tan-example (used to crash)

Percentage Accurate: 79.6% → 99.7%

Time: 43.6s

Alternatives: 14

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: 79.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 0.4× speedup

Math

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

FPCore

(FPCore (x y z a)
 :precision binary64
 (+ x (- (* (+ (tan y) (tan z)) (/ 1.0 (- 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 / (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 / (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 / (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 / (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(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 / (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[(1.0 - N[(N[Tan[y], $MachinePrecision] * N[Tan[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 80.4%

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

   1. tan-sum 99.7%

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

   2. div-inv 99.7%

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

   3. fmm-def 99.7%

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

4. Applied egg-rr 99.7%

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

   1. fmm-undef 99.7%

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

6. Simplified 99.7%

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

Alternative 2: 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 80.4%

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

   1. +-commutative 80.4%

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

   2. sub-neg 80.4%

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

   3. associate-+l+ 80.3%

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

   4. tan-sum 99.6%

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

   5. div-inv 99.6%

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

   6. fma-define 99.6%

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

   7. neg-mul-1 99.6%

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

   8. fma-define 99.6%

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

4. Applied egg-rr 99.6%

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

   1. fma-undefine 99.6%

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

   2. fma-undefine 99.6%

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

   3. neg-mul-1 99.6%

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

   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/ 99.7%

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

   7. *-rgt-identity 99.7%

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

6. Simplified 99.7%

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

7. Final simplification 99.7%

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

Alternative 3: 89.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -0.00018:\\ \;\;\;\;x + \left(\tan y + \left(\tan z - \tan a\right)\right)\\ \mathbf{elif}\;a \leq 0.00115:\\ \;\;\;\;x + \left(\left(\tan y + \tan z\right) \cdot \frac{1}{1 - \tan y \cdot \tan z} - a\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(\tan \left(y + z\right) + \frac{-1}{\frac{\cos a}{\sin a}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (if (<= a -0.00018)
   (+ x (+ (tan y) (- (tan z) (tan a))))
   (if (<= a 0.00115)
     (+ x (- (* (+ (tan y) (tan z)) (/ 1.0 (- 1.0 (* (tan y) (tan z))))) a))
     (+ x (+ (tan (+ y z)) (/ -1.0 (/ (cos a) (sin a))))))))

C

double code(double x, double y, double z, double a) {
	double tmp;
	if (a <= -0.00018) {
		tmp = x + (tan(y) + (tan(z) - tan(a)));
	} else if (a <= 0.00115) {
		tmp = x + (((tan(y) + tan(z)) * (1.0 / (1.0 - (tan(y) * tan(z))))) - a);
	} else {
		tmp = x + (tan((y + z)) + (-1.0 / (cos(a) / sin(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 <= (-0.00018d0)) then
        tmp = x + (tan(y) + (tan(z) - tan(a)))
    else if (a <= 0.00115d0) then
        tmp = x + (((tan(y) + tan(z)) * (1.0d0 / (1.0d0 - (tan(y) * tan(z))))) - a)
    else
        tmp = x + (tan((y + z)) + ((-1.0d0) / (cos(a) / sin(a))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double a) {
	double tmp;
	if (a <= -0.00018) {
		tmp = x + (Math.tan(y) + (Math.tan(z) - Math.tan(a)));
	} else if (a <= 0.00115) {
		tmp = x + (((Math.tan(y) + Math.tan(z)) * (1.0 / (1.0 - (Math.tan(y) * Math.tan(z))))) - a);
	} else {
		tmp = x + (Math.tan((y + z)) + (-1.0 / (Math.cos(a) / Math.sin(a))));
	}
	return tmp;
}

Python

def code(x, y, z, a):
	tmp = 0
	if a <= -0.00018:
		tmp = x + (math.tan(y) + (math.tan(z) - math.tan(a)))
	elif a <= 0.00115:
		tmp = x + (((math.tan(y) + math.tan(z)) * (1.0 / (1.0 - (math.tan(y) * math.tan(z))))) - a)
	else:
		tmp = x + (math.tan((y + z)) + (-1.0 / (math.cos(a) / math.sin(a))))
	return tmp

