tan-example (used to crash)

Percentage Accurate: 79.7% → 99.7%

Time: 31.2s

Alternatives: 20

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: 79.7% 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

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

FPCore

NOTE: x, y, z, and a should be sorted in increasing order before calling this function.
(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 z) 2.0) (pow (tan y) 2.0)) t_0)))
     (fma (/ 1.0 (cos z)) (sin z) (tan y))
     (- (tan a))))))

C

assert(x < y && y < z && z < a);
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(z), 2.0) * pow(tan(y), 2.0)) / t_0))), fma((1.0 / cos(z)), sin(z), tan(y)), -tan(a));
}

Julia

x, y, z, a = sort([x, y, z, a])
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(z) ^ 2.0) * (tan(y) ^ 2.0)) / t_0))), fma(Float64(1.0 / cos(z)), sin(z), tan(y)), Float64(-tan(a))))
end

Wolfram

NOTE: x, y, z, and a should be sorted in increasing order before calling this function.
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[(N[Power[N[Tan[z], $MachinePrecision], 2.0], $MachinePrecision] * N[Power[N[Tan[y], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 / N[Cos[z], $MachinePrecision]), $MachinePrecision] * N[Sin[z], $MachinePrecision] + N[Tan[y], $MachinePrecision]), $MachinePrecision] + (-N[Tan[a], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 76.8%

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

   1. lift--.f64 N/A

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

   2. sub-neg N/A

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

   3. lift-tan.f64 N/A

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

   4. lift-+.f64 N/A

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

   5. tan-sum N/A

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

   6. clear-num N/A

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

   7. associate-/r/ N/A

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

   8. lower-fma.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lower--.f64 N/A

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

   11. lower-*.f64 N/A

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

   12. lower-tan.f64 N/A

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

   13. lower-tan.f64 N/A

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

   14. lower-+.f64 N/A

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

   15. lower-tan.f64 N/A

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

   16. lower-tan.f64 N/A

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

   17. 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 rewrites 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-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-tan.f64 N/A

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

   4. tan-quot N/A

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

   5. lift-sin.f64 N/A

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

   6. lift-cos.f64 N/A

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

   7. clear-num N/A

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

   8. associate-/r/ N/A

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

   9. lower-fma.f64 N/A

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

   10. lower-/.f64 99.7

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

6. Applied rewrites 99.7%

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

   1. lift--.f64 N/A

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

   2. flip-- N/A

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

   3. metadata-eval N/A

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

   4. div-sub N/A

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

   5. lower--.f64 N/A

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

   6. lower-/.f64 N/A

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

   7. +-commutative N/A

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

   8. lift-*.f64 N/A

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

   9. lower-fma.f64 N/A

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

   10. lower-/.f64 N/A

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

   11. pow2 N/A

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

   12. lower-pow.f64 N/A

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

   13. +-commutative N/A

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

8. Applied rewrites 99.7%

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

   1. lift-pow.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. unpow-prod-down N/A

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

   4. lift-tan.f64 N/A

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

   5. tan-quot N/A

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

   6. lift-sin.f64 N/A

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

   7. lift-cos.f64 N/A

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

   8. div-inv N/A

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

   9. lift-/.f64 N/A

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

   10. *-commutative N/A

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

   11. pow2 N/A

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

   12. pow2 N/A

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

   13. *-commutative N/A

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

   14. lower-*.f64 N/A

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

10. Applied rewrites 99.7%

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

Alternative 2: 99.7% accurate, 0.2× speedup

Math

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

FPCore

NOTE: x, y, z, and a should be sorted in increasing order before calling this function.
(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)))
     (fma (/ 1.0 (cos z)) (sin z) (tan y))
     (- (tan a))))))

C

assert(x < y && y < z && z < a);
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))), fma((1.0 / cos(z)), sin(z), tan(y)), -tan(a));
}

Julia

x, y, z, a = sort([x, y, z, a])
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))), fma(Float64(1.0 / cos(z)), sin(z), tan(y)), Float64(-tan(a))))
end

Wolfram

NOTE: x, y, z, and a should be sorted in increasing order before calling this function.
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[(1.0 / N[Cos[z], $MachinePrecision]), $MachinePrecision] * N[Sin[z], $MachinePrecision] + N[Tan[y], $MachinePrecision]), $MachinePrecision] + (-N[Tan[a], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 76.8%

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

   1. lift--.f64 N/A

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

   2. sub-neg N/A

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

   3. lift-tan.f64 N/A

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

   4. lift-+.f64 N/A

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

   5. tan-sum N/A

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

   6. clear-num N/A

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

   7. associate-/r/ N/A

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

   8. lower-fma.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lower--.f64 N/A

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

   11. lower-*.f64 N/A

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

   12. lower-tan.f64 N/A

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

   13. lower-tan.f64 N/A

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

   14. lower-+.f64 N/A

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

   15. lower-tan.f64 N/A

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

   16. lower-tan.f64 N/A

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

   17. 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 rewrites 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-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-tan.f64 N/A

