tan-example (used to crash)

Percentage Accurate: 78.4% → 99.7%

Time: 38.4s

Alternatives: 14

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 78.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan y \cdot \tan z\\ x + \left(\frac{\tan y + \tan z}{1 - {t\_0}^{3}} \cdot \left(1 + t\_0 \cdot \left(1 + t\_0\right)\right) - \tan a\right) \end{array} \end{array} \]

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 3.0)))
      (+ 1.0 (* t_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, 3.0))) * (1.0 + (t_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 ** 3.0d0))) * (1.0d0 + (t_0 * (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, 3.0))) * (1.0 + (t_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, 3.0))) * (1.0 + (t_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(Float64(tan(y) + tan(z)) / Float64(1.0 - (t_0 ^ 3.0))) * Float64(1.0 + Float64(t_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 ^ 3.0))) * (1.0 + (t_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[(N[Tan[y], $MachinePrecision] + N[Tan[z], $MachinePrecision]), $MachinePrecision] / N[(1.0 - N[Power[t$95$0, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(t$95$0 * N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 79.3%

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

   1. tan-sum 99.7%

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

   2. div-inv 99.7%

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

4. Applied egg-rr 99.7%

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

   1. associate-*r/ 99.7%

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

   2. *-rgt-identity 99.7%

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

6. Simplified 99.7%

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

   1. flip3-- 99.7%

      \[\leadsto x + \left(\frac{\tan y + \tan z}{\color{blue}{\frac{{1}^{3} - {\left(\tan y \cdot \tan z\right)}^{3}}{1 \cdot 1 + \left(\left(\tan y \cdot \tan z\right) \cdot \left(\tan y \cdot \tan z\right) + 1 \cdot \left(\tan y \cdot \tan z\right)\right)}}} - \tan a\right) \]

   2. associate-/r/ 99.7%

      \[\leadsto x + \left(\color{blue}{\frac{\tan y + \tan z}{{1}^{3} - {\left(\tan y \cdot \tan z\right)}^{3}} \cdot \left(1 \cdot 1 + \left(\left(\tan y \cdot \tan z\right) \cdot \left(\tan y \cdot \tan z\right) + 1 \cdot \left(\tan y \cdot \tan z\right)\right)\right)} - \tan a\right) \]

   3. metadata-eval 99.7%

      \[\leadsto x + \left(\frac{\tan y + \tan z}{\color{blue}{1} - {\left(\tan y \cdot \tan z\right)}^{3}} \cdot \left(1 \cdot 1 + \left(\left(\tan y \cdot \tan z\right) \cdot \left(\tan y \cdot \tan z\right) + 1 \cdot \left(\tan y \cdot \tan z\right)\right)\right) - \tan a\right) \]

   4. metadata-eval 99.7%

      \[\leadsto x + \left(\frac{\tan y + \tan z}{1 - {\left(\tan y \cdot \tan z\right)}^{3}} \cdot \left(\color{blue}{1} + \left(\left(\tan y \cdot \tan z\right) \cdot \left(\tan y \cdot \tan z\right) + 1 \cdot \left(\tan y \cdot \tan z\right)\right)\right) - \tan a\right) \]

   5. distribute-rgt-out 99.7%

      \[\leadsto x + \left(\frac{\tan y + \tan z}{1 - {\left(\tan y \cdot \tan z\right)}^{3}} \cdot \left(1 + \color{blue}{\left(\tan y \cdot \tan z\right) \cdot \left(\tan y \cdot \tan z + 1\right)}\right) - \tan a\right) \]

8. Applied egg-rr 99.7%

   \[\leadsto x + \left(\color{blue}{\frac{\tan y + \tan z}{1 - {\left(\tan y \cdot \tan z\right)}^{3}} \cdot \left(1 + \left(\tan y \cdot \tan z\right) \cdot \left(\tan y \cdot \tan z + 1\right)\right)} - \tan a\right) \]

9. Final simplification 99.7%

   \[\leadsto x + \left(\frac{\tan y + \tan z}{1 - {\left(\tan y \cdot \tan z\right)}^{3}} \cdot \left(1 + \left(\tan y \cdot \tan z\right) \cdot \left(1 + \tan y \cdot \tan z\right)\right) - \tan a\right) \]
10. Add Preprocessing