Julia

function code(x, y, z, a)
	tmp = 0.0
	if (a <= -0.00018)
		tmp = Float64(x + Float64(tan(y) + Float64(tan(z) - tan(a))));
	elseif (a <= 0.00115)
		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(x + Float64(tan(Float64(y + z)) + Float64(-1.0 / Float64(cos(a) / sin(a)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, a)
	tmp = 0.0;
	if (a <= -0.00018)
		tmp = x + (tan(y) + (tan(z) - tan(a)));
	elseif (a <= 0.00115)
		tmp = x + (((tan(y) + tan(z)) * (1.0 / (1.0 - (tan(y) * tan(z))))) - a);
	else
		tmp = x + (tan((y + z)) + (-1.0 / (cos(a) / sin(a))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, a_] := If[LessEqual[a, -0.00018], N[(x + N[(N[Tan[y], $MachinePrecision] + N[(N[Tan[z], $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 0.00115], 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[(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. Split input into 3 regimes

2. if a < -1.80000000000000011e-4

   1. Initial program 78.7%

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

      1. tan-sum 99.7%

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

      2. clear-num 99.6%

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

   4. Applied egg-rr 99.6%

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

   5. Taylor expanded in y around 0 79.4%

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

      1. remove-double-div 79.5%

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

      2. associate-+r- 79.4%

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

   7. Applied egg-rr 79.4%

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

      1. associate-+r- 79.5%

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

      2. associate--l+ 79.5%

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

   9. Simplified 79.5%

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

   if -1.80000000000000011e-4 < a < 0.00115

   1. Initial program 79.1%

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

   3. Taylor expanded in a around 0 79.1%

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

      1. tan-sum 99.7%

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

      2. div-inv 99.7%

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

   5. Applied egg-rr 99.7%

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

   if 0.00115 < a

   1. Initial program 85.2%

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

      1. tan-quot 85.2%

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

      2. clear-num 85.2%

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

   4. Applied egg-rr 85.2%

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

4. Final simplification 90.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -0.00018:\\ \;\;\;\;x + \left(\tan y + \left(\tan z - \tan a\right)\right)\\ \mathbf{elif}\;a \leq 0.00115:\\ \;\;\;\;x + \left(\left(\tan y + \tan z\right) \cdot \frac{1}{1 - \tan y \cdot \tan z} - a\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(\tan \left(y + z\right) + \frac{-1}{\frac{\cos a}{\sin a}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 89.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -0.00038:\\ \;\;\;\;x + \left(\tan y + \left(\tan z - \tan a\right)\right)\\ \mathbf{elif}\;a \leq 0.00065:\\ \;\;\;\;x + \left(\frac{\tan y + \tan z}{1 - \tan y \cdot \tan z} - a\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(\tan \left(y + z\right) + \frac{-1}{\frac{\cos a}{\sin a}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (if (<= a -0.00038)
   (+ x (+ (tan y) (- (tan z) (tan a))))
   (if (<= a 0.00065)
     (+ x (- (/ (+ (tan y) (tan z)) (- 1.0 (* (tan y) (tan z)))) a))
     (+ x (+ (tan (+ y z)) (/ -1.0 (/ (cos a) (sin a))))))))

C

double code(double x, double y, double z, double a) {
	double tmp;
	if (a <= -0.00038) {
		tmp = x + (tan(y) + (tan(z) - tan(a)));
	} else if (a <= 0.00065) {
		tmp = x + (((tan(y) + tan(z)) / (1.0 - (tan(y) * tan(z)))) - a);
	} else {
		tmp = x + (tan((y + z)) + (-1.0 / (cos(a) / sin(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 <= (-0.00038d0)) then
        tmp = x + (tan(y) + (tan(z) - tan(a)))
    else if (a <= 0.00065d0) then
        tmp = x + (((tan(y) + tan(z)) / (1.0d0 - (tan(y) * tan(z)))) - a)
    else
        tmp = x + (tan((y + z)) + ((-1.0d0) / (cos(a) / sin(a))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double a) {
	double tmp;
	if (a <= -0.00038) {
		tmp = x + (Math.tan(y) + (Math.tan(z) - Math.tan(a)));
	} else if (a <= 0.00065) {
		tmp = x + (((Math.tan(y) + Math.tan(z)) / (1.0 - (Math.tan(y) * Math.tan(z)))) - a);
	} else {
		tmp = x + (Math.tan((y + z)) + (-1.0 / (Math.cos(a) / Math.sin(a))));
	}
	return tmp;
}

Python

def code(x, y, z, a):
	tmp = 0
	if a <= -0.00038:
		tmp = x + (math.tan(y) + (math.tan(z) - math.tan(a)))
	elif a <= 0.00065:
		tmp = x + (((math.tan(y) + math.tan(z)) / (1.0 - (math.tan(y) * math.tan(z)))) - a)
	else:
		tmp = x + (math.tan((y + z)) + (-1.0 / (math.cos(a) / math.sin(a))))
	return tmp

