tan-example (used to crash)

Percentage Accurate: 79.5% → 99.7%

Time: 27.1s

Alternatives: 11

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: 79.5% 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, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 78.9%

   \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
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. frac-2neg N/A

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

   7. div-inv N/A

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

   8. lower-fma.f64 N/A

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

4. Applied rewrites 99.7%

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

   1. lift-neg.f64 N/A

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

   2. neg-sub0 N/A

      \[\leadsto x + \mathsf{fma}\left(-\left(\tan z + \tan y\right), \frac{1}{\color{blue}{0 - \mathsf{fma}\left(-\tan z, \tan y, 1\right)}}, -\tan a\right) \]

   3. lift-fma.f64 N/A

      \[\leadsto x + \mathsf{fma}\left(-\left(\tan z + \tan y\right), \frac{1}{0 - \color{blue}{\left(\left(-\tan z\right) \cdot \tan y + 1\right)}}, -\tan a\right) \]

   4. associate--r+ N/A

      \[\leadsto x + \mathsf{fma}\left(-\left(\tan z + \tan y\right), \frac{1}{\color{blue}{\left(0 - \left(-\tan z\right) \cdot \tan y\right) - 1}}, -\tan a\right) \]

   5. neg-sub0 N/A

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

   6. lift-neg.f64 N/A

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

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

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

   8. lift-tan.f64 N/A

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

   9. lift-tan.f64 N/A

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

   10. remove-double-neg N/A

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

   11. lower--.f64 N/A

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

   12. lift-tan.f64 N/A

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

   13. lift-tan.f64 N/A

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

   14. *-commutative N/A

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

   15. lower-*.f64 99.7

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

6. Applied rewrites 99.7%

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

7. Final simplification 99.7%

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

Alternative 2: 89.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan z + \tan y\\ \mathbf{if}\;\tan a \leq -0.02:\\ \;\;\;\;x + \left(\tan \left(y + z\right) - {\left({\tan a}^{-1}\right)}^{-1}\right)\\ \mathbf{elif}\;\tan a \leq 5 \cdot 10^{-12}:\\ \;\;\;\;x + \mathsf{fma}\left(t\_0, \frac{-1}{\mathsf{fma}\left(\tan z, \tan y, -1\right)}, -\mathsf{fma}\left(0.3333333333333333, a \cdot a, 1\right) \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;x + \mathsf{fma}\left(-t\_0, {-1}^{-1}, -\tan a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (let* ((t_0 (+ (tan z) (tan y))))
   (if (<= (tan a) -0.02)
     (+ x (- (tan (+ y z)) (pow (pow (tan a) -1.0) -1.0)))
     (if (<= (tan a) 5e-12)
       (+
        x
        (fma
         t_0
         (/ -1.0 (fma (tan z) (tan y) -1.0))
         (- (* (fma 0.3333333333333333 (* a a) 1.0) a))))
       (+ x (fma (- t_0) (pow -1.0 -1.0) (- (tan a))))))))

C

double code(double x, double y, double z, double a) {
	double t_0 = tan(z) + tan(y);
	double tmp;
	if (tan(a) <= -0.02) {
		tmp = x + (tan((y + z)) - pow(pow(tan(a), -1.0), -1.0));
	} else if (tan(a) <= 5e-12) {
		tmp = x + fma(t_0, (-1.0 / fma(tan(z), tan(y), -1.0)), -(fma(0.3333333333333333, (a * a), 1.0) * a));
	} else {
		tmp = x + fma(-t_0, pow(-1.0, -1.0), -tan(a));
	}
	return tmp;
}

