tan-example (used to crash)

Percentage Accurate: 79.1% → 99.7%

Time: 43.4s

Alternatives: 9

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 79.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.0%

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

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

Alternative 2: 99.7% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.0%

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

   1. tan-sum 99.7%

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

   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 3: 59.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan \left(y + z\right)\\ \mathbf{if}\;t\_0 \leq -0.02 \lor \neg \left(t\_0 \leq 5 \cdot 10^{-13}\right):\\ \;\;\;\;x + t\_0\\ \mathbf{else}:\\ \;\;\;\;x + \left(y - \tan a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (let* ((t_0 (tan (+ y z))))
   (if (or (<= t_0 -0.02) (not (<= t_0 5e-13)))
     (+ x t_0)
     (+ x (- y (tan a))))))

C

double code(double x, double y, double z, double a) {
	double t_0 = tan((y + z));
	double tmp;
	if ((t_0 <= -0.02) || !(t_0 <= 5e-13)) {
		tmp = x + t_0;
	} else {
		tmp = x + (y - 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 ((t_0 <= (-0.02d0)) .or. (.not. (t_0 <= 5d-13))) then
        tmp = x + t_0
    else
        tmp = x + (y - 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 ((t_0 <= -0.02) || !(t_0 <= 5e-13)) {
		tmp = x + t_0;
	} else {
		tmp = x + (y - Math.tan(a));
	}
	return tmp;
}

Python

def code(x, y, z, a):
	t_0 = math.tan((y + z))
	tmp = 0
	if (t_0 <= -0.02) or not (t_0 <= 5e-13):
		tmp = x + t_0
	else:
		tmp = x + (y - math.tan(a))
	return tmp

Julia

function code(x, y, z, a)
	t_0 = tan(Float64(y + z))
	tmp = 0.0
	if ((t_0 <= -0.02) || !(t_0 <= 5e-13))
		tmp = Float64(x + t_0);
	else
		tmp = Float64(x + Float64(y - tan(a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, a)
	t_0 = tan((y + z));
	tmp = 0.0;
	if ((t_0 <= -0.02) || ~((t_0 <= 5e-13)))
		tmp = x + t_0;
	else
		tmp = x + (y - 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[Or[LessEqual[t$95$0, -0.02], N[Not[LessEqual[t$95$0, 5e-13]], $MachinePrecision]], N[(x + t$95$0), $MachinePrecision], N[(x + N[(y - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (tan.f64 (+.f64 y z)) < -0.0200000000000000004 or 4.9999999999999999e-13 < (tan.f64 (+.f64 y z))

   1. Initial program 68.9%

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

      1. add-cbrt-cube 68.5%

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

      2. pow1/3 62.3%

         \[\leadsto \color{blue}{{\left(\left(\left(x + \left(\tan \left(y + z\right) - \tan a\right)\right) \cdot \left(x + \left(\tan \left(y + z\right) - \tan a\right)\right)\right) \cdot \left(x + \left(\tan \left(y + z\right) - \tan a\right)\right)\right)}^{0.3333333333333333}} \]

      3. pow3 62.2%

         \[\leadsto {\color{blue}{\left({\left(x + \left(\tan \left(y + z\right) - \tan a\right)\right)}^{3}\right)}}^{0.3333333333333333} \]

      4. +-commutative 62.2%

         \[\leadsto {\left({\color{blue}{\left(\left(\tan \left(y + z\right) - \tan a\right) + x\right)}}^{3}\right)}^{0.3333333333333333} \]

      5. associate-+l- 62.3%

         \[\leadsto {\left({\color{blue}{\left(\tan \left(y + z\right) - \left(\tan a - x\right)\right)}}^{3}\right)}^{0.3333333333333333} \]

   4. Applied egg-rr 62.3%

      \[\leadsto \color{blue}{{\left({\left(\tan \left(y + z\right) - \left(\tan a - x\right)\right)}^{3}\right)}^{0.3333333333333333}} \]

   5. Taylor expanded in a around 0 41.9%

      \[\leadsto {\color{blue}{\left({\left(x + \frac{\sin \left(y + z\right)}{\cos \left(y + z\right)}\right)}^{3}\right)}}^{0.3333333333333333} \]
   6. Step-by-step derivation

      1. unpow1/3 45.4%

         \[\leadsto \color{blue}{\sqrt[3]{{\left(x + \frac{\sin \left(y + z\right)}{\cos \left(y + z\right)}\right)}^{3}}} \]

