tan-example (used to crash)

Percentage Accurate: 79.5% → 99.7%

Time: 46.4s

Alternatives: 21

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: 79.5% 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, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 77.0%

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

   1. sub-neg N/A

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

   2. tan-sum N/A

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

   3. clear-num N/A

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

   4. associate-/r/ N/A

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

   5. accelerator-lowering-fma.f64 N/A

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

   6. /-lowering-/.f64 N/A

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

   7. --lowering--.f64 N/A

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

   8. *-lowering-*.f64 N/A

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

   9. tan-lowering-tan.f64 N/A

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

   10. tan-lowering-tan.f64 N/A

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

   11. +-lowering-+.f64 N/A

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

   12. tan-lowering-tan.f64 N/A

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

   13. tan-lowering-tan.f64 N/A

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

   14. neg-lowering-neg.f64 N/A

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

   15. tan-lowering-tan.f64 99.8

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

4. Applied egg-rr 99.8%

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

5. Final simplification 99.8%

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

Alternative 2: 88.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan y + \tan z\\ t_1 := x + \mathsf{fma}\left(1, t\_0, -\tan a\right)\\ \mathbf{if}\;\tan a \leq -0.1:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\tan a \leq 2 \cdot 10^{-37}:\\ \;\;\;\;x + \frac{t\_0}{1 - \tan y \cdot \tan z}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (let* ((t_0 (+ (tan y) (tan z))) (t_1 (+ x (fma 1.0 t_0 (- (tan a))))))
   (if (<= (tan a) -0.1)
     t_1
     (if (<= (tan a) 2e-37) (+ x (/ t_0 (- 1.0 (* (tan y) (tan z))))) t_1))))

C

double code(double x, double y, double z, double a) {
	double t_0 = tan(y) + tan(z);
	double t_1 = x + fma(1.0, t_0, -tan(a));
	double tmp;
	if (tan(a) <= -0.1) {
		tmp = t_1;
	} else if (tan(a) <= 2e-37) {
		tmp = x + (t_0 / (1.0 - (tan(y) * tan(z))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, a)
	t_0 = Float64(tan(y) + tan(z))
	t_1 = Float64(x + fma(1.0, t_0, Float64(-tan(a))))
	tmp = 0.0
	if (tan(a) <= -0.1)
		tmp = t_1;
	elseif (tan(a) <= 2e-37)
		tmp = Float64(x + Float64(t_0 / Float64(1.0 - Float64(tan(y) * tan(z)))));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, a_] := Block[{t$95$0 = N[(N[Tan[y], $MachinePrecision] + N[Tan[z], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x + N[(1.0 * t$95$0 + (-N[Tan[a], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Tan[a], $MachinePrecision], -0.1], t$95$1, If[LessEqual[N[Tan[a], $MachinePrecision], 2e-37], N[(x + N[(t$95$0 / N[(1.0 - N[(N[Tan[y], $MachinePrecision] * N[Tan[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (tan.f64 a) < -0.10000000000000001 or 2.00000000000000013e-37 < (tan.f64 a)

   1. Initial program 76.5%

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

      1. sub-neg N/A

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

      2. tan-sum N/A

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

      3. clear-num N/A

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

      4. associate-/r/ N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. /-lowering-/.f64 N/A

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

      7. --lowering--.f64 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. tan-lowering-tan.f64 N/A

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

      10. tan-lowering-tan.f64 N/A

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

      11. +-lowering-+.f64 N/A

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

      12. tan-lowering-tan.f64 N/A

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

      13. tan-lowering-tan.f64 N/A

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

      14. neg-lowering-neg.f64 N/A

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

      15. tan-lowering-tan.f64 99.6

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

   4. Applied egg-rr 99.6%

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

   5. Taylor expanded in y around 0

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

   7. Simplified 77.6%

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

   if -0.10000000000000001 < (tan.f64 a) < 2.00000000000000013e-37

   1. Initial program 77.6%

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

      1. +-commutative N/A

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

      2. associate-+l- N/A

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

      3. --lowering--.f64 N/A

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

      4. tan-lowering-tan.f64 N/A

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

      5. +-lowering-+.f64 N/A

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

      6. --lowering--.f64 N/A

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

      7. tan-lowering-tan.f64 77.6

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

   4. Applied egg-rr 77.6%

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

   5. Taylor expanded in a around 0

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

      1. mul-1-neg N/A

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

      2. neg-lowering-neg.f64 77.3

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

   7. Simplified 77.3%

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

      1. +-commutative N/A

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

      2. tan-sum N/A

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

      3. +-commutative N/A

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

      4. tan-quot N/A

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

      5. associate-/l* N/A

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

      6. /-lowering-/.f64 N/A

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

      7. +-lowering-+.f64 N/A

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

      8. tan-lowering-tan.f64 N/A

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

      9. tan-lowering-tan.f64 N/A

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

      10. --lowering--.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-*l/ N/A

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

      13. tan-quot N/A

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

      14. *-lowering-*.f64 N/A

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

      15. tan-lowering-tan.f64 N/A

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

      16. tan-lowering-tan.f64 98.5

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

   9. Applied egg-rr 98.5%

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

4. Final simplification 86.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\tan a \leq -0.1:\\ \;\;\;\;x + \mathsf{fma}\left(1, \tan y + \tan z, -\tan a\right)\\ \mathbf{elif}\;\tan a \leq 2 \cdot 10^{-37}:\\ \;\;\;\;x + \frac{\tan y + \tan z}{1 - \tan y \cdot \tan z}\\ \mathbf{else}:\\ \;\;\;\;x + \mathsf{fma}\left(1, \tan y + \tan z, -\tan a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.7% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 77.0%

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

   1. tan-sum N/A

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

   2. /-lowering-/.f64 N/A

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

   3. +-lowering-+.f64 N/A

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

   4. tan-lowering-tan.f64 N/A

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

   5. tan-lowering-tan.f64 N/A

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

   6. --lowering--.f64 N/A

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

   7. *-lowering-*.f64 N/A

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

   8. tan-lowering-tan.f64 N/A

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

   9. tan-lowering-tan.f64 99.7

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

4. Applied egg-rr 99.7%

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

   1. sub-neg N/A

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

   2. +-commutative N/A

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

   3. *-commutative N/A

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

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

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

   5. accelerator-lowering-fma.f64 N/A

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

   6. tan-lowering-tan.f64 N/A

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

   7. neg-lowering-neg.f64 N/A

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

   8. tan-lowering-tan.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 egg-rr 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. Final simplification 99.7%

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

Alternative 4: 99.7% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: a
    code = x + (((tan(y) + tan(z)) / (1.0d0 - (tan(y) * tan(z)))) - tan(a))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.0%

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

   1. tan-sum N/A

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

   2. /-lowering-/.f64 N/A

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

   3. +-lowering-+.f64 N/A

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

   4. tan-lowering-tan.f64 N/A

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

   5. tan-lowering-tan.f64 N/A

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

   6. --lowering--.f64 N/A

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

   7. *-lowering-*.f64 N/A

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

   8. tan-lowering-tan.f64 N/A

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

   9. tan-lowering-tan.f64 99.7

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

4. Applied egg-rr 99.7%

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

Alternative 5: 89.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan y + \tan z\\ t_1 := x + \mathsf{fma}\left(1, t\_0, -\tan a\right)\\ \mathbf{if}\;a \leq -0.075:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 0.47:\\ \;\;\;\;x + \left(\frac{t\_0}{1 - \tan y \cdot \tan z} - \mathsf{fma}\left(\mathsf{fma}\left(a \cdot a, \mathsf{fma}\left(a, a \cdot 0.05396825396825397, 0.13333333333333333\right), 0.3333333333333333\right), a \cdot \left(a \cdot a\right), a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (let* ((t_0 (+ (tan y) (tan z))) (t_1 (+ x (fma 1.0 t_0 (- (tan a))))))
   (if (<= a -0.075)
     t_1
     (if (<= a 0.47)
       (+
        x
        (-
         (/ t_0 (- 1.0 (* (tan y) (tan z))))
         (fma
          (fma
           (* a a)
           (fma a (* a 0.05396825396825397) 0.13333333333333333)
           0.3333333333333333)
          (* a (* a a))
          a)))
       t_1))))

