e+308 z) (<= z -8.599796e-310)) (and (<= 3.293145e-311 z) (<= z 1.725154e+308)))) (or (and (<= -1.796658e+308 a) (<= a -9.425585e-310)) (and (<= 1.284938e-309 a) (<= a 1.751224e+308))))
  (+ x (- (tan (+ y z)) (tan a))))