Julia

function code(x, y, z, a)
	t_0 = Float64(tan(z) + tan(y))
	tmp = 0.0
	if (tan(a) <= -0.02)
		tmp = Float64(x + Float64(tan(Float64(y + z)) - ((tan(a) ^ -1.0) ^ -1.0)));
	elseif (tan(a) <= 5e-12)
		tmp = Float64(x + fma(t_0, Float64(-1.0 / fma(tan(z), tan(y), -1.0)), Float64(-Float64(fma(0.3333333333333333, Float64(a * a), 1.0) * a))));
	else
		tmp = Float64(x + fma(Float64(-t_0), (-1.0 ^ -1.0), Float64(-tan(a))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, a_] := Block[{t$95$0 = N[(N[Tan[z], $MachinePrecision] + N[Tan[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Tan[a], $MachinePrecision], -0.02], N[(x + N[(N[Tan[N[(y + z), $MachinePrecision]], $MachinePrecision] - N[Power[N[Power[N[Tan[a], $MachinePrecision], -1.0], $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Tan[a], $MachinePrecision], 5e-12], N[(x + N[(t$95$0 * N[(-1.0 / N[(N[Tan[z], $MachinePrecision] * N[Tan[y], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] + (-N[(N[(0.3333333333333333 * N[(a * a), $MachinePrecision] + 1.0), $MachinePrecision] * a), $MachinePrecision])), $MachinePrecision]), $MachinePrecision], N[(x + N[((-t$95$0) * N[Power[-1.0, -1.0], $MachinePrecision] + (-N[Tan[a], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (tan.f64 a) < -0.0200000000000000004

   1. Initial program 79.7%

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

      1. lift-tan.f64 N/A

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

      2. tan-quot N/A

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

      3. clear-num N/A

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

      4. lower-/.f64 N/A

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

      5. clear-num N/A

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

      6. tan-quot N/A

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

      7. lift-tan.f64 N/A

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

      8. lower-/.f64 79.8

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

   4. Applied rewrites 79.8%

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

   if -0.0200000000000000004 < (tan.f64 a) < 4.9999999999999997e-12

   1. Initial program 79.7%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   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. frac-2neg N/A

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

      7. div-inv N/A

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

      8. lower-fma.f64 N/A

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 99.9

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

   7. Applied rewrites 99.9%

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

   9. Applied rewrites 99.9%

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

   if 4.9999999999999997e-12 < (tan.f64 a)

   1. Initial program 76.6%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   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. frac-2neg N/A

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

      7. div-inv N/A

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

      8. lower-fma.f64 N/A

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

   4. Applied rewrites 99.5%

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

   5. Taylor expanded in y around 0

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

   7. Applied rewrites 78.0%

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

4. Final simplification 89.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\tan a \leq -0.02:\\ \;\;\;\;x + \left(\tan \left(y + z\right) - {\left({\tan a}^{-1}\right)}^{-1}\right)\\ \mathbf{elif}\;\tan a \leq 5 \cdot 10^{-12}:\\ \;\;\;\;x + \mathsf{fma}\left(\tan z + \tan y, \frac{-1}{\mathsf{fma}\left(\tan z, \tan y, -1\right)}, -\mathsf{fma}\left(0.3333333333333333, a \cdot a, 1\right) \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;x + \mathsf{fma}\left(-\left(\tan z + \tan y\right), {-1}^{-1}, -\tan a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 89.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan z + \tan y\\ \mathbf{if}\;\tan a \leq -4 \cdot 10^{-11}:\\ \;\;\;\;x + \left(\tan \left(y + z\right) - {\left({\tan a}^{-1}\right)}^{-1}\right)\\ \mathbf{elif}\;\tan a \leq 2 \cdot 10^{-15}:\\ \;\;\;\;\mathsf{fma}\left(t\_0, \frac{-1}{\mathsf{fma}\left(\tan z, \tan y, -1\right)}, x\right)\\ \mathbf{else}:\\ \;\;\;\;x + \mathsf{fma}\left(-t\_0, {-1}^{-1}, -\tan a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (let* ((t_0 (+ (tan z) (tan y))))
   (if (<= (tan a) -4e-11)
     (+ x (- (tan (+ y z)) (pow (pow (tan a) -1.0) -1.0)))
     (if (<= (tan a) 2e-15)
       (fma t_0 (/ -1.0 (fma (tan z) (tan y) -1.0)) x)
       (+ x (fma (- t_0) (pow -1.0 -1.0) (- (tan a))))))))