C

double code(double x, double y, double z, double a) {
	double t_0 = tan(y) + tan(z);
	double t_1 = x + fma(1.0, t_0, -tan(a));
	double tmp;
	if (a <= -0.075) {
		tmp = t_1;
	} else if (a <= 0.47) {
		tmp = x + ((t_0 / (1.0 - (tan(y) * tan(z)))) - fma(fma((a * a), fma(a, (a * 0.05396825396825397), 0.13333333333333333), 0.3333333333333333), (a * (a * a)), a));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, a)
	t_0 = Float64(tan(y) + tan(z))
	t_1 = Float64(x + fma(1.0, t_0, Float64(-tan(a))))
	tmp = 0.0
	if (a <= -0.075)
		tmp = t_1;
	elseif (a <= 0.47)
		tmp = Float64(x + Float64(Float64(t_0 / Float64(1.0 - Float64(tan(y) * tan(z)))) - fma(fma(Float64(a * a), fma(a, Float64(a * 0.05396825396825397), 0.13333333333333333), 0.3333333333333333), Float64(a * Float64(a * a)), a)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, a_] := Block[{t$95$0 = N[(N[Tan[y], $MachinePrecision] + N[Tan[z], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x + N[(1.0 * t$95$0 + (-N[Tan[a], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -0.075], t$95$1, If[LessEqual[a, 0.47], N[(x + N[(N[(t$95$0 / N[(1.0 - N[(N[Tan[y], $MachinePrecision] * N[Tan[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(a * a), $MachinePrecision] * N[(a * N[(a * 0.05396825396825397), $MachinePrecision] + 0.13333333333333333), $MachinePrecision] + 0.3333333333333333), $MachinePrecision] * N[(a * N[(a * a), $MachinePrecision]), $MachinePrecision] + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if a < -0.0749999999999999972 or 0.46999999999999997 < a

   1. Initial program 75.2%

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

      1. sub-neg N/A

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

      2. tan-sum N/A

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

      3. clear-num N/A

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

      4. associate-/r/ N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. /-lowering-/.f64 N/A

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

      7. --lowering--.f64 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. tan-lowering-tan.f64 N/A

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

      10. tan-lowering-tan.f64 N/A

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

      11. +-lowering-+.f64 N/A

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

      12. tan-lowering-tan.f64 N/A

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

      13. tan-lowering-tan.f64 N/A

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

      14. neg-lowering-neg.f64 N/A

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

      15. tan-lowering-tan.f64 99.6

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

   4. Applied egg-rr 99.6%

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

   5. Taylor expanded in y around 0

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

   7. Simplified 76.3%

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

   if -0.0749999999999999972 < a < 0.46999999999999997

   1. Initial program 79.1%

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

      1. tan-sum N/A

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

      2. /-lowering-/.f64 N/A

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

      3. +-lowering-+.f64 N/A

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

      4. tan-lowering-tan.f64 N/A

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

      5. tan-lowering-tan.f64 N/A

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

      6. --lowering--.f64 N/A

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

      7. *-lowering-*.f64 N/A

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

      8. tan-lowering-tan.f64 N/A

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

      9. tan-lowering-tan.f64 99.9

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in a around 0

      \[\leadsto x + \left(\frac{\tan y + \tan z}{1 - \tan y \cdot \tan z} - \color{blue}{a \cdot \left(1 + {a}^{2} \cdot \left(\frac{1}{3} + {a}^{2} \cdot \left(\frac{2}{15} + \frac{17}{315} \cdot {a}^{2}\right)\right)\right)}\right) \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x + \left(\frac{\tan y + \tan z}{1 - \tan y \cdot \tan z} - a \cdot \color{blue}{\left({a}^{2} \cdot \left(\frac{1}{3} + {a}^{2} \cdot \left(\frac{2}{15} + \frac{17}{315} \cdot {a}^{2}\right)\right) + 1\right)}\right) \]

      2. distribute-rgt-in N/A

         \[\leadsto x + \left(\frac{\tan y + \tan z}{1 - \tan y \cdot \tan z} - \color{blue}{\left(\left({a}^{2} \cdot \left(\frac{1}{3} + {a}^{2} \cdot \left(\frac{2}{15} + \frac{17}{315} \cdot {a}^{2}\right)\right)\right) \cdot a + 1 \cdot a\right)}\right) \]

      3. *-lft-identity N/A

         \[\leadsto x + \left(\frac{\tan y + \tan z}{1 - \tan y \cdot \tan z} - \left(\left({a}^{2} \cdot \left(\frac{1}{3} + {a}^{2} \cdot \left(\frac{2}{15} + \frac{17}{315} \cdot {a}^{2}\right)\right)\right) \cdot a + \color{blue}{a}\right)\right) \]

      4. *-commutative N/A

         \[\leadsto x + \left(\frac{\tan y + \tan z}{1 - \tan y \cdot \tan z} - \left(\color{blue}{\left(\left(\frac{1}{3} + {a}^{2} \cdot \left(\frac{2}{15} + \frac{17}{315} \cdot {a}^{2}\right)\right) \cdot {a}^{2}\right)} \cdot a + a\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto x + \left(\frac{\tan y + \tan z}{1 - \tan y \cdot \tan z} - \left(\color{blue}{\left(\frac{1}{3} + {a}^{2} \cdot \left(\frac{2}{15} + \frac{17}{315} \cdot {a}^{2}\right)\right) \cdot \left({a}^{2} \cdot a\right)} + a\right)\right) \]

      6. pow-plus N/A

         \[\leadsto x + \left(\frac{\tan y + \tan z}{1 - \tan y \cdot \tan z} - \left(\left(\frac{1}{3} + {a}^{2} \cdot \left(\frac{2}{15} + \frac{17}{315} \cdot {a}^{2}\right)\right) \cdot \color{blue}{{a}^{\left(2 + 1\right)}} + a\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto x + \left(\frac{\tan y + \tan z}{1 - \tan y \cdot \tan z} - \left(\left(\frac{1}{3} + {a}^{2} \cdot \left(\frac{2}{15} + \frac{17}{315} \cdot {a}^{2}\right)\right) \cdot {a}^{\color{blue}{3}} + a\right)\right) \]

      8. cube-unmult N/A

         \[\leadsto x + \left(\frac{\tan y + \tan z}{1 - \tan y \cdot \tan z} - \left(\left(\frac{1}{3} + {a}^{2} \cdot \left(\frac{2}{15} + \frac{17}{315} \cdot {a}^{2}\right)\right) \cdot \color{blue}{\left(a \cdot \left(a \cdot a\right)\right)} + a\right)\right) \]

      9. unpow2 N/A

         \[\leadsto x + \left(\frac{\tan y + \tan z}{1 - \tan y \cdot \tan z} - \left(\left(\frac{1}{3} + {a}^{2} \cdot \left(\frac{2}{15} + \frac{17}{315} \cdot {a}^{2}\right)\right) \cdot \left(a \cdot \color{blue}{{a}^{2}}\right) + a\right)\right) \]

      10. accelerator-lowering-fma.f64 N/A

          \[\leadsto x + \left(\frac{\tan y + \tan z}{1 - \tan y \cdot \tan z} - \color{blue}{\mathsf{fma}\left(\frac{1}{3} + {a}^{2} \cdot \left(\frac{2}{15} + \frac{17}{315} \cdot {a}^{2}\right), a \cdot {a}^{2}, a\right)}\right) \]

   7. Simplified 99.5%

      \[\leadsto x + \left(\frac{\tan y + \tan z}{1 - \tan y \cdot \tan z} - \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(a \cdot a, \mathsf{fma}\left(a, a \cdot 0.05396825396825397, 0.13333333333333333\right), 0.3333333333333333\right), a \cdot \left(a \cdot a\right), a\right)}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 89.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan y + \tan z\\ t_1 := x + \mathsf{fma}\left(1, t\_0, -\tan a\right)\\ \mathbf{if}\;a \leq -0.048:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 0.47:\\ \;\;\;\;x + \left(\frac{t\_0}{1 - \tan y \cdot \tan z} - \mathsf{fma}\left(\mathsf{fma}\left(a \cdot a, 0.13333333333333333, 0.3333333333333333\right), a \cdot \left(a \cdot a\right), a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (let* ((t_0 (+ (tan y) (tan z))) (t_1 (+ x (fma 1.0 t_0 (- (tan a))))))
   (if (<= a -0.048)
     t_1
     (if (<= a 0.47)
       (+
        x
        (-
         (/ t_0 (- 1.0 (* (tan y) (tan z))))
         (fma
          (fma (* a a) 0.13333333333333333 0.3333333333333333)
          (* a (* a a))
          a)))
       t_1))))

