tan-example (used to crash)

Percentage Accurate: 79.1% → 99.5%

Time: 50.5s

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: 79.1% 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: 99.5% accurate, 0.2× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z, double a) {
	double t_0 = 1.0 - (tan(y) * tan(z));
	return x + (log(exp((((tan(y) + tan(z)) * cos(a)) - (t_0 * sin(a))))) / (cos(a) * 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 = 1.0d0 - (tan(y) * tan(z))
    code = x + (log(exp((((tan(y) + tan(z)) * cos(a)) - (t_0 * sin(a))))) / (cos(a) * t_0))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.1%

   \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
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. add-log-exp 99.7%

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

6. Applied egg-rr 99.7%

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

7. Final simplification 99.7%

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

Alternative 2: 99.7% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.1%

   \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
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. Final simplification 99.7%

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

Alternative 3: 89.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan \left(y + z\right)\\ \mathbf{if}\;\tan a \leq -0.002:\\ \;\;\;\;x + \left(t_0 - \frac{\sin a}{\cos a}\right)\\ \mathbf{elif}\;\tan a \leq 4 \cdot 10^{-10}:\\ \;\;\;\;x + \left(\frac{\tan y + \tan z}{1 - \tan y \cdot \tan z} - a\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(t_0 - \tan a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (let* ((t_0 (tan (+ y z))))
   (if (<= (tan a) -0.002)
     (+ x (- t_0 (/ (sin a) (cos a))))
     (if (<= (tan a) 4e-10)
       (+ x (- (/ (+ (tan y) (tan z)) (- 1.0 (* (tan y) (tan z)))) a))
       (+ x (- t_0 (tan a)))))))

C

double code(double x, double y, double z, double a) {
	double t_0 = tan((y + z));
	double tmp;
	if (tan(a) <= -0.002) {
		tmp = x + (t_0 - (sin(a) / cos(a)));
	} else if (tan(a) <= 4e-10) {
		tmp = x + (((tan(y) + tan(z)) / (1.0 - (tan(y) * tan(z)))) - a);
	} else {
		tmp = x + (t_0 - 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) <= (-0.002d0)) then
        tmp = x + (t_0 - (sin(a) / cos(a)))
    else if (tan(a) <= 4d-10) then
        tmp = x + (((tan(y) + tan(z)) / (1.0d0 - (tan(y) * tan(z)))) - a)
    else
        tmp = x + (t_0 - 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) <= -0.002) {
		tmp = x + (t_0 - (Math.sin(a) / Math.cos(a)));
	} else if (Math.tan(a) <= 4e-10) {
		tmp = x + (((Math.tan(y) + Math.tan(z)) / (1.0 - (Math.tan(y) * Math.tan(z)))) - a);
	} else {
		tmp = x + (t_0 - Math.tan(a));
	}
	return tmp;
}

Python

def code(x, y, z, a):
	t_0 = math.tan((y + z))
	tmp = 0
	if math.tan(a) <= -0.002:
		tmp = x + (t_0 - (math.sin(a) / math.cos(a)))
	elif math.tan(a) <= 4e-10:
		tmp = x + (((math.tan(y) + math.tan(z)) / (1.0 - (math.tan(y) * math.tan(z)))) - a)
	else:
		tmp = x + (t_0 - math.tan(a))
	return tmp

Julia

function code(x, y, z, a)
	t_0 = tan(Float64(y + z))
	tmp = 0.0
	if (tan(a) <= -0.002)
		tmp = Float64(x + Float64(t_0 - Float64(sin(a) / cos(a))));
	elseif (tan(a) <= 4e-10)
		tmp = Float64(x + Float64(Float64(Float64(tan(y) + tan(z)) / Float64(1.0 - Float64(tan(y) * tan(z)))) - a));
	else
		tmp = Float64(x + Float64(t_0 - 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) <= -0.002)
		tmp = x + (t_0 - (sin(a) / cos(a)));
	elseif (tan(a) <= 4e-10)
		tmp = x + (((tan(y) + tan(z)) / (1.0 - (tan(y) * tan(z)))) - a);
	else
		tmp = x + (t_0 - 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], -0.002], N[(x + N[(t$95$0 - N[(N[Sin[a], $MachinePrecision] / N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Tan[a], $MachinePrecision], 4e-10], 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[(t$95$0 - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (tan.f64 a) < -2e-3

