tan-example (used to crash)

Percentage Accurate: 79.8% → 99.7%

Time: 30.5s

Alternatives: 11

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 79.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 81.0%

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

   1. lift--.f64 N/A

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

   2. sub-neg N/A

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

   3. lift-tan.f64 N/A

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

   4. lift-+.f64 N/A

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

   5. tan-sum N/A

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

   6. flip-- N/A

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

   7. associate-/r/ N/A

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

   8. lower-fma.f64 N/A

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

4. Applied rewrites 99.7%

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

5. Final simplification 99.7%

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

Alternative 2: 99.7% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 81.0%

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

   1. lift--.f64 N/A

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

   2. sub-neg N/A

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

   3. lift-tan.f64 N/A

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

   4. lift-+.f64 N/A

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

   5. tan-sum N/A

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

   6. frac-2neg N/A

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

   7. div-inv N/A

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

   8. lower-fma.f64 N/A

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

4. Applied rewrites 99.7%

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

   1. lift-fma.f64 N/A

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

6. Applied rewrites 99.7%

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

7. Final simplification 99.7%

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

Alternative 3: 99.7% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 81.0%

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

   1. lift-tan.f64 N/A

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

   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%

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

Alternative 4: 89.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan y + \tan z\\ \mathbf{if}\;a \leq -1.1 \cdot 10^{-10}:\\ \;\;\;\;\tan \left(y + z\right) - \mathsf{fma}\left(\frac{\frac{\sin a}{x}}{\cos a}, x, -x\right)\\ \mathbf{elif}\;a \leq 2.1 \cdot 10^{-11}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-1}{\mathsf{fma}\left(\tan z, \tan y, -1\right)}, t\_0, -\left(-x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1, t\_0, -\tan a\right) + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (let* ((t_0 (+ (tan y) (tan z))))
   (if (<= a -1.1e-10)
     (- (tan (+ y z)) (fma (/ (/ (sin a) x) (cos a)) x (- x)))
     (if (<= a 2.1e-11)
       (fma (/ -1.0 (fma (tan z) (tan y) -1.0)) t_0 (- (- x)))
       (+ (fma 1.0 t_0 (- (tan a))) x)))))

C

double code(double x, double y, double z, double a) {
	double t_0 = tan(y) + tan(z);
	double tmp;
	if (a <= -1.1e-10) {
		tmp = tan((y + z)) - fma(((sin(a) / x) / cos(a)), x, -x);
	} else if (a <= 2.1e-11) {
		tmp = fma((-1.0 / fma(tan(z), tan(y), -1.0)), t_0, -(-x));
	} else {
		tmp = fma(1.0, t_0, -tan(a)) + x;
	}
	return tmp;
}

Julia

function code(x, y, z, a)
	t_0 = Float64(tan(y) + tan(z))
	tmp = 0.0
	if (a <= -1.1e-10)
		tmp = Float64(tan(Float64(y + z)) - fma(Float64(Float64(sin(a) / x) / cos(a)), x, Float64(-x)));
	elseif (a <= 2.1e-11)
		tmp = fma(Float64(-1.0 / fma(tan(z), tan(y), -1.0)), t_0, Float64(-Float64(-x)));
	else
		tmp = Float64(fma(1.0, t_0, Float64(-tan(a))) + x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, a_] := Block[{t$95$0 = N[(N[Tan[y], $MachinePrecision] + N[Tan[z], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -1.1e-10], N[(N[Tan[N[(y + z), $MachinePrecision]], $MachinePrecision] - N[(N[(N[(N[Sin[a], $MachinePrecision] / x), $MachinePrecision] / N[Cos[a], $MachinePrecision]), $MachinePrecision] * x + (-x)), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 2.1e-11], N[(N[(-1.0 / N[(N[Tan[z], $MachinePrecision] * N[Tan[y], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] * t$95$0 + (-(-x))), $MachinePrecision], N[(N[(1.0 * t$95$0 + (-N[Tan[a], $MachinePrecision])), $MachinePrecision] + x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if a < -1.09999999999999995e-10

   1. Initial program 78.5%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   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 78.4

