tan-example (used to crash)

Percentage Accurate: 80.0% → 99.7%

Time: 39.5s

Alternatives: 12

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: 80.0% 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

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: a
    real(8) :: t_0
    t_0 = tan(y) * tan(z)
    code = x + ((((tan(y) + tan(z)) * cos(a)) + (sin(a) * (t_0 + (-1.0d0)))) / (cos(a) * (1.0d0 - t_0)))
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)) * Math.cos(a)) + (Math.sin(a) * (t_0 + -1.0))) / (Math.cos(a) * (1.0 - t_0)));
}

Python

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

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)) * cos(a)) + Float64(sin(a) * Float64(t_0 + -1.0))) / Float64(cos(a) * Float64(1.0 - t_0))))
end

MATLAB

function tmp = code(x, y, z, a)
	t_0 = tan(y) * tan(z);
	tmp = x + ((((tan(y) + tan(z)) * cos(a)) + (sin(a) * (t_0 + -1.0))) / (cos(a) * (1.0 - t_0)));
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[Cos[a], $MachinePrecision]), $MachinePrecision] + N[(N[Sin[a], $MachinePrecision] * N[(t$95$0 + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Cos[a], $MachinePrecision] * N[(1.0 - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 76.2%

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

   1. tan-sum 99.7%

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

   2. tan-quot 99.7%

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

   3. frac-sub 99.7%

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

4. Applied egg-rr 99.7%

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

5. Final simplification 99.7%

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

Alternative 2: 99.7% accurate, 0.4× speedup

Math

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

FPCore

(FPCore (x y z a)
 :precision binary64
 (-
  x
  (-
   (tan a)
   (/ (+ (tan y) (tan z)) (+ 1.0 (+ 1.0 (- -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)) / (1.0 + (1.0 + (-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)) / (1.0d0 + (1.0d0 + ((-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)) / (1.0 + (1.0 + (-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)) / (1.0 + (1.0 + (-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)) / Float64(1.0 + Float64(1.0 + 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)) / (1.0 + (1.0 + (-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[(1.0 + N[(1.0 + N[(-1.0 - N[(N[Tan[y], $MachinePrecision] * N[Tan[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 76.2%

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

   1. tan-sum 99.7%

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

   2. tan-quot 99.7%

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

   3. frac-sub 99.7%

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

4. Applied egg-rr 99.7%

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

   1. div-sub 99.7%

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

   2. times-frac 99.7%

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

   3. *-inverses 99.7%

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

   4. cancel-sign-sub-inv 99.7%

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

6. Simplified 99.7%

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

   1. expm1-log1p-u 91.9%

      \[\leadsto x + \left(\frac{\tan y + \tan z}{1 - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\tan y \cdot \tan z\right)\right)}} + \frac{-\sin a}{\cos a}\right) \]

   2. expm1-udef 91.9%

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

   3. log1p-udef 91.9%

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

   4. add-exp-log 99.7%

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

8. Applied egg-rr 99.7%

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

   1. distribute-frac-neg 99.7%

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

   2. tan-quot 99.7%

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

10. Applied egg-rr 99.7%

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

11. Final simplification 99.7%

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

Alternative 3: 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 76.2%

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

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

Alternative 4: 89.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan \left(y + z\right)\\ \mathbf{if}\;a \leq -0.00142:\\ \;\;\;\;\left(x + t\_0\right) - \tan a\\ \mathbf{elif}\;a \leq 2.9 \cdot 10^{-12}:\\ \;\;\;\;x + \left(\frac{\tan y + \tan z}{1 - \tan y \cdot \tan z} - a\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(t\_0 - \tan a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (let* ((t_0 (tan (+ y z))))
   (if (<= a -0.00142)
     (- (+ x t_0) (tan a))
     (if (<= a 2.9e-12)
       (+ x (- (/ (+ (tan y) (tan z)) (- 1.0 (* (tan y) (tan z)))) a))
       (+ x (- t_0 (tan a)))))))

