tan-example (used to crash)

Percentage Accurate: 79.4% → 99.7%

Time: 39.5s

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: 79.4% 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, 0.4× 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 79.2%

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

   1. tan-sum 99.8%

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

   2. div-inv 99.8%

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

3. Applied egg-rr 99.8%

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

   1. associate-*r/ 99.8%

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

   2. *-rgt-identity 99.8%

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

5. Simplified 99.8%

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

6. Final simplification 99.8%

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

Alternative 2: 89.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -0.055 \lor \neg \left(a \leq 1.14 \cdot 10^{-7}\right):\\ \;\;\;\;x + \left(\tan \left(y + z\right) - \tan a\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1 - \tan y \cdot \tan z}{\tan y + \tan z}} + \left(x - a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (if (or (<= a -0.055) (not (<= a 1.14e-7)))
   (+ x (- (tan (+ y z)) (tan a)))
   (+ (/ 1.0 (/ (- 1.0 (* (tan y) (tan z))) (+ (tan y) (tan z)))) (- x a))))

C

double code(double x, double y, double z, double a) {
	double tmp;
	if ((a <= -0.055) || !(a <= 1.14e-7)) {
		tmp = x + (tan((y + z)) - tan(a));
	} else {
		tmp = (1.0 / ((1.0 - (tan(y) * tan(z))) / (tan(y) + tan(z)))) + (x - a);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: a
    real(8) :: tmp
    if ((a <= (-0.055d0)) .or. (.not. (a <= 1.14d-7))) then
        tmp = x + (tan((y + z)) - tan(a))
    else
        tmp = (1.0d0 / ((1.0d0 - (tan(y) * tan(z))) / (tan(y) + tan(z)))) + (x - a)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double a) {
	double tmp;
	if ((a <= -0.055) || !(a <= 1.14e-7)) {
		tmp = x + (Math.tan((y + z)) - Math.tan(a));
	} else {
		tmp = (1.0 / ((1.0 - (Math.tan(y) * Math.tan(z))) / (Math.tan(y) + Math.tan(z)))) + (x - a);
	}
	return tmp;
}

Python

def code(x, y, z, a):
	tmp = 0
	if (a <= -0.055) or not (a <= 1.14e-7):
		tmp = x + (math.tan((y + z)) - math.tan(a))
	else:
		tmp = (1.0 / ((1.0 - (math.tan(y) * math.tan(z))) / (math.tan(y) + math.tan(z)))) + (x - a)
	return tmp

Julia

function code(x, y, z, a)
	tmp = 0.0
	if ((a <= -0.055) || !(a <= 1.14e-7))
		tmp = Float64(x + Float64(tan(Float64(y + z)) - tan(a)));
	else
		tmp = Float64(Float64(1.0 / Float64(Float64(1.0 - Float64(tan(y) * tan(z))) / Float64(tan(y) + tan(z)))) + Float64(x - a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, a)
	tmp = 0.0;
	if ((a <= -0.055) || ~((a <= 1.14e-7)))
		tmp = x + (tan((y + z)) - tan(a));
	else
		tmp = (1.0 / ((1.0 - (tan(y) * tan(z))) / (tan(y) + tan(z)))) + (x - a);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, a_] := If[Or[LessEqual[a, -0.055], N[Not[LessEqual[a, 1.14e-7]], $MachinePrecision]], N[(x + N[(N[Tan[N[(y + z), $MachinePrecision]], $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(N[(1.0 - N[(N[Tan[y], $MachinePrecision] * N[Tan[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Tan[y], $MachinePrecision] + N[Tan[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x - a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -0.0550000000000000003 or 1.14000000000000002e-7 < a

   1. Initial program 79.6%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]

   if -0.0550000000000000003 < a < 1.14000000000000002e-7

   1. Initial program 78.8%

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

      1. associate-+r- 78.8%

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

      2. +-commutative 78.8%

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

      3. associate--l+ 78.8%

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

   3. Simplified 78.8%

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

   4. Taylor expanded in a around 0 78.8%

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

      1. +-commutative 78.8%

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

      2. mul-1-neg 78.8%

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

      3. unsub-neg 78.8%

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

   6. Simplified 78.8%

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

      1. tan-sum 99.9%

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

      2. div-inv 99.9%

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

   8. Applied egg-rr 99.5%

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

      1. *-commutative 99.5%

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

      2. associate-/r/ 99.5%

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

   10. Simplified 99.5%

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

4. Final simplification 89.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -0.055 \lor \neg \left(a \leq 1.14 \cdot 10^{-7}\right):\\ \;\;\;\;x + \left(\tan \left(y + z\right) - \tan a\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1 - \tan y \cdot \tan z}{\tan y + \tan z}} + \left(x - a\right)\\ \end{array} \]

