tan-example (used to crash)

Percentage Accurate: 79.4% → 99.7%

Time: 26.5s

Alternatives: 15

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: 79.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 80.1%

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

   1. tan-sum 99.8%

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

   2. tan-quot 99.8%

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

   3. frac-sub 99.8%

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

4. Applied egg-rr 99.8%

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

   1. +-commutative 99.8%

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

   2. div-inv 99.8%

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

   3. fma-define 99.8%

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

   4. *-commutative 99.8%

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

6. Applied egg-rr 99.8%

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

7. Final simplification 99.8%

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

Alternative 2: 99.7% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

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
    t_0 = tan(y) * tan(z)
    code = x + ((((tan(y) + tan(z)) * cos(a)) + (sin(a) * (t_0 + (-1.0d0)))) / (cos(a) * (1.0d0 - t_0)))
end function

Java

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

Python

def code(x, y, z, a):
	t_0 = math.tan(y) * math.tan(z)
	return x + ((((math.tan(y) + math.tan(z)) * math.cos(a)) + (math.sin(a) * (t_0 + -1.0))) / (math.cos(a) * (1.0 - t_0)))

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 80.1%

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

   1. tan-sum 99.8%

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

   2. tan-quot 99.8%

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

   3. frac-sub 99.8%

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

4. Applied egg-rr 99.8%

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

5. Final simplification 99.8%

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

Alternative 3: 89.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan \left(y + z\right)\\ \mathbf{if}\;\tan a \leq -0.0005:\\ \;\;\;\;x + \left(t\_0 + \frac{-1}{\frac{\cos a}{\sin a}}\right)\\ \mathbf{elif}\;\tan a \leq 4 \cdot 10^{-5}:\\ \;\;\;\;x + \left(\left(\tan y + \tan z\right) \cdot \frac{1}{1 - \tan y \cdot \tan z} - a\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 + \left(x - \tan a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (let* ((t_0 (tan (+ y z))))
   (if (<= (tan a) -0.0005)
     (+ x (+ t_0 (/ -1.0 (/ (cos a) (sin a)))))
     (if (<= (tan a) 4e-5)
       (+ x (- (* (+ (tan y) (tan z)) (/ 1.0 (- 1.0 (* (tan y) (tan z))))) a))
       (+ t_0 (- x (tan a)))))))

C

double code(double x, double y, double z, double a) {
	double t_0 = tan((y + z));
	double tmp;
	if (tan(a) <= -0.0005) {
		tmp = x + (t_0 + (-1.0 / (cos(a) / sin(a))));
	} else if (tan(a) <= 4e-5) {
		tmp = x + (((tan(y) + tan(z)) * (1.0 / (1.0 - (tan(y) * tan(z))))) - a);
	} else {
		tmp = t_0 + (x - 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) :: t_0
    real(8) :: tmp
    t_0 = tan((y + z))
    if (tan(a) <= (-0.0005d0)) then
        tmp = x + (t_0 + ((-1.0d0) / (cos(a) / sin(a))))
    else if (tan(a) <= 4d-5) then
        tmp = x + (((tan(y) + tan(z)) * (1.0d0 / (1.0d0 - (tan(y) * tan(z))))) - a)
    else
        tmp = t_0 + (x - tan(a))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double a) {
	double t_0 = Math.tan((y + z));
	double tmp;
	if (Math.tan(a) <= -0.0005) {
		tmp = x + (t_0 + (-1.0 / (Math.cos(a) / Math.sin(a))));
	} else if (Math.tan(a) <= 4e-5) {
		tmp = x + (((Math.tan(y) + Math.tan(z)) * (1.0 / (1.0 - (Math.tan(y) * Math.tan(z))))) - a);
	} else {
		tmp = t_0 + (x - Math.tan(a));
	}
	return tmp;
}

Python

def code(x, y, z, a):
	t_0 = math.tan((y + z))
	tmp = 0
	if math.tan(a) <= -0.0005:
		tmp = x + (t_0 + (-1.0 / (math.cos(a) / math.sin(a))))
	elif math.tan(a) <= 4e-5:
		tmp = x + (((math.tan(y) + math.tan(z)) * (1.0 / (1.0 - (math.tan(y) * math.tan(z))))) - a)
	else:
		tmp = t_0 + (x - math.tan(a))
	return tmp

