tan-example (used to crash)

Percentage Accurate: 79.2% → 99.7%

Time: 31.1s

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: 79.2% 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

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 74.5%

   \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
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 99.8%

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

6. Applied rewrites 99.8%

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

7. Final simplification 99.8%

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

Alternative 2: 88.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan z + \tan y\\ t_1 := x - \left(\frac{t\_0}{-1} + \tan a\right)\\ \mathbf{if}\;\tan a \leq -5 \cdot 10^{-13}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\tan a \leq 2 \cdot 10^{-36}:\\ \;\;\;\;\frac{-1}{\mathsf{fma}\left(\tan z, \tan y, -1\right)} \cdot t\_0 - \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (let* ((t_0 (+ (tan z) (tan y))) (t_1 (- x (+ (/ t_0 -1.0) (tan a)))))
   (if (<= (tan a) -5e-13)
     t_1
     (if (<= (tan a) 2e-36)
       (- (* (/ -1.0 (fma (tan z) (tan y) -1.0)) t_0) (- x))
       t_1))))

C

double code(double x, double y, double z, double a) {
	double t_0 = tan(z) + tan(y);
	double t_1 = x - ((t_0 / -1.0) + tan(a));
	double tmp;
	if (tan(a) <= -5e-13) {
		tmp = t_1;
	} else if (tan(a) <= 2e-36) {
		tmp = ((-1.0 / fma(tan(z), tan(y), -1.0)) * t_0) - -x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, a)
	t_0 = Float64(tan(z) + tan(y))
	t_1 = Float64(x - Float64(Float64(t_0 / -1.0) + tan(a)))
	tmp = 0.0
	if (tan(a) <= -5e-13)
		tmp = t_1;
	elseif (tan(a) <= 2e-36)
		tmp = Float64(Float64(Float64(-1.0 / fma(tan(z), tan(y), -1.0)) * t_0) - Float64(-x));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, a_] := Block[{t$95$0 = N[(N[Tan[z], $MachinePrecision] + N[Tan[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x - N[(N[(t$95$0 / -1.0), $MachinePrecision] + N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Tan[a], $MachinePrecision], -5e-13], t$95$1, If[LessEqual[N[Tan[a], $MachinePrecision], 2e-36], N[(N[(N[(-1.0 / N[(N[Tan[z], $MachinePrecision] * N[Tan[y], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] - (-x)), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (tan.f64 a) < -4.9999999999999999e-13 or 1.9999999999999999e-36 < (tan.f64 a)

   1. Initial program 74.8%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   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%

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

   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. Taylor expanded in z around 0

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

   9. Applied rewrites 75.6%

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

       1. lift-+.f64 N/A

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

       2. +-commutative N/A

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

       3. lower-+.f64 75.6

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

   11. Applied rewrites 75.6%

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

   if -4.9999999999999999e-13 < (tan.f64 a) < 1.9999999999999999e-36

   1. Initial program 74.3%

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

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

      3. lift--.f64 N/A

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

      4. associate-+l- N/A

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

      5. lower--.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. lower--.f64 74.3

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

   4. Applied rewrites 74.3%

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

   5. Taylor expanded in a around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 74.3

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

   7. Applied rewrites 74.3%

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

      1. lift-tan.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. tan-sum N/A

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

      4. lift-tan.f64 N/A

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

      5. lift-tan.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. div-inv N/A

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

      9. metadata-eval N/A

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

      10. lift-tan.f64 N/A

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

      11. lift-tan.f64 N/A

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

      12. *-commutative N/A

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

      13. sub-neg N/A

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

      14. metadata-eval N/A

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

      15. distribute-neg-in N/A

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

      16. +-commutative N/A

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

      17. lift-fma.f64 N/A

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

      18. frac-2neg N/A

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

      19. lift-/.f64 N/A

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

      20. *-commutative N/A

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

   9. Applied rewrites 99.8%

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

4. Final simplification 88.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\tan a \leq -5 \cdot 10^{-13}:\\ \;\;\;\;x - \left(\frac{\tan z + \tan y}{-1} + \tan a\right)\\ \mathbf{elif}\;\tan a \leq 2 \cdot 10^{-36}:\\ \;\;\;\;\frac{-1}{\mathsf{fma}\left(\tan z, \tan y, -1\right)} \cdot \left(\tan z + \tan y\right) - \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;x - \left(\frac{\tan z + \tan y}{-1} + \tan a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.7% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 74.5%

