tan-example (used to crash)

Percentage Accurate: 79.3% → 99.7%

Time: 33.6s

Alternatives: 18

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

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\tan y, \tan z, 1\right)\\ x + \mathsf{fma}\left(\frac{1}{\frac{1}{t\_0} - \frac{{\left(\tan y \cdot \tan z\right)}^{2}}{t\_0}}, \tan y + \tan z, -\tan a\right) \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (let* ((t_0 (fma (tan y) (tan z) 1.0)))
   (+
    x
    (fma
     (/ 1.0 (- (/ 1.0 t_0) (/ (pow (* (tan y) (tan z)) 2.0) t_0)))
     (+ (tan y) (tan z))
     (- (tan a))))))

C

double code(double x, double y, double z, double a) {
	double t_0 = fma(tan(y), tan(z), 1.0);
	return x + fma((1.0 / ((1.0 / t_0) - (pow((tan(y) * tan(z)), 2.0) / t_0))), (tan(y) + tan(z)), -tan(a));
}

Julia

function code(x, y, z, a)
	t_0 = fma(tan(y), tan(z), 1.0)
	return Float64(x + fma(Float64(1.0 / Float64(Float64(1.0 / t_0) - Float64((Float64(tan(y) * tan(z)) ^ 2.0) / t_0))), Float64(tan(y) + tan(z)), Float64(-tan(a))))
end

Wolfram

code[x_, y_, z_, a_] := Block[{t$95$0 = N[(N[Tan[y], $MachinePrecision] * N[Tan[z], $MachinePrecision] + 1.0), $MachinePrecision]}, N[(x + N[(N[(1.0 / N[(N[(1.0 / t$95$0), $MachinePrecision] - N[(N[Power[N[(N[Tan[y], $MachinePrecision] * N[Tan[z], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Tan[y], $MachinePrecision] + N[Tan[z], $MachinePrecision]), $MachinePrecision] + (-N[Tan[a], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 78.4%

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

   1. lift-+.f64 N/A

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

   2. lift-tan.f64 N/A

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

   3. lift-tan.f64 N/A

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

   4. sub-neg N/A

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

   5. lift-tan.f64 N/A

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

   6. lift-+.f64 N/A

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

   7. tan-sum N/A

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

   8. clear-num N/A

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

   9. associate-/r/ N/A

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

   10. lower-fma.f64 N/A

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

   11. lower-/.f64 N/A

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

   12. lower--.f64 N/A

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

   13. lower-*.f64 N/A

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

   14. lower-tan.f64 N/A

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

   15. lower-tan.f64 N/A

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

   16. lower-+.f64 N/A

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

   17. lower-tan.f64 N/A

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

   18. lower-tan.f64 N/A

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

   19. lower-neg.f64 99.7

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

4. Applied egg-rr 99.7%

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

   1. lift-tan.f64 N/A

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

   2. lift-tan.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. flip-- N/A

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

   5. metadata-eval N/A

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

   6. div-sub N/A

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

   7. lower--.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. +-commutative N/A