Julia

function code(x, y, z, a)
	t_0 = tan(Float64(y + z))
	tmp = 0.0
	if (tan(a) <= -0.0005)
		tmp = Float64(x + Float64(t_0 + Float64(-1.0 / Float64(cos(a) / sin(a)))));
	elseif (tan(a) <= 4e-5)
		tmp = Float64(x + Float64(Float64(Float64(tan(y) + tan(z)) * Float64(1.0 / Float64(1.0 - Float64(tan(y) * tan(z))))) - a));
	else
		tmp = Float64(t_0 + Float64(x - tan(a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, a)
	t_0 = tan((y + z));
	tmp = 0.0;
	if (tan(a) <= -0.0005)
		tmp = x + (t_0 + (-1.0 / (cos(a) / sin(a))));
	elseif (tan(a) <= 4e-5)
		tmp = x + (((tan(y) + tan(z)) * (1.0 / (1.0 - (tan(y) * tan(z))))) - a);
	else
		tmp = t_0 + (x - tan(a));
	end
	tmp_2 = 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], -0.0005], N[(x + N[(t$95$0 + N[(-1.0 / N[(N[Cos[a], $MachinePrecision] / N[Sin[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Tan[a], $MachinePrecision], 4e-5], N[(x + N[(N[(N[(N[Tan[y], $MachinePrecision] + N[Tan[z], $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(1.0 - N[(N[Tan[y], $MachinePrecision] * N[Tan[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision]), $MachinePrecision], N[(t$95$0 + N[(x - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (tan.f64 a) < -5.0000000000000001e-4

   1. Initial program 79.6%

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

      1. tan-quot 79.6%

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

      2. clear-num 79.6%

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

   4. Applied egg-rr 79.6%

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

   if -5.0000000000000001e-4 < (tan.f64 a) < 4.00000000000000033e-5

   1. Initial program 81.9%

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

   3. Taylor expanded in a around 0 81.9%

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

      1. tan-sum 99.8%

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

      2. div-inv 99.8%

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

   5. Applied egg-rr 99.8%

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

   if 4.00000000000000033e-5 < (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. +-commutative 76.5%

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

      2. associate-+l- 76.5%

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

   4. Applied egg-rr 76.5%

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

4. Final simplification 89.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\tan a \leq -0.0005:\\ \;\;\;\;x + \left(\tan \left(y + z\right) + \frac{-1}{\frac{\cos a}{\sin a}}\right)\\ \mathbf{elif}\;\tan a \leq 4 \cdot 10^{-5}:\\ \;\;\;\;x + \left(\left(\tan y + \tan z\right) \cdot \frac{1}{1 - \tan y \cdot \tan z} - a\right)\\ \mathbf{else}:\\ \;\;\;\;\tan \left(y + z\right) + \left(x - \tan a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 89.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan \left(y + z\right)\\ \mathbf{if}\;\tan a \leq -0.0005:\\ \;\;\;\;x + \left(t\_0 + \frac{-1}{\frac{\cos a}{\sin a}}\right)\\ \mathbf{elif}\;\tan a \leq 4 \cdot 10^{-5}:\\ \;\;\;\;x + \left(\frac{\tan y + \tan z}{1 - \tan y \cdot \tan z} - a\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 + \left(x - \tan a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (let* ((t_0 (tan (+ y z))))
   (if (<= (tan a) -0.0005)
     (+ x (+ t_0 (/ -1.0 (/ (cos a) (sin a)))))
     (if (<= (tan a) 4e-5)
       (+ x (- (/ (+ (tan y) (tan z)) (- 1.0 (* (tan y) (tan z)))) a))
       (+ t_0 (- x (tan a)))))))