C

double code(double x, double y, double z, double a) {
	double t_0 = tan(z) + tan(y);
	double tmp;
	if (tan(a) <= -4e-11) {
		tmp = x + (tan((y + z)) - pow(pow(tan(a), -1.0), -1.0));
	} else if (tan(a) <= 2e-15) {
		tmp = fma(t_0, (-1.0 / fma(tan(z), tan(y), -1.0)), x);
	} else {
		tmp = x + fma(-t_0, pow(-1.0, -1.0), -tan(a));
	}
	return tmp;
}

Julia

function code(x, y, z, a)
	t_0 = Float64(tan(z) + tan(y))
	tmp = 0.0
	if (tan(a) <= -4e-11)
		tmp = Float64(x + Float64(tan(Float64(y + z)) - ((tan(a) ^ -1.0) ^ -1.0)));
	elseif (tan(a) <= 2e-15)
		tmp = fma(t_0, Float64(-1.0 / fma(tan(z), tan(y), -1.0)), x);
	else
		tmp = Float64(x + fma(Float64(-t_0), (-1.0 ^ -1.0), Float64(-tan(a))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, a_] := Block[{t$95$0 = N[(N[Tan[z], $MachinePrecision] + N[Tan[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Tan[a], $MachinePrecision], -4e-11], N[(x + N[(N[Tan[N[(y + z), $MachinePrecision]], $MachinePrecision] - N[Power[N[Power[N[Tan[a], $MachinePrecision], -1.0], $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Tan[a], $MachinePrecision], 2e-15], N[(t$95$0 * N[(-1.0 / N[(N[Tan[z], $MachinePrecision] * N[Tan[y], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], N[(x + N[((-t$95$0) * N[Power[-1.0, -1.0], $MachinePrecision] + (-N[Tan[a], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (tan.f64 a) < -3.99999999999999976e-11

   1. Initial program 80.0%

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

      1. lift-tan.f64 N/A

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

      2. tan-quot N/A

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

      3. clear-num N/A

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

      4. lower-/.f64 N/A

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

      5. clear-num N/A

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

      6. tan-quot N/A

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

      7. lift-tan.f64 N/A

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

      8. lower-/.f64 80.1

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

   4. Applied rewrites 80.1%

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

   if -3.99999999999999976e-11 < (tan.f64 a) < 2.0000000000000002e-15

   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

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

      2. flip3-+ N/A

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

      3. clear-num N/A

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

      4. lower-/.f64 N/A

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

      5. clear-num N/A

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

      6. flip3-+ N/A

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

      7. lift-+.f64 N/A

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

      8. lower-/.f64 79.1

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

   4. Applied rewrites 79.1%

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

   5. Taylor expanded in a around 0

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

      1. +-commutative N/A

         \[\leadsto \frac{1}{\frac{1}{\color{blue}{\frac{\sin \left(y + z\right)}{\cos \left(y + z\right)} + x}}} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{1}{\frac{1}{\color{blue}{\frac{\sin \left(y + z\right)}{\cos \left(y + z\right)} + x}}} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{1}{\frac{1}{\color{blue}{\frac{\sin \left(y + z\right)}{\cos \left(y + z\right)}} + x}} \]

      4. lower-sin.f64 N/A

         \[\leadsto \frac{1}{\frac{1}{\frac{\color{blue}{\sin \left(y + z\right)}}{\cos \left(y + z\right)} + x}} \]

      5. lower-+.f64 N/A

         \[\leadsto \frac{1}{\frac{1}{\frac{\sin \color{blue}{\left(y + z\right)}}{\cos \left(y + z\right)} + x}} \]

      6. lower-cos.f64 N/A

         \[\leadsto \frac{1}{\frac{1}{\frac{\sin \left(y + z\right)}{\color{blue}{\cos \left(y + z\right)}} + x}} \]

