tan-example (used to crash)

Percentage Accurate: 79.7% → 99.7%

Time: 42.9s

Alternatives: 15

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 79.7% 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, 0.3× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: a
    real(8) :: t_0
    t_0 = tan(y) * tan(z)
    code = x + (((tan(y) + tan(z)) / ((1.0d0 - (t_0 ** 2.0d0)) / (1.0d0 + t_0))) - tan(a))
end function

Java

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

Python

def code(x, y, z, a):
	t_0 = math.tan(y) * math.tan(z)
	return x + (((math.tan(y) + math.tan(z)) / ((1.0 - math.pow(t_0, 2.0)) / (1.0 + t_0))) - math.tan(a))

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.7%

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

   1. tan-sum 99.8%

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

   2. div-inv 99.8%

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

4. Applied egg-rr 99.8%

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

   1. associate-*r/ 99.8%

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

   2. *-rgt-identity 99.8%

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

6. Simplified 99.8%

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

   1. flip-- 99.8%

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

   2. metadata-eval 99.8%

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

   3. pow2 99.8%

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

   4. +-commutative 99.8%

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

8. Applied egg-rr 99.8%

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

9. Final simplification 99.8%

   \[\leadsto x + \left(\frac{\tan y + \tan z}{\frac{1 - {\left(\tan y \cdot \tan z\right)}^{2}}{1 + \tan y \cdot \tan z}} - \tan a\right) \]
10. Add Preprocessing

Alternative 2: 99.6% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.7%

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

   1. tan-sum 99.8%

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

   2. div-inv 99.8%

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

4. Applied egg-rr 99.8%

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

   1. associate-*r/ 99.8%

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

   2. *-rgt-identity 99.8%

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

6. Simplified 99.8%

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

   1. add-log-exp 99.8%

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

8. Applied egg-rr 99.8%

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

9. Final simplification 99.8%

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

Alternative 3: 89.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan y + \tan z\\ \mathbf{if}\;\tan a \leq -0.005 \lor \neg \left(\tan a \leq 10^{-12}\right):\\ \;\;\;\;x + \left(t_0 - \tan a\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(\frac{t_0}{1 - \tan y \cdot \tan z} - a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (let* ((t_0 (+ (tan y) (tan z))))
   (if (or (<= (tan a) -0.005) (not (<= (tan a) 1e-12)))
     (+ x (- t_0 (tan a)))
     (+ x (- (/ t_0 (- 1.0 (* (tan y) (tan z)))) a)))))

C

double code(double x, double y, double z, double a) {
	double t_0 = tan(y) + tan(z);
	double tmp;
	if ((tan(a) <= -0.005) || !(tan(a) <= 1e-12)) {
		tmp = x + (t_0 - tan(a));
	} else {
		tmp = x + ((t_0 / (1.0 - (tan(y) * 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) :: t_0
    real(8) :: tmp
    t_0 = tan(y) + tan(z)
    if ((tan(a) <= (-0.005d0)) .or. (.not. (tan(a) <= 1d-12))) then
        tmp = x + (t_0 - tan(a))
    else
        tmp = x + ((t_0 / (1.0d0 - (tan(y) * tan(z)))) - 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) + Math.tan(z);
	double tmp;
	if ((Math.tan(a) <= -0.005) || !(Math.tan(a) <= 1e-12)) {
		tmp = x + (t_0 - Math.tan(a));
	} else {
		tmp = x + ((t_0 / (1.0 - (Math.tan(y) * Math.tan(z)))) - a);
	}
	return tmp;
}

Python

def code(x, y, z, a):
	t_0 = math.tan(y) + math.tan(z)
	tmp = 0
	if (math.tan(a) <= -0.005) or not (math.tan(a) <= 1e-12):
		tmp = x + (t_0 - math.tan(a))
	else:
		tmp = x + ((t_0 / (1.0 - (math.tan(y) * math.tan(z)))) - a)
	return tmp

Julia

function code(x, y, z, a)
	t_0 = Float64(tan(y) + tan(z))
	tmp = 0.0
	if ((tan(a) <= -0.005) || !(tan(a) <= 1e-12))
		tmp = Float64(x + Float64(t_0 - tan(a)));
	else
		tmp = Float64(x + Float64(Float64(t_0 / Float64(1.0 - Float64(tan(y) * tan(z)))) - a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, a)
	t_0 = tan(y) + tan(z);
	tmp = 0.0;
	if ((tan(a) <= -0.005) || ~((tan(a) <= 1e-12)))
		tmp = x + (t_0 - tan(a));
	else
		tmp = x + ((t_0 / (1.0 - (tan(y) * tan(z)))) - a);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, a_] := Block[{t$95$0 = N[(N[Tan[y], $MachinePrecision] + N[Tan[z], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[N[Tan[a], $MachinePrecision], -0.005], N[Not[LessEqual[N[Tan[a], $MachinePrecision], 1e-12]], $MachinePrecision]], N[(x + N[(t$95$0 - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[(t$95$0 / N[(1.0 - N[(N[Tan[y], $MachinePrecision] * N[Tan[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (tan.f64 a) < -0.0050000000000000001 or 9.9999999999999998e-13 < (tan.f64 a)

