tan-example (used to crash)

Percentage Accurate: 79.3% → 99.7%

Time: 31.9s

Alternatives: 10

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: 79.3% 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

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 79.6%

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

   1. lift-+.f64 N/A

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

   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. lift-tan.f64 N/A

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

   5. lift-+.f64 N/A

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

   6. tan-sum N/A

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

   7. lift-tan.f64 N/A

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

   8. tan-quot N/A

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

   9. frac-sub N/A

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

   10. div-inv N/A

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

   11. lower-fma.f64 N/A

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

4. Applied rewrites 99.7%

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

5. Taylor expanded in z around inf

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

   1. +-commutative N/A

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

   2. mul-1-neg N/A

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

   3. associate-/l* N/A

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

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

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

   5. mul-1-neg N/A

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

   6. lower-fma.f64 N/A

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

   7. mul-1-neg N/A

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

   8. lower-neg.f64 N/A

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

   9. lower-sin.f64 N/A

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

   10. lower-/.f64 N/A

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

   11. lower-sin.f64 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f64 N/A

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

   14. lower-cos.f64 N/A

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

   15. lower-cos.f64 99.7

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

7. Applied rewrites 99.7%

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

8. Final simplification 99.7%

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

Alternative 2: 99.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-\tan z, \tan y, 1\right)\\ \mathsf{fma}\left(\left(\tan y + \tan z\right) \cdot \cos a - \sin a \cdot t\_0, \frac{1}{t\_0 \cdot \cos a}, x\right) \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (let* ((t_0 (fma (- (tan z)) (tan y) 1.0)))
   (fma
    (- (* (+ (tan y) (tan z)) (cos a)) (* (sin a) t_0))
    (/ 1.0 (* t_0 (cos a)))
    x)))

C

double code(double x, double y, double z, double a) {
	double t_0 = fma(-tan(z), tan(y), 1.0);
	return fma((((tan(y) + tan(z)) * cos(a)) - (sin(a) * t_0)), (1.0 / (t_0 * cos(a))), x);
}

Julia

function code(x, y, z, a)
	t_0 = fma(Float64(-tan(z)), tan(y), 1.0)
	return fma(Float64(Float64(Float64(tan(y) + tan(z)) * cos(a)) - Float64(sin(a) * t_0)), Float64(1.0 / Float64(t_0 * cos(a))), x)
end

Wolfram

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

Derivation

1. Initial program 79.6%

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

   1. lift-+.f64 N/A

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

   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. lift-tan.f64 N/A

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

   5. lift-+.f64 N/A

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

   6. tan-sum N/A

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

   7. lift-tan.f64 N/A

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

   8. tan-quot N/A

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

   9. frac-sub N/A

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

   10. div-inv N/A

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

   11. lower-fma.f64 N/A

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

4. Applied rewrites 99.7%

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

5. Final simplification 99.7%

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

Alternative 3: 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 79.6%

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

   1. lift--.f64 N/A

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

   2. sub-neg N/A

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

   3. lift-tan.f64 N/A

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

   4. lift-+.f64 N/A

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

   5. tan-sum N/A

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

   6. 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 99.6%

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

5. Final simplification 99.6%

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

Alternative 4: 89.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan y + \tan z\\ \mathbf{if}\;a \leq -0.00045:\\ \;\;\;\;\frac{t\_0}{1} - \left(\tan a - x\right)\\ \mathbf{elif}\;a \leq 1.4 \cdot 10^{-15}:\\ \;\;\;\;\mathsf{fma}\left(t\_0, {\left(1 - \tan y \cdot \tan z\right)}^{-1}, -\left(-x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\tan \left(y + z\right) - \tan a\right) + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (let* ((t_0 (+ (tan y) (tan z))))
   (if (<= a -0.00045)
     (- (/ t_0 1.0) (- (tan a) x))
     (if (<= a 1.4e-15)
       (fma t_0 (pow (- 1.0 (* (tan y) (tan z))) -1.0) (- (- x)))
       (+ (- (tan (+ y z)) (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 <= -0.00045) {
		tmp = (t_0 / 1.0) - (tan(a) - x);
	} else if (a <= 1.4e-15) {
		tmp = fma(t_0, pow((1.0 - (tan(y) * tan(z))), -1.0), -(-x));
	} else {
		tmp = (tan((y + z)) - tan(a)) + x;
	}
	return tmp;
}