      7. lower-+.f64 79.1

         \[\leadsto \frac{1}{\frac{1}{\frac{\sin \left(y + z\right)}{\cos \color{blue}{\left(y + z\right)}} + x}} \]

   7. Applied rewrites 79.1%

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

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{1}{\frac{\sin \left(y + z\right)}{\cos \left(y + z\right)} + x}}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{1}{\frac{\sin \left(y + z\right)}{\cos \left(y + z\right)} + x}}} \]

      3. remove-double-div 79.3

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

   9. Applied rewrites 79.3%

      \[\leadsto \color{blue}{\tan \left(y + z\right) + x} \]
   10. Step-by-step derivation

   11. Applied rewrites 99.9%

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

   if 2.0000000000000002e-15 < (tan.f64 a)

   1. Initial program 77.0%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   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. frac-2neg N/A

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

      7. div-inv N/A

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

      8. lower-fma.f64 N/A

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

   4. Applied rewrites 99.5%

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

   5. Taylor expanded in y around 0

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

   7. Applied rewrites 78.5%

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

4. Final simplification 89.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\tan a \leq -4 \cdot 10^{-11}:\\ \;\;\;\;x + \left(\tan \left(y + z\right) - {\left({\tan a}^{-1}\right)}^{-1}\right)\\ \mathbf{elif}\;\tan a \leq 2 \cdot 10^{-15}:\\ \;\;\;\;\mathsf{fma}\left(\tan z + \tan y, \frac{-1}{\mathsf{fma}\left(\tan z, \tan y, -1\right)}, x\right)\\ \mathbf{else}:\\ \;\;\;\;x + \mathsf{fma}\left(-\left(\tan z + \tan y\right), {-1}^{-1}, -\tan a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 89.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan z + \tan y\\ \mathbf{if}\;\tan a \leq -4 \cdot 10^{-11}:\\ \;\;\;\;x + \left(\tan \left(y + z\right) - {\left({\tan a}^{-1}\right)}^{-1}\right)\\ \mathbf{elif}\;\tan a \leq 2 \cdot 10^{-15}:\\ \;\;\;\;\frac{t\_0}{-\mathsf{fma}\left(\tan z, \tan y, -1\right)} + x\\ \mathbf{else}:\\ \;\;\;\;x + \mathsf{fma}\left(-t\_0, {-1}^{-1}, -\tan a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (let* ((t_0 (+ (tan z) (tan y))))
   (if (<= (tan a) -4e-11)
     (+ x (- (tan (+ y z)) (pow (pow (tan a) -1.0) -1.0)))
     (if (<= (tan a) 2e-15)
       (+ (/ t_0 (- (fma (tan z) (tan y) -1.0))) x)
       (+ x (fma (- t_0) (pow -1.0 -1.0) (- (tan a))))))))

C

double code(double x, double y, double z, double a) {
	double t_0 = tan(z) + tan(y);
	double tmp;
	if (tan(a) <= -4e-11) {
		tmp = x + (tan((y + z)) - pow(pow(tan(a), -1.0), -1.0));
	} else if (tan(a) <= 2e-15) {
		tmp = (t_0 / -fma(tan(z), tan(y), -1.0)) + x;
	} else {
		tmp = x + fma(-t_0, pow(-1.0, -1.0), -tan(a));
	}
	return tmp;
}

Julia

function code(x, y, z, a)
	t_0 = Float64(tan(z) + tan(y))
	tmp = 0.0
	if (tan(a) <= -4e-11)
		tmp = Float64(x + Float64(tan(Float64(y + z)) - ((tan(a) ^ -1.0) ^ -1.0)));
	elseif (tan(a) <= 2e-15)
		tmp = Float64(Float64(t_0 / Float64(-fma(tan(z), tan(y), -1.0))) + x);
	else
		tmp = Float64(x + fma(Float64(-t_0), (-1.0 ^ -1.0), Float64(-tan(a))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, a_] := Block[{t$95$0 = N[(N[Tan[z], $MachinePrecision] + N[Tan[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Tan[a], $MachinePrecision], -4e-11], N[(x + N[(N[Tan[N[(y + z), $MachinePrecision]], $MachinePrecision] - N[Power[N[Power[N[Tan[a], $MachinePrecision], -1.0], $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Tan[a], $MachinePrecision], 2e-15], N[(N[(t$95$0 / (-N[(N[Tan[z], $MachinePrecision] * N[Tan[y], $MachinePrecision] + -1.0), $MachinePrecision])), $MachinePrecision] + x), $MachinePrecision], N[(x + N[((-t$95$0) * N[Power[-1.0, -1.0], $MachinePrecision] + (-N[Tan[a], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (tan.f64 a) < -3.99999999999999976e-11