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

   4. tan-quot N/A

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

   5. lift-sin.f64 N/A

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

   6. lift-cos.f64 N/A

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

   7. clear-num N/A

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

   8. associate-/r/ N/A

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

   9. lower-fma.f64 N/A

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

   10. lower-/.f64 99.7

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

6. Applied rewrites 99.7%

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

   1. lift--.f64 N/A

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

   2. flip-- N/A

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

   3. metadata-eval N/A

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

   4. div-sub N/A

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

   5. lower--.f64 N/A

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

   6. lower-/.f64 N/A

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

   7. +-commutative N/A

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

   8. lift-*.f64 N/A

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

   9. lower-fma.f64 N/A

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

   10. lower-/.f64 N/A

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

   11. pow2 N/A

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

   12. lower-pow.f64 N/A

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

   13. +-commutative N/A

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

8. Applied rewrites 99.7%

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

Alternative 3: 88.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y, z, a] = \mathsf{sort}([x, y, z, a])\\ \\ \begin{array}{l} t_0 := \tan y + \tan z\\ \mathbf{if}\;\tan a \leq -0.05:\\ \;\;\;\;\mathsf{fma}\left(1, t\_0, x\right) - \tan a\\ \mathbf{elif}\;\tan a \leq 10^{-14}:\\ \;\;\;\;x + \mathsf{fma}\left(\frac{1}{1 - \tan y \cdot \tan z}, t\_0, -\mathsf{fma}\left(a, a \cdot \left(a \cdot 0.3333333333333333\right), a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x + \mathsf{fma}\left(1, t\_0, -\tan a\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and a should be sorted in increasing order before calling this function.
(FPCore (x y z a)
 :precision binary64
 (let* ((t_0 (+ (tan y) (tan z))))
   (if (<= (tan a) -0.05)
     (- (fma 1.0 t_0 x) (tan a))
     (if (<= (tan a) 1e-14)
       (+
        x
        (fma
         (/ 1.0 (- 1.0 (* (tan y) (tan z))))
         t_0
         (- (fma a (* a (* a 0.3333333333333333)) a))))
       (+ x (fma 1.0 t_0 (- (tan a))))))))

C

assert(x < y && y < z && z < a);
double code(double x, double y, double z, double a) {
	double t_0 = tan(y) + tan(z);
	double tmp;
	if (tan(a) <= -0.05) {
		tmp = fma(1.0, t_0, x) - tan(a);
	} else if (tan(a) <= 1e-14) {
		tmp = x + fma((1.0 / (1.0 - (tan(y) * tan(z)))), t_0, -fma(a, (a * (a * 0.3333333333333333)), a));
	} else {
		tmp = x + fma(1.0, t_0, -tan(a));
	}
	return tmp;
}

Julia

x, y, z, a = sort([x, y, z, a])
function code(x, y, z, a)
	t_0 = Float64(tan(y) + tan(z))
	tmp = 0.0
	if (tan(a) <= -0.05)
		tmp = Float64(fma(1.0, t_0, x) - tan(a));
	elseif (tan(a) <= 1e-14)
		tmp = Float64(x + fma(Float64(1.0 / Float64(1.0 - Float64(tan(y) * tan(z)))), t_0, Float64(-fma(a, Float64(a * Float64(a * 0.3333333333333333)), a))));
	else
		tmp = Float64(x + fma(1.0, t_0, Float64(-tan(a))));
	end
	return tmp
end

Wolfram

NOTE: x, y, z, and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, a_] := Block[{t$95$0 = N[(N[Tan[y], $MachinePrecision] + N[Tan[z], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Tan[a], $MachinePrecision], -0.05], N[(N[(1.0 * t$95$0 + x), $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Tan[a], $MachinePrecision], 1e-14], N[(x + N[(N[(1.0 / N[(1.0 - N[(N[Tan[y], $MachinePrecision] * N[Tan[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0 + (-N[(a * N[(a * N[(a * 0.3333333333333333), $MachinePrecision]), $MachinePrecision] + a), $MachinePrecision])), $MachinePrecision]), $MachinePrecision], N[(x + N[(1.0 * t$95$0 + (-N[Tan[a], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (tan.f64 a) < -0.050000000000000003

   1. Initial program 79.0%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. lift-tan.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. tan-sum N/A

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

      6. clear-num N/A

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

      7. associate-/r/ N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-tan.f64 N/A

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

      13. lower-tan.f64 N/A

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

      14. lower-+.f64 N/A

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

      15. lower-tan.f64 N/A

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

      16. lower-tan.f64 N/A

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

      17. 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 rewrites 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. Applied rewrites 80.0%

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

      1. lift-+.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. associate-+r+ N/A

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

      4. lift-neg.f64 N/A

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

      5. unsub-neg N/A

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

      6. lower--.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 80.0

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

   9. Applied rewrites 80.0%

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

   if -0.050000000000000003 < (tan.f64 a) < 9.99999999999999999e-15

   1. Initial program 74.6%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   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 74.6

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

   5. Applied rewrites 74.6%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. lift-tan.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. tan-sum N/A

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

      6. lift-tan.f64 N/A

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

      7. lift-tan.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. lift-tan.f64 N/A

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

      10. lift-tan.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift--.f64 N/A

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

      13. div-inv N/A

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

      14. lift-/.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-fma.f64 N/A

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

   7. Applied rewrites 99.7%

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

   if 9.99999999999999999e-15 < (tan.f64 a)

   1. Initial program 79.0%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. lift-tan.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. tan-sum N/A

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

      6. clear-num N/A

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

      7. associate-/r/ N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-tan.f64 N/A

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

      13. lower-tan.f64 N/A

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

      14. lower-+.f64 N/A

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

      15. lower-tan.f64 N/A

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

      16. lower-tan.f64 N/A

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

      17. 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 rewrites 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. Applied rewrites 79.3%

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

Alternative 4: 88.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y, z, a] = \mathsf{sort}([x, y, z, a])\\ \\ \begin{array}{l} t_0 := \tan y + \tan z\\ \mathbf{if}\;\tan a \leq -0.05:\\ \;\;\;\;\mathsf{fma}\left(1, t\_0, x\right) - \tan a\\ \mathbf{elif}\;\tan a \leq 10^{-14}:\\ \;\;\;\;x + \left(\frac{t\_0}{1 - \tan y \cdot \tan z} - \mathsf{fma}\left(a \cdot a, a \cdot 0.3333333333333333, a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x + \mathsf{fma}\left(1, t\_0, -\tan a\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and a should be sorted in increasing order before calling this function.
(FPCore (x y z a)
 :precision binary64
 (let* ((t_0 (+ (tan y) (tan z))))
   (if (<= (tan a) -0.05)
     (- (fma 1.0 t_0 x) (tan a))
     (if (<= (tan a) 1e-14)
       (+
        x
        (-
         (/ t_0 (- 1.0 (* (tan y) (tan z))))
         (fma (* a a) (* a 0.3333333333333333) a)))
       (+ x (fma 1.0 t_0 (- (tan a))))))))

