tan-example (used to crash)

Percentage Accurate: 79.9% → 99.7%

Time: 30.5s

Alternatives: 8

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: 79.9% 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, 0.2× speedup

Math

\[\begin{array}{l} \\ \left(\frac{\frac{1}{\mathsf{fma}\left(-\tan z, \tan y, 1\right)} \cdot \left({\tan y}^{2} - {\tan z}^{2}\right)}{\tan y - \tan z} - \tan a\right) + x \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (+
  (-
   (/
    (*
     (/ 1.0 (fma (- (tan z)) (tan y) 1.0))
     (- (pow (tan y) 2.0) (pow (tan z) 2.0)))
    (- (tan y) (tan z)))
   (tan a))
  x))

C

double code(double x, double y, double z, double a) {
	return ((((1.0 / fma(-tan(z), tan(y), 1.0)) * (pow(tan(y), 2.0) - pow(tan(z), 2.0))) / (tan(y) - tan(z))) - tan(a)) + x;
}

Julia

function code(x, y, z, a)
	return Float64(Float64(Float64(Float64(Float64(1.0 / fma(Float64(-tan(z)), tan(y), 1.0)) * Float64((tan(y) ^ 2.0) - (tan(z) ^ 2.0))) / Float64(tan(y) - tan(z))) - tan(a)) + x)
end

Wolfram

code[x_, y_, z_, a_] := N[(N[(N[(N[(N[(1.0 / N[((-N[Tan[z], $MachinePrecision]) * N[Tan[y], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[Tan[y], $MachinePrecision], 2.0], $MachinePrecision] - N[Power[N[Tan[z], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Tan[y], $MachinePrecision] - N[Tan[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]

Derivation

1. Initial program 83.7%

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

   1. lift-tan.f64 N/A

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

   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. div-inv N/A

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

   5. flip-+ N/A

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

   6. associate-*l/ N/A

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

   7. lower-/.f64 N/A

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

4. Applied rewrites 99.7%

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

5. Final simplification 99.7%

   \[\leadsto \left(\frac{\frac{1}{\mathsf{fma}\left(-\tan z, \tan y, 1\right)} \cdot \left({\tan y}^{2} - {\tan z}^{2}\right)}{\tan y - \tan z} - \tan a\right) + x \]
6. Add Preprocessing

Alternative 2: 89.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan z + \tan y\\ t_1 := \left(\frac{t\_0}{1} - \tan a\right) + x\\ \mathbf{if}\;\tan a \leq -0.005:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\tan a \leq 10^{-22}:\\ \;\;\;\;\left(\frac{t\_0}{\mathsf{fma}\left(-\tan z, \tan y, 1\right)} - 1 \cdot a\right) + x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (let* ((t_0 (+ (tan z) (tan y))) (t_1 (+ (- (/ t_0 1.0) (tan a)) x)))
   (if (<= (tan a) -0.005)
     t_1
     (if (<= (tan a) 1e-22)
       (+ (- (/ t_0 (fma (- (tan z)) (tan y) 1.0)) (* 1.0 a)) x)
       t_1))))

C

double code(double x, double y, double z, double a) {
	double t_0 = tan(z) + tan(y);
	double t_1 = ((t_0 / 1.0) - tan(a)) + x;
	double tmp;
	if (tan(a) <= -0.005) {
		tmp = t_1;
	} else if (tan(a) <= 1e-22) {
		tmp = ((t_0 / fma(-tan(z), tan(y), 1.0)) - (1.0 * a)) + x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, a)
	t_0 = Float64(tan(z) + tan(y))
	t_1 = Float64(Float64(Float64(t_0 / 1.0) - tan(a)) + x)
	tmp = 0.0
	if (tan(a) <= -0.005)
		tmp = t_1;
	elseif (tan(a) <= 1e-22)
		tmp = Float64(Float64(Float64(t_0 / fma(Float64(-tan(z)), tan(y), 1.0)) - Float64(1.0 * a)) + 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[(N[(N[(t$95$0 / 1.0), $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[N[Tan[a], $MachinePrecision], -0.005], t$95$1, If[LessEqual[N[Tan[a], $MachinePrecision], 1e-22], N[(N[(N[(t$95$0 / N[((-N[Tan[z], $MachinePrecision]) * N[Tan[y], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] - N[(1.0 * a), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (tan.f64 a) < -0.0050000000000000001 or 1e-22 < (tan.f64 a)