Alternative 2: 88.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan \left(y + z\right)\\ \mathbf{if}\;\tan a \leq -0.002:\\ \;\;\;\;x + \left(t\_0 + \sin a \cdot \frac{-1}{\cos a}\right)\\ \mathbf{elif}\;\tan a \leq 2 \cdot 10^{-6}:\\ \;\;\;\;x + \left(\left(\tan y + \tan z\right) \cdot \frac{1}{1 - \tan y \cdot \tan z} - a\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(t\_0 - \log \left(e^{\tan a}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (let* ((t_0 (tan (+ y z))))
   (if (<= (tan a) -0.002)
     (+ x (+ t_0 (* (sin a) (/ -1.0 (cos a)))))
     (if (<= (tan a) 2e-6)
       (+ x (- (* (+ (tan y) (tan z)) (/ 1.0 (- 1.0 (* (tan y) (tan z))))) a))
       (+ x (- t_0 (log (exp (tan a)))))))))

C

double code(double x, double y, double z, double a) {
	double t_0 = tan((y + z));
	double tmp;
	if (tan(a) <= -0.002) {
		tmp = x + (t_0 + (sin(a) * (-1.0 / cos(a))));
	} else if (tan(a) <= 2e-6) {
		tmp = x + (((tan(y) + tan(z)) * (1.0 / (1.0 - (tan(y) * tan(z))))) - a);
	} else {
		tmp = x + (t_0 - log(exp(tan(a))));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: a
    real(8) :: t_0
    real(8) :: tmp
    t_0 = tan((y + z))
    if (tan(a) <= (-0.002d0)) then
        tmp = x + (t_0 + (sin(a) * ((-1.0d0) / cos(a))))
    else if (tan(a) <= 2d-6) then
        tmp = x + (((tan(y) + tan(z)) * (1.0d0 / (1.0d0 - (tan(y) * tan(z))))) - a)
    else
        tmp = x + (t_0 - log(exp(tan(a))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double a) {
	double t_0 = Math.tan((y + z));
	double tmp;
	if (Math.tan(a) <= -0.002) {
		tmp = x + (t_0 + (Math.sin(a) * (-1.0 / Math.cos(a))));
	} else if (Math.tan(a) <= 2e-6) {
		tmp = x + (((Math.tan(y) + Math.tan(z)) * (1.0 / (1.0 - (Math.tan(y) * Math.tan(z))))) - a);
	} else {
		tmp = x + (t_0 - Math.log(Math.exp(Math.tan(a))));
	}
	return tmp;
}

Python

def code(x, y, z, a):
	t_0 = math.tan((y + z))
	tmp = 0
	if math.tan(a) <= -0.002:
		tmp = x + (t_0 + (math.sin(a) * (-1.0 / math.cos(a))))
	elif math.tan(a) <= 2e-6:
		tmp = x + (((math.tan(y) + math.tan(z)) * (1.0 / (1.0 - (math.tan(y) * math.tan(z))))) - a)
	else:
		tmp = x + (t_0 - math.log(math.exp(math.tan(a))))
	return tmp

Julia

function code(x, y, z, a)
	t_0 = tan(Float64(y + z))
	tmp = 0.0
	if (tan(a) <= -0.002)
		tmp = Float64(x + Float64(t_0 + Float64(sin(a) * Float64(-1.0 / cos(a)))));
	elseif (tan(a) <= 2e-6)
		tmp = Float64(x + Float64(Float64(Float64(tan(y) + tan(z)) * Float64(1.0 / Float64(1.0 - Float64(tan(y) * tan(z))))) - a));
	else
		tmp = Float64(x + Float64(t_0 - log(exp(tan(a)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, a)
	t_0 = tan((y + z));
	tmp = 0.0;
	if (tan(a) <= -0.002)
		tmp = x + (t_0 + (sin(a) * (-1.0 / cos(a))));
	elseif (tan(a) <= 2e-6)
		tmp = x + (((tan(y) + tan(z)) * (1.0 / (1.0 - (tan(y) * tan(z))))) - a);
	else
		tmp = x + (t_0 - log(exp(tan(a))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, a_] := Block[{t$95$0 = N[Tan[N[(y + z), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[Tan[a], $MachinePrecision], -0.002], N[(x + N[(t$95$0 + N[(N[Sin[a], $MachinePrecision] * N[(-1.0 / N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Tan[a], $MachinePrecision], 2e-6], N[(x + N[(N[(N[(N[Tan[y], $MachinePrecision] + N[Tan[z], $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(1.0 - N[(N[Tan[y], $MachinePrecision] * N[Tan[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision]), $MachinePrecision], N[(x + N[(t$95$0 - N[Log[N[Exp[N[Tan[a], $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 82.7%