      2. rem-cbrt-cube 45.6%

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

      3. +-commutative 45.6%

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

      4. quot-tan 45.6%

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

   7. Applied egg-rr 45.6%

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

   if -0.0200000000000000004 < (tan.f64 (+.f64 y z)) < 4.9999999999999999e-13

   1. Initial program 99.9%

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

      1. add-sqr-sqrt 47.7%

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

      2. sqrt-unprod 99.3%

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

      3. pow2 99.3%

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

   4. Applied egg-rr 99.3%

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

   5. Taylor expanded in z around 0 99.3%

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

   6. Taylor expanded in y around 0 99.9%

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

4. Final simplification 59.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\tan \left(y + z\right) \leq -0.02 \lor \neg \left(\tan \left(y + z\right) \leq 5 \cdot 10^{-13}\right):\\ \;\;\;\;x + \tan \left(y + z\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(y - \tan a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 59.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y + z \leq 5 \cdot 10^{-13}:\\ \;\;\;\;x + \left(\tan y - \tan a\right)\\ \mathbf{else}:\\ \;\;\;\;x + \tan \left(y + z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (if (<= (+ y z) 5e-13) (+ x (- (tan y) (tan a))) (+ x (tan (+ y z)))))

C

double code(double x, double y, double z, double a) {
	double tmp;
	if ((y + z) <= 5e-13) {
		tmp = x + (tan(y) - tan(a));
	} else {
		tmp = x + tan((y + z));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: a
    real(8) :: tmp
    if ((y + z) <= 5d-13) then
        tmp = x + (tan(y) - tan(a))
    else
        tmp = x + tan((y + z))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double a) {
	double tmp;
	if ((y + z) <= 5e-13) {
		tmp = x + (Math.tan(y) - Math.tan(a));
	} else {
		tmp = x + Math.tan((y + z));
	}
	return tmp;
}

Python

def code(x, y, z, a):
	tmp = 0
	if (y + z) <= 5e-13:
		tmp = x + (math.tan(y) - math.tan(a))
	else:
		tmp = x + math.tan((y + z))
	return tmp

Julia

function code(x, y, z, a)
	tmp = 0.0
	if (Float64(y + z) <= 5e-13)
		tmp = Float64(x + Float64(tan(y) - tan(a)));
	else
		tmp = Float64(x + tan(Float64(y + z)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, a)
	tmp = 0.0;
	if ((y + z) <= 5e-13)
		tmp = x + (tan(y) - tan(a));
	else
		tmp = x + tan((y + z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, a_] := If[LessEqual[N[(y + z), $MachinePrecision], 5e-13], N[(x + N[(N[Tan[y], $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[Tan[N[(y + z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 y z) < 4.9999999999999999e-13

   1. Initial program 80.9%

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

      1. add-sqr-sqrt 36.7%

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

      2. sqrt-unprod 64.5%

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

      3. pow2 64.5%

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

   4. Applied egg-rr 64.5%

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

   5. Taylor expanded in z around 0 58.3%

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

      1. tan-quot 58.4%

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

      2. sqrt-pow1 66.1%

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

      3. metadata-eval 66.1%

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

      4. pow1 66.1%

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

      5. *-un-lft-identity 66.1%

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

   7. Applied egg-rr 66.1%

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

      1. *-lft-identity 66.1%

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

   9. Simplified 66.1%

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

   if 4.9999999999999999e-13 < (+.f64 y z)

   1. Initial program 69.9%

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

      1. add-cbrt-cube 69.5%

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

      2. pow1/3 62.3%

         \[\leadsto \color{blue}{{\left(\left(\left(x + \left(\tan \left(y + z\right) - \tan a\right)\right) \cdot \left(x + \left(\tan \left(y + z\right) - \tan a\right)\right)\right) \cdot \left(x + \left(\tan \left(y + z\right) - \tan a\right)\right)\right)}^{0.3333333333333333}} \]

      3. pow3 62.2%

         \[\leadsto {\color{blue}{\left({\left(x + \left(\tan \left(y + z\right) - \tan a\right)\right)}^{3}\right)}}^{0.3333333333333333} \]

      4. +-commutative 62.2%

         \[\leadsto {\left({\color{blue}{\left(\left(\tan \left(y + z\right) - \tan a\right) + x\right)}}^{3}\right)}^{0.3333333333333333} \]

      5. associate-+l- 62.3%

         \[\leadsto {\left({\color{blue}{\left(\tan \left(y + z\right) - \left(\tan a - x\right)\right)}}^{3}\right)}^{0.3333333333333333} \]

   4. Applied egg-rr 62.3%

      \[\leadsto \color{blue}{{\left({\left(\tan \left(y + z\right) - \left(\tan a - x\right)\right)}^{3}\right)}^{0.3333333333333333}} \]