Julia

function code(x, y, z, a)
	tmp = 0.0
	if (a <= -0.00038)
		tmp = Float64(x + Float64(tan(y) + Float64(tan(z) - tan(a))));
	elseif (a <= 0.00065)
		tmp = Float64(x + Float64(Float64(Float64(tan(y) + tan(z)) / Float64(1.0 - Float64(tan(y) * tan(z)))) - a));
	else
		tmp = Float64(x + Float64(tan(Float64(y + z)) + Float64(-1.0 / Float64(cos(a) / sin(a)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, a)
	tmp = 0.0;
	if (a <= -0.00038)
		tmp = x + (tan(y) + (tan(z) - tan(a)));
	elseif (a <= 0.00065)
		tmp = x + (((tan(y) + tan(z)) / (1.0 - (tan(y) * tan(z)))) - a);
	else
		tmp = x + (tan((y + z)) + (-1.0 / (cos(a) / sin(a))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, a_] := If[LessEqual[a, -0.00038], N[(x + N[(N[Tan[y], $MachinePrecision] + N[(N[Tan[z], $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 0.00065], 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[(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. Split input into 3 regimes

2. if a < -3.8000000000000002e-4

   1. Initial program 78.7%

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

      1. tan-sum 99.7%

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

      2. clear-num 99.6%

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

   4. Applied egg-rr 99.6%

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

   5. Taylor expanded in y around 0 79.4%

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

      1. remove-double-div 79.5%

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

      2. associate-+r- 79.4%

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

   7. Applied egg-rr 79.4%

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

      1. associate-+r- 79.5%

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

      2. associate--l+ 79.5%

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

   9. Simplified 79.5%

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

   if -3.8000000000000002e-4 < a < 6.4999999999999997e-4

   1. Initial program 79.1%

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

      1. tan-sum 99.7%

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

      2. tan-quot 99.7%

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

      3. frac-sub 99.7%

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

   4. Applied egg-rr 99.7%

      \[\leadsto x + \color{blue}{\frac{\left(\tan y + \tan z\right) \cdot \cos a - \left(1 - \tan y \cdot \tan z\right) \cdot \sin a}{\left(1 - \tan y \cdot \tan z\right) \cdot \cos a}} \]
   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 99.7%

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

   7. Taylor expanded in a around 0 99.7%

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

   if 6.4999999999999997e-4 < a

   1. Initial program 85.2%

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

      1. tan-quot 85.2%

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

      2. clear-num 85.2%

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

   4. Applied egg-rr 85.2%

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

4. Final simplification 90.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -0.00038:\\ \;\;\;\;x + \left(\tan y + \left(\tan z - \tan a\right)\right)\\ \mathbf{elif}\;a \leq 0.00065:\\ \;\;\;\;x + \left(\frac{\tan y + \tan z}{1 - \tan y \cdot \tan z} - a\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(\tan \left(y + z\right) + \frac{-1}{\frac{\cos a}{\sin a}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 69.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\tan a \leq -0.02 \lor \neg \left(\tan a \leq 10^{-8}\right):\\ \;\;\;\;x + \left(\tan y - \tan a\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(\tan \left(y + z\right) - a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (if (or (<= (tan a) -0.02) (not (<= (tan a) 1e-8)))
   (+ x (- (tan y) (tan a)))
   (+ x (- (tan (+ y z)) a))))

C

double code(double x, double y, double z, double a) {
	double tmp;
	if ((tan(a) <= -0.02) || !(tan(a) <= 1e-8)) {
		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 ((tan(a) <= (-0.02d0)) .or. (.not. (tan(a) <= 1d-8))) 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 ((Math.tan(a) <= -0.02) || !(Math.tan(a) <= 1e-8)) {
		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 (math.tan(a) <= -0.02) or not (math.tan(a) <= 1e-8):
		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 ((tan(a) <= -0.02) || !(tan(a) <= 1e-8))
		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 ((tan(a) <= -0.02) || ~((tan(a) <= 1e-8)))
		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[N[Tan[a], $MachinePrecision], -0.02], N[Not[LessEqual[N[Tan[a], $MachinePrecision], 1e-8]], $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 (tan.f64 a) < -0.0200000000000000004 or 1e-8 < (tan.f64 a)