   \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
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. Final simplification 99.7%

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

Alternative 4: 89.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x - \left(\frac{\tan z + \tan y}{-1} + \tan a\right)\\ t_1 := -\tan z\\ \mathbf{if}\;a \leq -0.027:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;a \leq 0.036:\\ \;\;\;\;\mathsf{fma}\left(t\_1 - \tan y, \frac{-1}{\mathsf{fma}\left(t\_1, \tan y, 1\right)}, -\mathsf{fma}\left(\mathsf{fma}\left(0.13333333333333333, a \cdot a, 0.3333333333333333\right), a \cdot a, 1\right) \cdot a\right) + x\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (let* ((t_0 (- x (+ (/ (+ (tan z) (tan y)) -1.0) (tan a)))) (t_1 (- (tan z))))
   (if (<= a -0.027)
     t_0
     (if (<= a 0.036)
       (+
        (fma
         (- t_1 (tan y))
         (/ -1.0 (fma t_1 (tan y) 1.0))
         (-
          (*
           (fma
            (fma 0.13333333333333333 (* a a) 0.3333333333333333)
            (* a a)
            1.0)
           a)))
        x)
       t_0))))

C

double code(double x, double y, double z, double a) {
	double t_0 = x - (((tan(z) + tan(y)) / -1.0) + tan(a));
	double t_1 = -tan(z);
	double tmp;
	if (a <= -0.027) {
		tmp = t_0;
	} else if (a <= 0.036) {
		tmp = fma((t_1 - tan(y)), (-1.0 / fma(t_1, tan(y), 1.0)), -(fma(fma(0.13333333333333333, (a * a), 0.3333333333333333), (a * a), 1.0) * a)) + x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z, a)
	t_0 = Float64(x - Float64(Float64(Float64(tan(z) + tan(y)) / -1.0) + tan(a)))
	t_1 = Float64(-tan(z))
	tmp = 0.0
	if (a <= -0.027)
		tmp = t_0;
	elseif (a <= 0.036)
		tmp = Float64(fma(Float64(t_1 - tan(y)), Float64(-1.0 / fma(t_1, tan(y), 1.0)), Float64(-Float64(fma(fma(0.13333333333333333, Float64(a * a), 0.3333333333333333), Float64(a * a), 1.0) * a))) + x);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, a_] := Block[{t$95$0 = N[(x - N[(N[(N[(N[Tan[z], $MachinePrecision] + N[Tan[y], $MachinePrecision]), $MachinePrecision] / -1.0), $MachinePrecision] + N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = (-N[Tan[z], $MachinePrecision])}, If[LessEqual[a, -0.027], t$95$0, If[LessEqual[a, 0.036], N[(N[(N[(t$95$1 - N[Tan[y], $MachinePrecision]), $MachinePrecision] * N[(-1.0 / N[(t$95$1 * N[Tan[y], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + (-N[(N[(N[(0.13333333333333333 * N[(a * a), $MachinePrecision] + 0.3333333333333333), $MachinePrecision] * N[(a * a), $MachinePrecision] + 1.0), $MachinePrecision] * a), $MachinePrecision])), $MachinePrecision] + x), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 2 regimes

2. if a < -0.0269999999999999997 or 0.0359999999999999973 < a

   1. Initial program 74.4%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   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%

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

   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. Taylor expanded in z around 0

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

   9. Applied rewrites 75.1%

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

       1. lift-+.f64 N/A

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

       2. +-commutative N/A

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

       3. lower-+.f64 75.1

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

   11. Applied rewrites 75.1%

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

   if -0.0269999999999999997 < a < 0.0359999999999999973

   1. Initial program 74.7%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   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 99.8%

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

   5. Taylor expanded in a around 0

      \[\leadsto x + \mathsf{fma}\left(-\left(\tan z + \tan y\right), \frac{1}{-\mathsf{fma}\left(-\tan z, \tan y, 1\right)}, -\color{blue}{a \cdot \left(1 + {a}^{2} \cdot \left(\frac{1}{3} + \frac{2}{15} \cdot {a}^{2}\right)\right)}\right) \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto x + \mathsf{fma}\left(-\left(\tan z + \tan y\right), \frac{1}{-\mathsf{fma}\left(-\tan z, \tan y, 1\right)}, -\color{blue}{\left(1 + {a}^{2} \cdot \left(\frac{1}{3} + \frac{2}{15} \cdot {a}^{2}\right)\right) \cdot a}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto x + \mathsf{fma}\left(-\left(\tan z + \tan y\right), \frac{1}{-\mathsf{fma}\left(-\tan z, \tan y, 1\right)}, -\color{blue}{\left(1 + {a}^{2} \cdot \left(\frac{1}{3} + \frac{2}{15} \cdot {a}^{2}\right)\right) \cdot a}\right) \]

