tan-example (used to crash)

Percentage Accurate: 80.0% → 99.7%

Time: 17.3s

Alternatives: 10

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 80.0% 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, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan z \cdot \tan y\\ t_1 := 1 - t\_0\\ t_2 := \frac{\tan y}{t\_1}\\ t_3 := \frac{\tan z}{t\_1}\\ t_4 := -1 + t\_0\\ x - \left(\frac{\frac{\tan y}{t\_4} \cdot t\_2 - \frac{\tan z}{t\_4} \cdot t\_3}{t\_2 - t\_3} + \tan a\right) \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (let* ((t_0 (* (tan z) (tan y)))
        (t_1 (- 1.0 t_0))
        (t_2 (/ (tan y) t_1))
        (t_3 (/ (tan z) t_1))
        (t_4 (+ -1.0 t_0)))
   (-
    x
    (+
     (/ (- (* (/ (tan y) t_4) t_2) (* (/ (tan z) t_4) t_3)) (- t_2 t_3))
     (tan a)))))

C

double code(double x, double y, double z, double a) {
	double t_0 = tan(z) * tan(y);
	double t_1 = 1.0 - t_0;
	double t_2 = tan(y) / t_1;
	double t_3 = tan(z) / t_1;
	double t_4 = -1.0 + t_0;
	return x - (((((tan(y) / t_4) * t_2) - ((tan(z) / t_4) * t_3)) / (t_2 - t_3)) + 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
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    t_0 = tan(z) * tan(y)
    t_1 = 1.0d0 - t_0
    t_2 = tan(y) / t_1
    t_3 = tan(z) / t_1
    t_4 = (-1.0d0) + t_0
    code = x - (((((tan(y) / t_4) * t_2) - ((tan(z) / t_4) * t_3)) / (t_2 - t_3)) + tan(a))
end function

Java

public static double code(double x, double y, double z, double a) {
	double t_0 = Math.tan(z) * Math.tan(y);
	double t_1 = 1.0 - t_0;
	double t_2 = Math.tan(y) / t_1;
	double t_3 = Math.tan(z) / t_1;
	double t_4 = -1.0 + t_0;
	return x - (((((Math.tan(y) / t_4) * t_2) - ((Math.tan(z) / t_4) * t_3)) / (t_2 - t_3)) + Math.tan(a));
}

Python

def code(x, y, z, a):
	t_0 = math.tan(z) * math.tan(y)
	t_1 = 1.0 - t_0
	t_2 = math.tan(y) / t_1
	t_3 = math.tan(z) / t_1
	t_4 = -1.0 + t_0
	return x - (((((math.tan(y) / t_4) * t_2) - ((math.tan(z) / t_4) * t_3)) / (t_2 - t_3)) + math.tan(a))

Julia

function code(x, y, z, a)
	t_0 = Float64(tan(z) * tan(y))
	t_1 = Float64(1.0 - t_0)
	t_2 = Float64(tan(y) / t_1)
	t_3 = Float64(tan(z) / t_1)
	t_4 = Float64(-1.0 + t_0)
	return Float64(x - Float64(Float64(Float64(Float64(Float64(tan(y) / t_4) * t_2) - Float64(Float64(tan(z) / t_4) * t_3)) / Float64(t_2 - t_3)) + tan(a)))
end

MATLAB

function tmp = code(x, y, z, a)
	t_0 = tan(z) * tan(y);
	t_1 = 1.0 - t_0;
	t_2 = tan(y) / t_1;
	t_3 = tan(z) / t_1;
	t_4 = -1.0 + t_0;
	tmp = x - (((((tan(y) / t_4) * t_2) - ((tan(z) / t_4) * t_3)) / (t_2 - t_3)) + tan(a));
end

Wolfram

code[x_, y_, z_, a_] := Block[{t$95$0 = N[(N[Tan[z], $MachinePrecision] * N[Tan[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 - t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(N[Tan[y], $MachinePrecision] / t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(N[Tan[z], $MachinePrecision] / t$95$1), $MachinePrecision]}, Block[{t$95$4 = N[(-1.0 + t$95$0), $MachinePrecision]}, N[(x - N[(N[(N[(N[(N[(N[Tan[y], $MachinePrecision] / t$95$4), $MachinePrecision] * t$95$2), $MachinePrecision] - N[(N[(N[Tan[z], $MachinePrecision] / t$95$4), $MachinePrecision] * t$95$3), $MachinePrecision]), $MachinePrecision] / N[(t$95$2 - t$95$3), $MachinePrecision]), $MachinePrecision] + N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Initial program 79.0%