   1. Initial program 81.0%

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

   3. Taylor expanded in y around inf 58.5%

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

   if -0.0200000000000000004 < (tan.f64 a) < 1e-8

   1. Initial program 79.8%

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

   3. Taylor expanded in a around 0 79.3%

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

4. Final simplification 68.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\tan a \leq -0.02 \lor \neg \left(\tan a \leq 10^{-8}\right):\\ \;\;\;\;x + \left(\tan y - \tan a\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(\tan \left(y + z\right) - a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 60.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\tan a \leq -0.02 \lor \neg \left(\tan a \leq 10^{-8}\right):\\ \;\;\;\;x + \left(\sin z - \tan a\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(\tan \left(y + z\right) - a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (if (or (<= (tan a) -0.02) (not (<= (tan a) 1e-8)))
   (+ x (- (sin z) (tan a)))
   (+ x (- (tan (+ y z)) a))))

C

double code(double x, double y, double z, double a) {
	double tmp;
	if ((tan(a) <= -0.02) || !(tan(a) <= 1e-8)) {
		tmp = x + (sin(z) - 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 ((tan(a) <= (-0.02d0)) .or. (.not. (tan(a) <= 1d-8))) then
        tmp = x + (sin(z) - 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 ((Math.tan(a) <= -0.02) || !(Math.tan(a) <= 1e-8)) {
		tmp = x + (Math.sin(z) - Math.tan(a));
	} else {
		tmp = x + (Math.tan((y + z)) - a);
	}
	return tmp;
}

Python

def code(x, y, z, a):
	tmp = 0
	if (math.tan(a) <= -0.02) or not (math.tan(a) <= 1e-8):
		tmp = x + (math.sin(z) - math.tan(a))
	else:
		tmp = x + (math.tan((y + z)) - a)
	return tmp

Julia

function code(x, y, z, a)
	tmp = 0.0
	if ((tan(a) <= -0.02) || !(tan(a) <= 1e-8))
		tmp = Float64(x + Float64(sin(z) - 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 ((tan(a) <= -0.02) || ~((tan(a) <= 1e-8)))
		tmp = x + (sin(z) - tan(a));
	else
		tmp = x + (tan((y + z)) - a);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, a_] := If[Or[LessEqual[N[Tan[a], $MachinePrecision], -0.02], N[Not[LessEqual[N[Tan[a], $MachinePrecision], 1e-8]], $MachinePrecision]], N[(x + N[(N[Sin[z], $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 (tan.f64 a) < -0.0200000000000000004 or 1e-8 < (tan.f64 a)

   1. Initial program 81.0%

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

      1. tan-quot 80.9%

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

      2. div-inv 81.0%

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

      3. fmm-def 81.0%

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

   4. Applied egg-rr 81.0%

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

      1. fmm-undef 81.0%

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

   6. Simplified 81.0%

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

   7. Taylor expanded in z around 0 59.4%

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

   8. Taylor expanded in y around 0 41.3%

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

   if -0.0200000000000000004 < (tan.f64 a) < 1e-8

   1. Initial program 79.8%

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

   3. Taylor expanded in a around 0 79.3%

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

4. Final simplification 60.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\tan a \leq -0.02 \lor \neg \left(\tan a \leq 10^{-8}\right):\\ \;\;\;\;x + \left(\sin z - \tan a\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(\tan \left(y + z\right) - a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 79.9% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 80.4%

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

   1. tan-sum 99.7%

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

   2. clear-num 99.6%

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

4. Applied egg-rr 99.6%

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

5. Taylor expanded in y around 0 80.7%

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

   1. remove-double-div 80.7%

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

   2. associate-+r- 80.7%

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

7. Applied egg-rr 80.7%

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

   1. associate-+r- 80.7%

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

   2. associate--l+ 80.7%

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

9. Simplified 80.7%

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

Alternative 8: 50.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.9:\\ \;\;\;\;x\\ \mathbf{elif}\;a \leq 7.2 \cdot 10^{-5}:\\ \;\;\;\;x + \left(\tan \left(y + z\right) - a\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{{x}^{3}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (if (<= a -1.9)
   x
   (if (<= a 7.2e-5) (+ x (- (tan (+ y z)) a)) (cbrt (pow x 3.0)))))

C

double code(double x, double y, double z, double a) {
	double tmp;
	if (a <= -1.9) {
		tmp = x;
	} else if (a <= 7.2e-5) {
		tmp = x + (tan((y + z)) - a);
	} else {
		tmp = cbrt(pow(x, 3.0));
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double a) {
	double tmp;
	if (a <= -1.9) {
		tmp = x;
	} else if (a <= 7.2e-5) {
		tmp = x + (Math.tan((y + z)) - a);
	} else {
		tmp = Math.cbrt(Math.pow(x, 3.0));
	}
	return tmp;
}