      3. +-commutative N/A

         \[\leadsto x + \mathsf{fma}\left(-\left(\tan z + \tan y\right), \frac{1}{-\mathsf{fma}\left(-\tan z, \tan y, 1\right)}, -\color{blue}{\left({a}^{2} \cdot \left(\frac{1}{3} + \frac{2}{15} \cdot {a}^{2}\right) + 1\right)} \cdot a\right) \]

      4. *-commutative N/A

         \[\leadsto x + \mathsf{fma}\left(-\left(\tan z + \tan y\right), \frac{1}{-\mathsf{fma}\left(-\tan z, \tan y, 1\right)}, -\left(\color{blue}{\left(\frac{1}{3} + \frac{2}{15} \cdot {a}^{2}\right) \cdot {a}^{2}} + 1\right) \cdot a\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto x + \mathsf{fma}\left(-\left(\tan z + \tan y\right), \frac{1}{-\mathsf{fma}\left(-\tan z, \tan y, 1\right)}, -\color{blue}{\mathsf{fma}\left(\frac{1}{3} + \frac{2}{15} \cdot {a}^{2}, {a}^{2}, 1\right)} \cdot a\right) \]

      6. +-commutative N/A

         \[\leadsto x + \mathsf{fma}\left(-\left(\tan z + \tan y\right), \frac{1}{-\mathsf{fma}\left(-\tan z, \tan y, 1\right)}, -\mathsf{fma}\left(\color{blue}{\frac{2}{15} \cdot {a}^{2} + \frac{1}{3}}, {a}^{2}, 1\right) \cdot a\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto x + \mathsf{fma}\left(-\left(\tan z + \tan y\right), \frac{1}{-\mathsf{fma}\left(-\tan z, \tan y, 1\right)}, -\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{2}{15}, {a}^{2}, \frac{1}{3}\right)}, {a}^{2}, 1\right) \cdot a\right) \]

      8. unpow2 N/A

         \[\leadsto x + \mathsf{fma}\left(-\left(\tan z + \tan y\right), \frac{1}{-\mathsf{fma}\left(-\tan z, \tan y, 1\right)}, -\mathsf{fma}\left(\mathsf{fma}\left(\frac{2}{15}, \color{blue}{a \cdot a}, \frac{1}{3}\right), {a}^{2}, 1\right) \cdot a\right) \]

      9. lower-*.f64 N/A

         \[\leadsto x + \mathsf{fma}\left(-\left(\tan z + \tan y\right), \frac{1}{-\mathsf{fma}\left(-\tan z, \tan y, 1\right)}, -\mathsf{fma}\left(\mathsf{fma}\left(\frac{2}{15}, \color{blue}{a \cdot a}, \frac{1}{3}\right), {a}^{2}, 1\right) \cdot a\right) \]

      10. unpow2 N/A

          \[\leadsto x + \mathsf{fma}\left(-\left(\tan z + \tan y\right), \frac{1}{-\mathsf{fma}\left(-\tan z, \tan y, 1\right)}, -\mathsf{fma}\left(\mathsf{fma}\left(\frac{2}{15}, a \cdot a, \frac{1}{3}\right), \color{blue}{a \cdot a}, 1\right) \cdot a\right) \]

      11. lower-*.f64 99.8

          \[\leadsto x + \mathsf{fma}\left(-\left(\tan z + \tan y\right), \frac{1}{-\mathsf{fma}\left(-\tan z, \tan y, 1\right)}, -\mathsf{fma}\left(\mathsf{fma}\left(0.13333333333333333, a \cdot a, 0.3333333333333333\right), \color{blue}{a \cdot a}, 1\right) \cdot a\right) \]