C

double code(double x, double y, double z, double a) {
	double t_0 = tan((y + z));
	double tmp;
	if (tan(a) <= -0.0005) {
		tmp = x + (t_0 + (-1.0 / (cos(a) / sin(a))));
	} else if (tan(a) <= 4e-5) {
		tmp = x + (((tan(y) + tan(z)) / (1.0 - (tan(y) * tan(z)))) - a);
	} else {
		tmp = t_0 + (x - 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) :: t_0
    real(8) :: tmp
    t_0 = tan((y + z))
    if (tan(a) <= (-0.0005d0)) then
        tmp = x + (t_0 + ((-1.0d0) / (cos(a) / sin(a))))
    else if (tan(a) <= 4d-5) then
        tmp = x + (((tan(y) + tan(z)) / (1.0d0 - (tan(y) * tan(z)))) - a)
    else
        tmp = t_0 + (x - tan(a))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double a) {
	double t_0 = Math.tan((y + z));
	double tmp;
	if (Math.tan(a) <= -0.0005) {
		tmp = x + (t_0 + (-1.0 / (Math.cos(a) / Math.sin(a))));
	} else if (Math.tan(a) <= 4e-5) {
		tmp = x + (((Math.tan(y) + Math.tan(z)) / (1.0 - (Math.tan(y) * Math.tan(z)))) - a);
	} else {
		tmp = t_0 + (x - Math.tan(a));
	}
	return tmp;
}

Python

def code(x, y, z, a):
	t_0 = math.tan((y + z))
	tmp = 0
	if math.tan(a) <= -0.0005:
		tmp = x + (t_0 + (-1.0 / (math.cos(a) / math.sin(a))))
	elif math.tan(a) <= 4e-5:
		tmp = x + (((math.tan(y) + math.tan(z)) / (1.0 - (math.tan(y) * math.tan(z)))) - a)
	else:
		tmp = t_0 + (x - math.tan(a))
	return tmp

Julia

function code(x, y, z, a)
	t_0 = tan(Float64(y + z))
	tmp = 0.0
	if (tan(a) <= -0.0005)
		tmp = Float64(x + Float64(t_0 + Float64(-1.0 / Float64(cos(a) / sin(a)))));
	elseif (tan(a) <= 4e-5)
		tmp = Float64(x + Float64(Float64(Float64(tan(y) + tan(z)) / Float64(1.0 - Float64(tan(y) * tan(z)))) - a));
	else
		tmp = Float64(t_0 + Float64(x - tan(a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, a)
	t_0 = tan((y + z));
	tmp = 0.0;
	if (tan(a) <= -0.0005)
		tmp = x + (t_0 + (-1.0 / (cos(a) / sin(a))));
	elseif (tan(a) <= 4e-5)
		tmp = x + (((tan(y) + tan(z)) / (1.0 - (tan(y) * tan(z)))) - a);
	else
		tmp = t_0 + (x - tan(a));
	end
	tmp_2 = 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], -0.0005], N[(x + N[(t$95$0 + N[(-1.0 / N[(N[Cos[a], $MachinePrecision] / N[Sin[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Tan[a], $MachinePrecision], 4e-5], 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] - a), $MachinePrecision]), $MachinePrecision], N[(t$95$0 + N[(x - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (tan.f64 a) < -5.0000000000000001e-4

   1. Initial program 79.6%

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

      1. tan-quot 79.6%

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

      2. clear-num 79.6%

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

   4. Applied egg-rr 79.6%

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

   if -5.0000000000000001e-4 < (tan.f64 a) < 4.00000000000000033e-5

   1. Initial program 81.9%

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

   3. Taylor expanded in a around 0 81.9%

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

      1. tan-sum 99.8%

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

      2. div-inv 99.8%

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

   5. Applied egg-rr 99.8%

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

      1. associate-*r/ 99.8%

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

      2. *-rgt-identity 99.8%

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

   7. Simplified 99.8%

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

   if 4.00000000000000033e-5 < (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. +-commutative 76.5%