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

   4. Applied rewrites 78.4%

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

   5. Taylor expanded in x around inf

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. distribute-rgt-in N/A

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

      4. mul-1-neg N/A

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

      5. lower-fma.f64 N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-sin.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. lower-neg.f64 78.6

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

   7. Applied rewrites 78.6%

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

   if -1.09999999999999995e-10 < a < 2.0999999999999999e-11

   1. Initial program 80.0%

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

      1. lift-+.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 80.0

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

   4. Applied rewrites 80.0%

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

   5. Taylor expanded in x around inf

      \[\leadsto \tan \left(z + y\right) - \color{blue}{-1 \cdot x} \]
   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 80.0

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

   7. Applied rewrites 80.0%

      \[\leadsto \tan \left(z + y\right) - \color{blue}{\left(-x\right)} \]
   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 99.4%

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

   if 2.0999999999999999e-11 < a

   1. Initial program 85.1%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   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 99.6%

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

      1. lift-fma.f64 N/A

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

   6. Applied rewrites 99.6%

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

   7. Taylor expanded in y around 0

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

   9. Applied rewrites 85.4%

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

4. Final simplification 89.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.1 \cdot 10^{-10}:\\ \;\;\;\;\tan \left(y + z\right) - \mathsf{fma}\left(\frac{\frac{\sin a}{x}}{\cos a}, x, -x\right)\\ \mathbf{elif}\;a \leq 2.1 \cdot 10^{-11}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-1}{\mathsf{fma}\left(\tan z, \tan y, -1\right)}, \tan y + \tan z, -\left(-x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1, \tan y + \tan z, -\tan a\right) + x\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 89.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan y + \tan z\\ \mathbf{if}\;a \leq -1.1 \cdot 10^{-10}:\\ \;\;\;\;\tan \left(y + z\right) - \mathsf{fma}\left(\frac{\frac{\sin a}{x}}{\cos a}, x, -x\right)\\ \mathbf{elif}\;a \leq 2.1 \cdot 10^{-11}:\\ \;\;\;\;\frac{t\_0}{-\mathsf{fma}\left(\tan z, \tan y, -1\right)} - \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1, t\_0, -\tan a\right) + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (let* ((t_0 (+ (tan y) (tan z))))
   (if (<= a -1.1e-10)
     (- (tan (+ y z)) (fma (/ (/ (sin a) x) (cos a)) x (- x)))
     (if (<= a 2.1e-11)
       (- (/ t_0 (- (fma (tan z) (tan y) -1.0))) (- x))
       (+ (fma 1.0 t_0 (- (tan a))) x)))))

C

double code(double x, double y, double z, double a) {
	double t_0 = tan(y) + tan(z);
	double tmp;
	if (a <= -1.1e-10) {
		tmp = tan((y + z)) - fma(((sin(a) / x) / cos(a)), x, -x);
	} else if (a <= 2.1e-11) {
		tmp = (t_0 / -fma(tan(z), tan(y), -1.0)) - -x;
	} else {
		tmp = fma(1.0, t_0, -tan(a)) + x;
	}
	return tmp;
}

Julia

function code(x, y, z, a)
	t_0 = Float64(tan(y) + tan(z))
	tmp = 0.0
	if (a <= -1.1e-10)
		tmp = Float64(tan(Float64(y + z)) - fma(Float64(Float64(sin(a) / x) / cos(a)), x, Float64(-x)));
	elseif (a <= 2.1e-11)
		tmp = Float64(Float64(t_0 / Float64(-fma(tan(z), tan(y), -1.0))) - Float64(-x));
	else
		tmp = Float64(fma(1.0, t_0, Float64(-tan(a))) + x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, a_] := Block[{t$95$0 = N[(N[Tan[y], $MachinePrecision] + N[Tan[z], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -1.1e-10], N[(N[Tan[N[(y + z), $MachinePrecision]], $MachinePrecision] - N[(N[(N[(N[Sin[a], $MachinePrecision] / x), $MachinePrecision] / N[Cos[a], $MachinePrecision]), $MachinePrecision] * x + (-x)), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 2.1e-11], N[(N[(t$95$0 / (-N[(N[Tan[z], $MachinePrecision] * N[Tan[y], $MachinePrecision] + -1.0), $MachinePrecision])), $MachinePrecision] - (-x)), $MachinePrecision], N[(N[(1.0 * t$95$0 + (-N[Tan[a], $MachinePrecision])), $MachinePrecision] + x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if a < -1.09999999999999995e-10