   1. Initial program 75.6%

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

      1. tan-sum 99.6%

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

      2. div-inv 99.6%

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

   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/ 99.6%

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

      2. *-rgt-identity 99.6%

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

   6. Simplified 99.6%

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

   7. Taylor expanded in y around 0 76.1%

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

   if -0.0050000000000000001 < (tan.f64 a) < 9.9999999999999998e-13

   1. Initial program 79.7%

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

   3. Taylor expanded in a around 0 79.7%

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

      1. tan-sum 99.9%

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

      2. div-inv 99.9%

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

      3. fma-neg 99.9%

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

   5. Applied egg-rr 99.9%

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

      1. fma-udef 99.9%

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

      2. unsub-neg 99.9%

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

      3. associate-*r/ 99.9%

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

      4. *-rgt-identity 99.9%

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

   7. Simplified 99.9%

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

4. Final simplification 88.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\tan a \leq -0.005 \lor \neg \left(\tan a \leq 10^{-12}\right):\\ \;\;\;\;x + \left(\left(\tan y + \tan z\right) - \tan a\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(\frac{\tan y + \tan z}{1 - \tan y \cdot \tan z} - a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 89.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan y + \tan z\\ \mathbf{if}\;\tan a \leq -0.005 \lor \neg \left(\tan a \leq 2 \cdot 10^{-6}\right):\\ \;\;\;\;x + \left(t_0 - \tan a\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{1 - \tan y \cdot \tan z} + \left(x - a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (let* ((t_0 (+ (tan y) (tan z))))
   (if (or (<= (tan a) -0.005) (not (<= (tan a) 2e-6)))
     (+ x (- t_0 (tan a)))
     (+ (/ t_0 (- 1.0 (* (tan y) (tan z)))) (- x a)))))

C

double code(double x, double y, double z, double a) {
	double t_0 = tan(y) + tan(z);
	double tmp;
	if ((tan(a) <= -0.005) || !(tan(a) <= 2e-6)) {
		tmp = x + (t_0 - tan(a));
	} else {
		tmp = (t_0 / (1.0 - (tan(y) * tan(z)))) + (x - 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) + tan(z)
    if ((tan(a) <= (-0.005d0)) .or. (.not. (tan(a) <= 2d-6))) then
        tmp = x + (t_0 - tan(a))
    else
        tmp = (t_0 / (1.0d0 - (tan(y) * tan(z)))) + (x - 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) + Math.tan(z);
	double tmp;
	if ((Math.tan(a) <= -0.005) || !(Math.tan(a) <= 2e-6)) {
		tmp = x + (t_0 - Math.tan(a));
	} else {
		tmp = (t_0 / (1.0 - (Math.tan(y) * Math.tan(z)))) + (x - a);
	}
	return tmp;
}

Python

def code(x, y, z, a):
	t_0 = math.tan(y) + math.tan(z)
	tmp = 0
	if (math.tan(a) <= -0.005) or not (math.tan(a) <= 2e-6):
		tmp = x + (t_0 - math.tan(a))
	else:
		tmp = (t_0 / (1.0 - (math.tan(y) * math.tan(z)))) + (x - a)
	return tmp

Julia

function code(x, y, z, a)
	t_0 = Float64(tan(y) + tan(z))
	tmp = 0.0
	if ((tan(a) <= -0.005) || !(tan(a) <= 2e-6))
		tmp = Float64(x + Float64(t_0 - tan(a)));
	else
		tmp = Float64(Float64(t_0 / Float64(1.0 - Float64(tan(y) * tan(z)))) + Float64(x - a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, a)
	t_0 = tan(y) + tan(z);
	tmp = 0.0;
	if ((tan(a) <= -0.005) || ~((tan(a) <= 2e-6)))
		tmp = x + (t_0 - tan(a));
	else
		tmp = (t_0 / (1.0 - (tan(y) * tan(z)))) + (x - a);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, a_] := Block[{t$95$0 = N[(N[Tan[y], $MachinePrecision] + N[Tan[z], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[N[Tan[a], $MachinePrecision], -0.005], N[Not[LessEqual[N[Tan[a], $MachinePrecision], 2e-6]], $MachinePrecision]], N[(x + N[(t$95$0 - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 / N[(1.0 - N[(N[Tan[y], $MachinePrecision] * N[Tan[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x - a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (tan.f64 a) < -0.0050000000000000001 or 1.99999999999999991e-6 < (tan.f64 a)