C

double code(double x, double y, double z, double a) {
	double t_0 = tan(y) + tan(z);
	double t_1 = x + fma(1.0, t_0, -tan(a));
	double tmp;
	if (a <= -0.048) {
		tmp = t_1;
	} else if (a <= 0.47) {
		tmp = x + ((t_0 / (1.0 - (tan(y) * tan(z)))) - fma(fma((a * a), 0.13333333333333333, 0.3333333333333333), (a * (a * a)), a));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, a)
	t_0 = Float64(tan(y) + tan(z))
	t_1 = Float64(x + fma(1.0, t_0, Float64(-tan(a))))
	tmp = 0.0
	if (a <= -0.048)
		tmp = t_1;
	elseif (a <= 0.47)
		tmp = Float64(x + Float64(Float64(t_0 / Float64(1.0 - Float64(tan(y) * tan(z)))) - fma(fma(Float64(a * a), 0.13333333333333333, 0.3333333333333333), Float64(a * Float64(a * a)), a)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, a_] := Block[{t$95$0 = N[(N[Tan[y], $MachinePrecision] + N[Tan[z], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x + N[(1.0 * t$95$0 + (-N[Tan[a], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -0.048], t$95$1, If[LessEqual[a, 0.47], N[(x + N[(N[(t$95$0 / N[(1.0 - N[(N[Tan[y], $MachinePrecision] * N[Tan[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(a * a), $MachinePrecision] * 0.13333333333333333 + 0.3333333333333333), $MachinePrecision] * N[(a * N[(a * a), $MachinePrecision]), $MachinePrecision] + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if a < -0.048000000000000001 or 0.46999999999999997 < a

   1. Initial program 75.2%

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

      1. sub-neg N/A

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

      2. tan-sum N/A

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

      3. clear-num N/A

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

      4. associate-/r/ N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. /-lowering-/.f64 N/A

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

      7. --lowering--.f64 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. tan-lowering-tan.f64 N/A

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

      10. tan-lowering-tan.f64 N/A

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

      11. +-lowering-+.f64 N/A

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

      12. tan-lowering-tan.f64 N/A

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

      13. tan-lowering-tan.f64 N/A

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

      14. neg-lowering-neg.f64 N/A

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

      15. tan-lowering-tan.f64 99.6

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

   4. Applied egg-rr 99.6%

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

   5. Taylor expanded in y around 0

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

   7. Simplified 76.3%

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

   if -0.048000000000000001 < a < 0.46999999999999997

   1. Initial program 79.1%

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

      1. tan-sum N/A

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

      2. /-lowering-/.f64 N/A

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

      3. +-lowering-+.f64 N/A

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

      4. tan-lowering-tan.f64 N/A

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

      5. tan-lowering-tan.f64 N/A

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

      6. --lowering--.f64 N/A

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

      7. *-lowering-*.f64 N/A

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

      8. tan-lowering-tan.f64 N/A

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

      9. tan-lowering-tan.f64 99.9

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. pow-plus N/A

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

      6. metadata-eval N/A

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

      7. cube-unmult N/A

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

      8. unpow2 N/A

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

      9. *-lft-identity N/A

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

      10. accelerator-lowering-fma.f64 N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. accelerator-lowering-fma.f64 N/A

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

      14. unpow2 N/A

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

      15. *-lowering-*.f64 N/A

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

      16. *-lowering-*.f64 N/A

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

      17. unpow2 N/A

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

      18. *-lowering-*.f64 99.4

          \[\leadsto x + \left(\frac{\tan y + \tan z}{1 - \tan y \cdot \tan z} - \mathsf{fma}\left(\mathsf{fma}\left(a \cdot a, 0.13333333333333333, 0.3333333333333333\right), a \cdot \color{blue}{\left(a \cdot a\right)}, a\right)\right) \]

   7. Simplified 99.4%

      \[\leadsto x + \left(\frac{\tan y + \tan z}{1 - \tan y \cdot \tan z} - \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(a \cdot a, 0.13333333333333333, 0.3333333333333333\right), a \cdot \left(a \cdot a\right), a\right)}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 89.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan y + \tan z\\ t_1 := x + \mathsf{fma}\left(1, t\_0, -\tan a\right)\\ \mathbf{if}\;a \leq -0.026:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 0.47:\\ \;\;\;\;x + \left(\frac{t\_0}{1 - \tan y \cdot \tan z} - \mathsf{fma}\left(0.3333333333333333, a \cdot \left(a \cdot a\right), a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (let* ((t_0 (+ (tan y) (tan z))) (t_1 (+ x (fma 1.0 t_0 (- (tan a))))))
   (if (<= a -0.026)
     t_1
     (if (<= a 0.47)
       (+
        x
        (-
         (/ t_0 (- 1.0 (* (tan y) (tan z))))
         (fma 0.3333333333333333 (* a (* a a)) a)))
       t_1))))

C

double code(double x, double y, double z, double a) {
	double t_0 = tan(y) + tan(z);
	double t_1 = x + fma(1.0, t_0, -tan(a));
	double tmp;
	if (a <= -0.026) {
		tmp = t_1;
	} else if (a <= 0.47) {
		tmp = x + ((t_0 / (1.0 - (tan(y) * tan(z)))) - fma(0.3333333333333333, (a * (a * a)), a));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, a)
	t_0 = Float64(tan(y) + tan(z))
	t_1 = Float64(x + fma(1.0, t_0, Float64(-tan(a))))
	tmp = 0.0
	if (a <= -0.026)
		tmp = t_1;
	elseif (a <= 0.47)
		tmp = Float64(x + Float64(Float64(t_0 / Float64(1.0 - Float64(tan(y) * tan(z)))) - fma(0.3333333333333333, Float64(a * Float64(a * a)), a)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, a_] := Block[{t$95$0 = N[(N[Tan[y], $MachinePrecision] + N[Tan[z], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x + N[(1.0 * t$95$0 + (-N[Tan[a], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -0.026], t$95$1, If[LessEqual[a, 0.47], N[(x + N[(N[(t$95$0 / N[(1.0 - N[(N[Tan[y], $MachinePrecision] * N[Tan[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(0.3333333333333333 * N[(a * N[(a * a), $MachinePrecision]), $MachinePrecision] + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if a < -0.0259999999999999988 or 0.46999999999999997 < a

   1. Initial program 75.2%

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

      1. sub-neg N/A

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

      2. tan-sum N/A

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

      3. clear-num N/A

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

      4. associate-/r/ N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. /-lowering-/.f64 N/A

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

      7. --lowering--.f64 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. tan-lowering-tan.f64 N/A

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

      10. tan-lowering-tan.f64 N/A

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

      11. +-lowering-+.f64 N/A

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

      12. tan-lowering-tan.f64 N/A

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

      13. tan-lowering-tan.f64 N/A

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

      14. neg-lowering-neg.f64 N/A

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

      15. tan-lowering-tan.f64 99.6

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

   4. Applied egg-rr 99.6%

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

   5. Taylor expanded in y around 0

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

   7. Simplified 76.3%

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

   if -0.0259999999999999988 < a < 0.46999999999999997

   1. Initial program 79.1%

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

      1. tan-sum N/A

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

      2. /-lowering-/.f64 N/A

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

      3. +-lowering-+.f64 N/A

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

      4. tan-lowering-tan.f64 N/A

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

      5. tan-lowering-tan.f64 N/A

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

      6. --lowering--.f64 N/A

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

      7. *-lowering-*.f64 N/A

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

      8. tan-lowering-tan.f64 N/A

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

      9. tan-lowering-tan.f64 99.9

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. associate-*l* N/A

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

      4. pow-plus N/A

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

      5. metadata-eval N/A

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

      6. cube-unmult N/A

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

      7. unpow2 N/A

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

      8. *-lft-identity N/A

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

      9. accelerator-lowering-fma.f64 N/A

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

      10. *-lowering-*.f64 N/A

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

      11. unpow2 N/A

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

      12. *-lowering-*.f64 99.3

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

   7. Simplified 99.3%

      \[\leadsto x + \left(\frac{\tan y + \tan z}{1 - \tan y \cdot \tan z} - \color{blue}{\mathsf{fma}\left(0.3333333333333333, a \cdot \left(a \cdot a\right), a\right)}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 89.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan y + \tan z\\ t_1 := x + \mathsf{fma}\left(1, t\_0, -\tan a\right)\\ \mathbf{if}\;a \leq -0.0051:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 2.6 \cdot 10^{-34}:\\ \;\;\;\;x + \mathsf{fma}\left(\frac{-1}{-1 + \tan y \cdot \tan z}, t\_0, -a\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (let* ((t_0 (+ (tan y) (tan z))) (t_1 (+ x (fma 1.0 t_0 (- (tan a))))))
   (if (<= a -0.0051)
     t_1
     (if (<= a 2.6e-34)
       (+ x (fma (/ -1.0 (+ -1.0 (* (tan y) (tan z)))) t_0 (- a)))
       t_1))))