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

   1. lift-tan.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. tan-sum N/A

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

   4. div-add N/A

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

   5. flip-+ N/A

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

   6. lower-/.f64 N/A

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

4. Applied rewrites 99.8%

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

5. Final simplification 99.8%

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

Alternative 2: 99.7% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 79.0%

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

   1. lift-tan.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. tan-sum N/A

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

   4. div-add N/A

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

   5. flip-+ N/A

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

   6. lower-/.f64 N/A

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

4. Applied rewrites 99.8%

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

   1. lift-/.f64 N/A

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

6. Applied rewrites 99.7%

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

Alternative 3: 88.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\tan a \leq -0.05:\\ \;\;\;\;x + \left(\tan \left(y + z\right) - \tan a\right)\\ \mathbf{elif}\;\tan a \leq 10^{-12}:\\ \;\;\;\;x - \left(\frac{\tan z + \tan y}{-1 + \tan z \cdot \tan y} + \mathsf{fma}\left(a \cdot a, 0.3333333333333333, 1\right) \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{\sin \left(z + y\right)}{\cos \left(z + y\right)} - \frac{\sin a}{\cos a}}{x}, x, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (if (<= (tan a) -0.05)
   (+ x (- (tan (+ y z)) (tan a)))
   (if (<= (tan a) 1e-12)
     (-
      x
      (+
       (/ (+ (tan z) (tan y)) (+ -1.0 (* (tan z) (tan y))))
       (* (fma (* a a) 0.3333333333333333 1.0) a)))
     (fma (/ (- (/ (sin (+ z y)) (cos (+ z y))) (/ (sin a) (cos a))) x) x x))))

C

double code(double x, double y, double z, double a) {
	double tmp;
	if (tan(a) <= -0.05) {
		tmp = x + (tan((y + z)) - tan(a));
	} else if (tan(a) <= 1e-12) {
		tmp = x - (((tan(z) + tan(y)) / (-1.0 + (tan(z) * tan(y)))) + (fma((a * a), 0.3333333333333333, 1.0) * a));
	} else {
		tmp = fma((((sin((z + y)) / cos((z + y))) - (sin(a) / cos(a))) / x), x, x);
	}
	return tmp;
}

Julia

function code(x, y, z, a)
	tmp = 0.0
	if (tan(a) <= -0.05)
		tmp = Float64(x + Float64(tan(Float64(y + z)) - tan(a)));
	elseif (tan(a) <= 1e-12)
		tmp = Float64(x - Float64(Float64(Float64(tan(z) + tan(y)) / Float64(-1.0 + Float64(tan(z) * tan(y)))) + Float64(fma(Float64(a * a), 0.3333333333333333, 1.0) * a)));
	else
		tmp = fma(Float64(Float64(Float64(sin(Float64(z + y)) / cos(Float64(z + y))) - Float64(sin(a) / cos(a))) / x), x, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, a_] := If[LessEqual[N[Tan[a], $MachinePrecision], -0.05], N[(x + N[(N[Tan[N[(y + z), $MachinePrecision]], $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Tan[a], $MachinePrecision], 1e-12], N[(x - N[(N[(N[(N[Tan[z], $MachinePrecision] + N[Tan[y], $MachinePrecision]), $MachinePrecision] / N[(-1.0 + N[(N[Tan[z], $MachinePrecision] * N[Tan[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(a * a), $MachinePrecision] * 0.3333333333333333 + 1.0), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[Sin[N[(z + y), $MachinePrecision]], $MachinePrecision] / N[Cos[N[(z + y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(N[Sin[a], $MachinePrecision] / N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] * x + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 74.9%

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

   if -0.050000000000000003 < (tan.f64 a) < 9.9999999999999998e-13

   1. Initial program 82.8%

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

      1. lift-tan.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. tan-sum N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 N/A

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

      7. lower-tan.f64 N/A

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

      8. lower-tan.f64 N/A

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

      9. lower--.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-tan.f64 N/A

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

      13. lower-tan.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 99.9

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

   7. Applied rewrites 99.9%

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

   if 9.9999999999999998e-13 < (tan.f64 a)