Julia

function code(x, y, z, a)
	tmp = 0.0
	if (a <= -1.9)
		tmp = x;
	elseif (a <= 7.2e-5)
		tmp = Float64(x + Float64(tan(Float64(y + z)) - a));
	else
		tmp = cbrt((x ^ 3.0));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, a_] := If[LessEqual[a, -1.9], x, If[LessEqual[a, 7.2e-5], N[(x + N[(N[Tan[N[(y + z), $MachinePrecision]], $MachinePrecision] - a), $MachinePrecision]), $MachinePrecision], N[Power[N[Power[x, 3.0], $MachinePrecision], 1/3], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -1.8999999999999999

   1. Initial program 78.1%

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

   3. Taylor expanded in x around inf 21.1%

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

   if -1.8999999999999999 < a < 7.20000000000000018e-5

   1. Initial program 79.8%

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

   3. Taylor expanded in a around 0 79.3%

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

   if 7.20000000000000018e-5 < a

   1. Initial program 84.2%

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

      1. add-cbrt-cube 83.8%

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

      2. pow3 83.7%

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

      3. +-commutative 83.7%

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

      4. associate-+l- 83.5%

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

   4. Applied egg-rr 83.5%

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

   5. Taylor expanded in x around inf 22.3%

      \[\leadsto \sqrt[3]{\color{blue}{{x}^{3}}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 9: 60.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y + z \leq -5 \cdot 10^{-12}:\\ \;\;\;\;x + \left(\tan y - \tan a\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(\tan z - \tan a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (if (<= (+ y z) -5e-12) (+ x (- (tan y) (tan a))) (+ x (- (tan z) (tan a)))))

C

double code(double x, double y, double z, double a) {
	double tmp;
	if ((y + z) <= -5e-12) {
		tmp = x + (tan(y) - tan(a));
	} else {
		tmp = x + (tan(z) - tan(a));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: a
    real(8) :: tmp
    if ((y + z) <= (-5d-12)) then
        tmp = x + (tan(y) - tan(a))
    else
        tmp = x + (tan(z) - tan(a))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double a) {
	double tmp;
	if ((y + z) <= -5e-12) {
		tmp = x + (Math.tan(y) - Math.tan(a));
	} else {
		tmp = x + (Math.tan(z) - Math.tan(a));
	}
	return tmp;
}

Python

def code(x, y, z, a):
	tmp = 0
	if (y + z) <= -5e-12:
		tmp = x + (math.tan(y) - math.tan(a))
	else:
		tmp = x + (math.tan(z) - math.tan(a))
	return tmp

Julia

function code(x, y, z, a)
	tmp = 0.0
	if (Float64(y + z) <= -5e-12)
		tmp = Float64(x + Float64(tan(y) - tan(a)));
	else
		tmp = Float64(x + Float64(tan(z) - tan(a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, a)
	tmp = 0.0;
	if ((y + z) <= -5e-12)
		tmp = x + (tan(y) - tan(a));
	else
		tmp = x + (tan(z) - tan(a));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, a_] := If[LessEqual[N[(y + z), $MachinePrecision], -5e-12], N[(x + N[(N[Tan[y], $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[Tan[z], $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 y z) < -4.9999999999999997e-12

   1. Initial program 76.0%

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

   3. Taylor expanded in y around inf 47.9%

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

   if -4.9999999999999997e-12 < (+.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 67.6%

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

Alternative 10: 79.6% 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 80.4%

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

Alternative 11: 50.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.95:\\ \;\;\;\;x\\ \mathbf{elif}\;a \leq 7.2 \cdot 10^{-5}:\\ \;\;\;\;x + \left(\tan \left(y + z\right) - a\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (if (<= a -1.95) x (if (<= a 7.2e-5) (+ x (- (tan (+ y z)) a)) x)))

C

double code(double x, double y, double z, double a) {
	double tmp;
	if (a <= -1.95) {
		tmp = x;
	} else if (a <= 7.2e-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 <= (-1.95d0)) then
        tmp = x
    else if (a <= 7.2d-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 <= -1.95) {
		tmp = x;
	} else if (a <= 7.2e-5) {
		tmp = x + (Math.tan((y + z)) - a);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, a):
	tmp = 0
	if a <= -1.95:
		tmp = x
	elif a <= 7.2e-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 <= -1.95)
		tmp = x;
	elseif (a <= 7.2e-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 <= -1.95)
		tmp = x;
	elseif (a <= 7.2e-5)
		tmp = x + (tan((y + z)) - a);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, a_] := If[LessEqual[a, -1.95], x, If[LessEqual[a, 7.2e-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 < -1.94999999999999996 or 7.20000000000000018e-5 < a