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

      1. tan-quot 82.7%

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

      2. div-inv 82.7%

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

   4. Applied egg-rr 82.7%

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

   if -2e-3 < (tan.f64 a) < 1.99999999999999991e-6

   1. Initial program 75.8%

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

   3. Taylor expanded in a around 0 75.8%

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

      1. tan-sum 99.7%

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

      2. div-inv 99.7%

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

   5. Applied egg-rr 99.5%

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

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

   1. Initial program 84.1%

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

      1. add-log-exp 84.1%

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

   4. Applied egg-rr 84.1%

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

4. Final simplification 91.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\tan a \leq -0.002:\\ \;\;\;\;x + \left(\tan \left(y + z\right) + \sin a \cdot \frac{-1}{\cos a}\right)\\ \mathbf{elif}\;\tan a \leq 2 \cdot 10^{-6}:\\ \;\;\;\;x + \left(\left(\tan y + \tan z\right) \cdot \frac{1}{1 - \tan y \cdot \tan z} - a\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(\tan \left(y + z\right) - \log \left(e^{\tan a}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 88.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan \left(y + z\right)\\ \mathbf{if}\;\tan a \leq -0.002:\\ \;\;\;\;x + \left(t\_0 + \sin a \cdot \frac{-1}{\cos a}\right)\\ \mathbf{elif}\;\tan a \leq 2 \cdot 10^{-6}:\\ \;\;\;\;x + \left(\frac{\tan y + \tan z}{1 - \tan y \cdot \tan z} - a\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(t\_0 - \log \left(e^{\tan a}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (let* ((t_0 (tan (+ y z))))
   (if (<= (tan a) -0.002)
     (+ x (+ t_0 (* (sin a) (/ -1.0 (cos a)))))
     (if (<= (tan a) 2e-6)
       (+ x (- (/ (+ (tan y) (tan z)) (- 1.0 (* (tan y) (tan z)))) a))
       (+ x (- t_0 (log (exp (tan a)))))))))

C

double code(double x, double y, double z, double a) {
	double t_0 = tan((y + z));
	double tmp;
	if (tan(a) <= -0.002) {
		tmp = x + (t_0 + (sin(a) * (-1.0 / cos(a))));
	} else if (tan(a) <= 2e-6) {
		tmp = x + (((tan(y) + tan(z)) / (1.0 - (tan(y) * tan(z)))) - a);
	} else {
		tmp = x + (t_0 - log(exp(tan(a))));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: a
    real(8) :: t_0
    real(8) :: tmp
    t_0 = tan((y + z))
    if (tan(a) <= (-0.002d0)) then
        tmp = x + (t_0 + (sin(a) * ((-1.0d0) / cos(a))))
    else if (tan(a) <= 2d-6) then
        tmp = x + (((tan(y) + tan(z)) / (1.0d0 - (tan(y) * tan(z)))) - a)
    else
        tmp = x + (t_0 - log(exp(tan(a))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double a) {
	double t_0 = Math.tan((y + z));
	double tmp;
	if (Math.tan(a) <= -0.002) {
		tmp = x + (t_0 + (Math.sin(a) * (-1.0 / Math.cos(a))));
	} else if (Math.tan(a) <= 2e-6) {
		tmp = x + (((Math.tan(y) + Math.tan(z)) / (1.0 - (Math.tan(y) * Math.tan(z)))) - a);
	} else {
		tmp = x + (t_0 - Math.log(Math.exp(Math.tan(a))));
	}
	return tmp;
}