Julia

function code(x, y, z, a)
	t_0 = Float64(tan(y) + tan(z))
	tmp = 0.0
	if (a <= -0.00045)
		tmp = Float64(Float64(t_0 / 1.0) - Float64(tan(a) - x));
	elseif (a <= 1.4e-15)
		tmp = fma(t_0, (Float64(1.0 - Float64(tan(y) * tan(z))) ^ -1.0), Float64(-Float64(-x)));
	else
		tmp = Float64(Float64(tan(Float64(y + z)) - 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, -0.00045], N[(N[(t$95$0 / 1.0), $MachinePrecision] - N[(N[Tan[a], $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 1.4e-15], N[(t$95$0 * N[Power[N[(1.0 - N[(N[Tan[y], $MachinePrecision] * N[Tan[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision] + (-(-x))), $MachinePrecision], N[(N[(N[Tan[N[(y + z), $MachinePrecision]], $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if a < -4.4999999999999999e-4

   1. Initial program 80.6%

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

      1. lift-+.f64 N/A

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

      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 80.4

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

   4. Applied rewrites 80.4%

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

      1. lift-tan.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. tan-sum N/A

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

      4. lift-tan.f64 N/A

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

      5. lift-tan.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. lift-tan.f64 N/A

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

      12. lift-tan.f64 N/A

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

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

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

      14. lift-neg.f64 N/A

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

      15. +-commutative N/A

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

      16. lift-fma.f64 99.3

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

   6. Applied rewrites 99.3%

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

   7. Taylor expanded in z around 0

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

   9. Applied rewrites 80.7%

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

   if -4.4999999999999999e-4 < a < 1.40000000000000007e-15

   1. Initial program 78.3%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   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 78.3

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

   4. Applied rewrites 78.3%

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

   5. Taylor expanded in a around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 78.3

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

   7. Applied rewrites 78.3%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. lift-tan.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. tan-sum N/A

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

      6. lift-tan.f64 N/A

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

      7. lift-tan.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. div-inv N/A

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

      10. lower-fma.f64 N/A

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

   9. Applied rewrites 99.2%

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

   if 1.40000000000000007e-15 < a

   1. Initial program 80.9%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   2. Add Preprocessing
3. Recombined 3 regimes into one program.

4. Final simplification 89.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -0.00045:\\ \;\;\;\;\frac{\tan y + \tan z}{1} - \left(\tan a - x\right)\\ \mathbf{elif}\;a \leq 1.4 \cdot 10^{-15}:\\ \;\;\;\;\mathsf{fma}\left(\tan y + \tan z, {\left(1 - \tan y \cdot \tan z\right)}^{-1}, -\left(-x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\tan \left(y + z\right) - \tan a\right) + x\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 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 79.6%

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

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

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. Final simplification 99.6%

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

Alternative 6: 89.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan y + \tan z\\ \mathbf{if}\;a \leq -0.00045:\\ \;\;\;\;\frac{t\_0}{1} - \left(\tan a - x\right)\\ \mathbf{elif}\;a \leq 1.4 \cdot 10^{-15}:\\ \;\;\;\;\frac{t\_0}{1 - \tan y \cdot \tan z} - \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\tan \left(y + z\right) - \tan a\right) + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (let* ((t_0 (+ (tan y) (tan z))))
   (if (<= a -0.00045)
     (- (/ t_0 1.0) (- (tan a) x))
     (if (<= a 1.4e-15)
       (- (/ t_0 (- 1.0 (* (tan y) (tan z)))) (- x))
       (+ (- (tan (+ y z)) (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 <= -0.00045) {
		tmp = (t_0 / 1.0) - (tan(a) - x);
	} else if (a <= 1.4e-15) {
		tmp = (t_0 / (1.0 - (tan(y) * tan(z)))) - -x;
	} else {
		tmp = (tan((y + 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) :: t_0
    real(8) :: tmp
    t_0 = tan(y) + tan(z)
    if (a <= (-0.00045d0)) then
        tmp = (t_0 / 1.0d0) - (tan(a) - x)
    else if (a <= 1.4d-15) then
        tmp = (t_0 / (1.0d0 - (tan(y) * tan(z)))) - -x
    else
        tmp = (tan((y + z)) - tan(a)) + x
    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) + Math.tan(z);
	double tmp;
	if (a <= -0.00045) {
		tmp = (t_0 / 1.0) - (Math.tan(a) - x);
	} else if (a <= 1.4e-15) {
		tmp = (t_0 / (1.0 - (Math.tan(y) * Math.tan(z)))) - -x;
	} else {
		tmp = (Math.tan((y + z)) - Math.tan(a)) + x;
	}
	return tmp;
}

Python

def code(x, y, z, a):
	t_0 = math.tan(y) + math.tan(z)
	tmp = 0
	if a <= -0.00045:
		tmp = (t_0 / 1.0) - (math.tan(a) - x)
	elif a <= 1.4e-15:
		tmp = (t_0 / (1.0 - (math.tan(y) * math.tan(z)))) - -x
	else:
		tmp = (math.tan((y + z)) - math.tan(a)) + x
	return tmp