C

assert(x < y && y < z && z < a);
double code(double x, double y, double z, double a) {
	double t_0 = tan(y) + tan(z);
	double tmp;
	if (tan(a) <= -0.05) {
		tmp = fma(1.0, t_0, x) - tan(a);
	} else if (tan(a) <= 1e-14) {
		tmp = x + ((t_0 / (1.0 - (tan(y) * tan(z)))) - fma((a * a), (a * 0.3333333333333333), a));
	} else {
		tmp = x + fma(1.0, t_0, -tan(a));
	}
	return tmp;
}

Julia

x, y, z, a = sort([x, y, z, a])
function code(x, y, z, a)
	t_0 = Float64(tan(y) + tan(z))
	tmp = 0.0
	if (tan(a) <= -0.05)
		tmp = Float64(fma(1.0, t_0, x) - tan(a));
	elseif (tan(a) <= 1e-14)
		tmp = Float64(x + Float64(Float64(t_0 / Float64(1.0 - Float64(tan(y) * tan(z)))) - fma(Float64(a * a), Float64(a * 0.3333333333333333), a)));
	else
		tmp = Float64(x + fma(1.0, t_0, Float64(-tan(a))));
	end
	return tmp
end

Wolfram

NOTE: x, y, z, and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, a_] := Block[{t$95$0 = N[(N[Tan[y], $MachinePrecision] + N[Tan[z], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Tan[a], $MachinePrecision], -0.05], N[(N[(1.0 * t$95$0 + x), $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Tan[a], $MachinePrecision], 1e-14], N[(x + N[(N[(t$95$0 / N[(1.0 - N[(N[Tan[y], $MachinePrecision] * N[Tan[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(a * a), $MachinePrecision] * N[(a * 0.3333333333333333), $MachinePrecision] + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(1.0 * t$95$0 + (-N[Tan[a], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (tan.f64 a) < -0.050000000000000003

   1. Initial program 79.0%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. lift-tan.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. tan-sum N/A

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

      6. clear-num N/A

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

      7. associate-/r/ N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-tan.f64 N/A

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

      13. lower-tan.f64 N/A

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

      14. lower-+.f64 N/A

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

      15. lower-tan.f64 N/A

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

      16. lower-tan.f64 N/A

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

      17. 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 rewrites 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. Applied rewrites 80.0%

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

      1. lift-+.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. associate-+r+ N/A

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

      4. lift-neg.f64 N/A

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

      5. unsub-neg N/A

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

      6. lower--.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 80.0

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

   9. Applied rewrites 80.0%

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

   if -0.050000000000000003 < (tan.f64 a) < 9.99999999999999999e-15

   1. Initial program 74.6%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   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 74.6

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

   5. Applied rewrites 74.6%

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

      1. lift-tan.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. tan-sum N/A

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

      4. lift-tan.f64 N/A

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

      5. lift-tan.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-tan.f64 N/A

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

      8. lift-tan.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift--.f64 N/A

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

      11. lower-/.f64 99.7

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

   7. Applied rewrites 99.7%

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

   if 9.99999999999999999e-15 < (tan.f64 a)

   1. Initial program 79.0%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. lift-tan.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. tan-sum N/A

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

      6. clear-num N/A

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

      7. associate-/r/ N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-tan.f64 N/A

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

      13. lower-tan.f64 N/A

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

      14. lower-+.f64 N/A

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

      15. lower-tan.f64 N/A

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

      16. lower-tan.f64 N/A

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

      17. 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 rewrites 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. Applied rewrites 79.3%

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

Alternative 5: 99.7% accurate, 0.3× speedup

Math

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

FPCore

NOTE: x, y, z, and a should be sorted in increasing order before calling this function.
(FPCore (x y z a)
 :precision binary64
 (+
  x
  (fma
   (/ 1.0 (- 1.0 (* (tan y) (tan z))))
   (fma (/ 1.0 (cos z)) (sin z) (tan y))
   (- (tan a)))))

C

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

Julia

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

Wolfram

NOTE: x, y, z, and a should be sorted in increasing order before calling this function.
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[(1.0 / N[Cos[z], $MachinePrecision]), $MachinePrecision] * N[Sin[z], $MachinePrecision] + N[Tan[y], $MachinePrecision]), $MachinePrecision] + (-N[Tan[a], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 76.8%

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

   1. lift--.f64 N/A

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

   2. sub-neg N/A

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

   3. lift-tan.f64 N/A

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

   4. lift-+.f64 N/A

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

   5. tan-sum N/A

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

   6. clear-num N/A

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

   7. associate-/r/ N/A

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

   8. lower-fma.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lower--.f64 N/A

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

   11. lower-*.f64 N/A

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

   12. lower-tan.f64 N/A

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

   13. lower-tan.f64 N/A

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

   14. lower-+.f64 N/A

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

   15. lower-tan.f64 N/A

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

   16. lower-tan.f64 N/A

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

   17. 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 rewrites 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-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-tan.f64 N/A

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

   4. tan-quot N/A

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

   5. lift-sin.f64 N/A

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

   6. lift-cos.f64 N/A

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

   7. clear-num N/A

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

   8. associate-/r/ N/A

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

   9. lower-fma.f64 N/A

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

   10. lower-/.f64 99.7

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

6. Applied rewrites 99.7%

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

Alternative 6: 99.7% accurate, 0.3× speedup

Math

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

FPCore

NOTE: x, y, z, and a should be sorted in increasing order before calling this function.
(FPCore (x y z a)
 :precision binary64
 (+
  x
  (-
   (/ (fma (/ 1.0 (cos z)) (sin z) (tan y)) (- 1.0 (* (tan y) (tan z))))
   (tan a))))