   1. Initial program 75.4%

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

      1. tan-sum 99.6%

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

      2. div-inv 99.6%

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

   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/ 99.6%

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

      2. *-rgt-identity 99.6%

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

   6. Simplified 99.6%

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

   7. Taylor expanded in y around 0 76.0%

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

   if -0.0050000000000000001 < (tan.f64 a) < 1.99999999999999991e-6

   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 79.8%

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

      1. +-commutative 79.8%

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

      2. associate-+l- 79.9%

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

   5. Applied egg-rr 79.9%

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

      1. tan-sum 99.9%

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

      2. div-inv 99.9%

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

   7. Applied egg-rr 99.9%

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

      1. associate-*r/ 99.9%

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

      2. *-rgt-identity 99.9%

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

   9. Simplified 99.9%

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

4. Final simplification 88.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\tan a \leq -0.005 \lor \neg \left(\tan a \leq 2 \cdot 10^{-6}\right):\\ \;\;\;\;x + \left(\left(\tan y + \tan z\right) - \tan a\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan y + \tan z}{1 - \tan y \cdot \tan z} + \left(x - a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 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 77.7%

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

   1. tan-sum 99.8%

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

   2. div-inv 99.8%

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

4. Applied egg-rr 99.8%

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

5. Final simplification 99.8%

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

Alternative 6: 99.7% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.7%

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

   1. tan-sum 99.8%

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

   2. div-inv 99.8%

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

4. Applied egg-rr 99.8%

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

   1. associate-*r/ 99.8%

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

   2. *-rgt-identity 99.8%

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

6. Simplified 99.8%

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

7. Final simplification 99.8%

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

Alternative 7: 70.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\tan a \leq -0.005 \lor \neg \left(\tan a \leq 2 \cdot 10^{-6}\right):\\ \;\;\;\;x + \left(\tan z - \tan a\right)\\ \mathbf{else}:\\ \;\;\;\;\tan \left(y + z\right) + \left(x - a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (if (or (<= (tan a) -0.005) (not (<= (tan a) 2e-6)))
   (+ x (- (tan z) (tan a)))
   (+ (tan (+ y z)) (- x a))))

C

double code(double x, double y, double z, double a) {
	double tmp;
	if ((tan(a) <= -0.005) || !(tan(a) <= 2e-6)) {
		tmp = x + (tan(z) - tan(a));
	} else {
		tmp = tan((y + z)) + (x - 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) <= (-0.005d0)) .or. (.not. (tan(a) <= 2d-6))) then
        tmp = x + (tan(z) - tan(a))
    else
        tmp = tan((y + z)) + (x - 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) <= -0.005) || !(Math.tan(a) <= 2e-6)) {
		tmp = x + (Math.tan(z) - Math.tan(a));
	} else {
		tmp = Math.tan((y + z)) + (x - a);
	}
	return tmp;
}

Python

def code(x, y, z, a):
	tmp = 0
	if (math.tan(a) <= -0.005) or not (math.tan(a) <= 2e-6):
		tmp = x + (math.tan(z) - math.tan(a))
	else:
		tmp = math.tan((y + z)) + (x - a)
	return tmp

Julia

function code(x, y, z, a)
	tmp = 0.0
	if ((tan(a) <= -0.005) || !(tan(a) <= 2e-6))
		tmp = Float64(x + Float64(tan(z) - tan(a)));
	else
		tmp = Float64(tan(Float64(y + z)) + Float64(x - a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, a)
	tmp = 0.0;
	if ((tan(a) <= -0.005) || ~((tan(a) <= 2e-6)))
		tmp = x + (tan(z) - tan(a));
	else
		tmp = tan((y + z)) + (x - a);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, a_] := If[Or[LessEqual[N[Tan[a], $MachinePrecision], -0.005], N[Not[LessEqual[N[Tan[a], $MachinePrecision], 2e-6]], $MachinePrecision]], N[(x + N[(N[Tan[z], $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Tan[N[(y + z), $MachinePrecision]], $MachinePrecision] + N[(x - a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (tan.f64 a) < -0.0050000000000000001 or 1.99999999999999991e-6 < (tan.f64 a)