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

      2. associate-+l- 76.5%

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

   4. Applied egg-rr 76.5%

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

4. Final simplification 89.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\tan a \leq -0.0005:\\ \;\;\;\;x + \left(\tan \left(y + z\right) + \frac{-1}{\frac{\cos a}{\sin a}}\right)\\ \mathbf{elif}\;\tan a \leq 4 \cdot 10^{-5}:\\ \;\;\;\;x + \left(\frac{\tan y + \tan z}{1 - \tan y \cdot \tan z} - a\right)\\ \mathbf{else}:\\ \;\;\;\;\tan \left(y + z\right) + \left(x - \tan a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.7% accurate, 0.3× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z, double a) {
	return x + (((tan(y) + tan(z)) / (1.0 - (tan(y) * tan(z)))) - (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(y) + tan(z)) / (1.0d0 - (tan(y) * tan(z)))) - (sin(a) / cos(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.sin(a) / Math.cos(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.sin(a) / math.cos(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)))) - Float64(sin(a) / cos(a))))
end

MATLAB

function tmp = code(x, y, z, a)
	tmp = x + (((tan(y) + tan(z)) / (1.0 - (tan(y) * tan(z)))) - (sin(a) / cos(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[(N[Sin[a], $MachinePrecision] / N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 80.1%

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

   1. tan-sum 99.8%

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

   2. tan-quot 99.8%

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

   3. frac-sub 99.8%

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

4. Applied egg-rr 99.8%

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

   1. div-sub 99.7%

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

6. Simplified 99.8%

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

Alternative 6: 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 80.1%

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

   1. +-commutative 80.1%

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

   2. sub-neg 80.1%

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

   3. associate-+l+ 80.1%

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

   4. tan-sum 99.7%

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

   5. div-inv 99.7%

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

   6. fma-define 99.7%

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

   7. neg-mul-1 99.7%

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

   8. fma-define 99.7%

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

4. Applied egg-rr 99.7%

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

   1. fma-undefine 99.7%

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

   2. fma-undefine 99.7%

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

   3. neg-mul-1 99.7%

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

   4. associate-+r+ 99.7%

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

   5. sub-neg 99.7%

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

   6. associate-*r/ 99.8%

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

   7. *-rgt-identity 99.8%

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

6. Simplified 99.8%

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

7. Final simplification 99.8%

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

Alternative 7: 79.4% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 80.1%

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

   1. tan-quot 80.1%

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

   2. clear-num 80.1%

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

4. Applied egg-rr 80.1%

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

5. Final simplification 80.1%

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

Alternative 8: 69.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -10000 \lor \neg \left(a \leq 1.08 \cdot 10^{-16}\right):\\ \;\;\;\;x + \left(\tan y - \tan a\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(\tan \left(y + z\right) - a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (if (or (<= a -10000.0) (not (<= a 1.08e-16)))
   (+ x (- (tan y) (tan a)))
   (+ x (- (tan (+ y z)) a))))

C

double code(double x, double y, double z, double a) {
	double tmp;
	if ((a <= -10000.0) || !(a <= 1.08e-16)) {
		tmp = x + (tan(y) - tan(a));
	} else {
		tmp = x + (tan((y + z)) - 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 ((a <= (-10000.0d0)) .or. (.not. (a <= 1.08d-16))) then
        tmp = x + (tan(y) - tan(a))
    else
        tmp = x + (tan((y + z)) - a)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double a) {
	double tmp;
	if ((a <= -10000.0) || !(a <= 1.08e-16)) {
		tmp = x + (Math.tan(y) - Math.tan(a));
	} else {
		tmp = x + (Math.tan((y + z)) - a);
	}
	return tmp;
}

Python

def code(x, y, z, a):
	tmp = 0
	if (a <= -10000.0) or not (a <= 1.08e-16):
		tmp = x + (math.tan(y) - math.tan(a))
	else:
		tmp = x + (math.tan((y + z)) - a)
	return tmp

Julia

function code(x, y, z, a)
	tmp = 0.0
	if ((a <= -10000.0) || !(a <= 1.08e-16))
		tmp = Float64(x + Float64(tan(y) - tan(a)));
	else
		tmp = Float64(x + Float64(tan(Float64(y + z)) - a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, a)
	tmp = 0.0;
	if ((a <= -10000.0) || ~((a <= 1.08e-16)))
		tmp = x + (tan(y) - tan(a));
	else
		tmp = x + (tan((y + z)) - a);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, a_] := If[Or[LessEqual[a, -10000.0], N[Not[LessEqual[a, 1.08e-16]], $MachinePrecision]], N[(x + N[(N[Tan[y], $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[Tan[N[(y + z), $MachinePrecision]], $MachinePrecision] - a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -1e4 or 1.08e-16 < a