Python

def code(x, y, z, a):
	t_0 = math.tan((y + z))
	tmp = 0
	if math.tan(a) <= -0.002:
		tmp = x + (t_0 + (math.sin(a) * (-1.0 / math.cos(a))))
	elif math.tan(a) <= 2e-6:
		tmp = x + (((math.tan(y) + math.tan(z)) / (1.0 - (math.tan(y) * math.tan(z)))) - a)
	else:
		tmp = x + (t_0 - math.log(math.exp(math.tan(a))))
	return tmp

Julia

function code(x, y, z, a)
	t_0 = tan(Float64(y + z))
	tmp = 0.0
	if (tan(a) <= -0.002)
		tmp = Float64(x + Float64(t_0 + Float64(sin(a) * Float64(-1.0 / cos(a)))));
	elseif (tan(a) <= 2e-6)
		tmp = Float64(x + Float64(Float64(Float64(tan(y) + tan(z)) / Float64(1.0 - Float64(tan(y) * tan(z)))) - a));
	else
		tmp = Float64(x + Float64(t_0 - log(exp(tan(a)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, a)
	t_0 = tan((y + z));
	tmp = 0.0;
	if (tan(a) <= -0.002)
		tmp = x + (t_0 + (sin(a) * (-1.0 / cos(a))));
	elseif (tan(a) <= 2e-6)
		tmp = x + (((tan(y) + tan(z)) / (1.0 - (tan(y) * tan(z)))) - a);
	else
		tmp = x + (t_0 - log(exp(tan(a))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, a_] := Block[{t$95$0 = N[Tan[N[(y + z), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[Tan[a], $MachinePrecision], -0.002], N[(x + N[(t$95$0 + N[(N[Sin[a], $MachinePrecision] * N[(-1.0 / N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Tan[a], $MachinePrecision], 2e-6], N[(x + N[(N[(N[(N[Tan[y], $MachinePrecision] + N[Tan[z], $MachinePrecision]), $MachinePrecision] / N[(1.0 - N[(N[Tan[y], $MachinePrecision] * N[Tan[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision]), $MachinePrecision], N[(x + N[(t$95$0 - N[Log[N[Exp[N[Tan[a], $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 82.7%

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

      1. tan-quot 82.7%

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

      2. div-inv 82.7%

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

   4. Applied egg-rr 82.7%

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

   if -2e-3 < (tan.f64 a) < 1.99999999999999991e-6

   1. Initial program 75.8%

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

   3. Taylor expanded in a around 0 75.8%

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

      1. tan-sum 99.7%

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

      2. div-inv 99.7%

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

   5. Applied egg-rr 99.5%

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

      1. associate-*r/ 99.7%

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

      2. *-rgt-identity 99.7%

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

   7. Simplified 99.5%

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

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

   1. Initial program 84.1%

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

      1. add-log-exp 84.1%

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

   4. Applied egg-rr 84.1%

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

4. Final simplification 91.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\tan a \leq -0.002:\\ \;\;\;\;x + \left(\tan \left(y + z\right) + \sin a \cdot \frac{-1}{\cos a}\right)\\ \mathbf{elif}\;\tan a \leq 2 \cdot 10^{-6}:\\ \;\;\;\;x + \left(\frac{\tan y + \tan z}{1 - \tan y \cdot \tan z} - a\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(\tan \left(y + z\right) - \log \left(e^{\tan a}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.7% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.3%

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

   1. tan-sum 99.7%

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

   2. div-inv 99.7%

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

4. Applied egg-rr 99.7%

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

   1. associate-*r/ 99.7%

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

   2. *-rgt-identity 99.7%

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

6. Simplified 99.7%

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

Alternative 5: 78.3% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.3%

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

   1. tan-quot 79.3%

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

   2. div-inv 79.3%

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

4. Applied egg-rr 79.3%

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

5. Final simplification 79.3%

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

Alternative 6: 68.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -0.0145 \lor \neg \left(a \leq 5 \cdot 10^{-17}\right):\\ \;\;\;\;x + \left(\tan z - \tan a\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(\tan \left(y + z\right) - a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (if (or (<= a -0.0145) (not (<= a 5e-17)))
   (+ x (- (tan z) (tan a)))
   (+ x (- (tan (+ y z)) a))))