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

   10. lift-*.f64 N/A

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

   11. lower-fma.f64 N/A

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

   12. lower-/.f64 N/A

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

6. Applied egg-rr 99.8%

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

Alternative 2: 99.7% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 78.4%

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

   1. lift-+.f64 N/A

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

   2. lift-tan.f64 N/A

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

   3. lift-tan.f64 N/A

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

   4. sub-neg N/A

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

   5. lift-tan.f64 N/A

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

   6. lift-+.f64 N/A

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

   7. tan-sum N/A

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

   8. clear-num N/A

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

   9. associate-/r/ N/A

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

   10. lower-fma.f64 N/A

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

   11. lower-/.f64 N/A

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

   12. lower--.f64 N/A

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

   13. lower-*.f64 N/A

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

   14. lower-tan.f64 N/A

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

   15. lower-tan.f64 N/A

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

   16. lower-+.f64 N/A

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

   17. lower-tan.f64 N/A

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

   18. lower-tan.f64 N/A

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

   19. lower-neg.f64 99.7

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

4. Applied egg-rr 99.7%

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

   1. lift-tan.f64 N/A

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

   2. lift-tan.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. flip-- N/A

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

   5. div-inv N/A

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

   6. lower-*.f64 N/A

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

   7. metadata-eval N/A

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

   8. lower--.f64 N/A

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

   9. pow2 N/A

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

   10. lower-pow.f64 N/A

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

   11. lower-/.f64 N/A

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

   12. +-commutative N/A

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

   13. lift-*.f64 N/A

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

   14. lower-fma.f64 99.8

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

6. Applied egg-rr 99.8%

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

7. Final simplification 99.8%

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

Alternative 3: 89.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan y + \tan z\\ t_1 := x + \mathsf{fma}\left(1, t\_0, -\tan a\right)\\ \mathbf{if}\;\tan a \leq -0.002:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\tan a \leq 10^{-19}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{1 - \tan y \cdot \tan z}, t\_0, x - a\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (let* ((t_0 (+ (tan y) (tan z))) (t_1 (+ x (fma 1.0 t_0 (- (tan a))))))
   (if (<= (tan a) -0.002)
     t_1
     (if (<= (tan a) 1e-19)
       (fma (/ 1.0 (- 1.0 (* (tan y) (tan z)))) t_0 (- x a))
       t_1))))

C

double code(double x, double y, double z, double a) {
	double t_0 = tan(y) + tan(z);
	double t_1 = x + fma(1.0, t_0, -tan(a));
	double tmp;
	if (tan(a) <= -0.002) {
		tmp = t_1;
	} else if (tan(a) <= 1e-19) {
		tmp = fma((1.0 / (1.0 - (tan(y) * tan(z)))), t_0, (x - a));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, a)
	t_0 = Float64(tan(y) + tan(z))
	t_1 = Float64(x + fma(1.0, t_0, Float64(-tan(a))))
	tmp = 0.0
	if (tan(a) <= -0.002)
		tmp = t_1;
	elseif (tan(a) <= 1e-19)
		tmp = fma(Float64(1.0 / Float64(1.0 - Float64(tan(y) * tan(z)))), t_0, Float64(x - a));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, a_] := Block[{t$95$0 = N[(N[Tan[y], $MachinePrecision] + N[Tan[z], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x + N[(1.0 * t$95$0 + (-N[Tan[a], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Tan[a], $MachinePrecision], -0.002], t$95$1, If[LessEqual[N[Tan[a], $MachinePrecision], 1e-19], N[(N[(1.0 / N[(1.0 - N[(N[Tan[y], $MachinePrecision] * N[Tan[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0 + N[(x - a), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (tan.f64 a) < -2e-3 or 9.9999999999999998e-20 < (tan.f64 a)

   1. Initial program 79.8%

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

      1. lift-+.f64 N/A

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

      2. lift-tan.f64 N/A

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

      3. lift-tan.f64 N/A

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

      4. sub-neg N/A

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

      5. lift-tan.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. tan-sum N/A

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

      8. clear-num N/A

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

      9. associate-/r/ N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-tan.f64 N/A

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

      15. lower-tan.f64 N/A

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

      16. lower-+.f64 N/A

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

      17. lower-tan.f64 N/A

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

      18. lower-tan.f64 N/A

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

      19. lower-neg.f64 99.6

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

   4. Applied egg-rr 99.6%

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

   5. Taylor expanded in y around 0

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

   7. Simplified 80.6%

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

   if -2e-3 < (tan.f64 a) < 9.9999999999999998e-20

   1. Initial program 77.0%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. *-rgt-identity N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 77.0

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

   5. Simplified 77.0%

      \[\leadsto x + \left(\tan \left(y + z\right) - \color{blue}{\mathsf{fma}\left(a \cdot a, a \cdot 0.3333333333333333, a\right)}\right) \]
   6. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. lift-tan.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-fma.f64 N/A

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

      6. lift--.f64 N/A

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

      7. +-commutative N/A

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

      8. lift--.f64 N/A

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

      9. sub-neg N/A

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

      10. associate-+l+ N/A

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

   7. Applied egg-rr 99.9%

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

   8. Taylor expanded in a around 0

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

      1. mul-1-neg N/A

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

      2. sub-neg N/A

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

      3. lower--.f64 99.9

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

   10. Simplified 99.9%

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

Alternative 4: 99.7% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 78.4%

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

   1. lift-+.f64 N/A

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

   2. lift-tan.f64 N/A

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

   3. lift-tan.f64 N/A

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

   4. sub-neg N/A

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

   5. lift-tan.f64 N/A

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

   6. lift-+.f64 N/A

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

   7. tan-sum N/A

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

   8. clear-num N/A

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

   9. associate-/r/ N/A

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

   10. lower-fma.f64 N/A

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

   11. lower-/.f64 N/A

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

   12. lower--.f64 N/A

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

   13. lower-*.f64 N/A

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

   14. lower-tan.f64 N/A

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

   15. lower-tan.f64 N/A

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

   16. lower-+.f64 N/A

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

   17. lower-tan.f64 N/A

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

   18. lower-tan.f64 N/A

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

   19. lower-neg.f64 99.7

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

4. Applied egg-rr 99.7%

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

Alternative 5: 99.7% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.4%

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

   1. tan-sum N/A

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

   2. lower-/.f64 N/A

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

   3. lower-+.f64 N/A

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

   4. lower-tan.f64 N/A

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

   5. lower-tan.f64 N/A

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

   6. lower--.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lower-tan.f64 N/A