   1. Initial program 75.4%

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

      1. tan-sum 99.6%

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

      2. div-inv 99.6%

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

   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. div-inv 99.6%

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

      2. tan-sum 75.4%

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

      3. tan-quot 75.4%

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

      4. un-div-inv 75.2%

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

      5. add-sqr-sqrt 38.0%

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

      6. sqrt-unprod 46.8%

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

      7. pow2 46.8%

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

      8. un-div-inv 46.8%

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

      9. tan-quot 46.8%

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

   6. Applied egg-rr 46.8%

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

      1. unpow2 46.8%

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

      2. rem-sqrt-square 46.8%

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

   8. Simplified 46.8%

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

   9. Taylor expanded in y around 0 35.0%

      \[\leadsto x + \left|\color{blue}{\frac{\sin z}{\cos z}} - \tan a\right| \]
   10. Step-by-step derivation

       1. expm1-log1p-u 34.8%

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

       2. expm1-udef 34.8%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(x + \left|\frac{\sin z}{\cos z} - \tan a\right|\right)} - 1} \]

       3. tan-quot 34.8%

          \[\leadsto e^{\mathsf{log1p}\left(x + \left|\color{blue}{\tan z} - \tan a\right|\right)} - 1 \]

       4. add-sqr-sqrt 25.1%

          \[\leadsto e^{\mathsf{log1p}\left(x + \left|\color{blue}{\sqrt{\tan z - \tan a} \cdot \sqrt{\tan z - \tan a}}\right|\right)} - 1 \]

       5. fabs-sqr 25.1%

          \[\leadsto e^{\mathsf{log1p}\left(x + \color{blue}{\sqrt{\tan z - \tan a} \cdot \sqrt{\tan z - \tan a}}\right)} - 1 \]

       6. add-sqr-sqrt 50.4%

          \[\leadsto e^{\mathsf{log1p}\left(x + \color{blue}{\left(\tan z - \tan a\right)}\right)} - 1 \]

   11. Applied egg-rr 50.4%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(x + \left(\tan z - \tan a\right)\right)} - 1} \]
   12. Step-by-step derivation

       1. expm1-def 50.4%

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

       2. expm1-log1p 55.2%

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

   13. Simplified 55.2%

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

   if -0.0050000000000000001 < (tan.f64 a) < 1.99999999999999991e-6

   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 79.8%

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

      1. +-commutative 79.8%

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

      2. associate-+l- 79.9%

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

   5. Applied egg-rr 79.9%

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

4. Final simplification 67.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\tan a \leq -0.005 \lor \neg \left(\tan a \leq 2 \cdot 10^{-6}\right):\\ \;\;\;\;x + \left(\tan z - \tan a\right)\\ \mathbf{else}:\\ \;\;\;\;\tan \left(y + z\right) + \left(x - a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 50.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\tan a \leq -0.005:\\ \;\;\;\;x\\ \mathbf{elif}\;\tan a \leq 2 \cdot 10^{-6}:\\ \;\;\;\;\tan \left(y + z\right) + \left(x - a\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{expm1}\left(\log x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (if (<= (tan a) -0.005)
   x
   (if (<= (tan a) 2e-6) (+ (tan (+ y z)) (- x a)) (expm1 (log x)))))

C

double code(double x, double y, double z, double a) {
	double tmp;
	if (tan(a) <= -0.005) {
		tmp = x;
	} else if (tan(a) <= 2e-6) {
		tmp = tan((y + z)) + (x - a);
	} else {
		tmp = expm1(log(x));
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double a) {
	double tmp;
	if (Math.tan(a) <= -0.005) {
		tmp = x;
	} else if (Math.tan(a) <= 2e-6) {
		tmp = Math.tan((y + z)) + (x - a);
	} else {
		tmp = Math.expm1(Math.log(x));
	}
	return tmp;
}

Python

def code(x, y, z, a):
	tmp = 0
	if math.tan(a) <= -0.005:
		tmp = x
	elif math.tan(a) <= 2e-6:
		tmp = math.tan((y + z)) + (x - a)
	else:
		tmp = math.expm1(math.log(x))
	return tmp

Julia

function code(x, y, z, a)
	tmp = 0.0
	if (tan(a) <= -0.005)
		tmp = x;
	elseif (tan(a) <= 2e-6)
		tmp = Float64(tan(Float64(y + z)) + Float64(x - a));
	else
		tmp = expm1(log(x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, a_] := If[LessEqual[N[Tan[a], $MachinePrecision], -0.005], x, If[LessEqual[N[Tan[a], $MachinePrecision], 2e-6], N[(N[Tan[N[(y + z), $MachinePrecision]], $MachinePrecision] + N[(x - a), $MachinePrecision]), $MachinePrecision], N[(Exp[N[Log[x], $MachinePrecision]] - 1), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (tan.f64 a) < -0.0050000000000000001