C

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

Julia

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

Wolfram

NOTE: x, y, z, and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, a_] := N[(x + N[(N[(N[(N[(1.0 / N[Cos[z], $MachinePrecision]), $MachinePrecision] * N[Sin[z], $MachinePrecision] + N[Tan[y], $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 76.8%

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

   1. lift-tan.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. tan-sum N/A

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

   4. lower-/.f64 N/A

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

   5. lower-+.f64 N/A

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

   6. lower-tan.f64 N/A

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

   7. lower-tan.f64 N/A

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

   8. lower--.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. lower-tan.f64 N/A

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

   11. lower-tan.f64 99.6

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

4. Applied rewrites 99.6%

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

   1. lift-+.f64 N/A

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

   2. lift-tan.f64 N/A

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

   3. tan-quot N/A

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

   4. lift-sin.f64 N/A

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

   5. lift-cos.f64 N/A

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

   6. div-inv N/A

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

   7. lift-/.f64 N/A

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

   8. *-commutative N/A

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

   9. +-commutative N/A

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

   10. lift-fma.f64 99.6

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

6. Applied rewrites 99.6%

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

Alternative 7: 99.7% accurate, 0.4× speedup

Math

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

FPCore

NOTE: x, y, z, and a should be sorted in increasing order before calling this function.
(FPCore (x y z a)
 :precision binary64
 (+
  x
  (fma (/ 1.0 (- 1.0 (* (tan y) (tan z)))) (+ (tan y) (tan z)) (- (tan a)))))

C

assert(x < y && y < z && z < a);
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

x, y, z, a = sort([x, y, z, a])
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

NOTE: x, y, z, and a should be sorted in increasing order before calling this function.
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 76.8%

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

   1. lift--.f64 N/A

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

   2. sub-neg N/A

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

   3. lift-tan.f64 N/A

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

   4. lift-+.f64 N/A

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

   5. tan-sum N/A

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

   6. clear-num N/A

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

   7. associate-/r/ N/A

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

   8. lower-fma.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lower--.f64 N/A

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

   11. lower-*.f64 N/A

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

   12. lower-tan.f64 N/A

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

   13. lower-tan.f64 N/A

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

   14. lower-+.f64 N/A

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

   15. lower-tan.f64 N/A

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

   16. lower-tan.f64 N/A

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

   17. 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 rewrites 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 8: 99.7% accurate, 0.4× speedup

Math

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

FPCore

NOTE: x, y, z, and a should be sorted in increasing order before calling this function.
(FPCore (x y z a)
 :precision binary64
 (+ x (- (/ (+ (tan y) (tan z)) (fma (tan z) (- (tan y)) 1.0)) (tan a))))

C

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

Julia

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

Wolfram

NOTE: x, y, z, and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, a_] := N[(x + N[(N[(N[(N[Tan[y], $MachinePrecision] + N[Tan[z], $MachinePrecision]), $MachinePrecision] / N[(N[Tan[z], $MachinePrecision] * (-N[Tan[y], $MachinePrecision]) + 1.0), $MachinePrecision]), $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 76.8%

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

   1. lift-tan.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. tan-sum N/A

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

   4. lower-/.f64 N/A

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

   5. lower-+.f64 N/A

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

   6. lower-tan.f64 N/A

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

   7. lower-tan.f64 N/A

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

   8. lower--.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. lower-tan.f64 N/A

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

   11. lower-tan.f64 99.6

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

4. Applied rewrites 99.6%

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

   1. lift--.f64 N/A

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

   2. sub-neg N/A

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

   3. +-commutative N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. distribute-rgt-neg-in N/A

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

   7. lower-fma.f64 N/A

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

   8. lower-neg.f64 99.7

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

6. Applied rewrites 99.7%

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

Alternative 9: 99.7% accurate, 0.4× speedup

Math

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

FPCore

NOTE: x, y, z, and a should be sorted in increasing order before calling this function.
(FPCore (x y z a)
 :precision binary64
 (+ x (- (/ (+ (tan y) (tan z)) (- 1.0 (* (tan y) (tan z)))) (tan a))))

C

assert(x < y && y < z && z < a);
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

NOTE: x, y, z, and a should be sorted in increasing order before calling this function.
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

assert x < y && y < z && z < a;
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

[x, y, z, a] = sort([x, y, z, a])
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

x, y, z, a = sort([x, y, z, a])
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

x, y, z, a = num2cell(sort([x, y, z, a])){:}
function tmp = code(x, y, z, a)
	tmp = x + (((tan(y) + tan(z)) / (1.0 - (tan(y) * tan(z)))) - tan(a));
end

Wolfram

NOTE: x, y, z, and a should be sorted in increasing order before calling this function.
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 76.8%

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

   1. lift-tan.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. tan-sum N/A

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

   4. lower-/.f64 N/A

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

   5. lower-+.f64 N/A

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

   6. lower-tan.f64 N/A

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

   7. lower-tan.f64 N/A

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

   8. lower--.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. lower-tan.f64 N/A