Julia

function code(x, y, z, a)
	t_0 = Float64(tan(y) + tan(z))
	tmp = 0.0
	if (a <= -0.00045)
		tmp = Float64(Float64(t_0 / 1.0) - Float64(tan(a) - x));
	elseif (a <= 1.4e-15)
		tmp = Float64(Float64(t_0 / Float64(1.0 - Float64(tan(y) * tan(z)))) - Float64(-x));
	else
		tmp = Float64(Float64(tan(Float64(y + z)) - tan(a)) + x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, a)
	t_0 = tan(y) + tan(z);
	tmp = 0.0;
	if (a <= -0.00045)
		tmp = (t_0 / 1.0) - (tan(a) - x);
	elseif (a <= 1.4e-15)
		tmp = (t_0 / (1.0 - (tan(y) * tan(z)))) - -x;
	else
		tmp = (tan((y + z)) - tan(a)) + x;
	end
	tmp_2 = 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, -0.00045], N[(N[(t$95$0 / 1.0), $MachinePrecision] - N[(N[Tan[a], $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 1.4e-15], N[(N[(t$95$0 / N[(1.0 - N[(N[Tan[y], $MachinePrecision] * N[Tan[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - (-x)), $MachinePrecision], N[(N[(N[Tan[N[(y + z), $MachinePrecision]], $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if a < -4.4999999999999999e-4

   1. Initial program 80.6%

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

      1. lift-+.f64 N/A

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

      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 80.4

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

   4. Applied rewrites 80.4%

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

      1. lift-tan.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. tan-sum N/A

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

      4. lift-tan.f64 N/A

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

      5. lift-tan.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. lift-tan.f64 N/A

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

      12. lift-tan.f64 N/A

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

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

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

      14. lift-neg.f64 N/A

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

      15. +-commutative N/A

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

      16. lift-fma.f64 99.3

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

   6. Applied rewrites 99.3%

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

   7. Taylor expanded in z around 0

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

   9. Applied rewrites 80.7%

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

   if -4.4999999999999999e-4 < a < 1.40000000000000007e-15

   1. Initial program 78.3%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   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 78.3

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

   4. Applied rewrites 78.3%

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

   5. Taylor expanded in a around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 78.3

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

   7. Applied rewrites 78.3%

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

      1. lift-tan.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. tan-sum N/A

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

      4. lift-tan.f64 N/A

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

      5. lift-tan.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-+.f64 N/A

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

      11. lift-tan.f64 N/A

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

      12. lift-tan.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lower--.f64 99.2

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 99.2

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

   9. Applied rewrites 99.2%

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

   if 1.40000000000000007e-15 < a

   1. Initial program 80.9%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   2. Add Preprocessing
3. Recombined 3 regimes into one program.

4. Final simplification 89.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -0.00045:\\ \;\;\;\;\frac{\tan y + \tan z}{1} - \left(\tan a - x\right)\\ \mathbf{elif}\;a \leq 1.4 \cdot 10^{-15}:\\ \;\;\;\;\frac{\tan y + \tan z}{1 - \tan y \cdot \tan z} - \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\tan \left(y + z\right) - \tan a\right) + x\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 79.6% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.6%

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

   1. lift-+.f64 N/A

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

   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 79.5

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

4. Applied rewrites 79.5%

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

   1. lift-tan.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. tan-sum N/A

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

   4. lift-tan.f64 N/A

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

   5. lift-tan.f64 N/A

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

   6. lift-+.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. lift-+.f64 N/A

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

   9. +-commutative N/A

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

   10. lower-+.f64 N/A

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

   11. lift-tan.f64 N/A

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

   12. lift-tan.f64 N/A

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

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

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

   14. lift-neg.f64 N/A

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

   15. +-commutative N/A

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

   16. lift-fma.f64 99.5

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

6. Applied rewrites 99.5%

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

7. Taylor expanded in z around 0

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

9. Applied rewrites 79.8%

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

Alternative 8: 79.3% 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 79.6%

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

3. Final simplification 79.6%

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

Alternative 9: 50.4% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.6%

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

   1. lift-+.f64 N/A

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

   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 79.5

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

4. Applied rewrites 79.5%

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

5. Taylor expanded in a around 0

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 48.7

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

7. Applied rewrites 48.7%

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

8. Final simplification 48.7%

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

Alternative 10: 31.7% 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 79.6%

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

   1. lift-+.f64 N/A

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

   2. flip3-+ N/A

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

   3. clear-num N/A

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

   4. lower-/.f64 N/A

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

   5. clear-num N/A

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

   6. flip3-+ N/A

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

   7. lift-+.f64 N/A

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

   8. lower-/.f64 79.4

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

4. Applied rewrites 79.4%

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

5. Taylor expanded in x around inf

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

   1. lower-/.f64 29.4

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

7. Applied rewrites 29.4%

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

Reproduce

herbie shell --seed 2024241 
(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))))