   1. Initial program 80.0%

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

      1. lift-tan.f64 N/A

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

      2. tan-quot N/A

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

      3. clear-num N/A

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

      4. lower-/.f64 N/A

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

      5. clear-num N/A

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

      6. tan-quot N/A

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

      7. lift-tan.f64 N/A

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

      8. lower-/.f64 80.1

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

   4. Applied rewrites 80.1%

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

   if -3.99999999999999976e-11 < (tan.f64 a) < 2.0000000000000002e-15

   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

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

      2. flip3-+ N/A

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

      3. clear-num N/A

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

      4. lower-/.f64 N/A

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

      5. clear-num N/A

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

      6. flip3-+ N/A

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

      7. lift-+.f64 N/A

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

      8. lower-/.f64 79.1

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

   4. Applied rewrites 79.1%

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

   5. Taylor expanded in a around 0

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

      1. +-commutative N/A

         \[\leadsto \frac{1}{\frac{1}{\color{blue}{\frac{\sin \left(y + z\right)}{\cos \left(y + z\right)} + x}}} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{1}{\frac{1}{\color{blue}{\frac{\sin \left(y + z\right)}{\cos \left(y + z\right)} + x}}} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{1}{\frac{1}{\color{blue}{\frac{\sin \left(y + z\right)}{\cos \left(y + z\right)}} + x}} \]

      4. lower-sin.f64 N/A

         \[\leadsto \frac{1}{\frac{1}{\frac{\color{blue}{\sin \left(y + z\right)}}{\cos \left(y + z\right)} + x}} \]

      5. lower-+.f64 N/A

         \[\leadsto \frac{1}{\frac{1}{\frac{\sin \color{blue}{\left(y + z\right)}}{\cos \left(y + z\right)} + x}} \]

      6. lower-cos.f64 N/A

         \[\leadsto \frac{1}{\frac{1}{\frac{\sin \left(y + z\right)}{\color{blue}{\cos \left(y + z\right)}} + x}} \]

      7. lower-+.f64 79.1

         \[\leadsto \frac{1}{\frac{1}{\frac{\sin \left(y + z\right)}{\cos \color{blue}{\left(y + z\right)}} + x}} \]

   7. Applied rewrites 79.1%

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

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{1}{\frac{\sin \left(y + z\right)}{\cos \left(y + z\right)} + x}}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{1}{\frac{\sin \left(y + z\right)}{\cos \left(y + z\right)} + x}}} \]

      3. remove-double-div 79.3

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

   9. Applied rewrites 79.3%

      \[\leadsto \color{blue}{\tan \left(y + z\right) + x} \]
   10. Step-by-step derivation

   11. Applied rewrites 99.9%

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

   if 2.0000000000000002e-15 < (tan.f64 a)

   1. Initial program 77.0%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   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. frac-2neg N/A

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

      7. div-inv N/A

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

      8. lower-fma.f64 N/A

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

   4. Applied rewrites 99.5%

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

   5. Taylor expanded in y around 0

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

   7. Applied rewrites 78.5%

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

4. Final simplification 89.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\tan a \leq -4 \cdot 10^{-11}:\\ \;\;\;\;x + \left(\tan \left(y + z\right) - {\left({\tan a}^{-1}\right)}^{-1}\right)\\ \mathbf{elif}\;\tan a \leq 2 \cdot 10^{-15}:\\ \;\;\;\;\frac{\tan z + \tan y}{-\mathsf{fma}\left(\tan z, \tan y, -1\right)} + x\\ \mathbf{else}:\\ \;\;\;\;x + \mathsf{fma}\left(-\left(\tan z + \tan y\right), {-1}^{-1}, -\tan a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.7% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 78.9%