   7. Applied rewrites 99.8%

      \[\leadsto x + \mathsf{fma}\left(-\left(\tan z + \tan y\right), \frac{1}{-\mathsf{fma}\left(-\tan z, \tan y, 1\right)}, -\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.13333333333333333, a \cdot a, 0.3333333333333333\right), a \cdot a, 1\right) \cdot a}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 88.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -0.027:\\ \;\;\;\;x - \left(\frac{\tan z + \tan y}{-1} + \tan a\right)\\ \mathbf{elif}\;a \leq 0.036:\\ \;\;\;\;\mathsf{fma}\left(\left(-\tan z\right) - \tan y, \frac{-1}{\mathsf{fma}\left(-\tan z, \tan y, 1\right)}, -\mathsf{fma}\left(\mathsf{fma}\left(0.13333333333333333, a \cdot a, 0.3333333333333333\right), a \cdot a, 1\right) \cdot a\right) + x\\ \mathbf{else}:\\ \;\;\;\;x - \left(\frac{\tan z + \tan y}{-1} + \tan a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 89.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x - \left(\frac{\tan z + \tan y}{-1} + \tan a\right)\\ t_1 := -\tan z\\ \mathbf{if}\;a \leq -0.0185:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;a \leq 0.0165:\\ \;\;\;\;\mathsf{fma}\left(t\_1 - \tan y, \frac{-1}{\mathsf{fma}\left(t\_1, \tan y, 1\right)}, -\mathsf{fma}\left(0.3333333333333333, a \cdot a, 1\right) \cdot a\right) + x\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (let* ((t_0 (- x (+ (/ (+ (tan z) (tan y)) -1.0) (tan a)))) (t_1 (- (tan z))))
   (if (<= a -0.0185)
     t_0
     (if (<= a 0.0165)
       (+
        (fma
         (- t_1 (tan y))
         (/ -1.0 (fma t_1 (tan y) 1.0))
         (- (* (fma 0.3333333333333333 (* a a) 1.0) a)))
        x)
       t_0))))

C

double code(double x, double y, double z, double a) {
	double t_0 = x - (((tan(z) + tan(y)) / -1.0) + tan(a));
	double t_1 = -tan(z);
	double tmp;
	if (a <= -0.0185) {
		tmp = t_0;
	} else if (a <= 0.0165) {
		tmp = fma((t_1 - tan(y)), (-1.0 / fma(t_1, tan(y), 1.0)), -(fma(0.3333333333333333, (a * a), 1.0) * a)) + x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z, a)
	t_0 = Float64(x - Float64(Float64(Float64(tan(z) + tan(y)) / -1.0) + tan(a)))
	t_1 = Float64(-tan(z))
	tmp = 0.0
	if (a <= -0.0185)
		tmp = t_0;
	elseif (a <= 0.0165)
		tmp = Float64(fma(Float64(t_1 - tan(y)), Float64(-1.0 / fma(t_1, tan(y), 1.0)), Float64(-Float64(fma(0.3333333333333333, Float64(a * a), 1.0) * a))) + x);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, a_] := Block[{t$95$0 = N[(x - N[(N[(N[(N[Tan[z], $MachinePrecision] + N[Tan[y], $MachinePrecision]), $MachinePrecision] / -1.0), $MachinePrecision] + N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = (-N[Tan[z], $MachinePrecision])}, If[LessEqual[a, -0.0185], t$95$0, If[LessEqual[a, 0.0165], N[(N[(N[(t$95$1 - N[Tan[y], $MachinePrecision]), $MachinePrecision] * N[(-1.0 / N[(t$95$1 * N[Tan[y], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + (-N[(N[(0.3333333333333333 * N[(a * a), $MachinePrecision] + 1.0), $MachinePrecision] * a), $MachinePrecision])), $MachinePrecision] + x), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 2 regimes

2. if a < -0.0184999999999999991 or 0.016500000000000001 < a

   1. Initial program 74.4%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   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%

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

   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. Taylor expanded in z around 0

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

   9. Applied rewrites 75.1%

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

       1. lift-+.f64 N/A

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

       2. +-commutative N/A

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

       3. lower-+.f64 75.1

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

   11. Applied rewrites 75.1%

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

   if -0.0184999999999999991 < a < 0.016500000000000001

   1. Initial program 74.7%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   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 99.8%

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

   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. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 99.7