   1. Initial program 73.8%

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

   3. Taylor expanded in a around inf 73.9%

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

   if -2e-3 < (tan.f64 a) < 4.00000000000000015e-10

   1. Initial program 77.5%

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

   3. Taylor expanded in a around 0 77.5%

      \[\leadsto x + \left(\tan \left(y + z\right) - \color{blue}{a}\right) \]
   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 4.00000000000000015e-10 < (tan.f64 a)

   1. Initial program 79.4%

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

4. Final simplification 87.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\tan a \leq -0.002:\\ \;\;\;\;x + \left(\tan \left(y + z\right) - \frac{\sin a}{\cos a}\right)\\ \mathbf{elif}\;\tan a \leq 4 \cdot 10^{-10}:\\ \;\;\;\;x + \left(\frac{\tan y + \tan z}{1 - \tan y \cdot \tan z} - a\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(\tan \left(y + z\right) - \tan a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.7% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.1%

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

   1. tan-sum 47.1%

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

   2. div-inv 47.1%

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

4. Applied egg-rr 99.6%

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

   1. associate-*r/ 47.1%

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

   2. *-rgt-identity 47.1%

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

6. Simplified 99.7%

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

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 5: 79.1% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.1%

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

3. Taylor expanded in a around inf 77.1%

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

4. Final simplification 77.1%

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

Alternative 6: 68.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -3.5 \cdot 10^{-48} \lor \neg \left(a \leq 2.3 \cdot 10^{-10}\right):\\ \;\;\;\;x + \left(\tan y - \tan a\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(\tan \left(y + z\right) - a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (if (or (<= a -3.5e-48) (not (<= a 2.3e-10)))
   (+ 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 <= -3.5e-48) || !(a <= 2.3e-10)) {
		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 <= (-3.5d-48)) .or. (.not. (a <= 2.3d-10))) 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 <= -3.5e-48) || !(a <= 2.3e-10)) {
		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 <= -3.5e-48) or not (a <= 2.3e-10):
		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 <= -3.5e-48) || !(a <= 2.3e-10))
		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 <= -3.5e-48) || ~((a <= 2.3e-10)))
		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, -3.5e-48], N[Not[LessEqual[a, 2.3e-10]], $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 < -3.49999999999999991e-48 or 2.30000000000000007e-10 < a

   1. Initial program 76.2%

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

      1. add-sqr-sqrt 35.2%

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

      2. sqrt-unprod 46.3%

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

      3. pow2 46.3%

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

   4. Applied egg-rr 46.3%

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

   5. Taylor expanded in z around 0 41.3%

      \[\leadsto x + \sqrt{{\left(\color{blue}{\frac{\sin y}{\cos y}} - \tan a\right)}^{2}} \]
   6. Step-by-step derivation

      1. sqrt-pow1 60.6%

         \[\leadsto x + \color{blue}{{\left(\frac{\sin y}{\cos y} - \tan a\right)}^{\left(\frac{2}{2}\right)}} \]

      2. tan-quot 60.6%

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

      3. metadata-eval 60.6%

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

      4. pow1 60.6%

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

      5. sub-neg 60.6%

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

   7. Applied egg-rr 60.6%

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

      1. sub-neg 60.6%

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

   9. Simplified 60.6%

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

   if -3.49999999999999991e-48 < a < 2.30000000000000007e-10

   1. Initial program 78.2%

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

   3. Taylor expanded in a around 0 78.2%

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

4. Final simplification 68.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -3.5 \cdot 10^{-48} \lor \neg \left(a \leq 2.3 \cdot 10^{-10}\right):\\ \;\;\;\;x + \left(\tan y - \tan a\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(\tan \left(y + z\right) - a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 50.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -11:\\ \;\;\;\;\mathsf{expm1}\left(\mathsf{log1p}\left(x\right)\right)\\ \mathbf{elif}\;a \leq 1.56:\\ \;\;\;\;\left(x + \tan \left(y + z\right)\right) - a\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{{x}^{3}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (if (<= a -11.0)
   (expm1 (log1p x))
   (if (<= a 1.56) (- (+ 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 <= -11.0) {
		tmp = expm1(log1p(x));
	} else if (a <= 1.56) {
		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 <= -11.0) {
		tmp = Math.expm1(Math.log1p(x));
	} else if (a <= 1.56) {
		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 <= -11.0)
		tmp = expm1(log1p(x));
	elseif (a <= 1.56)
		tmp = Float64(Float64(x + tan(Float64(y + z))) - a);
	else
		tmp = cbrt((x ^ 3.0));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, a_] := If[LessEqual[a, -11.0], N[(Exp[N[Log[1 + x], $MachinePrecision]] - 1), $MachinePrecision], If[LessEqual[a, 1.56], N[(N[(x + N[Tan[N[(y + z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision], N[Power[N[Power[x, 3.0], $MachinePrecision], 1/3], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -11