   \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
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. frac-2neg N/A

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

   7. div-inv N/A

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

   8. lower-fma.f64 N/A

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

4. Applied rewrites 99.7%

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

   1. lift-fma.f64 N/A

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

6. Applied rewrites 99.7%

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

Alternative 6: 99.7% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_, z_, a_] := N[(x + N[(N[(N[(N[Tan[z], $MachinePrecision] + N[Tan[y], $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 78.9%

   \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
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. +-commutative N/A

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

   6. lower-+.f64 N/A

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

   7. lower-tan.f64 N/A

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

   8. lower-tan.f64 N/A

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

   9. sub-neg N/A

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

   10. +-commutative N/A

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

   11. *-commutative N/A

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

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

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

   13. lower-fma.f64 N/A

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

   14. lower-neg.f64 N/A

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

   15. lower-tan.f64 N/A

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

   16. lower-tan.f64 99.7

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

4. Applied rewrites 99.7%

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

Alternative 7: 99.6% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 78.9%

   \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
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. frac-2neg N/A

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

   7. div-inv N/A

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

   8. lower-fma.f64 N/A

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

4. Applied rewrites 99.7%

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

   1. lift-neg.f64 N/A

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

   2. neg-sub0 N/A

      \[\leadsto x + \mathsf{fma}\left(-\left(\tan z + \tan y\right), \frac{1}{\color{blue}{0 - \mathsf{fma}\left(-\tan z, \tan y, 1\right)}}, -\tan a\right) \]

   3. lift-fma.f64 N/A

      \[\leadsto x + \mathsf{fma}\left(-\left(\tan z + \tan y\right), \frac{1}{0 - \color{blue}{\left(\left(-\tan z\right) \cdot \tan y + 1\right)}}, -\tan a\right) \]

   4. associate--r+ N/A

      \[\leadsto x + \mathsf{fma}\left(-\left(\tan z + \tan y\right), \frac{1}{\color{blue}{\left(0 - \left(-\tan z\right) \cdot \tan y\right) - 1}}, -\tan a\right) \]

   5. neg-sub0 N/A

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

   6. lift-neg.f64 N/A

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

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

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

   8. lift-tan.f64 N/A

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

   9. lift-tan.f64 N/A

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

   10. remove-double-neg N/A

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

   11. lower--.f64 N/A

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

   12. lift-tan.f64 N/A

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

   13. lift-tan.f64 N/A

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

   14. *-commutative N/A

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

   15. lower-*.f64 99.7

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

6. Applied rewrites 99.7%

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

   1. lift-+.f64 N/A

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

   2. lift-fma.f64 N/A

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

   3. associate-+r+ N/A

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

   4. lift-neg.f64 N/A

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

   5. unsub-neg N/A

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

   6. lower--.f64 N/A

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

8. Applied rewrites 99.7%

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

Alternative 8: 79.9% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 78.9%

   \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
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. frac-2neg N/A

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

   7. div-inv N/A

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

   8. lower-fma.f64 N/A

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

4. Applied rewrites 99.7%

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

5. Taylor expanded in y around 0

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

7. Applied rewrites 79.2%

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

8. Final simplification 79.2%

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

Alternative 9: 79.5% 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 78.9%

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

Alternative 10: 31.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ {\left({x}^{-1}\right)}^{-1} \end{array} \]

FPCore

(FPCore (x y z a) :precision binary64 (pow (pow x -1.0) -1.0))

C

double code(double x, double y, double z, double a) {
	return pow(pow(x, -1.0), -1.0);
}

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 ** (-1.0d0)) ** (-1.0d0)
end function

Java

public static double code(double x, double y, double z, double a) {
	return Math.pow(Math.pow(x, -1.0), -1.0);
}