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

   11. lower-tan.f64 99.6

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

4. Applied rewrites 99.6%

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

Alternative 10: 60.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y, z, a] = \mathsf{sort}([x, y, z, a])\\ \\ \begin{array}{l} \mathbf{if}\;\tan a \leq -0.05:\\ \;\;\;\;x + \left(\frac{1}{\frac{\mathsf{fma}\left(z, z \cdot -0.3333333333333333, 1\right)}{z}} - \tan a\right)\\ \mathbf{elif}\;\tan a \leq 5 \cdot 10^{-7}:\\ \;\;\;\;x + \left(\tan \left(y + z\right) - \mathsf{fma}\left(\mathsf{fma}\left(a, a \cdot 0.13333333333333333, 0.3333333333333333\right), a \cdot \left(a \cdot a\right), a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(\frac{1}{\frac{\mathsf{fma}\left(z \cdot z, \mathsf{fma}\left(z \cdot z, -0.022222222222222223, -0.3333333333333333\right), 1\right)}{z}} - \tan a\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and a should be sorted in increasing order before calling this function.
(FPCore (x y z a)
 :precision binary64
 (if (<= (tan a) -0.05)
   (+ x (- (/ 1.0 (/ (fma z (* z -0.3333333333333333) 1.0) z)) (tan a)))
   (if (<= (tan a) 5e-7)
     (+
      x
      (-
       (tan (+ y z))
       (fma
        (fma a (* a 0.13333333333333333) 0.3333333333333333)
        (* a (* a a))
        a)))
     (+
      x
      (-
       (/
        1.0
        (/
         (fma
          (* z z)
          (fma (* z z) -0.022222222222222223 -0.3333333333333333)
          1.0)
         z))
       (tan a))))))

C

assert(x < y && y < z && z < a);
double code(double x, double y, double z, double a) {
	double tmp;
	if (tan(a) <= -0.05) {
		tmp = x + ((1.0 / (fma(z, (z * -0.3333333333333333), 1.0) / z)) - tan(a));
	} else if (tan(a) <= 5e-7) {
		tmp = x + (tan((y + z)) - fma(fma(a, (a * 0.13333333333333333), 0.3333333333333333), (a * (a * a)), a));
	} else {
		tmp = x + ((1.0 / (fma((z * z), fma((z * z), -0.022222222222222223, -0.3333333333333333), 1.0) / z)) - tan(a));
	}
	return tmp;
}

Julia

x, y, z, a = sort([x, y, z, a])
function code(x, y, z, a)
	tmp = 0.0
	if (tan(a) <= -0.05)
		tmp = Float64(x + Float64(Float64(1.0 / Float64(fma(z, Float64(z * -0.3333333333333333), 1.0) / z)) - tan(a)));
	elseif (tan(a) <= 5e-7)
		tmp = Float64(x + Float64(tan(Float64(y + z)) - fma(fma(a, Float64(a * 0.13333333333333333), 0.3333333333333333), Float64(a * Float64(a * a)), a)));
	else
		tmp = Float64(x + Float64(Float64(1.0 / Float64(fma(Float64(z * z), fma(Float64(z * z), -0.022222222222222223, -0.3333333333333333), 1.0) / z)) - tan(a)));
	end
	return tmp
end

Wolfram

NOTE: x, y, z, and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, a_] := If[LessEqual[N[Tan[a], $MachinePrecision], -0.05], N[(x + N[(N[(1.0 / N[(N[(z * N[(z * -0.3333333333333333), $MachinePrecision] + 1.0), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Tan[a], $MachinePrecision], 5e-7], N[(x + N[(N[Tan[N[(y + z), $MachinePrecision]], $MachinePrecision] - N[(N[(a * N[(a * 0.13333333333333333), $MachinePrecision] + 0.3333333333333333), $MachinePrecision] * N[(a * N[(a * a), $MachinePrecision]), $MachinePrecision] + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[(1.0 / N[(N[(N[(z * z), $MachinePrecision] * N[(N[(z * z), $MachinePrecision] * -0.022222222222222223 + -0.3333333333333333), $MachinePrecision] + 1.0), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (tan.f64 a) < -0.050000000000000003

   1. Initial program 79.0%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   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 65.4

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

   5. Applied rewrites 65.4%

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

   7. Applied rewrites 65.3%

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

   8. Taylor expanded in z around 0

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

   10. Applied rewrites 43.6%

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

   if -0.050000000000000003 < (tan.f64 a) < 4.99999999999999977e-7

   1. Initial program 75.2%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. *-rgt-identity N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. unpow2 N/A

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

      10. associate-*l* N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 75.2

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

   5. Applied rewrites 75.2%

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

   if 4.99999999999999977e-7 < (tan.f64 a)

   1. Initial program 78.0%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   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 57.6

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

   5. Applied rewrites 57.6%

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

   7. Applied rewrites 57.6%

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

   8. Taylor expanded in z around 0

      \[\leadsto x + \left(\frac{1}{\frac{1 + {z}^{2} \cdot \left(\frac{-1}{45} \cdot {z}^{2} - \frac{1}{3}\right)}{\color{blue}{z}}} - \tan a\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 42.2%

       \[\leadsto x + \left(\frac{1}{\frac{\mathsf{fma}\left(z \cdot z, \mathsf{fma}\left(z \cdot z, -0.022222222222222223, -0.3333333333333333\right), 1\right)}{\color{blue}{z}}} - \tan a\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 11: 60.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y, z, a] = \mathsf{sort}([x, y, z, a])\\ \\ \begin{array}{l} t_0 := x + \left(\frac{1}{\frac{\mathsf{fma}\left(z, z \cdot -0.3333333333333333, 1\right)}{z}} - \tan a\right)\\ \mathbf{if}\;\tan a \leq -0.05:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\tan a \leq 5 \cdot 10^{-7}:\\ \;\;\;\;x + \left(\tan \left(y + z\right) - \mathsf{fma}\left(\mathsf{fma}\left(a, a \cdot 0.13333333333333333, 0.3333333333333333\right), a \cdot \left(a \cdot a\right), a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and a should be sorted in increasing order before calling this function.
(FPCore (x y z a)
 :precision binary64
 (let* ((t_0
         (+
          x
          (- (/ 1.0 (/ (fma z (* z -0.3333333333333333) 1.0) z)) (tan a)))))
   (if (<= (tan a) -0.05)
     t_0
     (if (<= (tan a) 5e-7)
       (+
        x
        (-
         (tan (+ y z))
         (fma
          (fma a (* a 0.13333333333333333) 0.3333333333333333)
          (* a (* a a))
          a)))
       t_0))))