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

   7. Applied rewrites 99.7%

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

4. Final simplification 88.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -0.0185:\\ \;\;\;\;x - \left(\frac{\tan z + \tan y}{-1} + \tan a\right)\\ \mathbf{elif}\;a \leq 0.0165:\\ \;\;\;\;\mathsf{fma}\left(\left(-\tan z\right) - \tan y, \frac{-1}{\mathsf{fma}\left(-\tan z, \tan y, 1\right)}, -\mathsf{fma}\left(0.3333333333333333, a \cdot a, 1\right) \cdot a\right) + x\\ \mathbf{else}:\\ \;\;\;\;x - \left(\frac{\tan z + \tan y}{-1} + \tan a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 88.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan z + \tan y\\ t_1 := x - \left(\frac{t\_0}{-1} + \tan a\right)\\ \mathbf{if}\;a \leq -9 \cdot 10^{-13}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 1.45 \cdot 10^{-35}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-1}{\mathsf{fma}\left(\tan z, \tan y, -1\right)}, t\_0, -\left(-x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (let* ((t_0 (+ (tan z) (tan y))) (t_1 (- x (+ (/ t_0 -1.0) (tan a)))))
   (if (<= a -9e-13)
     t_1
     (if (<= a 1.45e-35)
       (fma (/ -1.0 (fma (tan z) (tan y) -1.0)) t_0 (- (- x)))
       t_1))))

C

double code(double x, double y, double z, double a) {
	double t_0 = tan(z) + tan(y);
	double t_1 = x - ((t_0 / -1.0) + tan(a));
	double tmp;
	if (a <= -9e-13) {
		tmp = t_1;
	} else if (a <= 1.45e-35) {
		tmp = fma((-1.0 / fma(tan(z), tan(y), -1.0)), t_0, -(-x));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, a)
	t_0 = Float64(tan(z) + tan(y))
	t_1 = Float64(x - Float64(Float64(t_0 / -1.0) + tan(a)))
	tmp = 0.0
	if (a <= -9e-13)
		tmp = t_1;
	elseif (a <= 1.45e-35)
		tmp = fma(Float64(-1.0 / fma(tan(z), tan(y), -1.0)), t_0, Float64(-Float64(-x)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, a_] := Block[{t$95$0 = N[(N[Tan[z], $MachinePrecision] + N[Tan[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x - N[(N[(t$95$0 / -1.0), $MachinePrecision] + N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -9e-13], t$95$1, If[LessEqual[a, 1.45e-35], N[(N[(-1.0 / N[(N[Tan[z], $MachinePrecision] * N[Tan[y], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] * t$95$0 + (-(-x))), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if a < -9e-13 or 1.4500000000000001e-35 < a

   1. Initial program 74.8%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   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%

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

   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. Taylor expanded in z around 0

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

   9. Applied rewrites 75.6%

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

       1. lift-+.f64 N/A

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

       2. +-commutative N/A

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

       3. lower-+.f64 75.6

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

   11. Applied rewrites 75.6%

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

   if -9e-13 < a < 1.4500000000000001e-35

   1. Initial program 74.3%

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

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

      3. lift--.f64 N/A

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

      4. associate-+l- N/A

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

      5. lower--.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. lower--.f64 74.3

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

   4. Applied rewrites 74.3%

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

   5. Taylor expanded in a around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 74.3

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

   7. Applied rewrites 74.3%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

   9. Applied rewrites 99.8%

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

4. Final simplification 88.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -9 \cdot 10^{-13}:\\ \;\;\;\;x - \left(\frac{\tan z + \tan y}{-1} + \tan a\right)\\ \mathbf{elif}\;a \leq 1.45 \cdot 10^{-35}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-1}{\mathsf{fma}\left(\tan z, \tan y, -1\right)}, \tan z + \tan y, -\left(-x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x - \left(\frac{\tan z + \tan y}{-1} + \tan a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 89.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x - \left(\frac{\tan z + \tan y}{-1} + \tan a\right)\\ \mathbf{if}\;a \leq -0.00032:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;a \leq 0.0098:\\ \;\;\;\;\frac{\left(-\tan z\right) - \tan y}{\mathsf{fma}\left(\tan y, \tan z, -1\right)} - \left(a - x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (let* ((t_0 (- x (+ (/ (+ (tan z) (tan y)) -1.0) (tan a)))))
   (if (<= a -0.00032)
     t_0
     (if (<= a 0.0098)
       (- (/ (- (- (tan z)) (tan y)) (fma (tan y) (tan z) -1.0)) (- a x))
       t_0))))