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

   9. lower-tan.f64 99.7

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

4. Applied egg-rr 99.7%

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

Alternative 6: 69.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x + \left(\tan z - \tan a\right)\\ \mathbf{if}\;\tan a \leq -0.002:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\tan a \leq 5 \cdot 10^{-13}:\\ \;\;\;\;x + \left(\tan \left(y + z\right) - \mathsf{fma}\left(a \cdot a, a \cdot 0.3333333333333333, a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (let* ((t_0 (+ x (- (tan z) (tan a)))))
   (if (<= (tan a) -0.002)
     t_0
     (if (<= (tan a) 5e-13)
       (+ x (- (tan (+ y z)) (fma (* a a) (* a 0.3333333333333333) a)))
       t_0))))

C

double code(double x, double y, double z, double a) {
	double t_0 = x + (tan(z) - tan(a));
	double tmp;
	if (tan(a) <= -0.002) {
		tmp = t_0;
	} else if (tan(a) <= 5e-13) {
		tmp = x + (tan((y + z)) - fma((a * a), (a * 0.3333333333333333), a));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z, a)
	t_0 = Float64(x + Float64(tan(z) - tan(a)))
	tmp = 0.0
	if (tan(a) <= -0.002)
		tmp = t_0;
	elseif (tan(a) <= 5e-13)
		tmp = Float64(x + Float64(tan(Float64(y + z)) - fma(Float64(a * a), Float64(a * 0.3333333333333333), a)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, a_] := Block[{t$95$0 = N[(x + N[(N[Tan[z], $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Tan[a], $MachinePrecision], -0.002], t$95$0, If[LessEqual[N[Tan[a], $MachinePrecision], 5e-13], N[(x + N[(N[Tan[N[(y + z), $MachinePrecision]], $MachinePrecision] - N[(N[(a * a), $MachinePrecision] * N[(a * 0.3333333333333333), $MachinePrecision] + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if (tan.f64 a) < -2e-3 or 4.9999999999999999e-13 < (tan.f64 a)

   1. Initial program 79.5%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-cos.f64 64.1

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

   5. Simplified 64.1%

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

      1. lift-sin.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-tan.f64 N/A

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

      5. lift--.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 64.1

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

      8. lift-/.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. lift-cos.f64 N/A

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

      11. tan-quot N/A

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

      12. lift-tan.f64 64.1

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

   7. Applied egg-rr 64.1%

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

   if -2e-3 < (tan.f64 a) < 4.9999999999999999e-13

   1. Initial program 77.4%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. *-rgt-identity N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 77.4

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

   5. Simplified 77.4%

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

4. Final simplification 70.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\tan a \leq -0.002:\\ \;\;\;\;x + \left(\tan z - \tan a\right)\\ \mathbf{elif}\;\tan a \leq 5 \cdot 10^{-13}:\\ \;\;\;\;x + \left(\tan \left(y + z\right) - \mathsf{fma}\left(a \cdot a, a \cdot 0.3333333333333333, a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(\tan z - \tan a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 79.7% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 78.4%

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

   1. lift-+.f64 N/A

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

   2. lift-tan.f64 N/A

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

   3. lift-tan.f64 N/A

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

   4. sub-neg N/A

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

   5. lift-tan.f64 N/A

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

   6. lift-+.f64 N/A

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

   7. tan-sum N/A

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

   8. clear-num N/A

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

   9. associate-/r/ N/A

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

   10. lower-fma.f64 N/A

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

   11. lower-/.f64 N/A

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

   12. lower--.f64 N/A

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

   13. lower-*.f64 N/A

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

   14. lower-tan.f64 N/A

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

   15. lower-tan.f64 N/A

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

   16. lower-+.f64 N/A

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

   17. lower-tan.f64 N/A

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

   18. lower-tan.f64 N/A

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

   19. lower-neg.f64 99.7

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

4. Applied egg-rr 99.7%

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

5. Taylor expanded in y around 0

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

7. Simplified 78.9%

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

Alternative 8: 79.2% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 78.4%

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

3. Taylor expanded in x around inf

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

   1. associate--l+ N/A

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

   2. distribute-lft-in N/A

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

   3. *-rgt-identity N/A

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

   4. +-commutative N/A

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

   5. associate-/l/ N/A

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

   6. associate-/l/ N/A

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

   7. div-sub N/A

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

   8. lower-fma.f64 N/A

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

5. Simplified 78.4%

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

   1. lift-+.f64 N/A

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

   2. lift-sin.f64 N/A

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

   3. lift-+.f64 N/A

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

   4. lift-cos.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-sin.f64 N/A

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

   7. lift-cos.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. associate-/r* N/A