C

double code(double x, double y, double z, double a) {
	double t_0 = tan(y) + tan(z);
	double t_1 = x + fma(1.0, t_0, -tan(a));
	double tmp;
	if (a <= -0.0051) {
		tmp = t_1;
	} else if (a <= 2.6e-34) {
		tmp = x + fma((-1.0 / (-1.0 + (tan(y) * tan(z)))), t_0, -a);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, a)
	t_0 = Float64(tan(y) + tan(z))
	t_1 = Float64(x + fma(1.0, t_0, Float64(-tan(a))))
	tmp = 0.0
	if (a <= -0.0051)
		tmp = t_1;
	elseif (a <= 2.6e-34)
		tmp = Float64(x + fma(Float64(-1.0 / Float64(-1.0 + Float64(tan(y) * tan(z)))), t_0, Float64(-a)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, a_] := Block[{t$95$0 = N[(N[Tan[y], $MachinePrecision] + N[Tan[z], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x + N[(1.0 * t$95$0 + (-N[Tan[a], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -0.0051], t$95$1, If[LessEqual[a, 2.6e-34], N[(x + N[(N[(-1.0 / N[(-1.0 + N[(N[Tan[y], $MachinePrecision] * N[Tan[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0 + (-a)), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if a < -0.0051000000000000004 or 2.5999999999999999e-34 < a

   1. Initial program 76.0%

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

      1. sub-neg N/A

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

      2. tan-sum N/A

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

      3. clear-num N/A

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

      4. associate-/r/ N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. /-lowering-/.f64 N/A

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

      7. --lowering--.f64 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. tan-lowering-tan.f64 N/A

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

      10. tan-lowering-tan.f64 N/A

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

      11. +-lowering-+.f64 N/A

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

      12. tan-lowering-tan.f64 N/A

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

      13. tan-lowering-tan.f64 N/A

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

      14. neg-lowering-neg.f64 N/A

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

      15. tan-lowering-tan.f64 99.6

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

   4. Applied egg-rr 99.6%

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

   5. Taylor expanded in y around 0

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

   7. Simplified 77.1%

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

   if -0.0051000000000000004 < a < 2.5999999999999999e-34

   1. Initial program 78.3%

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

      1. sub-neg N/A

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

      2. tan-sum N/A

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

      3. clear-num N/A

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

      4. associate-/r/ N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. /-lowering-/.f64 N/A

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

      7. --lowering--.f64 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. tan-lowering-tan.f64 N/A

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

      10. tan-lowering-tan.f64 N/A

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

      11. +-lowering-+.f64 N/A

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

      12. tan-lowering-tan.f64 N/A

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

      13. tan-lowering-tan.f64 N/A

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

      14. neg-lowering-neg.f64 N/A

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

      15. tan-lowering-tan.f64 99.9

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in a around 0

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

   7. Simplified 99.8%

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

4. Final simplification 86.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -0.0051:\\ \;\;\;\;x + \mathsf{fma}\left(1, \tan y + \tan z, -\tan a\right)\\ \mathbf{elif}\;a \leq 2.6 \cdot 10^{-34}:\\ \;\;\;\;x + \mathsf{fma}\left(\frac{-1}{-1 + \tan y \cdot \tan z}, \tan y + \tan z, -a\right)\\ \mathbf{else}:\\ \;\;\;\;x + \mathsf{fma}\left(1, \tan y + \tan z, -\tan a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 89.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan y + \tan z\\ t_1 := x + \mathsf{fma}\left(1, t\_0, -\tan a\right)\\ \mathbf{if}\;a \leq -0.00033:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 2.6 \cdot 10^{-34}:\\ \;\;\;\;x + \left(\frac{t\_0}{1 - \tan y \cdot \tan z} - a\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (let* ((t_0 (+ (tan y) (tan z))) (t_1 (+ x (fma 1.0 t_0 (- (tan a))))))
   (if (<= a -0.00033)
     t_1
     (if (<= a 2.6e-34) (+ x (- (/ t_0 (- 1.0 (* (tan y) (tan z)))) a)) t_1))))

C

double code(double x, double y, double z, double a) {
	double t_0 = tan(y) + tan(z);
	double t_1 = x + fma(1.0, t_0, -tan(a));
	double tmp;
	if (a <= -0.00033) {
		tmp = t_1;
	} else if (a <= 2.6e-34) {
		tmp = x + ((t_0 / (1.0 - (tan(y) * tan(z)))) - a);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, a)
	t_0 = Float64(tan(y) + tan(z))
	t_1 = Float64(x + fma(1.0, t_0, Float64(-tan(a))))
	tmp = 0.0
	if (a <= -0.00033)
		tmp = t_1;
	elseif (a <= 2.6e-34)
		tmp = Float64(x + Float64(Float64(t_0 / Float64(1.0 - Float64(tan(y) * tan(z)))) - a));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, a_] := Block[{t$95$0 = N[(N[Tan[y], $MachinePrecision] + N[Tan[z], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x + N[(1.0 * t$95$0 + (-N[Tan[a], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -0.00033], t$95$1, If[LessEqual[a, 2.6e-34], N[(x + N[(N[(t$95$0 / N[(1.0 - N[(N[Tan[y], $MachinePrecision] * N[Tan[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if a < -3.3e-4 or 2.5999999999999999e-34 < a

   1. Initial program 76.0%

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

      1. sub-neg N/A

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

      2. tan-sum N/A

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

      3. clear-num N/A

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

      4. associate-/r/ N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. /-lowering-/.f64 N/A

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

      7. --lowering--.f64 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. tan-lowering-tan.f64 N/A

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

      10. tan-lowering-tan.f64 N/A

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

      11. +-lowering-+.f64 N/A

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

      12. tan-lowering-tan.f64 N/A

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

      13. tan-lowering-tan.f64 N/A

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

      14. neg-lowering-neg.f64 N/A

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

      15. tan-lowering-tan.f64 99.6

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

   4. Applied egg-rr 99.6%

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

   5. Taylor expanded in y around 0

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

   7. Simplified 77.1%

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

   if -3.3e-4 < a < 2.5999999999999999e-34

   1. Initial program 78.3%

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

      1. tan-sum N/A

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

      2. /-lowering-/.f64 N/A

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

      3. +-lowering-+.f64 N/A

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

      4. tan-lowering-tan.f64 N/A

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

      5. tan-lowering-tan.f64 N/A

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

      6. --lowering--.f64 N/A

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

      7. *-lowering-*.f64 N/A

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

      8. tan-lowering-tan.f64 N/A

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

      9. tan-lowering-tan.f64 99.9

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in a around 0

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

   7. Simplified 99.8%

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

Alternative 10: 79.9% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 77.0%

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

   1. sub-neg N/A

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

   2. tan-sum N/A

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

   3. clear-num N/A

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

   4. associate-/r/ N/A

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

   5. accelerator-lowering-fma.f64 N/A

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

   6. /-lowering-/.f64 N/A

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

   7. --lowering--.f64 N/A

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

   8. *-lowering-*.f64 N/A

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

   9. tan-lowering-tan.f64 N/A

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

   10. tan-lowering-tan.f64 N/A

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

   11. +-lowering-+.f64 N/A

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

   12. tan-lowering-tan.f64 N/A

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

   13. tan-lowering-tan.f64 N/A

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

   14. neg-lowering-neg.f64 N/A

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

   15. tan-lowering-tan.f64 99.8

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

4. Applied egg-rr 99.8%

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

5. Taylor expanded in y around 0

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

7. Simplified 78.0%

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

Alternative 11: 70.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 4.2 \cdot 10^{-14}:\\ \;\;\;\;x + \left(\tan y - \tan a\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(\tan z - \tan a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (if (<= z 4.2e-14) (+ x (- (tan y) (tan a))) (+ x (- (tan z) (tan a)))))