   1. Initial program 77.7%

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

   3. Taylor expanded in a around 0 4.8%

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

      1. expm1-log1p-u 4.8%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x + \left(\tan \left(y + z\right) - a\right)\right)\right)} \]

   5. Applied egg-rr 4.8%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x + \left(\tan \left(y + z\right) - a\right)\right)\right)} \]

   6. Taylor expanded in x around inf 23.0%

      \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{x}\right)\right) \]

   if -11 < a < 1.5600000000000001

   1. Initial program 78.2%

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

   3. Taylor expanded in a around 0 76.4%

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

      1. associate-+r- 76.4%

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

   5. Applied egg-rr 76.4%

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

   if 1.5600000000000001 < a

   1. Initial program 74.5%

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

      1. add-cbrt-cube 74.0%

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

      2. pow3 74.0%

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

      3. +-commutative 74.0%

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

      4. associate-+l- 74.1%

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

   4. Applied egg-rr 74.1%

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

   5. Taylor expanded in x around inf 23.2%

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

4. Final simplification 47.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -11:\\ \;\;\;\;\mathsf{expm1}\left(\mathsf{log1p}\left(x\right)\right)\\ \mathbf{elif}\;a \leq 1.56:\\ \;\;\;\;\left(x + \tan \left(y + z\right)\right) - a\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{{x}^{3}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 64.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 120:\\ \;\;\;\;x + \left(\tan y - \tan a\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(\tan a + \tan \left(y + z\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (if (<= z 120.0) (+ x (- (tan y) (tan a))) (+ x (+ (tan a) (tan (+ y z))))))

C

double code(double x, double y, double z, double a) {
	double tmp;
	if (z <= 120.0) {
		tmp = x + (tan(y) - tan(a));
	} else {
		tmp = x + (tan(a) + tan((y + z)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z <= 120.0d0) then
        tmp = x + (tan(y) - tan(a))
    else
        tmp = x + (tan(a) + tan((y + z)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double a) {
	double tmp;
	if (z <= 120.0) {
		tmp = x + (Math.tan(y) - Math.tan(a));
	} else {
		tmp = x + (Math.tan(a) + Math.tan((y + z)));
	}
	return tmp;
}

Python

def code(x, y, z, a):
	tmp = 0
	if z <= 120.0:
		tmp = x + (math.tan(y) - math.tan(a))
	else:
		tmp = x + (math.tan(a) + math.tan((y + z)))
	return tmp

Julia

function code(x, y, z, a)
	tmp = 0.0
	if (z <= 120.0)
		tmp = Float64(x + Float64(tan(y) - tan(a)));
	else
		tmp = Float64(x + Float64(tan(a) + tan(Float64(y + z))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, a)
	tmp = 0.0;
	if (z <= 120.0)
		tmp = x + (tan(y) - tan(a));
	else
		tmp = x + (tan(a) + tan((y + z)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, a_] := If[LessEqual[z, 120.0], N[(x + N[(N[Tan[y], $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[Tan[a], $MachinePrecision] + N[Tan[N[(y + z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 120

   1. Initial program 84.1%

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

      1. add-sqr-sqrt 42.9%

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

      2. sqrt-unprod 58.5%

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

      3. pow2 58.5%

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

   4. Applied egg-rr 58.5%

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

   5. Taylor expanded in z around 0 53.7%

      \[\leadsto x + \sqrt{{\left(\color{blue}{\frac{\sin y}{\cos y}} - \tan a\right)}^{2}} \]
   6. Step-by-step derivation

      1. sqrt-pow1 74.3%

         \[\leadsto x + \color{blue}{{\left(\frac{\sin y}{\cos y} - \tan a\right)}^{\left(\frac{2}{2}\right)}} \]

      2. tan-quot 74.3%

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

      3. metadata-eval 74.3%

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

      4. pow1 74.3%

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

      5. sub-neg 74.3%

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

   7. Applied egg-rr 74.3%

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

      1. sub-neg 74.3%

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

   9. Simplified 74.3%

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

   if 120 < z

   1. Initial program 58.3%

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

      1. sub-neg 58.3%

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

   4. Applied egg-rr 58.3%

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

      1. rem-square-sqrt 28.1%

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

      2. fabs-sqr 28.1%

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

      3. rem-square-sqrt 47.5%

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

      4. fabs-neg 47.5%

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

      5. rem-square-sqrt 19.4%

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

      6. fabs-sqr 19.4%

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

      7. rem-square-sqrt 40.7%

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

      8. +-commutative 40.7%

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

   6. Simplified 40.7%

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

4. Final simplification 65.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 120:\\ \;\;\;\;x + \left(\tan y - \tan a\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(\tan a + \tan \left(y + z\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 79.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.1%