Alternative 3: 89.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -0.055 \lor \neg \left(a \leq 1.14 \cdot 10^{-7}\right):\\ \;\;\;\;x + \left(\tan \left(y + z\right) - \tan a\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan y + \tan z}{1 - \tan y \cdot \tan z} + \left(x - a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (if (or (<= a -0.055) (not (<= a 1.14e-7)))
   (+ x (- (tan (+ y z)) (tan a)))
   (+ (/ (+ (tan y) (tan z)) (- 1.0 (* (tan y) (tan z)))) (- x a))))

C

double code(double x, double y, double z, double a) {
	double tmp;
	if ((a <= -0.055) || !(a <= 1.14e-7)) {
		tmp = x + (tan((y + z)) - tan(a));
	} else {
		tmp = ((tan(y) + tan(z)) / (1.0 - (tan(y) * tan(z)))) + (x - a);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: a
    real(8) :: tmp
    if ((a <= (-0.055d0)) .or. (.not. (a <= 1.14d-7))) then
        tmp = x + (tan((y + z)) - tan(a))
    else
        tmp = ((tan(y) + tan(z)) / (1.0d0 - (tan(y) * tan(z)))) + (x - a)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double a) {
	double tmp;
	if ((a <= -0.055) || !(a <= 1.14e-7)) {
		tmp = x + (Math.tan((y + z)) - Math.tan(a));
	} else {
		tmp = ((Math.tan(y) + Math.tan(z)) / (1.0 - (Math.tan(y) * Math.tan(z)))) + (x - a);
	}
	return tmp;
}

Python

def code(x, y, z, a):
	tmp = 0
	if (a <= -0.055) or not (a <= 1.14e-7):
		tmp = x + (math.tan((y + z)) - math.tan(a))
	else:
		tmp = ((math.tan(y) + math.tan(z)) / (1.0 - (math.tan(y) * math.tan(z)))) + (x - a)
	return tmp

Julia

function code(x, y, z, a)
	tmp = 0.0
	if ((a <= -0.055) || !(a <= 1.14e-7))
		tmp = Float64(x + Float64(tan(Float64(y + z)) - tan(a)));
	else
		tmp = Float64(Float64(Float64(tan(y) + tan(z)) / Float64(1.0 - Float64(tan(y) * tan(z)))) + Float64(x - a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, a)
	tmp = 0.0;
	if ((a <= -0.055) || ~((a <= 1.14e-7)))
		tmp = x + (tan((y + z)) - tan(a));
	else
		tmp = ((tan(y) + tan(z)) / (1.0 - (tan(y) * tan(z)))) + (x - a);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, a_] := If[Or[LessEqual[a, -0.055], N[Not[LessEqual[a, 1.14e-7]], $MachinePrecision]], N[(x + N[(N[Tan[N[(y + z), $MachinePrecision]], $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 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[(x - a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -0.0550000000000000003 or 1.14000000000000002e-7 < a

   1. Initial program 79.6%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]

   if -0.0550000000000000003 < a < 1.14000000000000002e-7

   1. Initial program 78.8%

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

      1. associate-+r- 78.8%

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

      2. +-commutative 78.8%

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

      3. associate--l+ 78.8%

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

   3. Simplified 78.8%

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

   4. Taylor expanded in a around 0 78.8%

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

      1. +-commutative 78.8%

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

      2. mul-1-neg 78.8%

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

      3. unsub-neg 78.8%

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

   6. Simplified 78.8%

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

      1. tan-sum 99.9%

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

      2. div-inv 99.9%

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

   8. Applied egg-rr 99.5%

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

      1. associate-*r/ 99.9%

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

      2. *-rgt-identity 99.9%

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

   10. Simplified 99.5%

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

4. Final simplification 89.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -0.055 \lor \neg \left(a \leq 1.14 \cdot 10^{-7}\right):\\ \;\;\;\;x + \left(\tan \left(y + z\right) - \tan a\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan y + \tan z}{1 - \tan y \cdot \tan z} + \left(x - a\right)\\ \end{array} \]