C

assert(x < y && y < z && z < a);
double code(double x, double y, double z, double a) {
	double t_0 = x + ((1.0 / (fma(z, (z * -0.3333333333333333), 1.0) / z)) - tan(a));
	double tmp;
	if (tan(a) <= -0.05) {
		tmp = t_0;
	} else if (tan(a) <= 5e-7) {
		tmp = x + (tan((y + z)) - fma(fma(a, (a * 0.13333333333333333), 0.3333333333333333), (a * (a * a)), a));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

x, y, z, a = sort([x, y, z, a])
function code(x, y, z, a)
	t_0 = Float64(x + Float64(Float64(1.0 / Float64(fma(z, Float64(z * -0.3333333333333333), 1.0) / z)) - tan(a)))
	tmp = 0.0
	if (tan(a) <= -0.05)
		tmp = t_0;
	elseif (tan(a) <= 5e-7)
		tmp = Float64(x + Float64(tan(Float64(y + z)) - fma(fma(a, Float64(a * 0.13333333333333333), 0.3333333333333333), Float64(a * Float64(a * a)), a)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

NOTE: x, y, z, and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, a_] := Block[{t$95$0 = N[(x + N[(N[(1.0 / N[(N[(z * N[(z * -0.3333333333333333), $MachinePrecision] + 1.0), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Tan[a], $MachinePrecision], -0.05], t$95$0, If[LessEqual[N[Tan[a], $MachinePrecision], 5e-7], N[(x + N[(N[Tan[N[(y + z), $MachinePrecision]], $MachinePrecision] - N[(N[(a * N[(a * 0.13333333333333333), $MachinePrecision] + 0.3333333333333333), $MachinePrecision] * N[(a * N[(a * a), $MachinePrecision]), $MachinePrecision] + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if (tan.f64 a) < -0.050000000000000003 or 4.99999999999999977e-7 < (tan.f64 a)

   1. Initial program 78.5%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   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 61.4

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

   5. Applied rewrites 61.4%

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

   7. Applied rewrites 61.4%

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

   8. Taylor expanded in z around 0

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

   10. Applied rewrites 42.9%

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

   if -0.050000000000000003 < (tan.f64 a) < 4.99999999999999977e-7

   1. Initial program 75.2%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. *-rgt-identity N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. unpow2 N/A

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

      10. associate-*l* N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 75.2

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

   5. Applied rewrites 75.2%

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

Alternative 12: 80.0% accurate, 0.7× speedup

Math

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

FPCore

NOTE: x, y, z, and a should be sorted in increasing order before calling this function.
(FPCore (x y z a)
 :precision binary64
 (+ x (fma 1.0 (+ (tan y) (tan z)) (- (tan a)))))

C

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

Julia

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

Wolfram

NOTE: x, y, z, and a should be sorted in increasing order before calling this function.
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 76.8%

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

   1. lift--.f64 N/A

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

   2. sub-neg N/A

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

   3. lift-tan.f64 N/A

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

   4. lift-+.f64 N/A

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

   5. tan-sum N/A

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

   6. clear-num N/A

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

   7. associate-/r/ N/A

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

   8. lower-fma.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lower--.f64 N/A

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

   11. lower-*.f64 N/A

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

   12. lower-tan.f64 N/A

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

   13. lower-tan.f64 N/A

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

   14. lower-+.f64 N/A

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

   15. lower-tan.f64 N/A

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

   16. lower-tan.f64 N/A

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

   17. 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 rewrites 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. Applied rewrites 76.9%

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

Alternative 13: 80.0% accurate, 0.7× speedup

Math

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

FPCore

NOTE: x, y, z, and a should be sorted in increasing order before calling this function.
(FPCore (x y z a)
 :precision binary64
 (fma 1.0 (+ (tan y) (tan z)) (- x (tan a))))

C

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

Julia

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

Wolfram

NOTE: x, y, z, and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, a_] := N[(1.0 * N[(N[Tan[y], $MachinePrecision] + N[Tan[z], $MachinePrecision]), $MachinePrecision] + N[(x - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 76.8%

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

   1. lift-+.f64 N/A

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

   2. lift--.f64 N/A

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

   3. associate-+r- N/A

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

   4. +-commutative N/A

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

   5. associate--l+ N/A

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

   6. lift-tan.f64 N/A

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

   7. lift-+.f64 N/A

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

   8. tan-sum N/A

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

   9. clear-num N/A

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

   10. associate-/r/ N/A

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

   11. lower-fma.f64 N/A

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

   12. lower-/.f64 N/A

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

   13. lower--.f64 N/A

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

   14. lower-*.f64 N/A

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

   15. lower-tan.f64 N/A

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

   16. lower-tan.f64 N/A

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

   17. lower-+.f64 N/A

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

   18. lower-tan.f64 N/A

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

   19. lower-tan.f64 N/A

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

   20. lower--.f64 99.6

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

4. Applied rewrites 99.6%

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

5. Taylor expanded in y around 0

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

7. Applied rewrites 76.9%

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

Alternative 14: 80.0% accurate, 0.7× speedup

Math

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

FPCore

NOTE: x, y, z, and a should be sorted in increasing order before calling this function.
(FPCore (x y z a)
 :precision binary64
 (- (fma 1.0 (+ (tan y) (tan z)) x) (tan a)))

C

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

Julia

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

Wolfram

NOTE: x, y, z, and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, a_] := N[(N[(1.0 * N[(N[Tan[y], $MachinePrecision] + N[Tan[z], $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 76.8%

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

   1. lift--.f64 N/A

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

   2. sub-neg N/A

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

   3. lift-tan.f64 N/A

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

   4. lift-+.f64 N/A

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

   5. tan-sum N/A

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

   6. clear-num N/A

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

   7. associate-/r/ N/A

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

   8. lower-fma.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lower--.f64 N/A