   1. Initial program 69.4%

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

   3. Taylor expanded in x around inf 22.8%

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

   if -0.0050000000000000001 < (tan.f64 a) < 1.99999999999999991e-6

   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 79.8%

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

      1. +-commutative 79.8%

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

      2. associate-+l- 79.9%

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

   5. Applied egg-rr 79.9%

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

   if 1.99999999999999991e-6 < (tan.f64 a)

   1. Initial program 81.2%

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

   3. Taylor expanded in a around 0 3.4%

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

      1. expm1-log1p-u 2.0%

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

      2. +-commutative 2.0%

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

      3. associate-+l- 2.0%

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

   5. Applied egg-rr 2.0%

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

   6. Taylor expanded in x around inf 19.8%

      \[\leadsto \mathsf{expm1}\left(\color{blue}{-1 \cdot \log \left(\frac{1}{x}\right)}\right) \]
   7. Step-by-step derivation

      1. mul-1-neg 19.8%

         \[\leadsto \mathsf{expm1}\left(\color{blue}{-\log \left(\frac{1}{x}\right)}\right) \]

      2. log-rec 19.8%

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

      3. remove-double-neg 19.8%

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

   8. Simplified 19.8%

      \[\leadsto \mathsf{expm1}\left(\color{blue}{\log x}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 51.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\tan a \leq -0.005:\\ \;\;\;\;x\\ \mathbf{elif}\;\tan a \leq 2 \cdot 10^{-6}:\\ \;\;\;\;\tan \left(y + z\right) + \left(x - a\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{expm1}\left(\log x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 80.1% accurate, 0.7× speedup

Math

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

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(Float64(tan(y) + 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[(N[Tan[y], $MachinePrecision] + N[Tan[z], $MachinePrecision]), $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 77.7%

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

   1. tan-sum 99.8%

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

   2. div-inv 99.8%

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

4. Applied egg-rr 99.8%

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

   1. associate-*r/ 99.8%

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

   2. *-rgt-identity 99.8%

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

6. Simplified 99.8%

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

7. Taylor expanded in y around 0 78.1%

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

8. Final simplification 78.1%

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

Alternative 10: 70.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -0.0018:\\ \;\;\;\;x + \left(\tan z - \tan a\right)\\ \mathbf{elif}\;a \leq 0.00175:\\ \;\;\;\;\tan \left(y + z\right) + \left(x - a\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x + \tan z\right) - \tan a\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (if (<= a -0.0018)
   (+ x (- (tan z) (tan a)))
   (if (<= a 0.00175) (+ (tan (+ y z)) (- x a)) (- (+ x (tan z)) (tan a)))))

C

double code(double x, double y, double z, double a) {
	double tmp;
	if (a <= -0.0018) {
		tmp = x + (tan(z) - tan(a));
	} else if (a <= 0.00175) {
		tmp = tan((y + z)) + (x - 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 (a <= (-0.0018d0)) then
        tmp = x + (tan(z) - tan(a))
    else if (a <= 0.00175d0) then
        tmp = tan((y + z)) + (x - 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 (a <= -0.0018) {
		tmp = x + (Math.tan(z) - Math.tan(a));
	} else if (a <= 0.00175) {
		tmp = Math.tan((y + z)) + (x - a);
	} else {
		tmp = (x + Math.tan(z)) - Math.tan(a);
	}
	return tmp;
}

Python

def code(x, y, z, a):
	tmp = 0
	if a <= -0.0018:
		tmp = x + (math.tan(z) - math.tan(a))
	elif a <= 0.00175:
		tmp = math.tan((y + z)) + (x - a)
	else:
		tmp = (x + math.tan(z)) - math.tan(a)
	return tmp

Julia

function code(x, y, z, a)
	tmp = 0.0
	if (a <= -0.0018)
		tmp = Float64(x + Float64(tan(z) - tan(a)));
	elseif (a <= 0.00175)
		tmp = Float64(tan(Float64(y + z)) + Float64(x - a));
	else
		tmp = Float64(Float64(x + tan(z)) - tan(a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, a)
	tmp = 0.0;
	if (a <= -0.0018)
		tmp = x + (tan(z) - tan(a));
	elseif (a <= 0.00175)
		tmp = tan((y + z)) + (x - a);
	else
		tmp = (x + tan(z)) - tan(a);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, a_] := If[LessEqual[a, -0.0018], N[(x + N[(N[Tan[z], $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 0.00175], N[(N[Tan[N[(y + z), $MachinePrecision]], $MachinePrecision] + N[(x - a), $MachinePrecision]), $MachinePrecision], N[(N[(x + N[Tan[z], $MachinePrecision]), $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -0.0018