C

double code(double x, double y, double z, double a) {
	double tmp;
	if (z <= 4.2e-14) {
		tmp = x + (tan(y) - tan(a));
	} else {
		tmp = x + (tan(z) - tan(a));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z <= 4.2d-14) then
        tmp = x + (tan(y) - tan(a))
    else
        tmp = x + (tan(z) - tan(a))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double a) {
	double tmp;
	if (z <= 4.2e-14) {
		tmp = x + (Math.tan(y) - Math.tan(a));
	} else {
		tmp = x + (Math.tan(z) - Math.tan(a));
	}
	return tmp;
}

Python

def code(x, y, z, a):
	tmp = 0
	if z <= 4.2e-14:
		tmp = x + (math.tan(y) - math.tan(a))
	else:
		tmp = x + (math.tan(z) - math.tan(a))
	return tmp

Julia

function code(x, y, z, a)
	tmp = 0.0
	if (z <= 4.2e-14)
		tmp = Float64(x + Float64(tan(y) - tan(a)));
	else
		tmp = Float64(x + Float64(tan(z) - tan(a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, a)
	tmp = 0.0;
	if (z <= 4.2e-14)
		tmp = x + (tan(y) - tan(a));
	else
		tmp = x + (tan(z) - tan(a));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, a_] := If[LessEqual[z, 4.2e-14], N[(x + N[(N[Tan[y], $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[Tan[z], $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 4.1999999999999998e-14

   1. Initial program 85.9%

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

   3. Taylor expanded in y around inf

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

   5. Simplified 74.0%

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

   if 4.1999999999999998e-14 < z

   1. Initial program 50.9%

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

   3. Taylor expanded in y around 0

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

   5. Simplified 50.9%

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

Alternative 12: 65.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 0.001:\\ \;\;\;\;x + \left(\tan y - \tan a\right)\\ \mathbf{else}:\\ \;\;\;\;x + \tan z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (if (<= z 0.001) (+ x (- (tan y) (tan a))) (+ x (tan z))))

C

double code(double x, double y, double z, double a) {
	double tmp;
	if (z <= 0.001) {
		tmp = x + (tan(y) - tan(a));
	} else {
		tmp = x + tan(z);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z <= 0.001d0) then
        tmp = x + (tan(y) - tan(a))
    else
        tmp = x + tan(z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double a) {
	double tmp;
	if (z <= 0.001) {
		tmp = x + (Math.tan(y) - Math.tan(a));
	} else {
		tmp = x + Math.tan(z);
	}
	return tmp;
}

Python

def code(x, y, z, a):
	tmp = 0
	if z <= 0.001:
		tmp = x + (math.tan(y) - math.tan(a))
	else:
		tmp = x + math.tan(z)
	return tmp

Julia

function code(x, y, z, a)
	tmp = 0.0
	if (z <= 0.001)
		tmp = Float64(x + Float64(tan(y) - tan(a)));
	else
		tmp = Float64(x + tan(z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, a)
	tmp = 0.0;
	if (z <= 0.001)
		tmp = x + (tan(y) - tan(a));
	else
		tmp = x + tan(z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, a_] := If[LessEqual[z, 0.001], N[(x + N[(N[Tan[y], $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[Tan[z], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 1e-3

   1. Initial program 85.9%

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

   3. Taylor expanded in y around inf

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

   5. Simplified 73.8%

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

   if 1e-3 < z

   1. Initial program 49.9%

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

      1. +-commutative N/A

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

      2. associate-+l- N/A

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

      3. --lowering--.f64 N/A

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

      4. tan-lowering-tan.f64 N/A

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

      5. +-lowering-+.f64 N/A

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

      6. --lowering--.f64 N/A

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

      7. tan-lowering-tan.f64 49.8

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

   4. Applied egg-rr 49.8%

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

   5. Taylor expanded in a around 0

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

      1. mul-1-neg N/A

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

      2. neg-lowering-neg.f64 34.3

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

   7. Simplified 34.3%

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

   8. Taylor expanded in y around 0

      \[\leadsto \tan \color{blue}{z} - \left(\mathsf{neg}\left(x\right)\right) \]
   9. Step-by-step derivation

   10. Simplified 34.8%

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

4. Final simplification 64.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 0.001:\\ \;\;\;\;x + \left(\tan y - \tan a\right)\\ \mathbf{else}:\\ \;\;\;\;x + \tan z\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 79.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.0%

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

Alternative 14: 50.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y + z \leq -50:\\ \;\;\;\;x + \tan y\\ \mathbf{elif}\;y + z \leq 5 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(y, \left(y \cdot y\right) \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.05396825396825397, 0.13333333333333333\right), 0.3333333333333333\right), y\right) + \left(x - \tan a\right)\\ \mathbf{else}:\\ \;\;\;\;x + \tan \left(\mathsf{fma}\left(z, \frac{y}{z}, z\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (if (<= (+ y z) -50.0)
   (+ x (tan y))
   (if (<= (+ y z) 5e-5)
     (+
      (fma
       y
       (*
        (* y y)
        (fma
         (* y y)
         (fma (* y y) 0.05396825396825397 0.13333333333333333)
         0.3333333333333333))
       y)
      (- x (tan a)))
     (+ x (tan (fma z (/ y z) z))))))

C

double code(double x, double y, double z, double a) {
	double tmp;
	if ((y + z) <= -50.0) {
		tmp = x + tan(y);
	} else if ((y + z) <= 5e-5) {
		tmp = fma(y, ((y * y) * fma((y * y), fma((y * y), 0.05396825396825397, 0.13333333333333333), 0.3333333333333333)), y) + (x - tan(a));
	} else {
		tmp = x + tan(fma(z, (y / z), z));
	}
	return tmp;
}

Julia

function code(x, y, z, a)
	tmp = 0.0
	if (Float64(y + z) <= -50.0)
		tmp = Float64(x + tan(y));
	elseif (Float64(y + z) <= 5e-5)
		tmp = Float64(fma(y, Float64(Float64(y * y) * fma(Float64(y * y), fma(Float64(y * y), 0.05396825396825397, 0.13333333333333333), 0.3333333333333333)), y) + Float64(x - tan(a)));
	else
		tmp = Float64(x + tan(fma(z, Float64(y / z), z)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, a_] := If[LessEqual[N[(y + z), $MachinePrecision], -50.0], N[(x + N[Tan[y], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(y + z), $MachinePrecision], 5e-5], N[(N[(y * N[(N[(y * y), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * 0.05396825396825397 + 0.13333333333333333), $MachinePrecision] + 0.3333333333333333), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision] + N[(x - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[Tan[N[(z * N[(y / z), $MachinePrecision] + z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 y z) < -50

   1. Initial program 69.4%

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

      1. +-commutative N/A

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

      2. associate-+l- N/A

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

      3. --lowering--.f64 N/A

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

      4. tan-lowering-tan.f64 N/A

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

      5. +-lowering-+.f64 N/A

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

      6. --lowering--.f64 N/A

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

      7. tan-lowering-tan.f64 69.4

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

   4. Applied egg-rr 69.4%

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

   5. Taylor expanded in a around 0

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

      1. mul-1-neg N/A

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

      2. neg-lowering-neg.f64 42.6

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

   7. Simplified 42.6%

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

   8. Taylor expanded in y around inf

      \[\leadsto \tan \color{blue}{y} - \left(\mathsf{neg}\left(x\right)\right) \]
   9. Step-by-step derivation