   1. Initial program 78.4%

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

   3. Taylor expanded in y around inf 60.6%

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

   if -1e4 < a < 1.08e-16

   1. Initial program 81.7%

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

   3. Taylor expanded in a around 0 81.1%

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

4. Final simplification 71.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -10000 \lor \neg \left(a \leq 1.08 \cdot 10^{-16}\right):\\ \;\;\;\;x + \left(\tan y - \tan a\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(\tan \left(y + z\right) - a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 51.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.35 \lor \neg \left(a \leq 2.3 \cdot 10^{-5}\right):\\ \;\;\;\;{\left(\sqrt[3]{x}\right)}^{3}\\ \mathbf{else}:\\ \;\;\;\;x + \left(\tan \left(y + z\right) - a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (if (or (<= a -1.35) (not (<= a 2.3e-5)))
   (pow (cbrt x) 3.0)
   (+ x (- (tan (+ y z)) a))))

C

double code(double x, double y, double z, double a) {
	double tmp;
	if ((a <= -1.35) || !(a <= 2.3e-5)) {
		tmp = pow(cbrt(x), 3.0);
	} else {
		tmp = x + (tan((y + z)) - a);
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double a) {
	double tmp;
	if ((a <= -1.35) || !(a <= 2.3e-5)) {
		tmp = Math.pow(Math.cbrt(x), 3.0);
	} else {
		tmp = x + (Math.tan((y + z)) - a);
	}
	return tmp;
}

Julia

function code(x, y, z, a)
	tmp = 0.0
	if ((a <= -1.35) || !(a <= 2.3e-5))
		tmp = cbrt(x) ^ 3.0;
	else
		tmp = Float64(x + Float64(tan(Float64(y + z)) - a));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, a_] := If[Or[LessEqual[a, -1.35], N[Not[LessEqual[a, 2.3e-5]], $MachinePrecision]], N[Power[N[Power[x, 1/3], $MachinePrecision], 3.0], $MachinePrecision], N[(x + N[(N[Tan[N[(y + z), $MachinePrecision]], $MachinePrecision] - a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -1.3500000000000001 or 2.3e-5 < a

   1. Initial program 77.6%

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

   3. Taylor expanded in y around 0 55.4%

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

      1. tan-quot 55.4%

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

      2. add-cube-cbrt 54.6%

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

      3. pow3 54.6%

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

      4. tan-quot 54.6%

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

      5. +-commutative 54.6%

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

   5. Applied egg-rr 54.6%

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

   6. Taylor expanded in x around inf 21.5%

      \[\leadsto {\color{blue}{\left(\sqrt[3]{x}\right)}}^{3} \]

   if -1.3500000000000001 < a < 2.3e-5

   1. Initial program 82.5%

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

   3. Taylor expanded in a around 0 81.9%

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

4. Final simplification 52.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.35 \lor \neg \left(a \leq 2.3 \cdot 10^{-5}\right):\\ \;\;\;\;{\left(\sqrt[3]{x}\right)}^{3}\\ \mathbf{else}:\\ \;\;\;\;x + \left(\tan \left(y + z\right) - a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 60.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y + z \leq -4 \cdot 10^{-14}:\\ \;\;\;\;x + \left(\tan y - \tan a\right)\\ \mathbf{else}:\\ \;\;\;\;x - \left(\tan a - \tan z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (if (<= (+ y z) -4e-14) (+ x (- (tan y) (tan a))) (- x (- (tan a) (tan z)))))

C

double code(double x, double y, double z, double a) {
	double tmp;
	if ((y + z) <= -4e-14) {
		tmp = x + (tan(y) - tan(a));
	} else {
		tmp = x - (tan(a) - 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 ((y + z) <= (-4d-14)) then
        tmp = x + (tan(y) - tan(a))
    else
        tmp = x - (tan(a) - tan(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) <= -4e-14) {
		tmp = x + (Math.tan(y) - Math.tan(a));
	} else {
		tmp = x - (Math.tan(a) - Math.tan(z));
	}
	return tmp;
}