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

3. Final simplification 77.1%

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

Alternative 10: 50.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -11:\\ \;\;\;\;\mathsf{expm1}\left(\mathsf{log1p}\left(x\right)\right)\\ \mathbf{elif}\;a \leq 1.56:\\ \;\;\;\;\left(x + \tan \left(y + z\right)\right) - a\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (if (<= a -11.0)
   (expm1 (log1p x))
   (if (<= a 1.56) (- (+ x (tan (+ y z))) a) x)))

C

double code(double x, double y, double z, double a) {
	double tmp;
	if (a <= -11.0) {
		tmp = expm1(log1p(x));
	} else if (a <= 1.56) {
		tmp = (x + tan((y + z))) - a;
	} else {
		tmp = x;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double a) {
	double tmp;
	if (a <= -11.0) {
		tmp = Math.expm1(Math.log1p(x));
	} else if (a <= 1.56) {
		tmp = (x + Math.tan((y + z))) - a;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, a):
	tmp = 0
	if a <= -11.0:
		tmp = math.expm1(math.log1p(x))
	elif a <= 1.56:
		tmp = (x + math.tan((y + z))) - a
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, a)
	tmp = 0.0
	if (a <= -11.0)
		tmp = expm1(log1p(x));
	elseif (a <= 1.56)
		tmp = Float64(Float64(x + tan(Float64(y + z))) - a);
	else
		tmp = x;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, a_] := If[LessEqual[a, -11.0], N[(Exp[N[Log[1 + x], $MachinePrecision]] - 1), $MachinePrecision], If[LessEqual[a, 1.56], N[(N[(x + N[Tan[N[(y + z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision], x]]

Derivation

1. Split input into 3 regimes

2. if a < -11

   1. Initial program 77.7%

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

   3. Taylor expanded in a around 0 4.8%

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

      1. expm1-log1p-u 4.8%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x + \left(\tan \left(y + z\right) - a\right)\right)\right)} \]

   5. Applied egg-rr 4.8%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x + \left(\tan \left(y + z\right) - a\right)\right)\right)} \]

   6. Taylor expanded in x around inf 23.0%

      \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{x}\right)\right) \]

   if -11 < a < 1.5600000000000001

   1. Initial program 78.2%

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

   3. Taylor expanded in a around 0 76.4%

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

      1. associate-+r- 76.4%

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

   5. Applied egg-rr 76.4%

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

   if 1.5600000000000001 < a

   1. Initial program 74.5%

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

   3. Taylor expanded in x around inf 23.2%

      \[\leadsto \color{blue}{x} \]
3. Recombined 3 regimes into one program.

4. Final simplification 47.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -11:\\ \;\;\;\;\mathsf{expm1}\left(\mathsf{log1p}\left(x\right)\right)\\ \mathbf{elif}\;a \leq 1.56:\\ \;\;\;\;\left(x + \tan \left(y + z\right)\right) - a\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 50.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -11:\\ \;\;\;\;x\\ \mathbf{elif}\;a \leq 1.56:\\ \;\;\;\;x + \left(\tan \left(y + z\right) - a\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (if (<= a -11.0) x (if (<= a 1.56) (+ x (- (tan (+ y z)) a)) x)))

C

double code(double x, double y, double z, double a) {
	double tmp;
	if (a <= -11.0) {
		tmp = x;
	} else if (a <= 1.56) {
		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 <= (-11.0d0)) then
        tmp = x
    else if (a <= 1.56d0) 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 <= -11.0) {
		tmp = x;
	} else if (a <= 1.56) {
		tmp = x + (Math.tan((y + z)) - a);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, a):
	tmp = 0
	if a <= -11.0:
		tmp = x
	elif a <= 1.56:
		tmp = x + (math.tan((y + z)) - a)
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, a)
	tmp = 0.0
	if (a <= -11.0)
		tmp = x;
	elseif (a <= 1.56)
		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 <= -11.0)
		tmp = x;
	elseif (a <= 1.56)
		tmp = x + (tan((y + z)) - a);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, a_] := If[LessEqual[a, -11.0], x, If[LessEqual[a, 1.56], N[(x + N[(N[Tan[N[(y + z), $MachinePrecision]], $MachinePrecision] - a), $MachinePrecision]), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if a < -11 or 1.5600000000000001 < a