   5. Taylor expanded in a around 0 38.1%

      \[\leadsto {\color{blue}{\left({\left(x + \frac{\sin \left(y + z\right)}{\cos \left(y + z\right)}\right)}^{3}\right)}}^{0.3333333333333333} \]
   6. Step-by-step derivation

      1. unpow1/3 42.5%

         \[\leadsto \color{blue}{\sqrt[3]{{\left(x + \frac{\sin \left(y + z\right)}{\cos \left(y + z\right)}\right)}^{3}}} \]

      2. rem-cbrt-cube 42.7%

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

      3. +-commutative 42.7%

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

      4. quot-tan 42.7%

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

   7. Applied egg-rr 42.7%

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

4. Final simplification 57.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y + z \leq 5 \cdot 10^{-13}:\\ \;\;\;\;x + \left(\tan y - \tan a\right)\\ \mathbf{else}:\\ \;\;\;\;x + \tan \left(y + z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 69.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 2.4 \cdot 10^{-7}:\\ \;\;\;\;x + \left(\tan y - \tan a\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(\tan z - \tan a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (if (<= z 2.4e-7) (+ x (- (tan y) (tan a))) (+ x (- (tan z) (tan a)))))

C

double code(double x, double y, double z, double a) {
	double tmp;
	if (z <= 2.4e-7) {
		tmp = x + (tan(y) - tan(a));
	} else {
		tmp = x + (tan(z) - tan(a));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z <= 2.4d-7) then
        tmp = x + (tan(y) - tan(a))
    else
        tmp = x + (tan(z) - tan(a))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double a) {
	double tmp;
	if (z <= 2.4e-7) {
		tmp = x + (Math.tan(y) - Math.tan(a));
	} else {
		tmp = x + (Math.tan(z) - Math.tan(a));
	}
	return tmp;
}

Python

def code(x, y, z, a):
	tmp = 0
	if z <= 2.4e-7:
		tmp = x + (math.tan(y) - math.tan(a))
	else:
		tmp = x + (math.tan(z) - math.tan(a))
	return tmp

Julia

function code(x, y, z, a)
	tmp = 0.0
	if (z <= 2.4e-7)
		tmp = Float64(x + Float64(tan(y) - tan(a)));
	else
		tmp = Float64(x + Float64(tan(z) - tan(a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, a)
	tmp = 0.0;
	if (z <= 2.4e-7)
		tmp = x + (tan(y) - tan(a));
	else
		tmp = x + (tan(z) - tan(a));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, a_] := If[LessEqual[z, 2.4e-7], N[(x + N[(N[Tan[y], $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[Tan[z], $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 2.39999999999999979e-7

   1. Initial program 84.6%

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

      1. add-sqr-sqrt 37.3%

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

      2. sqrt-unprod 64.0%

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

      3. pow2 64.0%

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

   4. Applied egg-rr 64.0%

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

   5. Taylor expanded in z around 0 58.5%

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

      1. tan-quot 58.5%

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

      2. sqrt-pow1 71.5%

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

      3. metadata-eval 71.5%

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

      4. pow1 71.5%

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

      5. *-un-lft-identity 71.5%

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

   7. Applied egg-rr 71.5%

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

      1. *-lft-identity 71.5%

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

   9. Simplified 71.5%

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

   if 2.39999999999999979e-7 < z

   1. Initial program 56.8%

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

   3. Taylor expanded in y around 0 56.9%

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

      1. tan-quot 56.9%

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

      2. tan-quot 56.9%

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

      3. associate-+r- 56.9%

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

      4. tan-quot 56.9%

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

      5. +-commutative 56.9%

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

      6. tan-quot 56.9%

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

   5. Applied egg-rr 56.9%

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

4. Final simplification 67.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 2.4 \cdot 10^{-7}:\\ \;\;\;\;x + \left(\tan y - \tan a\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(\tan z - \tan a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 69.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 6.8 \cdot 10^{-7}:\\ \;\;\;\;\left(x + \tan y\right) - \tan a\\ \mathbf{else}:\\ \;\;\;\;x + \left(\tan z - \tan a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (if (<= z 6.8e-7) (- (+ x (tan y)) (tan a)) (+ x (- (tan z) (tan a)))))