   1. Initial program 79.9%

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

      1. tan-sum 99.8%

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

      2. div-inv 99.8%

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

   4. Applied egg-rr 99.8%

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

      1. div-inv 99.8%

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

      2. tan-sum 79.9%

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

      3. tan-quot 79.9%

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

      4. un-div-inv 79.7%

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

      5. add-sqr-sqrt 40.7%

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

      6. sqrt-unprod 49.5%

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

      7. pow2 49.5%

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

      8. un-div-inv 49.6%

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

      9. tan-quot 49.6%

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

   6. Applied egg-rr 49.6%

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

      1. unpow2 49.6%

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

      2. rem-sqrt-square 49.6%

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

   8. Simplified 49.6%

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

   9. Taylor expanded in y around 0 36.1%

      \[\leadsto x + \left|\color{blue}{\frac{\sin z}{\cos z}} - \tan a\right| \]
   10. Step-by-step derivation

       1. expm1-log1p-u 35.9%

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

       2. expm1-udef 35.9%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(x + \left|\frac{\sin z}{\cos z} - \tan a\right|\right)} - 1} \]

       3. tan-quot 35.9%

          \[\leadsto e^{\mathsf{log1p}\left(x + \left|\color{blue}{\tan z} - \tan a\right|\right)} - 1 \]

       4. add-sqr-sqrt 25.2%

          \[\leadsto e^{\mathsf{log1p}\left(x + \left|\color{blue}{\sqrt{\tan z - \tan a} \cdot \sqrt{\tan z - \tan a}}\right|\right)} - 1 \]

       5. fabs-sqr 25.2%

          \[\leadsto e^{\mathsf{log1p}\left(x + \color{blue}{\sqrt{\tan z - \tan a} \cdot \sqrt{\tan z - \tan a}}\right)} - 1 \]

       6. add-sqr-sqrt 49.9%

          \[\leadsto e^{\mathsf{log1p}\left(x + \color{blue}{\left(\tan z - \tan a\right)}\right)} - 1 \]

   11. Applied egg-rr 49.9%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(x + \left(\tan z - \tan a\right)\right)} - 1} \]
   12. Step-by-step derivation

       1. expm1-def 49.9%

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

       2. expm1-log1p 58.1%

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

   13. Simplified 58.1%

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

   if -0.0018 < a < 0.00175000000000000004

   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 79.8%

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

      1. +-commutative 79.8%

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

      2. associate-+l- 79.9%

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

   5. Applied egg-rr 79.9%

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

   if 0.00175000000000000004 < a

   1. Initial program 72.0%

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

      1. tan-sum 99.5%

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

      2. div-inv 99.5%

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

   4. Applied egg-rr 99.5%

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

      1. div-inv 99.5%

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

      2. tan-sum 72.0%

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

      3. tan-quot 72.0%

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

      4. un-div-inv 71.9%

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

      5. add-sqr-sqrt 35.9%

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

      6. sqrt-unprod 44.7%

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

      7. pow2 44.7%

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

      8. un-div-inv 44.6%

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

      9. tan-quot 44.7%

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

   6. Applied egg-rr 44.7%

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

      1. unpow2 44.7%

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

      2. rem-sqrt-square 44.7%

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

   8. Simplified 44.7%

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

   9. Taylor expanded in y around 0 34.1%

      \[\leadsto x + \left|\color{blue}{\frac{\sin z}{\cos z}} - \tan a\right| \]
   10. Step-by-step derivation

       1. expm1-log1p-u 33.9%

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

       2. expm1-udef 33.9%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(x + \left|\frac{\sin z}{\cos z} - \tan a\right|\right)} - 1} \]

       3. tan-quot 33.9%

          \[\leadsto e^{\mathsf{log1p}\left(x + \left|\color{blue}{\tan z} - \tan a\right|\right)} - 1 \]

       4. add-sqr-sqrt 24.9%

          \[\leadsto e^{\mathsf{log1p}\left(x + \left|\color{blue}{\sqrt{\tan z - \tan a} \cdot \sqrt{\tan z - \tan a}}\right|\right)} - 1 \]

       5. fabs-sqr 24.9%

          \[\leadsto e^{\mathsf{log1p}\left(x + \color{blue}{\sqrt{\tan z - \tan a} \cdot \sqrt{\tan z - \tan a}}\right)} - 1 \]