C

double code(double x, double y, double z, double a) {
	double t_0 = x - (((tan(z) + tan(y)) / -1.0) + tan(a));
	double tmp;
	if (a <= -0.00032) {
		tmp = t_0;
	} else if (a <= 0.0098) {
		tmp = ((-tan(z) - tan(y)) / fma(tan(y), tan(z), -1.0)) - (a - x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z, a)
	t_0 = Float64(x - Float64(Float64(Float64(tan(z) + tan(y)) / -1.0) + tan(a)))
	tmp = 0.0
	if (a <= -0.00032)
		tmp = t_0;
	elseif (a <= 0.0098)
		tmp = Float64(Float64(Float64(Float64(-tan(z)) - tan(y)) / fma(tan(y), tan(z), -1.0)) - Float64(a - x));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, a_] := Block[{t$95$0 = N[(x - N[(N[(N[(N[Tan[z], $MachinePrecision] + N[Tan[y], $MachinePrecision]), $MachinePrecision] / -1.0), $MachinePrecision] + N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -0.00032], t$95$0, If[LessEqual[a, 0.0098], N[(N[(N[((-N[Tan[z], $MachinePrecision]) - N[Tan[y], $MachinePrecision]), $MachinePrecision] / N[(N[Tan[y], $MachinePrecision] * N[Tan[z], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] - N[(a - x), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if a < -3.20000000000000026e-4 or 0.0097999999999999997 < a

   1. Initial program 74.4%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   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%

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

   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. Taylor expanded in z around 0

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

   9. Applied rewrites 75.1%

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

       1. lift-+.f64 N/A

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

       2. +-commutative N/A

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

       3. lower-+.f64 75.1

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

   11. Applied rewrites 75.1%

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

   if -3.20000000000000026e-4 < a < 0.0097999999999999997

   1. Initial program 74.7%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift--.f64 N/A

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

      4. associate-+l- N/A

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

      5. lower--.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. lower--.f64 74.7

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

   4. Applied rewrites 74.7%

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

   5. Taylor expanded in a around 0

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

      1. lower--.f64 74.5

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

   7. Applied rewrites 74.5%

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

      1. lift-tan.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. tan-sum N/A

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

      4. lift-tan.f64 N/A

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

      5. lift-tan.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-tan.f64 N/A

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

      8. lift-tan.f64 N/A

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

      9. cancel-sign-sub-inv N/A

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

      10. lift-neg.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-fma.f64 N/A

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

      13. lower-/.f64 99.3

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lift-+.f64 99.3

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

      17. remove-double-neg N/A

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

      18. lift-neg.f64 N/A

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

      19. lower-neg.f64 99.3

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

      20. lift-neg.f64 N/A

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

   9. Applied rewrites 99.3%

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

4. Final simplification 88.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -0.00032:\\ \;\;\;\;x - \left(\frac{\tan z + \tan y}{-1} + \tan a\right)\\ \mathbf{elif}\;a \leq 0.0098:\\ \;\;\;\;\frac{\left(-\tan z\right) - \tan y}{\mathsf{fma}\left(\tan y, \tan z, -1\right)} - \left(a - x\right)\\ \mathbf{else}:\\ \;\;\;\;x - \left(\frac{\tan z + \tan y}{-1} + \tan a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 88.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x - \left(\frac{\tan z + \tan y}{-1} + \tan a\right)\\ \mathbf{if}\;a \leq -9 \cdot 10^{-13}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;a \leq 1.45 \cdot 10^{-35}:\\ \;\;\;\;\frac{\left(-\tan z\right) - \tan y}{\mathsf{fma}\left(\tan z, \tan y, -1\right)} - \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (let* ((t_0 (- x (+ (/ (+ (tan z) (tan y)) -1.0) (tan a)))))
   (if (<= a -9e-13)
     t_0
     (if (<= a 1.45e-35)
       (- (/ (- (- (tan z)) (tan y)) (fma (tan z) (tan y) -1.0)) (- x))
       t_0))))

C

double code(double x, double y, double z, double a) {
	double t_0 = x - (((tan(z) + tan(y)) / -1.0) + tan(a));
	double tmp;
	if (a <= -9e-13) {
		tmp = t_0;
	} else if (a <= 1.45e-35) {
		tmp = ((-tan(z) - tan(y)) / fma(tan(z), tan(y), -1.0)) - -x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z, a)
	t_0 = Float64(x - Float64(Float64(Float64(tan(z) + tan(y)) / -1.0) + tan(a)))
	tmp = 0.0
	if (a <= -9e-13)
		tmp = t_0;
	elseif (a <= 1.45e-35)
		tmp = Float64(Float64(Float64(Float64(-tan(z)) - tan(y)) / fma(tan(z), tan(y), -1.0)) - Float64(-x));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, a_] := Block[{t$95$0 = N[(x - N[(N[(N[(N[Tan[z], $MachinePrecision] + N[Tan[y], $MachinePrecision]), $MachinePrecision] / -1.0), $MachinePrecision] + N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -9e-13], t$95$0, If[LessEqual[a, 1.45e-35], 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] - (-x)), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if a < -9e-13 or 1.4500000000000001e-35 < a