   1. Initial program 81.0%

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

   3. Taylor expanded in x around inf 21.6%

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

   if -1.94999999999999996 < a < 7.20000000000000018e-5

   1. Initial program 79.8%

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

   3. Taylor expanded in a around 0 79.3%

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

Alternative 12: 40.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.5 \cdot 10^{-15}:\\ \;\;\;\;x\\ \mathbf{elif}\;a \leq 4.4 \cdot 10^{-31}:\\ \;\;\;\;x + \left(\tan y - a\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (if (<= a -1.5e-15) x (if (<= a 4.4e-31) (+ x (- (tan y) a)) x)))

C

double code(double x, double y, double z, double a) {
	double tmp;
	if (a <= -1.5e-15) {
		tmp = x;
	} else if (a <= 4.4e-31) {
		tmp = x + (tan(y) - 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 <= (-1.5d-15)) then
        tmp = x
    else if (a <= 4.4d-31) then
        tmp = x + (tan(y) - 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 <= -1.5e-15) {
		tmp = x;
	} else if (a <= 4.4e-31) {
		tmp = x + (Math.tan(y) - a);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, a):
	tmp = 0
	if a <= -1.5e-15:
		tmp = x
	elif a <= 4.4e-31:
		tmp = x + (math.tan(y) - a)
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, a)
	tmp = 0.0
	if (a <= -1.5e-15)
		tmp = x;
	elseif (a <= 4.4e-31)
		tmp = Float64(x + Float64(tan(y) - a));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, a)
	tmp = 0.0;
	if (a <= -1.5e-15)
		tmp = x;
	elseif (a <= 4.4e-31)
		tmp = x + (tan(y) - a);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, a_] := If[LessEqual[a, -1.5e-15], x, If[LessEqual[a, 4.4e-31], N[(x + N[(N[Tan[y], $MachinePrecision] - a), $MachinePrecision]), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if a < -1.5e-15 or 4.40000000000000019e-31 < a

   1. Initial program 80.2%

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

   3. Taylor expanded in x around inf 22.4%

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

   if -1.5e-15 < a < 4.40000000000000019e-31

   1. Initial program 80.6%

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

   3. Taylor expanded in a around 0 80.6%

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

   4. Taylor expanded in y around inf 59.3%

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

Alternative 13: 36.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 1.6 \cdot 10^{-7}:\\ \;\;\;\;x + \left(\tan y - a\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(\tan z - a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (if (<= z 1.6e-7) (+ x (- (tan y) a)) (+ x (- (tan z) a))))

C

double code(double x, double y, double z, double a) {
	double tmp;
	if (z <= 1.6e-7) {
		tmp = x + (tan(y) - a);
	} 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 (z <= 1.6d-7) then
        tmp = x + (tan(y) - a)
    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 (z <= 1.6e-7) {
		tmp = x + (Math.tan(y) - a);
	} else {
		tmp = x + (Math.tan(z) - a);
	}
	return tmp;
}

Python

def code(x, y, z, a):
	tmp = 0
	if z <= 1.6e-7:
		tmp = x + (math.tan(y) - a)
	else:
		tmp = x + (math.tan(z) - a)
	return tmp

Julia

function code(x, y, z, a)
	tmp = 0.0
	if (z <= 1.6e-7)
		tmp = Float64(x + Float64(tan(y) - a));
	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 (z <= 1.6e-7)
		tmp = x + (tan(y) - a);
	else
		tmp = x + (tan(z) - a);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, a_] := If[LessEqual[z, 1.6e-7], N[(x + N[(N[Tan[y], $MachinePrecision] - a), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[Tan[z], $MachinePrecision] - a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 1.6e-7

   1. Initial program 88.6%

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

   3. Taylor expanded in a around 0 45.7%

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

   4. Taylor expanded in y around inf 38.1%

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

   if 1.6e-7 < z

   1. Initial program 59.9%

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

   3. Taylor expanded in a around 0 28.6%

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

   4. Taylor expanded in y around 0 28.6%

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

Alternative 14: 31.4% 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 80.4%

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

3. Taylor expanded in x around inf 30.4%

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

Reproduce

herbie shell --seed 2024160 
(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))))