C

double code(double x, double y, double z, double a) {
	double t_0 = tan((y + z));
	double tmp;
	if (a <= -0.00142) {
		tmp = (x + t_0) - tan(a);
	} else if (a <= 2.9e-12) {
		tmp = x + (((tan(y) + tan(z)) / (1.0 - (tan(y) * tan(z)))) - a);
	} else {
		tmp = x + (t_0 - tan(a));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: a
    real(8) :: t_0
    real(8) :: tmp
    t_0 = tan((y + z))
    if (a <= (-0.00142d0)) then
        tmp = (x + t_0) - tan(a)
    else if (a <= 2.9d-12) then
        tmp = x + (((tan(y) + tan(z)) / (1.0d0 - (tan(y) * tan(z)))) - a)
    else
        tmp = x + (t_0 - tan(a))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double a) {
	double t_0 = Math.tan((y + z));
	double tmp;
	if (a <= -0.00142) {
		tmp = (x + t_0) - Math.tan(a);
	} else if (a <= 2.9e-12) {
		tmp = x + (((Math.tan(y) + Math.tan(z)) / (1.0 - (Math.tan(y) * Math.tan(z)))) - a);
	} else {
		tmp = x + (t_0 - Math.tan(a));
	}
	return tmp;
}

Python

def code(x, y, z, a):
	t_0 = math.tan((y + z))
	tmp = 0
	if a <= -0.00142:
		tmp = (x + t_0) - math.tan(a)
	elif a <= 2.9e-12:
		tmp = x + (((math.tan(y) + math.tan(z)) / (1.0 - (math.tan(y) * math.tan(z)))) - a)
	else:
		tmp = x + (t_0 - math.tan(a))
	return tmp

Julia

function code(x, y, z, a)
	t_0 = tan(Float64(y + z))
	tmp = 0.0
	if (a <= -0.00142)
		tmp = Float64(Float64(x + t_0) - tan(a));
	elseif (a <= 2.9e-12)
		tmp = Float64(x + Float64(Float64(Float64(tan(y) + tan(z)) / Float64(1.0 - Float64(tan(y) * tan(z)))) - a));
	else
		tmp = Float64(x + Float64(t_0 - tan(a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, a)
	t_0 = tan((y + z));
	tmp = 0.0;
	if (a <= -0.00142)
		tmp = (x + t_0) - tan(a);
	elseif (a <= 2.9e-12)
		tmp = x + (((tan(y) + tan(z)) / (1.0 - (tan(y) * tan(z)))) - a);
	else
		tmp = x + (t_0 - tan(a));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, a_] := Block[{t$95$0 = N[Tan[N[(y + z), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[a, -0.00142], N[(N[(x + t$95$0), $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 2.9e-12], N[(x + N[(N[(N[(N[Tan[y], $MachinePrecision] + N[Tan[z], $MachinePrecision]), $MachinePrecision] / N[(1.0 - N[(N[Tan[y], $MachinePrecision] * N[Tan[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision]), $MachinePrecision], N[(x + N[(t$95$0 - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if a < -0.00142000000000000004

   1. Initial program 76.6%

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

      1. associate-+r- 76.7%

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

      2. +-commutative 76.7%

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

   4. Applied egg-rr 76.7%

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

   if -0.00142000000000000004 < a < 2.9000000000000002e-12

   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 99.8%

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

      2. tan-quot 99.8%

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

      3. frac-sub 99.8%

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

   4. Applied egg-rr 99.8%

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

      1. div-sub 99.8%

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

      2. times-frac 99.8%

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

      3. *-inverses 99.8%

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

      4. cancel-sign-sub-inv 99.8%

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

   6. Simplified 99.8%

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

   7. Taylor expanded in a around 0 99.8%

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

      1. mul-1-neg 99.8%

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

   9. Simplified 99.8%

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

   if 2.9000000000000002e-12 < a

   1. Initial program 77.1%

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

4. Final simplification 89.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -0.00142:\\ \;\;\;\;\left(x + \tan \left(y + z\right)\right) - \tan a\\ \mathbf{elif}\;a \leq 2.9 \cdot 10^{-12}:\\ \;\;\;\;x + \left(\frac{\tan y + \tan z}{1 - \tan y \cdot \tan z} - a\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(\tan \left(y + z\right) - \tan a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 42.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -0.7:\\ \;\;\;\;e^{\log x}\\ \mathbf{elif}\;y \leq 3.8 \cdot 10^{-6}:\\ \;\;\;\;x + \left(y - \tan a\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{expm1}\left(\mathsf{log1p}\left(x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (if (<= y -0.7)
   (exp (log x))
   (if (<= y 3.8e-6) (+ x (- y (tan a))) (expm1 (log1p x)))))