   1. Initial program 76.8%

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

   3. Taylor expanded in x around inf

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. distribute-rgt-in N/A

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

      4. *-lft-identity N/A

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

      5. lower-fma.f64 N/A

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

   5. Applied rewrites 76.8%

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

4. Final simplification 87.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\tan a \leq -0.05:\\ \;\;\;\;x + \left(\tan \left(y + z\right) - \tan a\right)\\ \mathbf{elif}\;\tan a \leq 10^{-12}:\\ \;\;\;\;x - \left(\frac{\tan z + \tan y}{-1 + \tan z \cdot \tan y} + \mathsf{fma}\left(a \cdot a, 0.3333333333333333, 1\right) \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{\sin \left(z + y\right)}{\cos \left(z + y\right)} - \frac{\sin a}{\cos a}}{x}, x, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 88.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan \left(y + z\right)\\ \mathbf{if}\;\tan a \leq -0.0002:\\ \;\;\;\;x + \left(t\_0 - \tan a\right)\\ \mathbf{elif}\;\tan a \leq 0.05:\\ \;\;\;\;\mathsf{fma}\left(\frac{\tan y + \tan z}{\mathsf{fma}\left(-\tan z, \tan y, 1\right) \cdot x}, x, x\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(t\_0 - \frac{\sin a}{\cos a}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (let* ((t_0 (tan (+ y z))))
   (if (<= (tan a) -0.0002)
     (+ x (- t_0 (tan a)))
     (if (<= (tan a) 0.05)
       (fma (/ (+ (tan y) (tan z)) (* (fma (- (tan z)) (tan y) 1.0) x)) x x)
       (+ x (- t_0 (/ (sin a) (cos a))))))))

C

double code(double x, double y, double z, double a) {
	double t_0 = tan((y + z));
	double tmp;
	if (tan(a) <= -0.0002) {
		tmp = x + (t_0 - tan(a));
	} else if (tan(a) <= 0.05) {
		tmp = fma(((tan(y) + tan(z)) / (fma(-tan(z), tan(y), 1.0) * x)), x, x);
	} else {
		tmp = x + (t_0 - (sin(a) / cos(a)));
	}
	return tmp;
}

Julia

function code(x, y, z, a)
	t_0 = tan(Float64(y + z))
	tmp = 0.0
	if (tan(a) <= -0.0002)
		tmp = Float64(x + Float64(t_0 - tan(a)));
	elseif (tan(a) <= 0.05)
		tmp = fma(Float64(Float64(tan(y) + tan(z)) / Float64(fma(Float64(-tan(z)), tan(y), 1.0) * x)), x, x);
	else
		tmp = Float64(x + Float64(t_0 - Float64(sin(a) / cos(a))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, a_] := Block[{t$95$0 = N[Tan[N[(y + z), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[Tan[a], $MachinePrecision], -0.0002], N[(x + N[(t$95$0 - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Tan[a], $MachinePrecision], 0.05], N[(N[(N[(N[Tan[y], $MachinePrecision] + N[Tan[z], $MachinePrecision]), $MachinePrecision] / N[(N[((-N[Tan[z], $MachinePrecision]) * N[Tan[y], $MachinePrecision] + 1.0), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] * x + x), $MachinePrecision], N[(x + N[(t$95$0 - N[(N[Sin[a], $MachinePrecision] / N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (tan.f64 a) < -2.0000000000000001e-4

   1. Initial program 75.5%

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

   if -2.0000000000000001e-4 < (tan.f64 a) < 0.050000000000000003

   1. Initial program 81.3%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 81.1

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

   5. Applied rewrites 81.1%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 81.1%

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

   10. Applied rewrites 98.0%

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

   if 0.050000000000000003 < (tan.f64 a)

   1. Initial program 78.8%

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

      1. lift-tan.f64 N/A

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

      2. tan-quot N/A

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

      3. lower-/.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-cos.f64 78.8

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

   4. Applied rewrites 78.8%

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

Alternative 5: 99.7% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.0%

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

   1. lift-tan.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. tan-sum N/A

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

   4. lower-/.f64 N/A

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

   5. +-commutative N/A

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

   6. lower-+.f64 N/A

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

   7. lower-tan.f64 N/A

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

   8. lower-tan.f64 N/A

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

   9. lower--.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 N/A

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

   12. lower-tan.f64 N/A

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

   13. lower-tan.f64 99.8

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

4. Applied rewrites 99.8%

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

   1. lift-tan.f64 N/A

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

   2. tan-quot N/A

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

   3. lift-cos.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. lower-sin.f64 99.8

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

6. Applied rewrites 99.8%

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

7. Final simplification 99.8%

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

Alternative 6: 99.7% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.0%

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

   1. lift-tan.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. tan-sum N/A

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

   4. lower-/.f64 N/A

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

   5. +-commutative N/A

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

   6. lower-+.f64 N/A

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

   7. lower-tan.f64 N/A

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

   8. lower-tan.f64 N/A

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

   9. lower--.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 N/A

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

   12. lower-tan.f64 N/A

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

   13. lower-tan.f64 99.8

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

4. Applied rewrites 99.8%

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

Alternative 7: 89.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan \left(y + z\right)\\ \mathbf{if}\;a \leq -0.0046:\\ \;\;\;\;x + \left(t\_0 - \tan a\right)\\ \mathbf{elif}\;a \leq 0.0072:\\ \;\;\;\;x - \left(\frac{\tan z + \tan y}{-1 + \tan z \cdot \tan y} + \mathsf{fma}\left(a \cdot a, 0.3333333333333333, 1\right) \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(t\_0 - \frac{\sin a}{\cos a}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (let* ((t_0 (tan (+ y z))))
   (if (<= a -0.0046)
     (+ x (- t_0 (tan a)))
     (if (<= a 0.0072)
       (-
        x
        (+
         (/ (+ (tan z) (tan y)) (+ -1.0 (* (tan z) (tan y))))
         (* (fma (* a a) 0.3333333333333333 1.0) a)))
       (+ x (- t_0 (/ (sin a) (cos a))))))))