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

   11. lower-*.f64 N/A

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

   12. lower-tan.f64 N/A

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

   13. lower-tan.f64 N/A

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

   14. lower-+.f64 N/A

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

   15. lower-tan.f64 N/A

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

   16. lower-tan.f64 N/A

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

   17. 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 rewrites 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. Applied rewrites 76.9%

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

   1. lift-+.f64 N/A

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

   2. lift-fma.f64 N/A

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

   3. associate-+r+ N/A

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

   4. lift-neg.f64 N/A

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

   5. unsub-neg N/A

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

   6. lower--.f64 N/A

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

   7. +-commutative N/A

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

   8. lower-fma.f64 76.9

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

9. Applied rewrites 76.9%

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

Alternative 15: 79.7% accurate, 0.9× speedup

Math

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

FPCore

NOTE: x, y, z, and a should be sorted in increasing order before calling this function.
(FPCore (x y z a)
 :precision binary64
 (+ x (- (tan (* (- z y) (/ (+ y z) (- z y)))) (tan a))))

C

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

Fortran

NOTE: x, y, z, and a should be sorted in increasing order before calling this function.
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 - y) * ((y + z) / (z - y)))) - tan(a))
end function

Java

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

Python

[x, y, z, a] = sort([x, y, z, a])
def code(x, y, z, a):
	return x + (math.tan(((z - y) * ((y + z) / (z - y)))) - math.tan(a))

Julia

x, y, z, a = sort([x, y, z, a])
function code(x, y, z, a)
	return Float64(x + Float64(tan(Float64(Float64(z - y) * Float64(Float64(y + z) / Float64(z - y)))) - tan(a)))
end

MATLAB

x, y, z, a = num2cell(sort([x, y, z, a])){:}
function tmp = code(x, y, z, a)
	tmp = x + (tan(((z - y) * ((y + z) / (z - y)))) - tan(a));
end

Wolfram

NOTE: x, y, z, and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, a_] := N[(x + N[(N[Tan[N[(N[(z - y), $MachinePrecision] * N[(N[(y + z), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 76.8%

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

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

   3. flip-+ N/A

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

   4. lower-/.f64 N/A

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

   5. difference-of-squares N/A

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

   6. +-commutative N/A

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

   7. lift-+.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. lower--.f64 N/A

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

   10. lower--.f64 46.7

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

4. Applied rewrites 46.7%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. associate-/l* N/A

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

   5. lower-*.f64 N/A

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

   6. lower-/.f64 76.9

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

6. Applied rewrites 76.9%

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

Alternative 16: 69.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y, z, a] = \mathsf{sort}([x, y, z, a])\\ \\ \begin{array}{l} t_0 := x + \left(\tan z - \tan a\right)\\ \mathbf{if}\;a \leq -0.019:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;a \leq 0.039:\\ \;\;\;\;x + \left(\tan \left(y + z\right) - \mathsf{fma}\left(a \cdot a, a \cdot 0.3333333333333333, a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and a should be sorted in increasing order before calling this function.
(FPCore (x y z a)
 :precision binary64
 (let* ((t_0 (+ x (- (tan z) (tan a)))))
   (if (<= a -0.019)
     t_0
     (if (<= a 0.039)
       (+ x (- (tan (+ y z)) (fma (* a a) (* a 0.3333333333333333) a)))
       t_0))))

C

assert(x < y && y < z && z < a);
double code(double x, double y, double z, double a) {
	double t_0 = x + (tan(z) - tan(a));
	double tmp;
	if (a <= -0.019) {
		tmp = t_0;
	} else if (a <= 0.039) {
		tmp = x + (tan((y + z)) - fma((a * a), (a * 0.3333333333333333), a));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

x, y, z, a = sort([x, y, z, a])
function code(x, y, z, a)
	t_0 = Float64(x + Float64(tan(z) - tan(a)))
	tmp = 0.0
	if (a <= -0.019)
		tmp = t_0;
	elseif (a <= 0.039)
		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

NOTE: x, y, z, and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, a_] := Block[{t$95$0 = N[(x + N[(N[Tan[z], $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -0.019], t$95$0, If[LessEqual[a, 0.039], 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 a < -0.0189999999999999995 or 0.0389999999999999999 < a

   1. Initial program 78.5%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   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 61.4

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

   5. Applied rewrites 61.4%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 61.4

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

   7. Applied rewrites 61.4%

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

   if -0.0189999999999999995 < a < 0.0389999999999999999

   1. Initial program 75.2%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   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 75.2

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

   5. Applied rewrites 75.2%

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

4. Final simplification 68.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -0.019:\\ \;\;\;\;x + \left(\tan z - \tan a\right)\\ \mathbf{elif}\;a \leq 0.039:\\ \;\;\;\;x + \left(\tan \left(y + z\right) - \mathsf{fma}\left(a \cdot a, a \cdot 0.3333333333333333, a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(\tan z - \tan a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 79.7% accurate, 1.0× speedup

Math

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

FPCore

NOTE: x, y, z, and a should be sorted in increasing order before calling this function.
(FPCore (x y z a) :precision binary64 (+ x (- (tan (+ y z)) (tan a))))

C

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

Fortran

NOTE: x, y, z, and a should be sorted in increasing order before calling this function.
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

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

Python

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

Julia

x, y, z, a = sort([x, y, z, a])
function code(x, y, z, a)
	return Float64(x + Float64(tan(Float64(y + z)) - tan(a)))
end

MATLAB

x, y, z, a = num2cell(sort([x, y, z, a])){:}
function tmp = code(x, y, z, a)
	tmp = x + (tan((y + z)) - tan(a));
end

Wolfram

NOTE: x, y, z, and a should be sorted in increasing order before calling this function.
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 76.8%