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

   11. lift-sin.f64 N/A

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

   12. lift-cos.f64 N/A

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

   13. tan-quot N/A

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

   14. lift-tan.f64 N/A

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

   15. lift-*.f64 N/A

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

7. Applied egg-rr 78.4%

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

8. Final simplification 78.4%

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

Alternative 9: 79.3% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 78.4%

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

3. Taylor expanded in x around inf

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

   1. associate--l+ N/A

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

   2. distribute-lft-in N/A

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

   3. *-rgt-identity N/A

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

   4. +-commutative N/A

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

   5. associate-/l/ N/A

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

   6. associate-/l/ N/A

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

   7. div-sub N/A

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

   8. lower-fma.f64 N/A

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

5. Simplified 78.4%

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

   1. lift-+.f64 N/A

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

   2. lift-sin.f64 N/A

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

   3. lift-+.f64 N/A

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

   4. lift-cos.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-/.f64 N/A

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

   7. lift-sin.f64 N/A

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

   8. lift-cos.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. lift-/.f64 N/A

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

   11. lift--.f64 78.4

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

7. Applied egg-rr 78.4%

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

Alternative 10: 79.2% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 78.4%

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

3. Taylor expanded in x around inf

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

   1. associate--l+ N/A

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

   2. distribute-lft-in N/A

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

   3. *-rgt-identity N/A

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

   4. +-commutative N/A

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

   5. associate-/l/ N/A

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

   6. associate-/l/ N/A

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

   7. div-sub N/A

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

   8. lower-fma.f64 N/A

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

5. Simplified 78.4%

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

   1. lift-+.f64 N/A

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

   2. lift-sin.f64 N/A

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

   3. lift-+.f64 N/A

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

   4. lift-cos.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-sin.f64 N/A

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

   7. lift-cos.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. associate-/r* N/A

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

   11. lift-sin.f64 N/A

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

   12. lift-cos.f64 N/A

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

   13. tan-quot N/A

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

   14. lift-tan.f64 N/A

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

   15. lift-*.f64 N/A

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

7. Applied egg-rr 78.4%

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

9. Applied egg-rr 99.7%

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

11. Applied egg-rr 78.4%

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

Alternative 11: 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]

Derivation

1. Initial program 78.4%

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

Alternative 12: 50.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y + z \leq -2 \cdot 10^{-11}:\\ \;\;\;\;x + \left(\tan \left(y + z\right) - \mathsf{fma}\left(a \cdot a, a \cdot 0.3333333333333333, a\right)\right)\\ \mathbf{elif}\;y + z \leq 5 \cdot 10^{-29}:\\ \;\;\;\;x + \left(\mathsf{fma}\left(0.3333333333333333, z \cdot \left(z \cdot z\right), z\right) - \tan a\right)\\ \mathbf{else}:\\ \;\;\;\;x + \tan z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (if (<= (+ y z) -2e-11)
   (+ x (- (tan (+ y z)) (fma (* a a) (* a 0.3333333333333333) a)))
   (if (<= (+ y z) 5e-29)
     (+ x (- (fma 0.3333333333333333 (* z (* z z)) z) (tan a)))
     (+ x (tan z)))))

C

double code(double x, double y, double z, double a) {
	double tmp;
	if ((y + z) <= -2e-11) {
		tmp = x + (tan((y + z)) - fma((a * a), (a * 0.3333333333333333), a));
	} else if ((y + z) <= 5e-29) {
		tmp = x + (fma(0.3333333333333333, (z * (z * z)), z) - tan(a));
	} else {
		tmp = x + tan(z);
	}
	return tmp;
}

Julia

function code(x, y, z, a)
	tmp = 0.0
	if (Float64(y + z) <= -2e-11)
		tmp = Float64(x + Float64(tan(Float64(y + z)) - fma(Float64(a * a), Float64(a * 0.3333333333333333), a)));
	elseif (Float64(y + z) <= 5e-29)
		tmp = Float64(x + Float64(fma(0.3333333333333333, Float64(z * Float64(z * z)), z) - tan(a)));
	else
		tmp = Float64(x + tan(z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, a_] := If[LessEqual[N[(y + z), $MachinePrecision], -2e-11], N[(x + N[(N[Tan[N[(y + z), $MachinePrecision]], $MachinePrecision] - N[(N[(a * a), $MachinePrecision] * N[(a * 0.3333333333333333), $MachinePrecision] + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(y + z), $MachinePrecision], 5e-29], N[(x + N[(N[(0.3333333333333333 * N[(z * N[(z * z), $MachinePrecision]), $MachinePrecision] + z), $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[Tan[z], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 y z) < -1.99999999999999988e-11