C

double code(double x, double y, double z, double a) {
	double tmp;
	if (z <= 6.8e-7) {
		tmp = (x + tan(y)) - tan(a);
	} else {
		tmp = x + (tan(z) - tan(a));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z <= 6.8d-7) then
        tmp = (x + tan(y)) - tan(a)
    else
        tmp = x + (tan(z) - tan(a))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double a) {
	double tmp;
	if (z <= 6.8e-7) {
		tmp = (x + Math.tan(y)) - Math.tan(a);
	} else {
		tmp = x + (Math.tan(z) - Math.tan(a));
	}
	return tmp;
}

Python

def code(x, y, z, a):
	tmp = 0
	if z <= 6.8e-7:
		tmp = (x + math.tan(y)) - math.tan(a)
	else:
		tmp = x + (math.tan(z) - math.tan(a))
	return tmp

Julia

function code(x, y, z, a)
	tmp = 0.0
	if (z <= 6.8e-7)
		tmp = Float64(Float64(x + tan(y)) - tan(a));
	else
		tmp = Float64(x + Float64(tan(z) - tan(a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, a)
	tmp = 0.0;
	if (z <= 6.8e-7)
		tmp = (x + tan(y)) - tan(a);
	else
		tmp = x + (tan(z) - tan(a));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, a_] := If[LessEqual[z, 6.8e-7], N[(N[(x + N[Tan[y], $MachinePrecision]), $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision], N[(x + N[(N[Tan[z], $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 6.79999999999999948e-7

   1. Initial program 84.6%

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

      1. add-sqr-sqrt 37.3%

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

      2. sqrt-unprod 64.0%

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

      3. pow2 64.0%

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

   4. Applied egg-rr 64.0%

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

   5. Taylor expanded in z around 0 58.5%

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

      1. tan-quot 58.5%

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

      2. sqrt-pow1 71.5%

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

      3. metadata-eval 71.5%

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

      4. pow1 71.5%

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

      5. log1p-expm1-u 71.2%

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

   7. Applied egg-rr 71.2%

      \[\leadsto x + \left(\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\tan y\right)\right)} - \tan a\right) \]
   8. Step-by-step derivation

      1. log1p-expm1-u 71.5%

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

      2. associate-+r- 71.4%

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

   9. Applied egg-rr 71.4%

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

   if 6.79999999999999948e-7 < z

   1. Initial program 56.8%

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

   3. Taylor expanded in y around 0 56.9%

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

      1. tan-quot 56.9%

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

      2. tan-quot 56.9%

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

      3. associate-+r- 56.9%

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

      4. tan-quot 56.9%

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

      5. +-commutative 56.9%

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

      6. tan-quot 56.9%

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

   5. Applied egg-rr 56.9%

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

4. Final simplification 67.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 6.8 \cdot 10^{-7}:\\ \;\;\;\;\left(x + \tan y\right) - \tan a\\ \mathbf{else}:\\ \;\;\;\;x + \left(\tan z - \tan a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 79.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.0%

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

3. Final simplification 77.0%

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

Alternative 8: 37.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.15:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x + \left(y - \tan a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a) :precision binary64 (if (<= y -1.15) x (+ x (- y (tan a)))))

C

double code(double x, double y, double z, double a) {
	double tmp;
	if (y <= -1.15) {
		tmp = x;
	} else {
		tmp = x + (y - 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 <= (-1.15d0)) then
        tmp = x
    else
        tmp = x + (y - 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 <= -1.15) {
		tmp = x;
	} else {
		tmp = x + (y - Math.tan(a));
	}
	return tmp;
}

Python

def code(x, y, z, a):
	tmp = 0
	if y <= -1.15:
		tmp = x
	else:
		tmp = x + (y - math.tan(a))
	return tmp

Julia

function code(x, y, z, a)
	tmp = 0.0
	if (y <= -1.15)
		tmp = x;
	else
		tmp = Float64(x + Float64(y - tan(a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, a)
	tmp = 0.0;
	if (y <= -1.15)
		tmp = x;
	else
		tmp = x + (y - tan(a));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, a_] := If[LessEqual[y, -1.15], x, N[(x + N[(y - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.1499999999999999

   1. Initial program 51.8%

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

   3. Taylor expanded in x around inf 24.0%

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

   if -1.1499999999999999 < y

   1. Initial program 85.6%

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

      1. add-sqr-sqrt 40.3%

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

      2. sqrt-unprod 65.9%

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

      3. pow2 65.9%

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

   4. Applied egg-rr 65.9%

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

   5. Taylor expanded in z around 0 53.9%

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

   6. Taylor expanded in y around 0 44.1%

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

4. Final simplification 39.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.15:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x + \left(y - \tan a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 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 77.0%

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

3. Taylor expanded in x around inf 31.9%

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

4. Final simplification 31.9%

   \[\leadsto x \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024075 
(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))))