C

double code(double x, double y, double z, double a) {
	double tmp;
	if ((a <= -0.0145) || !(a <= 5e-17)) {
		tmp = x + (tan(z) - tan(a));
	} else {
		tmp = x + (tan((y + z)) - a);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: a
    real(8) :: tmp
    if ((a <= (-0.0145d0)) .or. (.not. (a <= 5d-17))) then
        tmp = x + (tan(z) - tan(a))
    else
        tmp = x + (tan((y + z)) - a)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double a) {
	double tmp;
	if ((a <= -0.0145) || !(a <= 5e-17)) {
		tmp = x + (Math.tan(z) - Math.tan(a));
	} else {
		tmp = x + (Math.tan((y + z)) - a);
	}
	return tmp;
}

Python

def code(x, y, z, a):
	tmp = 0
	if (a <= -0.0145) or not (a <= 5e-17):
		tmp = x + (math.tan(z) - math.tan(a))
	else:
		tmp = x + (math.tan((y + z)) - a)
	return tmp

Julia

function code(x, y, z, a)
	tmp = 0.0
	if ((a <= -0.0145) || !(a <= 5e-17))
		tmp = Float64(x + Float64(tan(z) - tan(a)));
	else
		tmp = Float64(x + Float64(tan(Float64(y + z)) - a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, a)
	tmp = 0.0;
	if ((a <= -0.0145) || ~((a <= 5e-17)))
		tmp = x + (tan(z) - tan(a));
	else
		tmp = x + (tan((y + z)) - a);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, a_] := If[Or[LessEqual[a, -0.0145], N[Not[LessEqual[a, 5e-17]], $MachinePrecision]], N[(x + N[(N[Tan[z], $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[Tan[N[(y + z), $MachinePrecision]], $MachinePrecision] - a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -0.0145000000000000007 or 4.9999999999999999e-17 < a

   1. Initial program 83.6%

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

   3. Taylor expanded in y around 0 66.1%

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

      1. tan-quot 66.1%

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

      2. *-un-lft-identity 66.1%

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

   5. Applied egg-rr 66.1%

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

      1. *-lft-identity 66.1%

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

   7. Simplified 66.1%

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

   if -0.0145000000000000007 < a < 4.9999999999999999e-17

   1. Initial program 75.4%

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

   3. Taylor expanded in a around 0 75.1%

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

4. Final simplification 70.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -0.0145 \lor \neg \left(a \leq 5 \cdot 10^{-17}\right):\\ \;\;\;\;x + \left(\tan z - \tan a\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(\tan \left(y + z\right) - a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 68.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -0.0145:\\ \;\;\;\;x + \left(\tan z - \tan a\right)\\ \mathbf{elif}\;a \leq 1.55 \cdot 10^{-16}:\\ \;\;\;\;x + \left(\tan \left(y + z\right) - a\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x + \tan z\right) - \tan a\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (if (<= a -0.0145)
   (+ x (- (tan z) (tan a)))
   (if (<= a 1.55e-16) (+ x (- (tan (+ y z)) a)) (- (+ x (tan z)) (tan a)))))

C

double code(double x, double y, double z, double a) {
	double tmp;
	if (a <= -0.0145) {
		tmp = x + (tan(z) - tan(a));
	} else if (a <= 1.55e-16) {
		tmp = x + (tan((y + z)) - 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 <= (-0.0145d0)) then
        tmp = x + (tan(z) - tan(a))
    else if (a <= 1.55d-16) then
        tmp = x + (tan((y + z)) - 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 <= -0.0145) {
		tmp = x + (Math.tan(z) - Math.tan(a));
	} else if (a <= 1.55e-16) {
		tmp = x + (Math.tan((y + z)) - a);
	} else {
		tmp = (x + Math.tan(z)) - Math.tan(a);
	}
	return tmp;
}

Python

def code(x, y, z, a):
	tmp = 0
	if a <= -0.0145:
		tmp = x + (math.tan(z) - math.tan(a))
	elif a <= 1.55e-16:
		tmp = x + (math.tan((y + z)) - a)
	else:
		tmp = (x + math.tan(z)) - math.tan(a)
	return tmp