C

double code(double x, double y, double z, double a) {
	double t_0 = tan((y + z));
	double tmp;
	if (a <= -0.0046) {
		tmp = x + (t_0 - tan(a));
	} else if (a <= 0.0072) {
		tmp = x - (((tan(z) + tan(y)) / (-1.0 + (tan(z) * tan(y)))) + (fma((a * a), 0.3333333333333333, 1.0) * a));
	} else {
		tmp = x + (t_0 - (sin(a) / cos(a)));
	}
	return tmp;
}

Julia

function code(x, y, z, a)
	t_0 = tan(Float64(y + z))
	tmp = 0.0
	if (a <= -0.0046)
		tmp = Float64(x + Float64(t_0 - tan(a)));
	elseif (a <= 0.0072)
		tmp = Float64(x - Float64(Float64(Float64(tan(z) + tan(y)) / Float64(-1.0 + Float64(tan(z) * tan(y)))) + Float64(fma(Float64(a * a), 0.3333333333333333, 1.0) * a)));
	else
		tmp = Float64(x + Float64(t_0 - Float64(sin(a) / cos(a))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, a_] := Block[{t$95$0 = N[Tan[N[(y + z), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[a, -0.0046], N[(x + N[(t$95$0 - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 0.0072], N[(x - N[(N[(N[(N[Tan[z], $MachinePrecision] + N[Tan[y], $MachinePrecision]), $MachinePrecision] / N[(-1.0 + N[(N[Tan[z], $MachinePrecision] * N[Tan[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(a * a), $MachinePrecision] * 0.3333333333333333 + 1.0), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(t$95$0 - N[(N[Sin[a], $MachinePrecision] / N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if a < -0.0045999999999999999

   1. Initial program 74.5%

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

   if -0.0045999999999999999 < a < 0.0071999999999999998

   1. Initial program 82.8%

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

      1. lift-tan.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. tan-sum N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 N/A

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

      7. lower-tan.f64 N/A

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

      8. lower-tan.f64 N/A

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

      9. lower--.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-tan.f64 N/A

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

      13. lower-tan.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 99.9

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

   7. Applied rewrites 99.9%

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

   if 0.0071999999999999998 < a

   1. Initial program 76.9%

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

      1. lift-tan.f64 N/A

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

      2. tan-quot N/A

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

      3. lower-/.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-cos.f64 76.9

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

   4. Applied rewrites 76.9%

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

4. Final simplification 87.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -0.0046:\\ \;\;\;\;x + \left(\tan \left(y + z\right) - \tan a\right)\\ \mathbf{elif}\;a \leq 0.0072:\\ \;\;\;\;x - \left(\frac{\tan z + \tan y}{-1 + \tan z \cdot \tan y} + \mathsf{fma}\left(a \cdot a, 0.3333333333333333, 1\right) \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(\tan \left(y + z\right) - \frac{\sin a}{\cos a}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 80.0% 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 79.0%

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

Alternative 9: 50.7% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 79.0%

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

3. Taylor expanded in a around 0

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

   1. +-commutative N/A

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

   2. lower-+.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. lower-sin.f64 N/A

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

   5. +-commutative N/A

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

   6. lower-+.f64 N/A

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

   7. lower-cos.f64 N/A

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

   8. +-commutative N/A

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

   9. lower-+.f64 50.6

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

5. Applied rewrites 50.6%

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

6. Taylor expanded in x around inf

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

8. Applied rewrites 50.6%

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

10. Applied rewrites 50.6%

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

Alternative 10: 50.7% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.0%

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

3. Taylor expanded in a around 0

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

   1. +-commutative N/A

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

   2. lower-+.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. lower-sin.f64 N/A

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

   5. +-commutative N/A

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

   6. lower-+.f64 N/A

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

   7. lower-cos.f64 N/A

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

   8. +-commutative N/A

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

   9. lower-+.f64 50.6

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

5. Applied rewrites 50.6%

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

7. Applied rewrites 50.6%

   \[\leadsto \tan \left(z + y\right) + \color{blue}{x} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024344 
(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))))