   1. Initial program 78.5%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   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 78.4

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

   4. Applied rewrites 78.4%

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

   5. Taylor expanded in x around inf

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. distribute-rgt-in N/A

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

      4. mul-1-neg N/A

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

      5. lower-fma.f64 N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-sin.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. lower-neg.f64 78.6

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

   7. Applied rewrites 78.6%

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

   if -1.09999999999999995e-10 < a < 2.0999999999999999e-11

   1. Initial program 80.0%

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

      1. lift-+.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 80.0

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

   4. Applied rewrites 80.0%

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

   5. Taylor expanded in x around inf

      \[\leadsto \tan \left(z + y\right) - \color{blue}{-1 \cdot x} \]
   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 80.0

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

   7. Applied rewrites 80.0%

      \[\leadsto \tan \left(z + y\right) - \color{blue}{\left(-x\right)} \]
   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. +-commutative N/A

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

      4. tan-sum N/A

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

      5. lift-tan.f64 N/A

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

      6. lift-tan.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-+.f64 N/A

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

      9. lift-tan.f64 N/A

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

      10. lift-tan.f64 N/A

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

      11. *-commutative N/A

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

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

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

      13. lift-neg.f64 N/A

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

      14. +-commutative N/A

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

      15. lift-fma.f64 N/A

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

      16. lift-/.f64 99.4

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

      17. lift-fma.f64 N/A

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

      18. lift-neg.f64 N/A

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

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

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

      20. *-commutative N/A

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

      21. metadata-eval N/A

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

   9. Applied rewrites 99.4%

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

   if 2.0999999999999999e-11 < a

   1. Initial program 85.1%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   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 99.6%

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

      1. lift-fma.f64 N/A

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

   6. Applied rewrites 99.6%

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

   7. Taylor expanded in y around 0

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

   9. Applied rewrites 85.4%

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

4. Final simplification 89.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.1 \cdot 10^{-10}:\\ \;\;\;\;\tan \left(y + z\right) - \mathsf{fma}\left(\frac{\frac{\sin a}{x}}{\cos a}, x, -x\right)\\ \mathbf{elif}\;a \leq 2.1 \cdot 10^{-11}:\\ \;\;\;\;\frac{\tan y + \tan z}{-\mathsf{fma}\left(\tan z, \tan y, -1\right)} - \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1, \tan y + \tan z, -\tan a\right) + x\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 80.1% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 81.0%

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

   1. lift--.f64 N/A

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

   2. sub-neg N/A

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

   3. lift-tan.f64 N/A

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

   4. lift-+.f64 N/A

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

   5. tan-sum N/A

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

   6. frac-2neg N/A

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

   7. div-inv N/A

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

   8. lower-fma.f64 N/A

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

4. Applied rewrites 99.7%

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

   1. lift-fma.f64 N/A

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

6. Applied rewrites 99.7%

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

7. Taylor expanded in y around 0

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

9. Applied rewrites 81.2%

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

10. Final simplification 81.2%

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

Alternative 7: 64.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -6.5 \cdot 10^{-11}:\\ \;\;\;\;\tan \left(y + z\right) - \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\tan z - \tan a\right) + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (if (<= y -6.5e-11) (- (tan (+ y z)) (- x)) (+ (- (tan z) (tan a)) x)))

C

double code(double x, double y, double z, double a) {
	double tmp;
	if (y <= -6.5e-11) {
		tmp = tan((y + z)) - -x;
	} else {
		tmp = (tan(z) - tan(a)) + x;
	}
	return tmp;
}

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
    real(8) :: tmp
    if (y <= (-6.5d-11)) then
        tmp = tan((y + z)) - -x
    else
        tmp = (tan(z) - tan(a)) + x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double a) {
	double tmp;
	if (y <= -6.5e-11) {
		tmp = Math.tan((y + z)) - -x;
	} else {
		tmp = (Math.tan(z) - Math.tan(a)) + x;
	}
	return tmp;
}