   1. Initial program 72.4%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. *-rgt-identity N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 40.0

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

   5. Simplified 40.0%

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

   if -1.99999999999999988e-11 < (+.f64 y z) < 4.99999999999999986e-29

   1. Initial program 99.6%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-cos.f64 99.6

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

   5. Simplified 99.6%

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

   6. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. associate-*l* N/A

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

      4. pow-plus N/A

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

      5. metadata-eval N/A

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

      6. cube-unmult N/A

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

      7. unpow2 N/A

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

      8. *-lft-identity N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 99.6

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

   8. Simplified 99.6%

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

   if 4.99999999999999986e-29 < (+.f64 y z)

   1. Initial program 73.7%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-cos.f64 51.4

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

   5. Simplified 51.4%

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

   6. Taylor expanded in a around 0

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{\sin z}{\cos z} + x} \]

      2. lower-+.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin z}{\cos z} + x} \]

      3. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin z}{\cos z}} + x \]

      4. lower-sin.f64 N/A

         \[\leadsto \frac{\color{blue}{\sin z}}{\cos z} + x \]

      5. lower-cos.f64 38.6

         \[\leadsto \frac{\sin z}{\color{blue}{\cos z}} + x \]

   8. Simplified 38.6%

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

      1. tan-quot N/A

         \[\leadsto \color{blue}{\tan z} + x \]

      2. lift-tan.f64 N/A

         \[\leadsto \color{blue}{\tan z} + x \]

      3. lower-+.f64 38.6

         \[\leadsto \color{blue}{\tan z + x} \]

   10. Applied egg-rr 38.6%

       \[\leadsto \color{blue}{\tan z + x} \]
3. Recombined 3 regimes into one program.

4. Final simplification 51.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y + z \leq -2 \cdot 10^{-11}:\\ \;\;\;\;x + \left(\tan \left(y + z\right) - \mathsf{fma}\left(a \cdot a, a \cdot 0.3333333333333333, a\right)\right)\\ \mathbf{elif}\;y + z \leq 5 \cdot 10^{-29}:\\ \;\;\;\;x + \left(\mathsf{fma}\left(0.3333333333333333, z \cdot \left(z \cdot z\right), z\right) - \tan a\right)\\ \mathbf{else}:\\ \;\;\;\;x + \tan z\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 50.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y + z \leq -2 \cdot 10^{-11}:\\ \;\;\;\;x + \left(\tan \left(y + z\right) - 0.3333333333333333 \cdot \left(a \cdot \left(a \cdot a\right)\right)\right)\\ \mathbf{elif}\;y + z \leq 5 \cdot 10^{-29}:\\ \;\;\;\;x + \left(\mathsf{fma}\left(0.3333333333333333, z \cdot \left(z \cdot z\right), z\right) - \tan a\right)\\ \mathbf{else}:\\ \;\;\;\;x + \tan z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (if (<= (+ y z) -2e-11)
   (+ x (- (tan (+ y z)) (* 0.3333333333333333 (* a (* a a)))))
   (if (<= (+ y z) 5e-29)
     (+ x (- (fma 0.3333333333333333 (* z (* z z)) z) (tan a)))
     (+ x (tan z)))))

C

double code(double x, double y, double z, double a) {
	double tmp;
	if ((y + z) <= -2e-11) {
		tmp = x + (tan((y + z)) - (0.3333333333333333 * (a * (a * a))));
	} else if ((y + z) <= 5e-29) {
		tmp = x + (fma(0.3333333333333333, (z * (z * z)), z) - tan(a));
	} else {
		tmp = x + tan(z);
	}
	return tmp;
}

Julia

function code(x, y, z, a)
	tmp = 0.0
	if (Float64(y + z) <= -2e-11)
		tmp = Float64(x + Float64(tan(Float64(y + z)) - Float64(0.3333333333333333 * Float64(a * Float64(a * a)))));
	elseif (Float64(y + z) <= 5e-29)
		tmp = Float64(x + Float64(fma(0.3333333333333333, Float64(z * Float64(z * z)), z) - tan(a)));
	else
		tmp = Float64(x + tan(z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, a_] := If[LessEqual[N[(y + z), $MachinePrecision], -2e-11], N[(x + N[(N[Tan[N[(y + z), $MachinePrecision]], $MachinePrecision] - N[(0.3333333333333333 * N[(a * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(y + z), $MachinePrecision], 5e-29], N[(x + N[(N[(0.3333333333333333 * N[(z * N[(z * z), $MachinePrecision]), $MachinePrecision] + z), $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[Tan[z], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 y z) < -1.99999999999999988e-11