C

double code(double x, double y, double z, double a) {
	double tmp;
	if (y <= -0.7) {
		tmp = exp(log(x));
	} else if (y <= 3.8e-6) {
		tmp = x + (y - tan(a));
	} else {
		tmp = expm1(log1p(x));
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double a) {
	double tmp;
	if (y <= -0.7) {
		tmp = Math.exp(Math.log(x));
	} else if (y <= 3.8e-6) {
		tmp = x + (y - Math.tan(a));
	} else {
		tmp = Math.expm1(Math.log1p(x));
	}
	return tmp;
}

Python

def code(x, y, z, a):
	tmp = 0
	if y <= -0.7:
		tmp = math.exp(math.log(x))
	elif y <= 3.8e-6:
		tmp = x + (y - math.tan(a))
	else:
		tmp = math.expm1(math.log1p(x))
	return tmp

Julia

function code(x, y, z, a)
	tmp = 0.0
	if (y <= -0.7)
		tmp = exp(log(x));
	elseif (y <= 3.8e-6)
		tmp = Float64(x + Float64(y - tan(a)));
	else
		tmp = expm1(log1p(x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, a_] := If[LessEqual[y, -0.7], N[Exp[N[Log[x], $MachinePrecision]], $MachinePrecision], If[LessEqual[y, 3.8e-6], N[(x + N[(y - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(Exp[N[Log[1 + x], $MachinePrecision]] - 1), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -0.69999999999999996

   1. Initial program 61.9%

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

      1. add-exp-log 56.5%

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

      2. +-commutative 56.5%

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

      3. associate-+l- 56.5%

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

   4. Applied egg-rr 56.5%

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

   5. Taylor expanded in x around inf 21.7%

      \[\leadsto e^{\color{blue}{-1 \cdot \log \left(\frac{1}{x}\right)}} \]
   6. Step-by-step derivation

      1. mul-1-neg 21.7%

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

      2. log-rec 21.7%

         \[\leadsto e^{-\color{blue}{\left(-\log x\right)}} \]

      3. remove-double-neg 21.7%

         \[\leadsto e^{\color{blue}{\log x}} \]

   7. Simplified 21.7%

      \[\leadsto e^{\color{blue}{\log x}} \]

   if -0.69999999999999996 < y < 3.8e-6

   1. Initial program 99.8%

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

   3. Taylor expanded in z around 0 61.3%

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

   4. Taylor expanded in y around 0 61.0%

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

   if 3.8e-6 < y

   1. Initial program 46.3%

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

      1. add-exp-log 44.2%

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

      2. +-commutative 44.2%

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

      3. associate-+l- 44.2%

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

   4. Applied egg-rr 44.2%

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

   5. Taylor expanded in x around inf 22.5%

      \[\leadsto e^{\color{blue}{-1 \cdot \log \left(\frac{1}{x}\right)}} \]
   6. Step-by-step derivation

      1. mul-1-neg 22.5%

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

      2. log-rec 22.5%

         \[\leadsto e^{-\color{blue}{\left(-\log x\right)}} \]

      3. remove-double-neg 22.5%

         \[\leadsto e^{\color{blue}{\log x}} \]

   7. Simplified 22.5%

      \[\leadsto e^{\color{blue}{\log x}} \]
   8. Step-by-step derivation

      1. rem-exp-log 22.5%

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

      2. expm1-log1p-u 22.5%

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

   9. Applied egg-rr 22.5%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 41.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.7:\\ \;\;\;\;e^{\log x}\\ \mathbf{elif}\;y \leq 3.8 \cdot 10^{-6}:\\ \;\;\;\;x + \left(y - \tan a\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{expm1}\left(\mathsf{log1p}\left(x\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 42.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -0.7:\\ \;\;\;\;e^{-\log \left(\frac{1}{x}\right)}\\ \mathbf{elif}\;y \leq 3.8 \cdot 10^{-6}:\\ \;\;\;\;x + \left(y - \tan a\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{expm1}\left(\mathsf{log1p}\left(x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (if (<= y -0.7)
   (exp (- (log (/ 1.0 x))))
   (if (<= y 3.8e-6) (+ x (- y (tan a))) (expm1 (log1p x)))))