   10. Simplified 30.5%

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

       1. sub-neg N/A

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

       2. remove-double-neg N/A

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

       3. +-lowering-+.f64 N/A

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

       4. tan-lowering-tan.f64 30.5

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

   12. Applied egg-rr 30.5%

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

   if -50 < (+.f64 y z) < 5.00000000000000024e-5

   1. Initial program 100.0%

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

      1. +-commutative N/A

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

      2. associate-+l- N/A

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

      3. --lowering--.f64 N/A

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

      4. tan-lowering-tan.f64 N/A

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

      5. +-lowering-+.f64 N/A

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

      6. --lowering--.f64 N/A

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

      7. tan-lowering-tan.f64 99.9

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in y around inf

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

   7. Simplified 97.8%

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

   8. Taylor expanded in y around 0

      \[\leadsto \color{blue}{y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{3} + {y}^{2} \cdot \left(\frac{2}{15} + \frac{17}{315} \cdot {y}^{2}\right)\right)\right)} - \left(\tan a - x\right) \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto y \cdot \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{3} + {y}^{2} \cdot \left(\frac{2}{15} + \frac{17}{315} \cdot {y}^{2}\right)\right) + 1\right)} - \left(\tan a - x\right) \]

      2. distribute-lft-in N/A

         \[\leadsto \color{blue}{\left(y \cdot \left({y}^{2} \cdot \left(\frac{1}{3} + {y}^{2} \cdot \left(\frac{2}{15} + \frac{17}{315} \cdot {y}^{2}\right)\right)\right) + y \cdot 1\right)} - \left(\tan a - x\right) \]

      3. *-rgt-identity N/A

         \[\leadsto \left(y \cdot \left({y}^{2} \cdot \left(\frac{1}{3} + {y}^{2} \cdot \left(\frac{2}{15} + \frac{17}{315} \cdot {y}^{2}\right)\right)\right) + \color{blue}{y}\right) - \left(\tan a - x\right) \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(y, {y}^{2} \cdot \left(\frac{1}{3} + {y}^{2} \cdot \left(\frac{2}{15} + \frac{17}{315} \cdot {y}^{2}\right)\right), y\right)} - \left(\tan a - x\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, \color{blue}{{y}^{2} \cdot \left(\frac{1}{3} + {y}^{2} \cdot \left(\frac{2}{15} + \frac{17}{315} \cdot {y}^{2}\right)\right)}, y\right) - \left(\tan a - x\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(y, \color{blue}{\left(y \cdot y\right)} \cdot \left(\frac{1}{3} + {y}^{2} \cdot \left(\frac{2}{15} + \frac{17}{315} \cdot {y}^{2}\right)\right), y\right) - \left(\tan a - x\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, \color{blue}{\left(y \cdot y\right)} \cdot \left(\frac{1}{3} + {y}^{2} \cdot \left(\frac{2}{15} + \frac{17}{315} \cdot {y}^{2}\right)\right), y\right) - \left(\tan a - x\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(y, \left(y \cdot y\right) \cdot \color{blue}{\left({y}^{2} \cdot \left(\frac{2}{15} + \frac{17}{315} \cdot {y}^{2}\right) + \frac{1}{3}\right)}, y\right) - \left(\tan a - x\right) \]

      9. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, \left(y \cdot y\right) \cdot \color{blue}{\mathsf{fma}\left({y}^{2}, \frac{2}{15} + \frac{17}{315} \cdot {y}^{2}, \frac{1}{3}\right)}, y\right) - \left(\tan a - x\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(y, \left(y \cdot y\right) \cdot \mathsf{fma}\left(\color{blue}{y \cdot y}, \frac{2}{15} + \frac{17}{315} \cdot {y}^{2}, \frac{1}{3}\right), y\right) - \left(\tan a - x\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(y, \left(y \cdot y\right) \cdot \mathsf{fma}\left(\color{blue}{y \cdot y}, \frac{2}{15} + \frac{17}{315} \cdot {y}^{2}, \frac{1}{3}\right), y\right) - \left(\tan a - x\right) \]

      12. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(y, \left(y \cdot y\right) \cdot \mathsf{fma}\left(y \cdot y, \color{blue}{\frac{17}{315} \cdot {y}^{2} + \frac{2}{15}}, \frac{1}{3}\right), y\right) - \left(\tan a - x\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(y, \left(y \cdot y\right) \cdot \mathsf{fma}\left(y \cdot y, \color{blue}{{y}^{2} \cdot \frac{17}{315}} + \frac{2}{15}, \frac{1}{3}\right), y\right) - \left(\tan a - x\right) \]

      14. accelerator-lowering-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(y, \left(y \cdot y\right) \cdot \mathsf{fma}\left(y \cdot y, \color{blue}{\mathsf{fma}\left({y}^{2}, \frac{17}{315}, \frac{2}{15}\right)}, \frac{1}{3}\right), y\right) - \left(\tan a - x\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(y, \left(y \cdot y\right) \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(\color{blue}{y \cdot y}, \frac{17}{315}, \frac{2}{15}\right), \frac{1}{3}\right), y\right) - \left(\tan a - x\right) \]

      16. *-lowering-*.f64 97.8

          \[\leadsto \mathsf{fma}\left(y, \left(y \cdot y\right) \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(\color{blue}{y \cdot y}, 0.05396825396825397, 0.13333333333333333\right), 0.3333333333333333\right), y\right) - \left(\tan a - x\right) \]

   10. Simplified 97.8%

       \[\leadsto \color{blue}{\mathsf{fma}\left(y, \left(y \cdot y\right) \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.05396825396825397, 0.13333333333333333\right), 0.3333333333333333\right), y\right)} - \left(\tan a - x\right) \]

   if 5.00000000000000024e-5 < (+.f64 y z)

   1. Initial program 67.3%

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

      1. +-commutative N/A

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

      2. associate-+l- N/A

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

      3. --lowering--.f64 N/A

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

      4. tan-lowering-tan.f64 N/A

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

      5. +-lowering-+.f64 N/A

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

      6. --lowering--.f64 N/A

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

      7. tan-lowering-tan.f64 67.2

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

   4. Applied egg-rr 67.2%

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

   5. Taylor expanded in a around 0

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

      1. mul-1-neg N/A

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

      2. neg-lowering-neg.f64 45.1

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

   7. Simplified 45.1%

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

   8. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. /-lowering-/.f64 29.1

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

   10. Simplified 29.1%

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

4. Final simplification 48.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y + z \leq -50:\\ \;\;\;\;x + \tan y\\ \mathbf{elif}\;y + z \leq 5 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(y, \left(y \cdot y\right) \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.05396825396825397, 0.13333333333333333\right), 0.3333333333333333\right), y\right) + \left(x - \tan a\right)\\ \mathbf{else}:\\ \;\;\;\;x + \tan \left(\mathsf{fma}\left(z, \frac{y}{z}, z\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 50.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y + z \leq -50:\\ \;\;\;\;x + \tan y\\ \mathbf{elif}\;y + z \leq 5 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(y, \left(y \cdot y\right) \cdot \mathsf{fma}\left(y \cdot y, 0.13333333333333333, 0.3333333333333333\right), y\right) + \left(x - \tan a\right)\\ \mathbf{else}:\\ \;\;\;\;x + \tan \left(\mathsf{fma}\left(z, \frac{y}{z}, z\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (if (<= (+ y z) -50.0)
   (+ x (tan y))
   (if (<= (+ y z) 5e-5)
     (+
      (fma
       y
       (* (* y y) (fma (* y y) 0.13333333333333333 0.3333333333333333))
       y)
      (- x (tan a)))
     (+ x (tan (fma z (/ y z) z))))))

C

double code(double x, double y, double z, double a) {
	double tmp;
	if ((y + z) <= -50.0) {
		tmp = x + tan(y);
	} else if ((y + z) <= 5e-5) {
		tmp = fma(y, ((y * y) * fma((y * y), 0.13333333333333333, 0.3333333333333333)), y) + (x - tan(a));
	} else {
		tmp = x + tan(fma(z, (y / z), z));
	}
	return tmp;
}

Julia

function code(x, y, z, a)
	tmp = 0.0
	if (Float64(y + z) <= -50.0)
		tmp = Float64(x + tan(y));
	elseif (Float64(y + z) <= 5e-5)
		tmp = Float64(fma(y, Float64(Float64(y * y) * fma(Float64(y * y), 0.13333333333333333, 0.3333333333333333)), y) + Float64(x - tan(a)));
	else
		tmp = Float64(x + tan(fma(z, Float64(y / z), z)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, a_] := If[LessEqual[N[(y + z), $MachinePrecision], -50.0], N[(x + N[Tan[y], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(y + z), $MachinePrecision], 5e-5], N[(N[(y * N[(N[(y * y), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * 0.13333333333333333 + 0.3333333333333333), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision] + N[(x - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[Tan[N[(z * N[(y / z), $MachinePrecision] + z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 y z) < -50