   1. Initial program 72.4%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. *-rgt-identity N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 40.0

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

   5. Simplified 40.0%

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

   6. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. cube-mult N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 39.9

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

   8. Simplified 39.9%

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

   if -1.99999999999999988e-11 < (+.f64 y z) < 4.99999999999999986e-29

   1. Initial program 99.6%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-cos.f64 99.6

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

   5. Simplified 99.6%

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

   6. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. associate-*l* N/A

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

      4. pow-plus N/A

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

      5. metadata-eval N/A

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

      6. cube-unmult N/A

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

      7. unpow2 N/A

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

      8. *-lft-identity N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 99.6

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

   8. Simplified 99.6%

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

   if 4.99999999999999986e-29 < (+.f64 y z)

   1. Initial program 73.7%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-cos.f64 51.4

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

   5. Simplified 51.4%

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

   6. Taylor expanded in a around 0

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{\sin z}{\cos z} + x} \]

      2. lower-+.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin z}{\cos z} + x} \]

      3. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin z}{\cos z}} + x \]

      4. lower-sin.f64 N/A

         \[\leadsto \frac{\color{blue}{\sin z}}{\cos z} + x \]

      5. lower-cos.f64 38.6

         \[\leadsto \frac{\sin z}{\color{blue}{\cos z}} + x \]

   8. Simplified 38.6%

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

      1. tan-quot N/A

         \[\leadsto \color{blue}{\tan z} + x \]

      2. lift-tan.f64 N/A

         \[\leadsto \color{blue}{\tan z} + x \]

      3. lower-+.f64 38.6

         \[\leadsto \color{blue}{\tan z + x} \]

   10. Applied egg-rr 38.6%

       \[\leadsto \color{blue}{\tan z + x} \]
3. Recombined 3 regimes into one program.

4. Final simplification 51.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y + z \leq -2 \cdot 10^{-11}:\\ \;\;\;\;x + \left(\tan \left(y + z\right) - 0.3333333333333333 \cdot \left(a \cdot \left(a \cdot a\right)\right)\right)\\ \mathbf{elif}\;y + z \leq 5 \cdot 10^{-29}:\\ \;\;\;\;x + \left(\mathsf{fma}\left(0.3333333333333333, z \cdot \left(z \cdot z\right), z\right) - \tan a\right)\\ \mathbf{else}:\\ \;\;\;\;x + \tan z\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 45.2% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.15 \cdot 10^{-47}:\\ \;\;\;\;\frac{1}{\frac{1}{x}}\\ \mathbf{elif}\;z \leq 1.45 \cdot 10^{-9}:\\ \;\;\;\;x + \left(\mathsf{fma}\left(0.3333333333333333, z \cdot \left(z \cdot z\right), z\right) - \tan a\right)\\ \mathbf{else}:\\ \;\;\;\;x + \tan z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (if (<= z -2.15e-47)
   (/ 1.0 (/ 1.0 x))
   (if (<= z 1.45e-9)
     (+ x (- (fma 0.3333333333333333 (* z (* z z)) z) (tan a)))
     (+ x (tan z)))))

C

double code(double x, double y, double z, double a) {
	double tmp;
	if (z <= -2.15e-47) {
		tmp = 1.0 / (1.0 / x);
	} else if (z <= 1.45e-9) {
		tmp = x + (fma(0.3333333333333333, (z * (z * z)), z) - tan(a));
	} else {
		tmp = x + tan(z);
	}
	return tmp;
}

Julia

function code(x, y, z, a)
	tmp = 0.0
	if (z <= -2.15e-47)
		tmp = Float64(1.0 / Float64(1.0 / x));
	elseif (z <= 1.45e-9)
		tmp = Float64(x + Float64(fma(0.3333333333333333, Float64(z * Float64(z * z)), z) - tan(a)));
	else
		tmp = Float64(x + tan(z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, a_] := If[LessEqual[z, -2.15e-47], N[(1.0 / N[(1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.45e-9], N[(x + N[(N[(0.3333333333333333 * N[(z * N[(z * z), $MachinePrecision]), $MachinePrecision] + z), $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[Tan[z], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -2.1499999999999999e-47