Python

def code(x, y, z, a):
	tmp = 0
	if y <= -6.5e-11:
		tmp = math.tan((y + z)) - -x
	else:
		tmp = (math.tan(z) - math.tan(a)) + x
	return tmp

Julia

function code(x, y, z, a)
	tmp = 0.0
	if (y <= -6.5e-11)
		tmp = Float64(tan(Float64(y + z)) - Float64(-x));
	else
		tmp = Float64(Float64(tan(z) - tan(a)) + x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, a)
	tmp = 0.0;
	if (y <= -6.5e-11)
		tmp = tan((y + z)) - -x;
	else
		tmp = (tan(z) - tan(a)) + x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, a_] := If[LessEqual[y, -6.5e-11], N[(N[Tan[N[(y + z), $MachinePrecision]], $MachinePrecision] - (-x)), $MachinePrecision], N[(N[(N[Tan[z], $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -6.49999999999999953e-11

   1. Initial program 65.3%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   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 65.2

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

   4. Applied rewrites 65.2%

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

   5. Taylor expanded in x around inf

      \[\leadsto \tan \left(z + y\right) - \color{blue}{-1 \cdot x} \]
   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 42.5

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

   7. Applied rewrites 42.5%

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

   if -6.49999999999999953e-11 < y

   1. Initial program 87.6%

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

   3. Taylor expanded in y around 0

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-sin.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. lower-cos.f64 74.6

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

   5. Applied rewrites 74.6%

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

   7. Applied rewrites 74.7%

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

4. Final simplification 65.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -6.5 \cdot 10^{-11}:\\ \;\;\;\;\tan \left(y + z\right) - \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\tan z - \tan a\right) + x\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 79.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.0%

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

3. Final simplification 81.0%

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

Alternative 9: 59.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan \left(y + z\right) - \left(-x\right)\\ \mathbf{if}\;y + z \leq -5 \cdot 10^{-11}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y + z \leq 1:\\ \;\;\;\;\left(z + x\right) - \tan a\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (let* ((t_0 (- (tan (+ y z)) (- x))))
   (if (<= (+ y z) -5e-11) t_0 (if (<= (+ y z) 1.0) (- (+ z x) (tan a)) t_0))))

C

double code(double x, double y, double z, double a) {
	double t_0 = tan((y + z)) - -x;
	double tmp;
	if ((y + z) <= -5e-11) {
		tmp = t_0;
	} else if ((y + z) <= 1.0) {
		tmp = (z + x) - tan(a);
	} else {
		tmp = t_0;
	}
	return tmp;
}

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
    real(8) :: t_0
    real(8) :: tmp
    t_0 = tan((y + z)) - -x
    if ((y + z) <= (-5d-11)) then
        tmp = t_0
    else if ((y + z) <= 1.0d0) then
        tmp = (z + x) - tan(a)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double a) {
	double t_0 = Math.tan((y + z)) - -x;
	double tmp;
	if ((y + z) <= -5e-11) {
		tmp = t_0;
	} else if ((y + z) <= 1.0) {
		tmp = (z + x) - Math.tan(a);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z, a):
	t_0 = math.tan((y + z)) - -x
	tmp = 0
	if (y + z) <= -5e-11:
		tmp = t_0
	elif (y + z) <= 1.0:
		tmp = (z + x) - math.tan(a)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z, a)
	t_0 = Float64(tan(Float64(y + z)) - Float64(-x))
	tmp = 0.0
	if (Float64(y + z) <= -5e-11)
		tmp = t_0;
	elseif (Float64(y + z) <= 1.0)
		tmp = Float64(Float64(z + x) - tan(a));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, a)
	t_0 = tan((y + z)) - -x;
	tmp = 0.0;
	if ((y + z) <= -5e-11)
		tmp = t_0;
	elseif ((y + z) <= 1.0)
		tmp = (z + x) - tan(a);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, a_] := Block[{t$95$0 = N[(N[Tan[N[(y + z), $MachinePrecision]], $MachinePrecision] - (-x)), $MachinePrecision]}, If[LessEqual[N[(y + z), $MachinePrecision], -5e-11], t$95$0, If[LessEqual[N[(y + z), $MachinePrecision], 1.0], N[(N[(z + x), $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 y z) < -5.00000000000000018e-11 or 1 < (+.f64 y z)