Julia

function code(x, y, z, a)
	tmp = 0.0
	if (a <= -0.0145)
		tmp = Float64(x + Float64(tan(z) - tan(a)));
	elseif (a <= 1.55e-16)
		tmp = Float64(x + Float64(tan(Float64(y + z)) - 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 <= -0.0145)
		tmp = x + (tan(z) - tan(a));
	elseif (a <= 1.55e-16)
		tmp = x + (tan((y + z)) - a);
	else
		tmp = (x + tan(z)) - tan(a);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, a_] := If[LessEqual[a, -0.0145], N[(x + N[(N[Tan[z], $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 1.55e-16], N[(x + N[(N[Tan[N[(y + z), $MachinePrecision]], $MachinePrecision] - 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 < -0.0145000000000000007

   1. Initial program 83.5%

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

   3. Taylor expanded in y around 0 69.5%

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

      1. tan-quot 69.5%

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

      2. *-un-lft-identity 69.5%

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

   5. Applied egg-rr 69.5%

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

      1. *-lft-identity 69.5%

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

   7. Simplified 69.5%

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

   if -0.0145000000000000007 < a < 1.55e-16

   1. Initial program 75.4%

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

   3. Taylor expanded in a around 0 75.1%

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

   if 1.55e-16 < a

   1. Initial program 83.6%

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

   3. Taylor expanded in y around 0 62.1%

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

      1. tan-quot 62.0%

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

      2. associate-+r- 62.0%

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

   5. Applied egg-rr 62.0%

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

Alternative 8: 59.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y + z \leq -2 \cdot 10^{-7}:\\ \;\;\;\;x + \left(\tan a + \tan \left(y + z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(\tan z - \tan a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (if (<= (+ y z) -2e-7)
   (+ x (+ (tan a) (tan (+ y z))))
   (+ x (- (tan z) (tan a)))))

C

double code(double x, double y, double z, double a) {
	double tmp;
	if ((y + z) <= -2e-7) {
		tmp = x + (tan(a) + tan((y + z)));
	} 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) <= (-2d-7)) then
        tmp = x + (tan(a) + tan((y + z)))
    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) <= -2e-7) {
		tmp = x + (Math.tan(a) + Math.tan((y + z)));
	} else {
		tmp = x + (Math.tan(z) - Math.tan(a));
	}
	return tmp;
}

Python

def code(x, y, z, a):
	tmp = 0
	if (y + z) <= -2e-7:
		tmp = x + (math.tan(a) + math.tan((y + z)))
	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) <= -2e-7)
		tmp = Float64(x + Float64(tan(a) + tan(Float64(y + z))));
	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) <= -2e-7)
		tmp = x + (tan(a) + tan((y + z)));
	else
		tmp = x + (tan(z) - tan(a));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, a_] := If[LessEqual[N[(y + z), $MachinePrecision], -2e-7], N[(x + N[(N[Tan[a], $MachinePrecision] + N[Tan[N[(y + z), $MachinePrecision]], $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) < -1.9999999999999999e-7

   1. Initial program 71.2%

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

      1. sub-neg 71.2%

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

   4. Applied egg-rr 71.2%

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

      1. rem-square-sqrt 38.0%

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

      2. fabs-sqr 38.0%

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

      3. rem-square-sqrt 59.4%

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

      4. fabs-neg 59.4%

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

      5. rem-square-sqrt 21.4%

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

      6. fabs-sqr 21.4%

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

      7. rem-square-sqrt 46.6%

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

      8. +-commutative 46.6%

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

   6. Simplified 46.6%

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

   if -1.9999999999999999e-7 < (+.f64 y z)

   1. Initial program 83.6%

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

   3. Taylor expanded in y around 0 70.5%

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

      1. tan-quot 70.5%

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

      2. *-un-lft-identity 70.5%

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

   5. Applied egg-rr 70.5%

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

      1. *-lft-identity 70.5%

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

   7. Simplified 70.5%

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

4. Final simplification 62.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y + z \leq -2 \cdot 10^{-7}:\\ \;\;\;\;x + \left(\tan a + \tan \left(y + z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(\tan z - \tan a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 64.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -9.2 \cdot 10^{-8}:\\ \;\;\;\;x + \tan \left(y + z\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(\tan z - \tan a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (if (<= y -9.2e-8) (+ x (tan (+ y z))) (+ x (- (tan z) (tan a)))))