   1. Initial program 67.6%

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

      1. lift-+.f64 N/A

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

      2. lift-tan.f64 N/A

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

      3. lift-tan.f64 N/A

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

      4. lift--.f64 N/A

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

      5. flip3-+ N/A

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

      6. clear-num N/A

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

      7. lower-/.f64 N/A

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

      8. clear-num N/A

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

   4. Applied egg-rr 67.4%

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

   5. Taylor expanded in x around inf

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

      1. lower-/.f64 23.4

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

   7. Simplified 23.4%

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

   if -2.1499999999999999e-47 < z < 1.44999999999999996e-9

   1. Initial program 99.4%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-cos.f64 61.2

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

   5. Simplified 61.2%

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

   6. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. associate-*l* N/A

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

      4. pow-plus N/A

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

      5. metadata-eval N/A

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

      6. cube-unmult N/A

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

      7. unpow2 N/A

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

      8. *-lft-identity N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 61.2

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

   8. Simplified 61.2%

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

   if 1.44999999999999996e-9 < z

   1. Initial program 58.2%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-cos.f64 58.2

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

   5. Simplified 58.2%

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

   6. Taylor expanded in a around 0

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{\sin z}{\cos z} + x} \]

      2. lower-+.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin z}{\cos z} + x} \]

      3. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin z}{\cos z}} + x \]

      4. lower-sin.f64 N/A

         \[\leadsto \frac{\color{blue}{\sin z}}{\cos z} + x \]

      5. lower-cos.f64 40.1

         \[\leadsto \frac{\sin z}{\color{blue}{\cos z}} + x \]

   8. Simplified 40.1%

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

      1. tan-quot N/A

         \[\leadsto \color{blue}{\tan z} + x \]

      2. lift-tan.f64 N/A

         \[\leadsto \color{blue}{\tan z} + x \]

      3. lower-+.f64 40.1

         \[\leadsto \color{blue}{\tan z + x} \]

   10. Applied egg-rr 40.1%

       \[\leadsto \color{blue}{\tan z + x} \]
3. Recombined 3 regimes into one program.

4. Final simplification 43.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.15 \cdot 10^{-47}:\\ \;\;\;\;\frac{1}{\frac{1}{x}}\\ \mathbf{elif}\;z \leq 1.45 \cdot 10^{-9}:\\ \;\;\;\;x + \left(\mathsf{fma}\left(0.3333333333333333, z \cdot \left(z \cdot z\right), z\right) - \tan a\right)\\ \mathbf{else}:\\ \;\;\;\;x + \tan z\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 41.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ x + \tan z \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.4%

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

3. Taylor expanded in y around 0

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

   1. lower-/.f64 N/A

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

   2. lower-sin.f64 N/A

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

   3. lower-cos.f64 61.3

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

5. Simplified 61.3%

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

6. Taylor expanded in a around 0

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

   1. +-commutative N/A

      \[\leadsto \color{blue}{\frac{\sin z}{\cos z} + x} \]

   2. lower-+.f64 N/A

      \[\leadsto \color{blue}{\frac{\sin z}{\cos z} + x} \]

   3. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\sin z}{\cos z}} + x \]

   4. lower-sin.f64 N/A

      \[\leadsto \frac{\color{blue}{\sin z}}{\cos z} + x \]

   5. lower-cos.f64 40.0

      \[\leadsto \frac{\sin z}{\color{blue}{\cos z}} + x \]

8. Simplified 40.0%

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

   1. tan-quot N/A

      \[\leadsto \color{blue}{\tan z} + x \]

   2. lift-tan.f64 N/A

      \[\leadsto \color{blue}{\tan z} + x \]

   3. lower-+.f64 40.0

      \[\leadsto \color{blue}{\tan z + x} \]

10. Applied egg-rr 40.0%

    \[\leadsto \color{blue}{\tan z + x} \]

11. Final simplification 40.0%

    \[\leadsto x + \tan z \]
12. Add Preprocessing

Alternative 16: 23.8% accurate, 8.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 9:\\ \;\;\;\;\mathsf{fma}\left(z, \mathsf{fma}\left(z, z \cdot 0.3333333333333333, 1\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;x + a \cdot \left(\left(a \cdot a\right) \cdot -0.3333333333333333\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (if (<= z 9.0)
   (fma z (fma z (* z 0.3333333333333333) 1.0) x)
   (+ x (* a (* (* a a) -0.3333333333333333)))))

C

double code(double x, double y, double z, double a) {
	double tmp;
	if (z <= 9.0) {
		tmp = fma(z, fma(z, (z * 0.3333333333333333), 1.0), x);
	} else {
		tmp = x + (a * ((a * a) * -0.3333333333333333));
	}
	return tmp;
}