   1. Initial program 82.7%

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

      1. lift-tan.f64 N/A

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

      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

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

      15. lower-tan.f64 N/A

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

      16. lower-tan.f64 99.6

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in z around 0

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

   7. Applied rewrites 83.0%

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

   if -0.0050000000000000001 < (tan.f64 a) < 1e-22

   1. Initial program 84.7%

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

   3. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 84.7

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

   5. Applied rewrites 84.7%

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

   6. Taylor expanded in a around 0

      \[\leadsto x + \left(\tan \left(y + z\right) - 1 \cdot a\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 84.7%

      \[\leadsto x + \left(\tan \left(y + z\right) - 1 \cdot a\right) \]
   9. Step-by-step derivation

      1. lift-tan.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. tan-sum N/A

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

      5. lift-tan.f64 N/A

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

      6. lift-tan.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-tan.f64 N/A

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

      9. lift-tan.f64 N/A

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

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

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

      11. lift-neg.f64 N/A

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

      12. +-commutative N/A

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

      13. lift-fma.f64 N/A

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

      14. lift-/.f64 99.8

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

      15. lift-+.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 99.8

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

   10. Applied rewrites 99.8%

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

4. Final simplification 91.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\tan a \leq -0.005:\\ \;\;\;\;\left(\frac{\tan z + \tan y}{1} - \tan a\right) + x\\ \mathbf{elif}\;\tan a \leq 10^{-22}:\\ \;\;\;\;\left(\frac{\tan z + \tan y}{\mathsf{fma}\left(-\tan z, \tan y, 1\right)} - 1 \cdot a\right) + x\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\tan z + \tan y}{1} - \tan a\right) + x\\ \end{array} \]
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 83.7%

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

   1. lift-tan.f64 N/A

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

   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

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

   15. lower-tan.f64 N/A

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

   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: 99.7% accurate, 0.4× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z, double a) {
	return (((tan(z) + tan(y)) / (1.0 - (tan(z) * tan(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) + tan(y)) / (1.0d0 - (tan(z) * tan(y)))) - tan(a)) + x
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 83.7%

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

   1. lift-tan.f64 N/A

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

   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

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

   15. lower-tan.f64 N/A

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

   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. Step-by-step derivation

   1. lift-fma.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-neg.f64 N/A

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

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

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

   5. lift-tan.f64 N/A

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

   6. lift-tan.f64 N/A

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

   7. lower--.f64 N/A

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

   8. lift-tan.f64 N/A

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

   9. lift-tan.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 99.7

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

6. Applied rewrites 99.7%

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

7. Final simplification 99.7%

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

Alternative 5: 80.2% accurate, 0.7× speedup

Math

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

FPCore

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

Java

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

Python

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

Julia

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

MATLAB

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

Derivation

1. Initial program 83.7%

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

   1. lift-tan.f64 N/A

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

   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

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

   15. lower-tan.f64 N/A

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

   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. Taylor expanded in z around 0

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

7. Applied rewrites 84.0%

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

8. Final simplification 84.0%

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

Alternative 6: 80.2% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 83.7%

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

   1. lift-tan.f64 N/A

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

   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. div-inv N/A

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

   5. flip-+ N/A

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

   6. associate-*l/ N/A

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

   7. lower-/.f64 N/A

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

4. Applied rewrites 99.7%

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

   1. lift-+.f64 N/A

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

   2. lift--.f64 N/A

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

   3. associate-+r- N/A

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

   4. lower--.f64 N/A

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

6. Applied rewrites 99.7%

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

7. Taylor expanded in z around 0

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

9. Applied rewrites 84.0%

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

Alternative 7: 79.9% 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 83.7%

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

3. Final simplification 83.7%

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

Alternative 8: 41.8% accurate, 1.8× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z, double a) {
	return (tan((z + y)) - (1.0 * 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)) - (1.0d0 * a)) + x
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 83.7%

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

3. Taylor expanded in a around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. lower-fma.f64 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f64 47.3

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

5. Applied rewrites 47.3%

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

6. Taylor expanded in a around 0

   \[\leadsto x + \left(\tan \left(y + z\right) - 1 \cdot a\right) \]
7. Step-by-step derivation

8. Applied rewrites 47.7%

   \[\leadsto x + \left(\tan \left(y + z\right) - 1 \cdot a\right) \]

9. Final simplification 47.7%

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

Reproduce

herbie shell --seed 2024235 
(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))))