   1. Initial program 76.1%

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

   3. Taylor expanded in x around inf 23.1%

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

   if -11 < a < 1.5600000000000001

   1. Initial program 78.2%

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

   3. Taylor expanded in a around 0 76.4%

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

4. Final simplification 47.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -11:\\ \;\;\;\;x\\ \mathbf{elif}\;a \leq 1.56:\\ \;\;\;\;x + \left(\tan \left(y + z\right) - a\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 50.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -11:\\ \;\;\;\;x\\ \mathbf{elif}\;a \leq 1.56:\\ \;\;\;\;\left(x + \tan \left(y + z\right)\right) - a\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (if (<= a -11.0) x (if (<= a 1.56) (- (+ x (tan (+ y z))) a) x)))

C

double code(double x, double y, double z, double a) {
	double tmp;
	if (a <= -11.0) {
		tmp = x;
	} else if (a <= 1.56) {
		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 <= (-11.0d0)) then
        tmp = x
    else if (a <= 1.56d0) 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 <= -11.0) {
		tmp = x;
	} else if (a <= 1.56) {
		tmp = (x + Math.tan((y + z))) - a;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, a):
	tmp = 0
	if a <= -11.0:
		tmp = x
	elif a <= 1.56:
		tmp = (x + math.tan((y + z))) - a
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, a)
	tmp = 0.0
	if (a <= -11.0)
		tmp = x;
	elseif (a <= 1.56)
		tmp = Float64(Float64(x + 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 <= -11.0)
		tmp = x;
	elseif (a <= 1.56)
		tmp = (x + tan((y + z))) - a;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, a_] := If[LessEqual[a, -11.0], x, If[LessEqual[a, 1.56], N[(N[(x + N[Tan[N[(y + z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if a < -11 or 1.5600000000000001 < a

   1. Initial program 76.1%

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

   3. Taylor expanded in x around inf 23.1%

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

   if -11 < a < 1.5600000000000001

   1. Initial program 78.2%

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

   3. Taylor expanded in a around 0 76.4%

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

      1. associate-+r- 76.4%

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

   5. Applied egg-rr 76.4%

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

4. Final simplification 47.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -11:\\ \;\;\;\;x\\ \mathbf{elif}\;a \leq 1.56:\\ \;\;\;\;\left(x + \tan \left(y + z\right)\right) - a\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 41.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -11:\\ \;\;\;\;x\\ \mathbf{elif}\;a \leq 1.56:\\ \;\;\;\;x + \left(\tan z - a\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (if (<= a -11.0) x (if (<= a 1.56) (+ x (- (tan z) a)) x)))

C

double code(double x, double y, double z, double a) {
	double tmp;
	if (a <= -11.0) {
		tmp = x;
	} else if (a <= 1.56) {
		tmp = x + (tan(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 <= (-11.0d0)) then
        tmp = x
    else if (a <= 1.56d0) then
        tmp = x + (tan(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 <= -11.0) {
		tmp = x;
	} else if (a <= 1.56) {
		tmp = x + (Math.tan(z) - a);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, a):
	tmp = 0
	if a <= -11.0:
		tmp = x
	elif a <= 1.56:
		tmp = x + (math.tan(z) - a)
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, a)
	tmp = 0.0
	if (a <= -11.0)
		tmp = x;
	elseif (a <= 1.56)
		tmp = Float64(x + Float64(tan(z) - a));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, a)
	tmp = 0.0;
	if (a <= -11.0)
		tmp = x;
	elseif (a <= 1.56)
		tmp = x + (tan(z) - a);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, a_] := If[LessEqual[a, -11.0], x, If[LessEqual[a, 1.56], N[(x + N[(N[Tan[z], $MachinePrecision] - a), $MachinePrecision]), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if a < -11 or 1.5600000000000001 < a

   1. Initial program 76.1%

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

   3. Taylor expanded in x around inf 23.1%

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

   if -11 < a < 1.5600000000000001

   1. Initial program 78.2%

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

   3. Taylor expanded in a around 0 76.4%

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

   4. Taylor expanded in y around 0 54.6%

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

      1. tan-quot 54.6%

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

      2. associate--l+ 54.6%

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

   6. Applied egg-rr 54.6%

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

4. Final simplification 37.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -11:\\ \;\;\;\;x\\ \mathbf{elif}\;a \leq 1.56:\\ \;\;\;\;x + \left(\tan z - a\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 31.6% 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 77.1%

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

3. Taylor expanded in x around inf 30.0%

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

4. Final simplification 30.0%

   \[\leadsto x \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024024 
(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))))