Julia

function code(x, y, z, a)
	tmp = 0.0
	if (z <= 9.0)
		tmp = fma(z, fma(z, Float64(z * 0.3333333333333333), 1.0), x);
	else
		tmp = Float64(x + Float64(a * Float64(Float64(a * a) * -0.3333333333333333)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, a_] := If[LessEqual[z, 9.0], N[(z * N[(z * N[(z * 0.3333333333333333), $MachinePrecision] + 1.0), $MachinePrecision] + x), $MachinePrecision], N[(x + N[(a * N[(N[(a * a), $MachinePrecision] * -0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 9

   1. Initial program 86.1%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-cos.f64 62.6

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

   5. Simplified 62.6%

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

   6. Taylor expanded in a around 0

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{\sin z}{\cos z} + x} \]

      2. lower-+.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin z}{\cos z} + x} \]

      3. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin z}{\cos z}} + x \]

      4. lower-sin.f64 N/A

         \[\leadsto \frac{\color{blue}{\sin z}}{\cos z} + x \]

      5. lower-cos.f64 40.3

         \[\leadsto \frac{\sin z}{\color{blue}{\cos z}} + x \]

   8. Simplified 40.3%

      \[\leadsto \color{blue}{\frac{\sin z}{\cos z} + x} \]

   9. Taylor expanded in z around 0

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

       1. +-commutative N/A

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

       2. lower-fma.f64 N/A

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

       3. +-commutative N/A

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

       4. *-commutative N/A

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

       5. unpow2 N/A

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

       6. associate-*l* N/A

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

       7. lower-fma.f64 N/A

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

       8. lower-*.f64 26.1

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

   11. Simplified 26.1%

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

   if 9 < z

   1. Initial program 57.6%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. *-rgt-identity N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 30.7

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

   5. Simplified 30.7%

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

   6. Taylor expanded in a around inf

      \[\leadsto x + \color{blue}{\frac{-1}{3} \cdot {a}^{3}} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto x + \color{blue}{{a}^{3} \cdot \frac{-1}{3}} \]

      2. cube-mult N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 13.7

         \[\leadsto x + a \cdot \left(-0.3333333333333333 \cdot \color{blue}{\left(a \cdot a\right)}\right) \]

   8. Simplified 13.7%

      \[\leadsto x + \color{blue}{a \cdot \left(-0.3333333333333333 \cdot \left(a \cdot a\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 22.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 9:\\ \;\;\;\;\mathsf{fma}\left(z, \mathsf{fma}\left(z, z \cdot 0.3333333333333333, 1\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;x + a \cdot \left(\left(a \cdot a\right) \cdot -0.3333333333333333\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 31.5% 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 78.4%

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

   1. lift-+.f64 N/A

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

   2. lift-tan.f64 N/A

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

   3. lift-tan.f64 N/A

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

   4. lift--.f64 N/A

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

   5. flip3-+ N/A

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

   6. clear-num N/A

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

   7. lower-/.f64 N/A

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

   8. clear-num N/A

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

4. Applied egg-rr 78.2%

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

5. Taylor expanded in x around inf

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

   1. lower-/.f64 30.7

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

7. Simplified 30.7%

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

Alternative 18: 21.9% accurate, 52.5× speedup

Math

\[\begin{array}{l} \\ x + z \end{array} \]

FPCore

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

C

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

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 + z
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.4%

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

3. Taylor expanded in y around 0

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

   1. lower-/.f64 N/A

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

   2. lower-sin.f64 N/A

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

   3. lower-cos.f64 61.3

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

5. Simplified 61.3%

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

6. Taylor expanded in a around 0

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

   1. +-commutative N/A

      \[\leadsto \color{blue}{\frac{\sin z}{\cos z} + x} \]

   2. lower-+.f64 N/A

      \[\leadsto \color{blue}{\frac{\sin z}{\cos z} + x} \]

   3. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\sin z}{\cos z}} + x \]

   4. lower-sin.f64 N/A

      \[\leadsto \frac{\color{blue}{\sin z}}{\cos z} + x \]

   5. lower-cos.f64 40.0

      \[\leadsto \frac{\sin z}{\color{blue}{\cos z}} + x \]

8. Simplified 40.0%

   \[\leadsto \color{blue}{\frac{\sin z}{\cos z} + x} \]

9. Taylor expanded in z around 0

   \[\leadsto \color{blue}{x + z} \]
10. Step-by-step derivation

    1. +-commutative N/A

       \[\leadsto \color{blue}{z + x} \]

    2. lower-+.f64 20.5

       \[\leadsto \color{blue}{z + x} \]

11. Simplified 20.5%

    \[\leadsto \color{blue}{z + x} \]

12. Final simplification 20.5%

    \[\leadsto x + z \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2024207 
(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))))