C

double code(double x, double y, double z, double a) {
	double tmp;
	if (y <= -9.2e-8) {
		tmp = x + tan((y + z));
	} 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 <= (-9.2d-8)) then
        tmp = x + tan((y + z))
    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 <= -9.2e-8) {
		tmp = x + Math.tan((y + z));
	} else {
		tmp = x + (Math.tan(z) - Math.tan(a));
	}
	return tmp;
}

Python

def code(x, y, z, a):
	tmp = 0
	if y <= -9.2e-8:
		tmp = x + math.tan((y + z))
	else:
		tmp = x + (math.tan(z) - math.tan(a))
	return tmp

Julia

function code(x, y, z, a)
	tmp = 0.0
	if (y <= -9.2e-8)
		tmp = Float64(x + tan(Float64(y + z)));
	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 <= -9.2e-8)
		tmp = x + tan((y + z));
	else
		tmp = x + (tan(z) - tan(a));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, a_] := If[LessEqual[y, -9.2e-8], N[(x + N[Tan[N[(y + z), $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 y < -9.2000000000000003e-8

   1. Initial program 57.0%

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

      1. add-log-exp 57.0%

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

   4. Applied egg-rr 57.0%

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

   5. Taylor expanded in a around 0 44.2%

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

   if -9.2000000000000003e-8 < y

   1. Initial program 85.5%

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

   3. Taylor expanded in y around 0 74.6%

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

      1. tan-quot 74.6%

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

      2. *-un-lft-identity 74.6%

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

   5. Applied egg-rr 74.6%

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

      1. *-lft-identity 74.6%

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

   7. Simplified 74.6%

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

4. Final simplification 67.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -9.2 \cdot 10^{-8}:\\ \;\;\;\;x + \tan \left(y + z\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(\tan z - \tan a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 78.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.3%

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

Alternative 11: 55.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -400 \lor \neg \left(a \leq 31\right):\\ \;\;\;\;x + \left(z - \tan a\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x + \tan \left(y + z\right)\right) - a\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (if (or (<= a -400.0) (not (<= a 31.0)))
   (+ x (- z (tan a)))
   (- (+ x (tan (+ y z))) a)))

C

double code(double x, double y, double z, double a) {
	double tmp;
	if ((a <= -400.0) || !(a <= 31.0)) {
		tmp = x + (z - tan(a));
	} else {
		tmp = (x + tan((y + z))) - a;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: a
    real(8) :: tmp
    if ((a <= (-400.0d0)) .or. (.not. (a <= 31.0d0))) then
        tmp = x + (z - tan(a))
    else
        tmp = (x + tan((y + z))) - a
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double a) {
	double tmp;
	if ((a <= -400.0) || !(a <= 31.0)) {
		tmp = x + (z - Math.tan(a));
	} else {
		tmp = (x + Math.tan((y + z))) - a;
	}
	return tmp;
}

Python

def code(x, y, z, a):
	tmp = 0
	if (a <= -400.0) or not (a <= 31.0):
		tmp = x + (z - math.tan(a))
	else:
		tmp = (x + math.tan((y + z))) - a
	return tmp

Julia

function code(x, y, z, a)
	tmp = 0.0
	if ((a <= -400.0) || !(a <= 31.0))
		tmp = Float64(x + Float64(z - tan(a)));
	else
		tmp = Float64(Float64(x + tan(Float64(y + z))) - a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, a)
	tmp = 0.0;
	if ((a <= -400.0) || ~((a <= 31.0)))
		tmp = x + (z - tan(a));
	else
		tmp = (x + tan((y + z))) - a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, a_] := If[Or[LessEqual[a, -400.0], N[Not[LessEqual[a, 31.0]], $MachinePrecision]], N[(x + N[(z - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x + N[Tan[N[(y + z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -400 or 31 < a