C

double code(double x, double y, double z, double a) {
	double tmp;
	if (y <= -0.7) {
		tmp = exp(-log((1.0 / x)));
	} else if (y <= 3.8e-6) {
		tmp = x + (y - tan(a));
	} else {
		tmp = expm1(log1p(x));
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double a) {
	double tmp;
	if (y <= -0.7) {
		tmp = Math.exp(-Math.log((1.0 / x)));
	} else if (y <= 3.8e-6) {
		tmp = x + (y - Math.tan(a));
	} else {
		tmp = Math.expm1(Math.log1p(x));
	}
	return tmp;
}

Python

def code(x, y, z, a):
	tmp = 0
	if y <= -0.7:
		tmp = math.exp(-math.log((1.0 / x)))
	elif y <= 3.8e-6:
		tmp = x + (y - math.tan(a))
	else:
		tmp = math.expm1(math.log1p(x))
	return tmp

Julia

function code(x, y, z, a)
	tmp = 0.0
	if (y <= -0.7)
		tmp = exp(Float64(-log(Float64(1.0 / x))));
	elseif (y <= 3.8e-6)
		tmp = Float64(x + Float64(y - tan(a)));
	else
		tmp = expm1(log1p(x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, a_] := If[LessEqual[y, -0.7], N[Exp[(-N[Log[N[(1.0 / x), $MachinePrecision]], $MachinePrecision])], $MachinePrecision], If[LessEqual[y, 3.8e-6], N[(x + N[(y - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(Exp[N[Log[1 + x], $MachinePrecision]] - 1), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -0.69999999999999996

   1. Initial program 61.9%

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

      1. add-exp-log 56.5%

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

      2. +-commutative 56.5%

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

      3. associate-+l- 56.5%

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

   4. Applied egg-rr 56.5%

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

   5. Taylor expanded in x around inf 21.7%

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

   if -0.69999999999999996 < y < 3.8e-6

   1. Initial program 99.8%

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

   3. Taylor expanded in z around 0 61.3%

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

   4. Taylor expanded in y around 0 61.0%

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

   if 3.8e-6 < y

   1. Initial program 46.3%

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

      1. add-exp-log 44.2%

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

      2. +-commutative 44.2%

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

      3. associate-+l- 44.2%

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

   4. Applied egg-rr 44.2%

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

   5. Taylor expanded in x around inf 22.5%

      \[\leadsto e^{\color{blue}{-1 \cdot \log \left(\frac{1}{x}\right)}} \]
   6. Step-by-step derivation

      1. mul-1-neg 22.5%

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

      2. log-rec 22.5%

         \[\leadsto e^{-\color{blue}{\left(-\log x\right)}} \]

      3. remove-double-neg 22.5%

         \[\leadsto e^{\color{blue}{\log x}} \]

   7. Simplified 22.5%

      \[\leadsto e^{\color{blue}{\log x}} \]
   8. Step-by-step derivation

      1. rem-exp-log 22.5%

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

      2. expm1-log1p-u 22.5%

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

   9. Applied egg-rr 22.5%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 41.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.7:\\ \;\;\;\;e^{-\log \left(\frac{1}{x}\right)}\\ \mathbf{elif}\;y \leq 3.8 \cdot 10^{-6}:\\ \;\;\;\;x + \left(y - \tan a\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{expm1}\left(\mathsf{log1p}\left(x\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 46.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -9.8 \cdot 10^{-6}:\\ \;\;\;\;x + \left(\tan y + \tan a\right)\\ \mathbf{elif}\;y \leq 3.8 \cdot 10^{-6}:\\ \;\;\;\;x + \left(y - \tan a\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{expm1}\left(\mathsf{log1p}\left(x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (if (<= y -9.8e-6)
   (+ x (+ (tan y) (tan a)))
   (if (<= y 3.8e-6) (+ x (- y (tan a))) (expm1 (log1p x)))))

C

double code(double x, double y, double z, double a) {
	double tmp;
	if (y <= -9.8e-6) {
		tmp = x + (tan(y) + tan(a));
	} else if (y <= 3.8e-6) {
		tmp = x + (y - tan(a));
	} else {
		tmp = expm1(log1p(x));
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double a) {
	double tmp;
	if (y <= -9.8e-6) {
		tmp = x + (Math.tan(y) + Math.tan(a));
	} else if (y <= 3.8e-6) {
		tmp = x + (y - Math.tan(a));
	} else {
		tmp = Math.expm1(Math.log1p(x));
	}
	return tmp;
}