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

Alternative 18: 56.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} [x, y, z, a] = \mathsf{sort}([x, y, z, a])\\ \\ \begin{array}{l} t_0 := x + \left(\frac{1}{\frac{1}{z}} - \tan a\right)\\ \mathbf{if}\;a \leq -0.46:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;a \leq 0.156:\\ \;\;\;\;x + \left(\tan \left(y + z\right) - \mathsf{fma}\left(a \cdot a, a \cdot 0.3333333333333333, a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and a should be sorted in increasing order before calling this function.
(FPCore (x y z a)
 :precision binary64
 (let* ((t_0 (+ x (- (/ 1.0 (/ 1.0 z)) (tan a)))))
   (if (<= a -0.46)
     t_0
     (if (<= a 0.156)
       (+ x (- (tan (+ y z)) (fma (* a a) (* a 0.3333333333333333) a)))
       t_0))))

C

assert(x < y && y < z && z < a);
double code(double x, double y, double z, double a) {
	double t_0 = x + ((1.0 / (1.0 / z)) - tan(a));
	double tmp;
	if (a <= -0.46) {
		tmp = t_0;
	} else if (a <= 0.156) {
		tmp = x + (tan((y + z)) - fma((a * a), (a * 0.3333333333333333), a));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

x, y, z, a = sort([x, y, z, a])
function code(x, y, z, a)
	t_0 = Float64(x + Float64(Float64(1.0 / Float64(1.0 / z)) - tan(a)))
	tmp = 0.0
	if (a <= -0.46)
		tmp = t_0;
	elseif (a <= 0.156)
		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

NOTE: x, y, z, and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, a_] := Block[{t$95$0 = N[(x + N[(N[(1.0 / N[(1.0 / z), $MachinePrecision]), $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -0.46], t$95$0, If[LessEqual[a, 0.156], 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 a < -0.46000000000000002 or 0.156 < a

   1. Initial program 78.5%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   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 61.4

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

   5. Applied rewrites 61.4%

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

   7. Applied rewrites 61.4%

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

   8. Taylor expanded in z around 0

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

   10. Applied rewrites 31.6%

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

   if -0.46000000000000002 < a < 0.156

   1. Initial program 75.2%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   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 75.2

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

   5. Applied rewrites 75.2%

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

Alternative 19: 56.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} [x, y, z, a] = \mathsf{sort}([x, y, z, a])\\ \\ \begin{array}{l} t_0 := x + \left(\mathsf{fma}\left(0.3333333333333333, z \cdot \left(z \cdot z\right), z\right) - \tan a\right)\\ \mathbf{if}\;a \leq -0.46:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;a \leq 0.156:\\ \;\;\;\;x + \left(\tan \left(y + z\right) - \mathsf{fma}\left(a \cdot a, a \cdot 0.3333333333333333, a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and a should be sorted in increasing order before calling this function.
(FPCore (x y z a)
 :precision binary64
 (let* ((t_0 (+ x (- (fma 0.3333333333333333 (* z (* z z)) z) (tan a)))))
   (if (<= a -0.46)
     t_0
     (if (<= a 0.156)
       (+ x (- (tan (+ y z)) (fma (* a a) (* a 0.3333333333333333) a)))
       t_0))))

C

assert(x < y && y < z && z < a);
double code(double x, double y, double z, double a) {
	double t_0 = x + (fma(0.3333333333333333, (z * (z * z)), z) - tan(a));
	double tmp;
	if (a <= -0.46) {
		tmp = t_0;
	} else if (a <= 0.156) {
		tmp = x + (tan((y + z)) - fma((a * a), (a * 0.3333333333333333), a));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

x, y, z, a = sort([x, y, z, a])
function code(x, y, z, a)
	t_0 = Float64(x + Float64(fma(0.3333333333333333, Float64(z * Float64(z * z)), z) - tan(a)))
	tmp = 0.0
	if (a <= -0.46)
		tmp = t_0;
	elseif (a <= 0.156)
		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

NOTE: x, y, z, and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, a_] := Block[{t$95$0 = N[(x + N[(N[(0.3333333333333333 * N[(z * N[(z * z), $MachinePrecision]), $MachinePrecision] + z), $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -0.46], t$95$0, If[LessEqual[a, 0.156], 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 a < -0.46000000000000002 or 0.156 < a

   1. Initial program 78.5%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   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 61.4

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

   5. Applied rewrites 61.4%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 31.3%

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

   if -0.46000000000000002 < a < 0.156

   1. Initial program 75.2%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   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 75.2

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

   5. Applied rewrites 75.2%

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

Alternative 20: 32.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} [x, y, z, a] = \mathsf{sort}([x, y, z, a])\\ \\ x + \left(\mathsf{fma}\left(0.3333333333333333, z \cdot \left(z \cdot z\right), z\right) - \tan a\right) \end{array} \]

FPCore

NOTE: x, y, z, and a should be sorted in increasing order before calling this function.
(FPCore (x y z a)
 :precision binary64
 (+ x (- (fma 0.3333333333333333 (* z (* z z)) z) (tan a))))

C

assert(x < y && y < z && z < a);
double code(double x, double y, double z, double a) {
	return x + (fma(0.3333333333333333, (z * (z * z)), z) - tan(a));
}

Julia

x, y, z, a = sort([x, y, z, a])
function code(x, y, z, a)
	return Float64(x + Float64(fma(0.3333333333333333, Float64(z * Float64(z * z)), z) - tan(a)))
end

Wolfram

NOTE: x, y, z, and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, a_] := N[(x + N[(N[(0.3333333333333333 * N[(z * N[(z * z), $MachinePrecision]), $MachinePrecision] + z), $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 76.8%

   \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
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 60.2

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

5. Applied rewrites 60.2%

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

6. Taylor expanded in z around 0

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

8. Applied rewrites 29.2%

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

Reproduce

herbie shell --seed 2024237 
(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))))