Python

def code(x, y, z, a):
	return math.pow(math.pow(x, -1.0), -1.0)

Julia

function code(x, y, z, a)
	return (x ^ -1.0) ^ -1.0
end

MATLAB

function tmp = code(x, y, z, a)
	tmp = (x ^ -1.0) ^ -1.0;
end

Wolfram

code[x_, y_, z_, a_] := N[Power[N[Power[x, -1.0], $MachinePrecision], -1.0], $MachinePrecision]

Derivation

1. Initial program 78.9%

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

   1. lift-+.f64 N/A

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

   2. flip3-+ N/A

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

   3. clear-num N/A

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

   4. lower-/.f64 N/A

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

   5. clear-num N/A

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

   6. flip3-+ N/A

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

   7. lift-+.f64 N/A

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

   8. lower-/.f64 78.7

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

4. Applied rewrites 78.7%

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

5. Taylor expanded in x around inf

   \[\leadsto \frac{1}{\color{blue}{\frac{1}{x}}} \]
6. Step-by-step derivation

   1. lower-/.f64 31.8

      \[\leadsto \frac{1}{\color{blue}{\frac{1}{x}}} \]

7. Applied rewrites 31.8%

   \[\leadsto \frac{1}{\color{blue}{\frac{1}{x}}} \]

8. Final simplification 31.8%

   \[\leadsto {\left({x}^{-1}\right)}^{-1} \]
9. Add Preprocessing

Alternative 11: 50.1% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

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 = tan((y + z)) + x
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.9%

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

   1. lift-+.f64 N/A

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

   2. flip3-+ N/A

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

   3. clear-num N/A

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

   4. lower-/.f64 N/A

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

   5. clear-num N/A

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

   6. flip3-+ N/A

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

   7. lift-+.f64 N/A

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

   8. lower-/.f64 78.7

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

4. Applied rewrites 78.7%

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

5. Taylor expanded in a around 0

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

   1. +-commutative N/A

      \[\leadsto \frac{1}{\frac{1}{\color{blue}{\frac{\sin \left(y + z\right)}{\cos \left(y + z\right)} + x}}} \]

   2. lower-+.f64 N/A

      \[\leadsto \frac{1}{\frac{1}{\color{blue}{\frac{\sin \left(y + z\right)}{\cos \left(y + z\right)} + x}}} \]

   3. lower-/.f64 N/A

      \[\leadsto \frac{1}{\frac{1}{\color{blue}{\frac{\sin \left(y + z\right)}{\cos \left(y + z\right)}} + x}} \]

   4. lower-sin.f64 N/A

      \[\leadsto \frac{1}{\frac{1}{\frac{\color{blue}{\sin \left(y + z\right)}}{\cos \left(y + z\right)} + x}} \]

   5. lower-+.f64 N/A

      \[\leadsto \frac{1}{\frac{1}{\frac{\sin \color{blue}{\left(y + z\right)}}{\cos \left(y + z\right)} + x}} \]

   6. lower-cos.f64 N/A

      \[\leadsto \frac{1}{\frac{1}{\frac{\sin \left(y + z\right)}{\color{blue}{\cos \left(y + z\right)}} + x}} \]

   7. lower-+.f64 50.3

      \[\leadsto \frac{1}{\frac{1}{\frac{\sin \left(y + z\right)}{\cos \color{blue}{\left(y + z\right)}} + x}} \]

7. Applied rewrites 50.3%

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

   1. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{1}{\frac{1}{\frac{\sin \left(y + z\right)}{\cos \left(y + z\right)} + x}}} \]

   2. lift-/.f64 N/A

      \[\leadsto \frac{1}{\color{blue}{\frac{1}{\frac{\sin \left(y + z\right)}{\cos \left(y + z\right)} + x}}} \]

   3. remove-double-div 50.4

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

9. Applied rewrites 50.4%

   \[\leadsto \color{blue}{\tan \left(y + z\right) + x} \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024313 
(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))))