   1. Initial program 69.4%

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

      1. +-commutative N/A

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

      2. associate-+l- N/A

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

      3. --lowering--.f64 N/A

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

      4. tan-lowering-tan.f64 N/A

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

      5. +-lowering-+.f64 N/A

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

      6. --lowering--.f64 N/A

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

      7. tan-lowering-tan.f64 69.4

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

   4. Applied egg-rr 69.4%

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

   5. Taylor expanded in a around 0

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

      1. mul-1-neg N/A

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

      2. neg-lowering-neg.f64 42.6

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

   7. Simplified 42.6%

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

   8. Taylor expanded in y around inf

      \[\leadsto \tan \color{blue}{y} - \left(\mathsf{neg}\left(x\right)\right) \]
   9. Step-by-step derivation

   10. Simplified 30.5%

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

       1. sub-neg N/A

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

       2. remove-double-neg N/A

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

       3. +-lowering-+.f64 N/A

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

       4. tan-lowering-tan.f64 30.5

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

   12. Applied egg-rr 30.5%

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

   if -50 < (+.f64 y z) < 5.00000000000000024e-5

   1. Initial program 100.0%

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

      1. +-commutative N/A

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

      2. associate-+l- N/A

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

      3. --lowering--.f64 N/A

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

      4. tan-lowering-tan.f64 N/A

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

      5. +-lowering-+.f64 N/A

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

      6. --lowering--.f64 N/A

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

      7. tan-lowering-tan.f64 99.9

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in y around inf

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

   7. Simplified 97.8%

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

   8. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. unpow2 N/A

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

      7. *-lowering-*.f64 N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. accelerator-lowering-fma.f64 N/A

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

      11. unpow2 N/A

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

      12. *-lowering-*.f64 97.8

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

   10. Simplified 97.8%

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

   if 5.00000000000000024e-5 < (+.f64 y z)

   1. Initial program 67.3%

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

      1. +-commutative N/A

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

      2. associate-+l- N/A

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

      3. --lowering--.f64 N/A

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

      4. tan-lowering-tan.f64 N/A

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

      5. +-lowering-+.f64 N/A

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

      6. --lowering--.f64 N/A

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

      7. tan-lowering-tan.f64 67.2

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

   4. Applied egg-rr 67.2%

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

   5. Taylor expanded in a around 0

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

      1. mul-1-neg N/A

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

      2. neg-lowering-neg.f64 45.1

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

   7. Simplified 45.1%

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

   8. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. /-lowering-/.f64 29.1

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

   10. Simplified 29.1%

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

4. Final simplification 48.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y + z \leq -50:\\ \;\;\;\;x + \tan y\\ \mathbf{elif}\;y + z \leq 5 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(y, \left(y \cdot y\right) \cdot \mathsf{fma}\left(y \cdot y, 0.13333333333333333, 0.3333333333333333\right), y\right) + \left(x - \tan a\right)\\ \mathbf{else}:\\ \;\;\;\;x + \tan \left(\mathsf{fma}\left(z, \frac{y}{z}, z\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 50.8% accurate, 1.5× speedup

Math

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

FPCore

(FPCore (x y z a)
 :precision binary64
 (if (<= (+ y z) -0.002)
   (+ x (tan y))
   (if (<= (+ y z) 5e-5) (+ y (- x (tan a))) (+ x (tan (fma z (/ y z) z))))))

C

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

Julia

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

Wolfram

code[x_, y_, z_, a_] := If[LessEqual[N[(y + z), $MachinePrecision], -0.002], N[(x + N[Tan[y], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(y + z), $MachinePrecision], 5e-5], N[(y + N[(x - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[Tan[N[(z * N[(y / z), $MachinePrecision] + z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 y z) < -2e-3

   1. Initial program 70.1%

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

      1. +-commutative N/A

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

      2. associate-+l- N/A

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

      3. --lowering--.f64 N/A

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

      4. tan-lowering-tan.f64 N/A

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

      5. +-lowering-+.f64 N/A

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

      6. --lowering--.f64 N/A

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

      7. tan-lowering-tan.f64 70.0

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

   4. Applied egg-rr 70.0%

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

   5. Taylor expanded in a around 0

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

      1. mul-1-neg N/A

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

      2. neg-lowering-neg.f64 43.1

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

   7. Simplified 43.1%

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

   8. Taylor expanded in y around inf

      \[\leadsto \tan \color{blue}{y} - \left(\mathsf{neg}\left(x\right)\right) \]
   9. Step-by-step derivation

   10. Simplified 31.2%

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

       1. sub-neg N/A

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

       2. remove-double-neg N/A

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

       3. +-lowering-+.f64 N/A

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

       4. tan-lowering-tan.f64 31.2

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

   12. Applied egg-rr 31.2%

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

   if -2e-3 < (+.f64 y z) < 5.00000000000000024e-5

   1. Initial program 100.0%

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

      1. +-commutative N/A

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

      2. associate-+l- N/A

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

      3. --lowering--.f64 N/A

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

      4. tan-lowering-tan.f64 N/A

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

      5. +-lowering-+.f64 N/A

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

      6. --lowering--.f64 N/A

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

      7. tan-lowering-tan.f64 100.0

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in y around inf

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

   7. Simplified 98.7%

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

   8. Taylor expanded in y around 0

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

   10. Simplified 98.7%

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

   if 5.00000000000000024e-5 < (+.f64 y z)

   1. Initial program 67.3%

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

      1. +-commutative N/A

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

      2. associate-+l- N/A

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

      3. --lowering--.f64 N/A

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

      4. tan-lowering-tan.f64 N/A

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

      5. +-lowering-+.f64 N/A

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

      6. --lowering--.f64 N/A

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

      7. tan-lowering-tan.f64 67.2

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

   4. Applied egg-rr 67.2%

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

   5. Taylor expanded in a around 0

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

      1. mul-1-neg N/A

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

      2. neg-lowering-neg.f64 45.1

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

   7. Simplified 45.1%

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

   8. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. /-lowering-/.f64 29.1

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

   10. Simplified 29.1%

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

4. Final simplification 48.4%

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

Alternative 17: 54.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y + z \leq -0.002:\\ \;\;\;\;x + \tan y\\ \mathbf{elif}\;y + z \leq 5 \cdot 10^{-5}:\\ \;\;\;\;y + \left(x - \tan a\right)\\ \mathbf{else}:\\ \;\;\;\;x + \tan \left(y + z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (if (<= (+ y z) -0.002)
   (+ x (tan y))
   (if (<= (+ y z) 5e-5) (+ y (- x (tan a))) (+ x (tan (+ y z))))))

C

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

Fortran

real(8) function code(x, y, z, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: a
    real(8) :: tmp
    if ((y + z) <= (-0.002d0)) then
        tmp = x + tan(y)
    else if ((y + z) <= 5d-5) then
        tmp = y + (x - tan(a))
    else
        tmp = x + tan((y + z))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double a) {
	double tmp;
	if ((y + z) <= -0.002) {
		tmp = x + Math.tan(y);
	} else if ((y + z) <= 5e-5) {
		tmp = y + (x - Math.tan(a));
	} else {
		tmp = x + Math.tan((y + z));
	}
	return tmp;
}

Python

def code(x, y, z, a):
	tmp = 0
	if (y + z) <= -0.002:
		tmp = x + math.tan(y)
	elif (y + z) <= 5e-5:
		tmp = y + (x - math.tan(a))
	else:
		tmp = x + math.tan((y + z))
	return tmp

Julia

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

MATLAB

function tmp_2 = code(x, y, z, a)
	tmp = 0.0;
	if ((y + z) <= -0.002)
		tmp = x + tan(y);
	elseif ((y + z) <= 5e-5)
		tmp = y + (x - tan(a));
	else
		tmp = x + tan((y + z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, a_] := If[LessEqual[N[(y + z), $MachinePrecision], -0.002], N[(x + N[Tan[y], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(y + z), $MachinePrecision], 5e-5], N[(y + N[(x - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[Tan[N[(y + z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 y z) < -2e-3

