tan-example (used to crash)

Percentage Accurate: 79.3% → 99.7%

Time: 32.8s

Alternatives: 7

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: 79.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 0.4× speedup

Math

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

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 77.0%

   \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
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 2: 89.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan y + \tan z\\ \mathbf{if}\;\tan a \leq -5 \cdot 10^{-12} \lor \neg \left(\tan a \leq 2 \cdot 10^{-16}\right):\\ \;\;\;\;x + \left(t_0 - \tan a\right)\\ \mathbf{else}:\\ \;\;\;\;x + t_0 \cdot \frac{1}{1 - \tan y \cdot \tan z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (let* ((t_0 (+ (tan y) (tan z))))
   (if (or (<= (tan a) -5e-12) (not (<= (tan a) 2e-16)))
     (+ x (- t_0 (tan a)))
     (+ x (* t_0 (/ 1.0 (- 1.0 (* (tan y) (tan z)))))))))

C

double code(double x, double y, double z, double a) {
	double t_0 = tan(y) + tan(z);
	double tmp;
	if ((tan(a) <= -5e-12) || !(tan(a) <= 2e-16)) {
		tmp = x + (t_0 - tan(a));
	} else {
		tmp = x + (t_0 * (1.0 / (1.0 - (tan(y) * tan(z)))));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: a
    real(8) :: t_0
    real(8) :: tmp
    t_0 = tan(y) + tan(z)
    if ((tan(a) <= (-5d-12)) .or. (.not. (tan(a) <= 2d-16))) then
        tmp = x + (t_0 - tan(a))
    else
        tmp = x + (t_0 * (1.0d0 / (1.0d0 - (tan(y) * tan(z)))))
    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) <= -5e-12) || !(Math.tan(a) <= 2e-16)) {
		tmp = x + (t_0 - Math.tan(a));
	} else {
		tmp = x + (t_0 * (1.0 / (1.0 - (Math.tan(y) * Math.tan(z)))));
	}
	return tmp;
}

Python

def code(x, y, z, a):
	t_0 = math.tan(y) + math.tan(z)
	tmp = 0
	if (math.tan(a) <= -5e-12) or not (math.tan(a) <= 2e-16):
		tmp = x + (t_0 - math.tan(a))
	else:
		tmp = x + (t_0 * (1.0 / (1.0 - (math.tan(y) * math.tan(z)))))
	return tmp

Julia

function code(x, y, z, a)
	t_0 = Float64(tan(y) + tan(z))
	tmp = 0.0
	if ((tan(a) <= -5e-12) || !(tan(a) <= 2e-16))
		tmp = Float64(x + Float64(t_0 - tan(a)));
	else
		tmp = Float64(x + Float64(t_0 * Float64(1.0 / Float64(1.0 - Float64(tan(y) * tan(z))))));
	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) <= -5e-12) || ~((tan(a) <= 2e-16)))
		tmp = x + (t_0 - tan(a));
	else
		tmp = x + (t_0 * (1.0 / (1.0 - (tan(y) * tan(z)))));
	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], -5e-12], N[Not[LessEqual[N[Tan[a], $MachinePrecision], 2e-16]], $MachinePrecision]], N[(x + N[(t$95$0 - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(t$95$0 * N[(1.0 / N[(1.0 - N[(N[Tan[y], $MachinePrecision] * N[Tan[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (tan.f64 a) < -4.9999999999999997e-12 or 2e-16 < (tan.f64 a)

   1. Initial program 81.4%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   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. Taylor expanded in y around 0 81.6%

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

   if -4.9999999999999997e-12 < (tan.f64 a) < 2e-16

   1. Initial program 72.8%

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

      1. +-commutative 72.8%

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

      2. associate-+l- 72.8%

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

   4. Applied egg-rr 72.8%

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

   5. Taylor expanded in a around 0 72.8%

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

      1. neg-mul-1 72.8%

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

   7. Simplified 72.8%

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

      1. sub-neg 72.8%

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

      2. +-commutative 72.8%

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

   9. Applied egg-rr 72.8%

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

       1. remove-double-neg 72.8%

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

       2. +-commutative 72.8%

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

   11. Simplified 72.8%

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

       1. +-commutative 72.8%

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

       2. tan-sum 99.7%

          \[\leadsto x + \color{blue}{\frac{\tan y + \tan z}{1 - \tan y \cdot \tan z}} \]

       3. div-inv 99.8%

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

   13. Applied egg-rr 99.8%

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

       1. *-commutative 99.8%

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

   15. Simplified 99.8%

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

4. Final simplification 90.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\tan a \leq -5 \cdot 10^{-12} \lor \neg \left(\tan a \leq 2 \cdot 10^{-16}\right):\\ \;\;\;\;x + \left(\left(\tan y + \tan z\right) - \tan a\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(\tan y + \tan z\right) \cdot \frac{1}{1 - \tan y \cdot \tan z}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 89.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan y + \tan z\\ \mathbf{if}\;\tan a \leq -5 \cdot 10^{-12} \lor \neg \left(\tan a \leq 2 \cdot 10^{-16}\right):\\ \;\;\;\;x + \left(t_0 - \tan a\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{t_0}{1 - \tan y \cdot \tan z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (let* ((t_0 (+ (tan y) (tan z))))
   (if (or (<= (tan a) -5e-12) (not (<= (tan a) 2e-16)))
     (+ x (- t_0 (tan a)))
     (+ x (/ t_0 (- 1.0 (* (tan y) (tan z))))))))