   1. Initial program 74.8%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   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%

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

   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. Taylor expanded in z around 0

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

   9. Applied rewrites 75.6%

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

       1. lift-+.f64 N/A

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

       2. +-commutative N/A

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

       3. lower-+.f64 75.6

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

   11. Applied rewrites 75.6%

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

   if -9e-13 < a < 1.4500000000000001e-35

   1. Initial program 74.3%

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

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

      3. lift--.f64 N/A

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

      4. associate-+l- N/A

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

      5. lower--.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. lower--.f64 74.3

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

   4. Applied rewrites 74.3%

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

   5. Taylor expanded in a around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 74.3

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

   7. Applied rewrites 74.3%

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

      1. lift-tan.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. tan-sum N/A

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

      4. lift-tan.f64 N/A

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

      5. lift-tan.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. lift-tan.f64 N/A

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

      9. lift-tan.f64 N/A

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

      10. *-commutative N/A

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

      11. sub-neg N/A

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

      12. metadata-eval N/A

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

      13. distribute-neg-in N/A

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

      14. +-commutative N/A

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

      15. lift-fma.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 N/A

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

      20. lower-neg.f64 99.8

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

      21. lift-fma.f64 N/A

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

      22. *-commutative N/A

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

      23. lower-fma.f64 99.8

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

   9. Applied rewrites 99.8%

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

4. Final simplification 88.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -9 \cdot 10^{-13}:\\ \;\;\;\;x - \left(\frac{\tan z + \tan y}{-1} + \tan a\right)\\ \mathbf{elif}\;a \leq 1.45 \cdot 10^{-35}:\\ \;\;\;\;\frac{\left(-\tan z\right) - \tan y}{\mathsf{fma}\left(\tan z, \tan y, -1\right)} - \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;x - \left(\frac{\tan z + \tan y}{-1} + \tan a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 79.6% accurate, 0.7× speedup

Math

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

FPCore

(FPCore (x y z a)
 :precision binary64
 (- x (+ (/ (+ (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)) / -1.0) + 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(z) + tan(y)) / (-1.0d0)) + tan(a))
end function

Java

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

Python

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

Julia

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

MATLAB

function tmp = code(x, y, z, a)
	tmp = x - (((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] / -1.0), $MachinePrecision] + N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 74.5%

   \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
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 99.8%

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

6. Applied rewrites 99.8%

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

7. Taylor expanded in z around 0

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

9. Applied rewrites 75.4%

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

    1. lift-+.f64 N/A

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

    2. +-commutative N/A

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

    3. lower-+.f64 75.4

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

11. Applied rewrites 75.4%

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

12. Final simplification 75.4%

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

Alternative 10: 79.2% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 74.5%

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

   1. lift-+.f64 N/A

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

   2. flip-+ N/A

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

   3. div-sub N/A

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

   4. sub-neg N/A

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

   5. associate-/l* N/A

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

   6. lower-fma.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. lower--.f64 N/A

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

   9. lower-neg.f64 N/A

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

   10. associate-/l* N/A

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

   11. lower-*.f64 N/A

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

   12. lower-/.f64 N/A

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

   13. lower--.f64 74.6

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

4. Applied rewrites 74.6%

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

5. Final simplification 74.6%

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

Alternative 11: 79.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 74.5%

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

3. Final simplification 74.5%

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

Alternative 12: 50.8% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 74.5%

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

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

   3. lift--.f64 N/A

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

   4. associate-+l- N/A

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

   5. lower--.f64 N/A

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

   6. lift-+.f64 N/A

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

   7. +-commutative N/A

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

   8. lower-+.f64 N/A

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

   9. lower--.f64 74.5

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

4. Applied rewrites 74.5%

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

5. Taylor expanded in a around 0

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 50.3

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

7. Applied rewrites 50.3%

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

Reproduce

herbie shell --seed 2024284 
(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))))