Python

def code(x, y, z, a):
	tmp = 0
	if (y + z) <= -4e-14:
		tmp = x + (math.tan(y) - math.tan(a))
	else:
		tmp = x - (math.tan(a) - math.tan(z))
	return tmp

Julia

function code(x, y, z, a)
	tmp = 0.0
	if (Float64(y + z) <= -4e-14)
		tmp = Float64(x + Float64(tan(y) - tan(a)));
	else
		tmp = Float64(x - Float64(tan(a) - tan(z)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, a)
	tmp = 0.0;
	if ((y + z) <= -4e-14)
		tmp = x + (tan(y) - tan(a));
	else
		tmp = x - (tan(a) - tan(z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, a_] := If[LessEqual[N[(y + z), $MachinePrecision], -4e-14], N[(x + N[(N[Tan[y], $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x - N[(N[Tan[a], $MachinePrecision] - N[Tan[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 y z) < -4e-14

   1. Initial program 75.1%

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

   3. Taylor expanded in y around inf 52.7%

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

   if -4e-14 < (+.f64 y z)

   1. Initial program 83.3%

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

   3. Taylor expanded in y around 0 68.5%

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

4. Final simplification 62.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y + z \leq -4 \cdot 10^{-14}:\\ \;\;\;\;x + \left(\tan y - \tan a\right)\\ \mathbf{else}:\\ \;\;\;\;x - \left(\tan a - \tan z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 79.4% 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 80.1%

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

Alternative 12: 51.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.5:\\ \;\;\;\;x\\ \mathbf{elif}\;a \leq 2.3 \cdot 10^{-5}:\\ \;\;\;\;x + \left(\tan \left(y + z\right) - a\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (if (<= a -1.5) x (if (<= a 2.3e-5) (+ x (- (tan (+ y z)) a)) x)))

C

double code(double x, double y, double z, double a) {
	double tmp;
	if (a <= -1.5) {
		tmp = x;
	} else if (a <= 2.3e-5) {
		tmp = x + (tan((y + z)) - a);
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: a
    real(8) :: tmp
    if (a <= (-1.5d0)) then
        tmp = x
    else if (a <= 2.3d-5) then
        tmp = x + (tan((y + z)) - a)
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double a) {
	double tmp;
	if (a <= -1.5) {
		tmp = x;
	} else if (a <= 2.3e-5) {
		tmp = x + (Math.tan((y + z)) - a);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, a):
	tmp = 0
	if a <= -1.5:
		tmp = x
	elif a <= 2.3e-5:
		tmp = x + (math.tan((y + z)) - a)
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, a)
	tmp = 0.0
	if (a <= -1.5)
		tmp = x;
	elseif (a <= 2.3e-5)
		tmp = Float64(x + Float64(tan(Float64(y + z)) - a));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, a)
	tmp = 0.0;
	if (a <= -1.5)
		tmp = x;
	elseif (a <= 2.3e-5)
		tmp = x + (tan((y + z)) - a);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, a_] := If[LessEqual[a, -1.5], x, If[LessEqual[a, 2.3e-5], N[(x + N[(N[Tan[N[(y + z), $MachinePrecision]], $MachinePrecision] - a), $MachinePrecision]), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if a < -1.5 or 2.3e-5 < a

   1. Initial program 77.6%

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

   3. Taylor expanded in x around inf 21.5%

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

   if -1.5 < a < 2.3e-5

   1. Initial program 82.5%

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

   3. Taylor expanded in a around 0 81.9%

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

Alternative 13: 41.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -4.4 \cdot 10^{-15}:\\ \;\;\;\;x\\ \mathbf{elif}\;a \leq 2.3 \cdot 10^{-5}:\\ \;\;\;\;x - \left(a - \tan y\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (if (<= a -4.4e-15) x (if (<= a 2.3e-5) (- x (- a (tan y))) x)))