   1. Initial program 76.5%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   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 76.4

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

   4. Applied rewrites 76.4%

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

   5. Taylor expanded in x around inf

      \[\leadsto \tan \left(z + y\right) - \color{blue}{-1 \cdot x} \]
   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 45.5

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

   7. Applied rewrites 45.5%

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

   if -5.00000000000000018e-11 < (+.f64 y z) < 1

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-sin.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. lower-cos.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 98.6%

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

   10. Applied rewrites 98.5%

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

4. Final simplification 55.6%

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

Alternative 10: 35.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 1.45:\\ \;\;\;\;\left(z + x\right) - \tan a\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (if (<= z 1.45) (- (+ z x) (tan a)) (/ 1.0 (/ 1.0 x))))

C

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

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
    real(8) :: tmp
    if (z <= 1.45d0) then
        tmp = (z + x) - tan(a)
    else
        tmp = 1.0d0 / (1.0d0 / x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double a) {
	double tmp;
	if (z <= 1.45) {
		tmp = (z + x) - Math.tan(a);
	} else {
		tmp = 1.0 / (1.0 / x);
	}
	return tmp;
}

Python

def code(x, y, z, a):
	tmp = 0
	if z <= 1.45:
		tmp = (z + x) - math.tan(a)
	else:
		tmp = 1.0 / (1.0 / x)
	return tmp

Julia

function code(x, y, z, a)
	tmp = 0.0
	if (z <= 1.45)
		tmp = Float64(Float64(z + x) - tan(a));
	else
		tmp = Float64(1.0 / Float64(1.0 / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, a)
	tmp = 0.0;
	if (z <= 1.45)
		tmp = (z + x) - tan(a);
	else
		tmp = 1.0 / (1.0 / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, a_] := If[LessEqual[z, 1.45], N[(N[(z + x), $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(1.0 / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 1.44999999999999996

   1. Initial program 83.6%

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

   3. Taylor expanded in y around 0

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-sin.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. lower-cos.f64 54.5

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

   5. Applied rewrites 54.5%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 35.5%

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

   10. Applied rewrites 35.4%

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

   if 1.44999999999999996 < z

   1. Initial program 73.1%

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

      1. lift-+.f64 N/A

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

      2. flip3-+ N/A

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

      3. clear-num N/A

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

      4. lower-/.f64 N/A

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

      5. clear-num N/A

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

      6. flip3-+ N/A

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

      7. lift-+.f64 N/A

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

      8. lower-/.f64 73.0

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

   4. Applied rewrites 72.9%

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

   5. Taylor expanded in x around inf

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

      1. lower-/.f64 21.4

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

   7. Applied rewrites 21.4%

      \[\leadsto \frac{1}{\color{blue}{\frac{1}{x}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 31.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1.45:\\ \;\;\;\;\left(z + x\right) - \tan a\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 31.4% accurate, 9.1× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\frac{1}{x}} \end{array} \]

FPCore

(FPCore (x y z a) :precision binary64 (/ 1.0 (/ 1.0 x)))

C

double code(double x, double y, double z, double a) {
	return 1.0 / (1.0 / 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 = 1.0d0 / (1.0d0 / x)
end function

Java

public static double code(double x, double y, double z, double a) {
	return 1.0 / (1.0 / x);
}

Python

def code(x, y, z, a):
	return 1.0 / (1.0 / x)

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, a_] := N[(1.0 / N[(1.0 / x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 81.0%

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

   1. lift-+.f64 N/A

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

   2. flip3-+ N/A

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

   3. clear-num N/A

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

   4. lower-/.f64 N/A

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

   5. clear-num N/A

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

   6. flip3-+ N/A

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

   7. lift-+.f64 N/A

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

   8. lower-/.f64 80.8

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

4. Applied rewrites 80.8%

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

5. Taylor expanded in x around inf

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

   1. lower-/.f64 30.3

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

7. Applied rewrites 30.3%

   \[\leadsto \frac{1}{\color{blue}{\frac{1}{x}}} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024283 
(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))))