C

double code(double x, double y, double z, double a) {
	double t_0 = tan(y) + tan(z);
	double tmp;
	if ((tan(a) <= -5e-12) || !(tan(a) <= 2e-16)) {
		tmp = x + (t_0 - tan(a));
	} else {
		tmp = x + (t_0 / (1.0 - (tan(y) * tan(z))));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: a
    real(8) :: t_0
    real(8) :: tmp
    t_0 = tan(y) + tan(z)
    if ((tan(a) <= (-5d-12)) .or. (.not. (tan(a) <= 2d-16))) then
        tmp = x + (t_0 - tan(a))
    else
        tmp = x + (t_0 / (1.0d0 - (tan(y) * tan(z))))
    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) <= -5e-12) || !(Math.tan(a) <= 2e-16)) {
		tmp = x + (t_0 - Math.tan(a));
	} else {
		tmp = x + (t_0 / (1.0 - (Math.tan(y) * Math.tan(z))));
	}
	return tmp;
}

Python

def code(x, y, z, a):
	t_0 = math.tan(y) + math.tan(z)
	tmp = 0
	if (math.tan(a) <= -5e-12) or not (math.tan(a) <= 2e-16):
		tmp = x + (t_0 - math.tan(a))
	else:
		tmp = x + (t_0 / (1.0 - (math.tan(y) * math.tan(z))))
	return tmp

Julia

function code(x, y, z, a)
	t_0 = Float64(tan(y) + tan(z))
	tmp = 0.0
	if ((tan(a) <= -5e-12) || !(tan(a) <= 2e-16))
		tmp = Float64(x + Float64(t_0 - tan(a)));
	else
		tmp = Float64(x + Float64(t_0 / Float64(1.0 - Float64(tan(y) * tan(z)))));
	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) <= -5e-12) || ~((tan(a) <= 2e-16)))
		tmp = x + (t_0 - tan(a));
	else
		tmp = x + (t_0 / (1.0 - (tan(y) * tan(z))));
	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], -5e-12], N[Not[LessEqual[N[Tan[a], $MachinePrecision], 2e-16]], $MachinePrecision]], N[(x + N[(t$95$0 - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(t$95$0 / N[(1.0 - N[(N[Tan[y], $MachinePrecision] * N[Tan[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (tan.f64 a) < -4.9999999999999997e-12 or 2e-16 < (tan.f64 a)

   1. Initial program 81.4%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   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. Taylor expanded in y around 0 81.6%

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

   if -4.9999999999999997e-12 < (tan.f64 a) < 2e-16

   1. Initial program 72.8%

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

      1. +-commutative 72.8%

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

      2. associate-+l- 72.8%

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

   4. Applied egg-rr 72.8%

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

   5. Taylor expanded in a around 0 72.8%

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

      1. neg-mul-1 72.8%

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

   7. Simplified 72.8%

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

      1. sub-neg 72.8%

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

      2. +-commutative 72.8%

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

   9. Applied egg-rr 72.8%

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

       1. remove-double-neg 72.8%

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

       2. +-commutative 72.8%

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

   11. Simplified 72.8%

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

       1. +-commutative 72.8%

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

       2. tan-sum 99.7%

          \[\leadsto x + \color{blue}{\frac{\tan y + \tan z}{1 - \tan y \cdot \tan z}} \]

       3. div-inv 99.8%

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

   13. Applied egg-rr 99.8%

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

       1. associate-*r/ 99.7%

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

       2. *-rgt-identity 99.7%

          \[\leadsto x + \frac{\color{blue}{\tan y + \tan z}}{1 - \tan y \cdot \tan z} \]

   15. Simplified 99.7%

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

4. Final simplification 90.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\tan a \leq -5 \cdot 10^{-12} \lor \neg \left(\tan a \leq 2 \cdot 10^{-16}\right):\\ \;\;\;\;x + \left(\left(\tan y + \tan z\right) - \tan a\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{\tan y + \tan z}{1 - \tan y \cdot \tan z}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 79.6% 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 77.0%

   \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
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. Taylor expanded in y around 0 77.6%

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

6. Final simplification 77.6%

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

Alternative 5: 79.3% 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 77.0%

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

3. Final simplification 77.0%

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

Alternative 6: 50.8% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.0%

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

   1. +-commutative 77.0%

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

   2. associate-+l- 77.0%

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

4. Applied egg-rr 77.0%

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

5. Taylor expanded in a around 0 49.7%

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

   1. neg-mul-1 49.7%

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

7. Simplified 49.7%

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

   1. sub-neg 49.7%

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

   2. +-commutative 49.7%

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

9. Applied egg-rr 49.7%

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

    1. remove-double-neg 49.7%

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

    2. +-commutative 49.7%

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

11. Simplified 49.7%

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

12. Final simplification 49.7%

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

Alternative 7: 31.3% 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 77.0%

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

3. Taylor expanded in x around inf 33.9%

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

4. Final simplification 33.9%

   \[\leadsto x \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024020 
(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))))