   1. Initial program 70.1%

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

      1. +-commutative N/A

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

      2. associate-+l- N/A

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

      3. --lowering--.f64 N/A

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

      4. tan-lowering-tan.f64 N/A

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

      5. +-lowering-+.f64 N/A

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

      6. --lowering--.f64 N/A

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

      7. tan-lowering-tan.f64 70.0

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

   4. Applied egg-rr 70.0%

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

   5. Taylor expanded in a around 0

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

      1. mul-1-neg N/A

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

      2. neg-lowering-neg.f64 43.1

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

   7. Simplified 43.1%

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

   8. Taylor expanded in y around inf

      \[\leadsto \tan \color{blue}{y} - \left(\mathsf{neg}\left(x\right)\right) \]
   9. Step-by-step derivation

   10. Simplified 31.2%

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

       1. sub-neg N/A

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

       2. remove-double-neg N/A

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

       3. +-lowering-+.f64 N/A

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

       4. tan-lowering-tan.f64 31.2

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

   12. Applied egg-rr 31.2%

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

   if -2e-3 < (+.f64 y z) < 5.00000000000000024e-5

   1. Initial program 100.0%

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

      1. +-commutative N/A

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

      2. associate-+l- N/A

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

      3. --lowering--.f64 N/A

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

      4. tan-lowering-tan.f64 N/A

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

      5. +-lowering-+.f64 N/A

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

      6. --lowering--.f64 N/A

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

      7. tan-lowering-tan.f64 100.0

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in y around inf

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

   7. Simplified 98.7%

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

   8. Taylor expanded in y around 0

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

   10. Simplified 98.7%

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

   if 5.00000000000000024e-5 < (+.f64 y z)

   1. Initial program 67.3%

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

      1. +-commutative N/A

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

      2. associate-+l- N/A

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

      3. --lowering--.f64 N/A

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

      4. tan-lowering-tan.f64 N/A

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

      5. +-lowering-+.f64 N/A

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

      6. --lowering--.f64 N/A

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

      7. tan-lowering-tan.f64 67.2

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

   4. Applied egg-rr 67.2%

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

   5. Taylor expanded in a around 0

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

      1. mul-1-neg N/A

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

      2. neg-lowering-neg.f64 45.1

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

   7. Simplified 45.1%

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

      1. sub-neg N/A

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

      2. remove-double-neg N/A

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

      3. +-lowering-+.f64 N/A

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

      4. tan-lowering-tan.f64 N/A

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

      5. +-lowering-+.f64 45.1

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

   9. Applied egg-rr 45.1%

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

4. Final simplification 54.1%

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

Alternative 18: 45.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 4.2 \cdot 10^{-14}:\\ \;\;\;\;x + \tan y\\ \mathbf{else}:\\ \;\;\;\;x + \tan z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (if (<= z 4.2e-14) (+ x (tan y)) (+ x (tan z))))

C

double code(double x, double y, double z, double a) {
	double tmp;
	if (z <= 4.2e-14) {
		tmp = x + tan(y);
	} else {
		tmp = x + tan(z);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z <= 4.2d-14) then
        tmp = x + tan(y)
    else
        tmp = x + tan(z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double a) {
	double tmp;
	if (z <= 4.2e-14) {
		tmp = x + Math.tan(y);
	} else {
		tmp = x + Math.tan(z);
	}
	return tmp;
}

Python

def code(x, y, z, a):
	tmp = 0
	if z <= 4.2e-14:
		tmp = x + math.tan(y)
	else:
		tmp = x + math.tan(z)
	return tmp

Julia

function code(x, y, z, a)
	tmp = 0.0
	if (z <= 4.2e-14)
		tmp = Float64(x + tan(y));
	else
		tmp = Float64(x + tan(z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, a)
	tmp = 0.0;
	if (z <= 4.2e-14)
		tmp = x + tan(y);
	else
		tmp = x + tan(z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, a_] := If[LessEqual[z, 4.2e-14], N[(x + N[Tan[y], $MachinePrecision]), $MachinePrecision], N[(x + N[Tan[z], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 4.1999999999999998e-14

   1. Initial program 85.9%

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

      1. +-commutative N/A

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

      2. associate-+l- N/A

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

      3. --lowering--.f64 N/A

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

      4. tan-lowering-tan.f64 N/A

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

      5. +-lowering-+.f64 N/A

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

      6. --lowering--.f64 N/A

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

      7. tan-lowering-tan.f64 85.9

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

   4. Applied egg-rr 85.9%

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

   5. Taylor expanded in a around 0

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

      1. mul-1-neg N/A

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

      2. neg-lowering-neg.f64 53.2

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

   7. Simplified 53.2%

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

   8. Taylor expanded in y around inf

      \[\leadsto \tan \color{blue}{y} - \left(\mathsf{neg}\left(x\right)\right) \]
   9. Step-by-step derivation

   10. Simplified 47.0%

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

       1. sub-neg N/A

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

       2. remove-double-neg N/A

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

       3. +-lowering-+.f64 N/A

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

       4. tan-lowering-tan.f64 47.0

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

   12. Applied egg-rr 47.0%

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

   if 4.1999999999999998e-14 < z

   1. Initial program 50.9%

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

      1. +-commutative N/A

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

      2. associate-+l- N/A

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

      3. --lowering--.f64 N/A

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

      4. tan-lowering-tan.f64 N/A

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

      5. +-lowering-+.f64 N/A

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

      6. --lowering--.f64 N/A

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

      7. tan-lowering-tan.f64 50.8

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

   4. Applied egg-rr 50.8%

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

   5. Taylor expanded in a around 0

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

      1. mul-1-neg N/A

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

      2. neg-lowering-neg.f64 33.6

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

   7. Simplified 33.6%

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

   8. Taylor expanded in y around 0

      \[\leadsto \tan \color{blue}{z} - \left(\mathsf{neg}\left(x\right)\right) \]
   9. Step-by-step derivation

   10. Simplified 34.1%

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

4. Final simplification 43.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 4.2 \cdot 10^{-14}:\\ \;\;\;\;x + \tan y\\ \mathbf{else}:\\ \;\;\;\;x + \tan z\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 49.7% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.0%

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

   1. +-commutative N/A

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

   2. associate-+l- N/A

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

   3. --lowering--.f64 N/A

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

   4. tan-lowering-tan.f64 N/A

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

   5. +-lowering-+.f64 N/A

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

   6. --lowering--.f64 N/A

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

   7. tan-lowering-tan.f64 77.0

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

4. Applied egg-rr 77.0%

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

5. Taylor expanded in a around 0

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

   1. mul-1-neg N/A

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

   2. neg-lowering-neg.f64 48.2

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

7. Simplified 48.2%

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

   1. sub-neg N/A

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

   2. remove-double-neg N/A

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

   3. +-lowering-+.f64 N/A

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

   4. tan-lowering-tan.f64 N/A

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

   5. +-lowering-+.f64 48.2

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

9. Applied egg-rr 48.2%

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

10. Final simplification 48.2%

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

Alternative 20: 40.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

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)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.0%

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

   1. +-commutative N/A

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

   2. associate-+l- N/A

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

   3. --lowering--.f64 N/A

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

   4. tan-lowering-tan.f64 N/A

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

   5. +-lowering-+.f64 N/A

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

   6. --lowering--.f64 N/A

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

   7. tan-lowering-tan.f64 77.0

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

4. Applied egg-rr 77.0%

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

5. Taylor expanded in a around 0

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

   1. mul-1-neg N/A

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

   2. neg-lowering-neg.f64 48.2

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

7. Simplified 48.2%

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

8. Taylor expanded in y around inf

   \[\leadsto \tan \color{blue}{y} - \left(\mathsf{neg}\left(x\right)\right) \]
9. Step-by-step derivation

10. Simplified 40.8%

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

    1. sub-neg N/A

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

    2. remove-double-neg N/A

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

    3. +-lowering-+.f64 N/A

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

    4. tan-lowering-tan.f64 40.8

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

12. Applied egg-rr 40.8%

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

13. Final simplification 40.8%

    \[\leadsto x + \tan y \]
14. Add Preprocessing

Alternative 21: 31.8% accurate, 210.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, a_] := x

Derivation

1. Initial program 77.0%

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

3. Taylor expanded in x around inf

   \[\leadsto \color{blue}{x} \]
4. Step-by-step derivation

5. Simplified 32.2%

   \[\leadsto \color{blue}{x} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024204 
(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))))