Python

def code(x, y, z, a):
	tmp = 0
	if y <= -9.8e-6:
		tmp = x + (math.tan(y) + math.tan(a))
	elif y <= 3.8e-6:
		tmp = x + (y - math.tan(a))
	else:
		tmp = math.expm1(math.log1p(x))
	return tmp

Julia

function code(x, y, z, a)
	tmp = 0.0
	if (y <= -9.8e-6)
		tmp = Float64(x + Float64(tan(y) + tan(a)));
	elseif (y <= 3.8e-6)
		tmp = Float64(x + Float64(y - tan(a)));
	else
		tmp = expm1(log1p(x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, a_] := If[LessEqual[y, -9.8e-6], N[(x + N[(N[Tan[y], $MachinePrecision] + N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 3.8e-6], N[(x + N[(y - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(Exp[N[Log[1 + x], $MachinePrecision]] - 1), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -9.79999999999999934e-6

   1. Initial program 62.5%

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

   3. Taylor expanded in z around 0 61.2%

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

      1. tan-quot 61.2%

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

      2. sub-neg 61.2%

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

      3. tan-quot 61.2%

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

      4. distribute-frac-neg 61.2%

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

      5. add-sqr-sqrt 26.4%

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

      6. sqrt-unprod 47.5%

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

      7. sqr-neg 47.5%

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

      8. sqrt-unprod 21.1%

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

      9. add-sqr-sqrt 40.0%

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

      10. tan-quot 40.0%

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

   5. Applied egg-rr 40.0%

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

   if -9.79999999999999934e-6 < y < 3.8e-6

   1. Initial program 99.8%

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

   3. Taylor expanded in z around 0 61.0%

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

   4. Taylor expanded in y around 0 61.0%

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

   if 3.8e-6 < y

   1. Initial program 46.3%

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

      1. add-exp-log 44.2%

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

      2. +-commutative 44.2%

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

      3. associate-+l- 44.2%

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

   4. Applied egg-rr 44.2%

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

   5. Taylor expanded in x around inf 22.5%

      \[\leadsto e^{\color{blue}{-1 \cdot \log \left(\frac{1}{x}\right)}} \]
   6. Step-by-step derivation

      1. mul-1-neg 22.5%

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

      2. log-rec 22.5%

         \[\leadsto e^{-\color{blue}{\left(-\log x\right)}} \]

      3. remove-double-neg 22.5%

         \[\leadsto e^{\color{blue}{\log x}} \]

   7. Simplified 22.5%

      \[\leadsto e^{\color{blue}{\log x}} \]
   8. Step-by-step derivation

      1. rem-exp-log 22.5%

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

      2. expm1-log1p-u 22.5%

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

   9. Applied egg-rr 22.5%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 45.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -9.8 \cdot 10^{-6}:\\ \;\;\;\;x + \left(\tan y + \tan a\right)\\ \mathbf{elif}\;y \leq 3.8 \cdot 10^{-6}:\\ \;\;\;\;x + \left(y - \tan a\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{expm1}\left(\mathsf{log1p}\left(x\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 80.0% 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 76.2%

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

3. Final simplification 76.2%

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

Alternative 9: 42.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -0.7:\\ \;\;\;\;e^{\log x}\\ \mathbf{elif}\;y \leq 3.8 \cdot 10^{-6}:\\ \;\;\;\;x + \left(y - \tan a\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (if (<= y -0.7) (exp (log x)) (if (<= y 3.8e-6) (+ x (- y (tan a))) x)))

C

double code(double x, double y, double z, double a) {
	double tmp;
	if (y <= -0.7) {
		tmp = exp(log(x));
	} else if (y <= 3.8e-6) {
		tmp = x + (y - 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 (y <= (-0.7d0)) then
        tmp = exp(log(x))
    else if (y <= 3.8d-6) then
        tmp = x + (y - 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 (y <= -0.7) {
		tmp = Math.exp(Math.log(x));
	} else if (y <= 3.8e-6) {
		tmp = x + (y - Math.tan(a));
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, a):
	tmp = 0
	if y <= -0.7:
		tmp = math.exp(math.log(x))
	elif y <= 3.8e-6:
		tmp = x + (y - math.tan(a))
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, a)
	tmp = 0.0
	if (y <= -0.7)
		tmp = exp(log(x));
	elseif (y <= 3.8e-6)
		tmp = Float64(x + Float64(y - tan(a)));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, a)
	tmp = 0.0;
	if (y <= -0.7)
		tmp = exp(log(x));
	elseif (y <= 3.8e-6)
		tmp = x + (y - tan(a));
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, a_] := If[LessEqual[y, -0.7], N[Exp[N[Log[x], $MachinePrecision]], $MachinePrecision], If[LessEqual[y, 3.8e-6], N[(x + N[(y - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], x]]