       6. add-sqr-sqrt 50.7%

          \[\leadsto e^{\mathsf{log1p}\left(x + \color{blue}{\left(\tan z - \tan a\right)}\right)} - 1 \]

   11. Applied egg-rr 50.7%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(x + \left(\tan z - \tan a\right)\right)} - 1} \]
   12. Step-by-step derivation

       1. expm1-def 50.8%

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

       2. expm1-log1p 53.0%

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

       3. associate-+r- 53.0%

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

   13. Simplified 53.0%

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

4. Final simplification 67.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -0.0018:\\ \;\;\;\;x + \left(\tan z - \tan a\right)\\ \mathbf{elif}\;a \leq 0.00175:\\ \;\;\;\;\tan \left(y + z\right) + \left(x - a\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x + \tan z\right) - \tan a\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 60.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y + z \leq -5 \cdot 10^{+20}:\\ \;\;\;\;x + \left(\tan \left(y + z\right) + \tan a\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(\tan z - \tan a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (if (<= (+ y z) -5e+20)
   (+ x (+ (tan (+ y z)) (tan a)))
   (+ x (- (tan z) (tan a)))))

C

double code(double x, double y, double z, double a) {
	double tmp;
	if ((y + z) <= -5e+20) {
		tmp = x + (tan((y + z)) + 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+20)) then
        tmp = x + (tan((y + z)) + 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+20) {
		tmp = x + (Math.tan((y + z)) + 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+20:
		tmp = x + (math.tan((y + z)) + 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+20)
		tmp = Float64(x + Float64(tan(Float64(y + z)) + 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+20)
		tmp = x + (tan((y + z)) + 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+20], N[(x + N[(N[Tan[N[(y + z), $MachinePrecision]], $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) < -5e20

   1. Initial program 73.1%

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

      1. sub-neg 73.1%

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

   4. Applied egg-rr 73.1%

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

      1. rem-square-sqrt 37.9%

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

      2. fabs-sqr 37.9%

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

      3. rem-square-sqrt 57.6%

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

      4. fabs-neg 57.6%

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

      5. rem-square-sqrt 19.7%

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

      6. fabs-sqr 19.7%

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

      7. rem-square-sqrt 45.2%

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

      8. +-commutative 45.2%

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

   6. Simplified 45.2%

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

   if -5e20 < (+.f64 y z)

   1. Initial program 80.3%

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

      1. tan-sum 99.8%

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

      2. div-inv 99.8%

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

   4. Applied egg-rr 99.8%

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

      1. div-inv 99.8%

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

      2. tan-sum 80.3%

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

      3. tan-quot 80.3%

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

      4. un-div-inv 80.2%

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

      5. add-sqr-sqrt 41.8%

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

      6. sqrt-unprod 60.7%

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

      7. pow2 60.7%

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

      8. un-div-inv 60.7%

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

      9. tan-quot 60.7%

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

   6. Applied egg-rr 60.7%

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

      1. unpow2 60.7%

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

      2. rem-sqrt-square 60.7%

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

   8. Simplified 60.7%

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

   9. Taylor expanded in y around 0 51.8%

      \[\leadsto x + \left|\color{blue}{\frac{\sin z}{\cos z}} - \tan a\right| \]
   10. Step-by-step derivation

       1. expm1-log1p-u 51.4%

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

       2. expm1-udef 51.4%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(x + \left|\frac{\sin z}{\cos z} - \tan a\right|\right)} - 1} \]

       3. tan-quot 51.4%

          \[\leadsto e^{\mathsf{log1p}\left(x + \left|\color{blue}{\tan z} - \tan a\right|\right)} - 1 \]

       4. add-sqr-sqrt 31.2%

          \[\leadsto e^{\mathsf{log1p}\left(x + \left|\color{blue}{\sqrt{\tan z - \tan a} \cdot \sqrt{\tan z - \tan a}}\right|\right)} - 1 \]

       5. fabs-sqr 31.2%

          \[\leadsto e^{\mathsf{log1p}\left(x + \color{blue}{\sqrt{\tan z - \tan a} \cdot \sqrt{\tan z - \tan a}}\right)} - 1 \]

       6. add-sqr-sqrt 62.8%

          \[\leadsto e^{\mathsf{log1p}\left(x + \color{blue}{\left(\tan z - \tan a\right)}\right)} - 1 \]

   11. Applied egg-rr 62.8%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(x + \left(\tan z - \tan a\right)\right)} - 1} \]
   12. Step-by-step derivation