C

double code(double x, double y, double z, double a) {
	double tmp;
	if (a <= -4.4e-15) {
		tmp = x;
	} else if (a <= 2.3e-5) {
		tmp = x - (a - tan(y));
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: a
    real(8) :: tmp
    if (a <= (-4.4d-15)) then
        tmp = x
    else if (a <= 2.3d-5) then
        tmp = x - (a - tan(y))
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double a) {
	double tmp;
	if (a <= -4.4e-15) {
		tmp = x;
	} else if (a <= 2.3e-5) {
		tmp = x - (a - Math.tan(y));
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, a):
	tmp = 0
	if a <= -4.4e-15:
		tmp = x
	elif a <= 2.3e-5:
		tmp = x - (a - math.tan(y))
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, a)
	tmp = 0.0
	if (a <= -4.4e-15)
		tmp = x;
	elseif (a <= 2.3e-5)
		tmp = Float64(x - Float64(a - tan(y)));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, a)
	tmp = 0.0;
	if (a <= -4.4e-15)
		tmp = x;
	elseif (a <= 2.3e-5)
		tmp = x - (a - tan(y));
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, a_] := If[LessEqual[a, -4.4e-15], x, If[LessEqual[a, 2.3e-5], N[(x - N[(a - N[Tan[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if a < -4.39999999999999971e-15 or 2.3e-5 < a

   1. Initial program 77.2%

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

   3. Taylor expanded in x around inf 22.2%

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

   if -4.39999999999999971e-15 < a < 2.3e-5

   1. Initial program 83.0%

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

   3. Taylor expanded in a around 0 83.0%

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

   4. Taylor expanded in y around inf 65.2%

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

4. Final simplification 43.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -4.4 \cdot 10^{-15}:\\ \;\;\;\;x\\ \mathbf{elif}\;a \leq 2.3 \cdot 10^{-5}:\\ \;\;\;\;x - \left(a - \tan y\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 36.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4.6 \cdot 10^{-14}:\\ \;\;\;\;x - \left(a - \tan y\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(\tan z - a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (if (<= y -4.6e-14) (- x (- a (tan y))) (+ x (- (tan z) a))))

C

double code(double x, double y, double z, double a) {
	double tmp;
	if (y <= -4.6e-14) {
		tmp = x - (a - tan(y));
	} else {
		tmp = x + (tan(z) - 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 (y <= (-4.6d-14)) then
        tmp = x - (a - tan(y))
    else
        tmp = x + (tan(z) - a)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double a) {
	double tmp;
	if (y <= -4.6e-14) {
		tmp = x - (a - Math.tan(y));
	} else {
		tmp = x + (Math.tan(z) - a);
	}
	return tmp;
}

Python

def code(x, y, z, a):
	tmp = 0
	if y <= -4.6e-14:
		tmp = x - (a - math.tan(y))
	else:
		tmp = x + (math.tan(z) - a)
	return tmp

Julia

function code(x, y, z, a)
	tmp = 0.0
	if (y <= -4.6e-14)
		tmp = Float64(x - Float64(a - tan(y)));
	else
		tmp = Float64(x + Float64(tan(z) - a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, a)
	tmp = 0.0;
	if (y <= -4.6e-14)
		tmp = x - (a - tan(y));
	else
		tmp = x + (tan(z) - a);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, a_] := If[LessEqual[y, -4.6e-14], N[(x - N[(a - N[Tan[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[Tan[z], $MachinePrecision] - a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -4.59999999999999996e-14

   1. Initial program 67.0%

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

   3. Taylor expanded in a around 0 30.7%

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

   4. Taylor expanded in y around inf 30.9%

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

   if -4.59999999999999996e-14 < y

   1. Initial program 84.7%

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

   3. Taylor expanded in a around 0 48.3%

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

   4. Taylor expanded in y around 0 42.4%

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

4. Final simplification 39.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.6 \cdot 10^{-14}:\\ \;\;\;\;x - \left(a - \tan y\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(\tan z - a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 32.5% accurate, 207.0× 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 80.1%

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

3. Taylor expanded in x around inf 34.6%

   \[\leadsto \color{blue}{x} \]
4. Add Preprocessing

Reproduce

herbie shell --seed 2024163 
(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))))