   1. Initial program 84.0%

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

   3. Taylor expanded in y around 0 68.3%

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

   4. Taylor expanded in z around 0 40.2%

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

   if -400 < a < 31

   1. Initial program 75.6%

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

   3. Taylor expanded in a around 0 72.9%

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

      1. associate-+r- 72.9%

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

   5. Applied egg-rr 72.9%

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

4. Final simplification 58.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -400 \lor \neg \left(a \leq 31\right):\\ \;\;\;\;x + \left(z - \tan a\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x + \tan \left(y + z\right)\right) - a\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 55.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -400 \lor \neg \left(a \leq 31\right):\\ \;\;\;\;x + \left(z - \tan a\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(\tan \left(y + z\right) - a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (if (or (<= a -400.0) (not (<= a 31.0)))
   (+ x (- z (tan a)))
   (+ x (- (tan (+ y z)) a))))

C

double code(double x, double y, double z, double a) {
	double tmp;
	if ((a <= -400.0) || !(a <= 31.0)) {
		tmp = x + (z - tan(a));
	} else {
		tmp = x + (tan((y + z)) - a);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: a
    real(8) :: tmp
    if ((a <= (-400.0d0)) .or. (.not. (a <= 31.0d0))) then
        tmp = x + (z - tan(a))
    else
        tmp = x + (tan((y + z)) - a)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double a) {
	double tmp;
	if ((a <= -400.0) || !(a <= 31.0)) {
		tmp = x + (z - Math.tan(a));
	} else {
		tmp = x + (Math.tan((y + z)) - a);
	}
	return tmp;
}

Python

def code(x, y, z, a):
	tmp = 0
	if (a <= -400.0) or not (a <= 31.0):
		tmp = x + (z - math.tan(a))
	else:
		tmp = x + (math.tan((y + z)) - a)
	return tmp

Julia

function code(x, y, z, a)
	tmp = 0.0
	if ((a <= -400.0) || !(a <= 31.0))
		tmp = Float64(x + Float64(z - tan(a)));
	else
		tmp = Float64(x + Float64(tan(Float64(y + z)) - a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, a)
	tmp = 0.0;
	if ((a <= -400.0) || ~((a <= 31.0)))
		tmp = x + (z - tan(a));
	else
		tmp = x + (tan((y + z)) - a);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, a_] := If[Or[LessEqual[a, -400.0], N[Not[LessEqual[a, 31.0]], $MachinePrecision]], N[(x + N[(z - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[Tan[N[(y + z), $MachinePrecision]], $MachinePrecision] - a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -400 or 31 < a

   1. Initial program 84.0%

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

   3. Taylor expanded in y around 0 68.3%

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

   4. Taylor expanded in z around 0 40.2%

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

   if -400 < a < 31

   1. Initial program 75.6%

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

   3. Taylor expanded in a around 0 72.9%

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

4. Final simplification 58.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -400 \lor \neg \left(a \leq 31\right):\\ \;\;\;\;x + \left(z - \tan a\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(\tan \left(y + z\right) - a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 36.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 1.2:\\ \;\;\;\;x + \left(z - \tan a\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a) :precision binary64 (if (<= z 1.2) (+ x (- z (tan a))) x))

C

double code(double x, double y, double z, double a) {
	double tmp;
	if (z <= 1.2) {
		tmp = x + (z - tan(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 (z <= 1.2d0) then
        tmp = x + (z - tan(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 (z <= 1.2) {
		tmp = x + (z - Math.tan(a));
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, a):
	tmp = 0
	if z <= 1.2:
		tmp = x + (z - math.tan(a))
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, a)
	tmp = 0.0
	if (z <= 1.2)
		tmp = Float64(x + Float64(z - tan(a)));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, a)
	tmp = 0.0;
	if (z <= 1.2)
		tmp = x + (z - tan(a));
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, a_] := If[LessEqual[z, 1.2], N[(x + N[(z - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if z < 1.19999999999999996

   1. Initial program 86.2%

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

   3. Taylor expanded in y around 0 65.4%

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

   4. Taylor expanded in z around 0 46.3%

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

   if 1.19999999999999996 < z

   1. Initial program 56.2%

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

   3. Taylor expanded in x around inf 23.1%

      \[\leadsto \color{blue}{x} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 32.0% 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 79.3%

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

3. Taylor expanded in x around inf 34.5%

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

Reproduce

herbie shell --seed 2024086 
(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))))