       1. expm1-def 62.8%

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

       2. expm1-log1p 65.5%

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

   13. Simplified 65.5%

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

4. Final simplification 58.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y + z \leq -5 \cdot 10^{+20}:\\ \;\;\;\;x + \left(\tan \left(y + z\right) + \tan a\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(\tan z - \tan a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 79.7% 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 77.7%

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

3. Final simplification 77.7%

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

Alternative 13: 50.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -8:\\ \;\;\;\;x\\ \mathbf{elif}\;a \leq 1.55:\\ \;\;\;\;x + \left(\tan \left(y + z\right) - a\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (if (<= a -8.0) x (if (<= a 1.55) (+ x (- (tan (+ y z)) a)) x)))

C

double code(double x, double y, double z, double a) {
	double tmp;
	if (a <= -8.0) {
		tmp = x;
	} else if (a <= 1.55) {
		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 <= (-8.0d0)) then
        tmp = x
    else if (a <= 1.55d0) 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 <= -8.0) {
		tmp = x;
	} else if (a <= 1.55) {
		tmp = x + (Math.tan((y + z)) - a);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, a):
	tmp = 0
	if a <= -8.0:
		tmp = x
	elif a <= 1.55:
		tmp = x + (math.tan((y + z)) - a)
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, a)
	tmp = 0.0
	if (a <= -8.0)
		tmp = x;
	elseif (a <= 1.55)
		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 <= -8.0)
		tmp = x;
	elseif (a <= 1.55)
		tmp = x + (tan((y + z)) - a);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, a_] := If[LessEqual[a, -8.0], x, If[LessEqual[a, 1.55], N[(x + N[(N[Tan[N[(y + z), $MachinePrecision]], $MachinePrecision] - a), $MachinePrecision]), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if a < -8 or 1.55000000000000004 < a

   1. Initial program 75.2%

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

   3. Taylor expanded in x around inf 20.9%

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

   if -8 < a < 1.55000000000000004

   1. Initial program 80.0%

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

   3. Taylor expanded in a around 0 79.4%

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

4. Final simplification 50.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -8:\\ \;\;\;\;x\\ \mathbf{elif}\;a \leq 1.55:\\ \;\;\;\;x + \left(\tan \left(y + z\right) - a\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 50.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.6:\\ \;\;\;\;x\\ \mathbf{elif}\;a \leq 1.55:\\ \;\;\;\;\tan \left(y + z\right) + \left(x - a\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (if (<= a -1.6) x (if (<= a 1.55) (+ (tan (+ y z)) (- x a)) x)))

C

double code(double x, double y, double z, double a) {
	double tmp;
	if (a <= -1.6) {
		tmp = x;
	} else if (a <= 1.55) {
		tmp = tan((y + z)) + (x - 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 <= (-1.6d0)) then
        tmp = x
    else if (a <= 1.55d0) then
        tmp = tan((y + z)) + (x - 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 <= -1.6) {
		tmp = x;
	} else if (a <= 1.55) {
		tmp = Math.tan((y + z)) + (x - a);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, a):
	tmp = 0
	if a <= -1.6:
		tmp = x
	elif a <= 1.55:
		tmp = math.tan((y + z)) + (x - a)
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, a)
	tmp = 0.0
	if (a <= -1.6)
		tmp = x;
	elseif (a <= 1.55)
		tmp = Float64(tan(Float64(y + z)) + Float64(x - a));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, a)
	tmp = 0.0;
	if (a <= -1.6)
		tmp = x;
	elseif (a <= 1.55)
		tmp = tan((y + z)) + (x - a);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, a_] := If[LessEqual[a, -1.6], x, If[LessEqual[a, 1.55], N[(N[Tan[N[(y + z), $MachinePrecision]], $MachinePrecision] + N[(x - a), $MachinePrecision]), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if a < -1.6000000000000001 or 1.55000000000000004 < a

   1. Initial program 75.2%

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

   3. Taylor expanded in x around inf 20.9%

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

   if -1.6000000000000001 < a < 1.55000000000000004

   1. Initial program 80.0%

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

   3. Taylor expanded in a around 0 79.4%

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

      1. +-commutative 79.4%

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

      2. associate-+l- 79.5%

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

   5. Applied egg-rr 79.5%

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

4. Final simplification 50.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.6:\\ \;\;\;\;x\\ \mathbf{elif}\;a \leq 1.55:\\ \;\;\;\;\tan \left(y + z\right) + \left(x - a\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 32.1% 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 77.7%

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

3. Taylor expanded in x around inf 34.5%

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

4. Final simplification 34.5%

   \[\leadsto x \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024017 
(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))))