Derivation

1. Split input into 3 regimes

2. if y < -0.69999999999999996

   1. Initial program 61.9%

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

      1. add-exp-log 56.5%

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

      2. +-commutative 56.5%

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

      3. associate-+l- 56.5%

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

   4. Applied egg-rr 56.5%

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

   5. Taylor expanded in x around inf 21.7%

      \[\leadsto e^{\color{blue}{-1 \cdot \log \left(\frac{1}{x}\right)}} \]
   6. Step-by-step derivation

      1. mul-1-neg 21.7%

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

      2. log-rec 21.7%

         \[\leadsto e^{-\color{blue}{\left(-\log x\right)}} \]

      3. remove-double-neg 21.7%

         \[\leadsto e^{\color{blue}{\log x}} \]

   7. Simplified 21.7%

      \[\leadsto e^{\color{blue}{\log x}} \]

   if -0.69999999999999996 < y < 3.8e-6

   1. Initial program 99.8%

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

   3. Taylor expanded in z around 0 61.3%

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

   4. Taylor expanded in y around 0 61.0%

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

   if 3.8e-6 < y

   1. Initial program 46.3%

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

   3. Taylor expanded in x around inf 22.5%

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

4. Final simplification 41.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.7:\\ \;\;\;\;e^{\log x}\\ \mathbf{elif}\;y \leq 3.8 \cdot 10^{-6}:\\ \;\;\;\;x + \left(y - \tan a\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 60.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 76.2%

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

3. Taylor expanded in z around 0 57.0%

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

   1. tan-quot 57.0%

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

   2. expm1-log1p-u 48.8%

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

   3. expm1-udef 48.8%

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

5. Applied egg-rr 48.8%

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

   1. expm1-def 48.8%

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

   2. expm1-log1p 57.0%

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

7. Simplified 57.0%

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

8. Final simplification 57.0%

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

Alternative 11: 42.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -0.7:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 3.8 \cdot 10^{-6}:\\ \;\;\;\;x + \left(y - \tan a\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (if (<= y -0.7) x (if (<= y 3.8e-6) (+ x (- y (tan a))) x)))

C

double code(double x, double y, double z, double a) {
	double tmp;
	if (y <= -0.7) {
		tmp = x;
	} else if (y <= 3.8e-6) {
		tmp = x + (y - 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 (y <= (-0.7d0)) then
        tmp = x
    else if (y <= 3.8d-6) then
        tmp = x + (y - 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 (y <= -0.7) {
		tmp = x;
	} else if (y <= 3.8e-6) {
		tmp = x + (y - Math.tan(a));
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, a):
	tmp = 0
	if y <= -0.7:
		tmp = x
	elif y <= 3.8e-6:
		tmp = x + (y - math.tan(a))
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, a)
	tmp = 0.0
	if (y <= -0.7)
		tmp = x;
	elseif (y <= 3.8e-6)
		tmp = Float64(x + Float64(y - tan(a)));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, a)
	tmp = 0.0;
	if (y <= -0.7)
		tmp = x;
	elseif (y <= 3.8e-6)
		tmp = x + (y - tan(a));
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, a_] := If[LessEqual[y, -0.7], x, If[LessEqual[y, 3.8e-6], N[(x + N[(y - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if y < -0.69999999999999996 or 3.8e-6 < y

   1. Initial program 53.7%

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

   3. Taylor expanded in x around inf 22.1%

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

   if -0.69999999999999996 < y < 3.8e-6

   1. Initial program 99.8%

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

   3. Taylor expanded in z around 0 61.3%

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

   4. Taylor expanded in y around 0 61.0%

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

4. Final simplification 41.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.7:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 3.8 \cdot 10^{-6}:\\ \;\;\;\;x + \left(y - \tan a\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 31.8% 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 76.2%

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

3. Taylor expanded in x around inf 31.1%

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

4. Final simplification 31.1%

   \[\leadsto x \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024031 
(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))))