Alternative 4: 69.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.1 \cdot 10^{-20} \lor \neg \left(a \leq 0.0047\right):\\ \;\;\;\;x + \left(\tan y - \tan a\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(\tan \left(y + z\right) - a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (if (or (<= a -1.1e-20) (not (<= a 0.0047)))
   (+ x (- (tan y) (tan a)))
   (+ x (- (tan (+ y z)) a))))

C

double code(double x, double y, double z, double a) {
	double tmp;
	if ((a <= -1.1e-20) || !(a <= 0.0047)) {
		tmp = x + (tan(y) - 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 <= (-1.1d-20)) .or. (.not. (a <= 0.0047d0))) then
        tmp = x + (tan(y) - 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 <= -1.1e-20) || !(a <= 0.0047)) {
		tmp = x + (Math.tan(y) - Math.tan(a));
	} else {
		tmp = x + (Math.tan((y + z)) - a);
	}
	return tmp;
}

Python

def code(x, y, z, a):
	tmp = 0
	if (a <= -1.1e-20) or not (a <= 0.0047):
		tmp = x + (math.tan(y) - 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 <= -1.1e-20) || !(a <= 0.0047))
		tmp = Float64(x + Float64(tan(y) - 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 <= -1.1e-20) || ~((a <= 0.0047)))
		tmp = x + (tan(y) - tan(a));
	else
		tmp = x + (tan((y + z)) - a);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, a_] := If[Or[LessEqual[a, -1.1e-20], N[Not[LessEqual[a, 0.0047]], $MachinePrecision]], N[(x + N[(N[Tan[y], $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 < -1.09999999999999995e-20 or 0.00470000000000000018 < a

   1. Initial program 77.6%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]

   2. Taylor expanded in z around 0 62.4%

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

      1. tan-quot 62.4%

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

      2. expm1-log1p-u 54.7%

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

      3. expm1-udef 54.7%

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

   4. Applied egg-rr 54.7%

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

      1. expm1-def 54.7%

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

      2. expm1-log1p 62.4%

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

   6. Simplified 62.4%

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

   if -1.09999999999999995e-20 < a < 0.00470000000000000018

   1. Initial program 80.6%

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

      1. associate-+r- 80.6%

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

      2. +-commutative 80.6%

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

      3. associate--l+ 80.6%

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

   3. Simplified 80.6%

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

   4. Taylor expanded in a around 0 80.2%

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

      1. +-commutative 80.2%

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

      2. mul-1-neg 80.2%

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

      3. unsub-neg 80.2%

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

   6. Simplified 80.2%

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

      1. associate-+r- 80.2%

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

   8. Applied egg-rr 80.2%

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

      1. +-commutative 80.2%

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

      2. associate--l+ 80.3%

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

      3. +-commutative 80.3%

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

   10. Simplified 80.3%

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

4. Final simplification 71.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.1 \cdot 10^{-20} \lor \neg \left(a \leq 0.0047\right):\\ \;\;\;\;x + \left(\tan y - \tan a\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(\tan \left(y + z\right) - a\right)\\ \end{array} \]

Alternative 5: 69.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.15 \cdot 10^{-11}:\\ \;\;\;\;x + \left(\tan y - \tan a\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(\tan z - \tan a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (if (<= y -2.15e-11) (+ x (- (tan y) (tan a))) (+ x (- (tan z) (tan a)))))

C

double code(double x, double y, double z, double a) {
	double tmp;
	if (y <= -2.15e-11) {
		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 (y <= (-2.15d-11)) 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 (y <= -2.15e-11) {
		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 y <= -2.15e-11:
		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 (y <= -2.15e-11)
		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 (y <= -2.15e-11)
		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[y, -2.15e-11], 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 y < -2.15000000000000001e-11

   1. Initial program 62.0%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]

   2. Taylor expanded in z around 0 62.2%

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

      1. tan-quot 62.3%

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

      2. expm1-log1p-u 49.8%

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

      3. expm1-udef 49.8%

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

   4. Applied egg-rr 49.8%

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

      1. expm1-def 49.8%

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

      2. expm1-log1p 62.3%

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

   6. Simplified 62.3%

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

   if -2.15000000000000001e-11 < y

   1. Initial program 84.1%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]

   2. Taylor expanded in y around 0 70.9%

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

      1. tan-quot 71.0%

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

      2. tan-quot 71.0%

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

      3. associate--l+ 71.0%

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

   4. Applied egg-rr 71.0%

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

      1. associate-+r- 71.0%

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

      2. +-commutative 71.0%

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

      3. associate--l+ 71.1%

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

   6. Simplified 71.1%

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

4. Final simplification 69.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.15 \cdot 10^{-11}:\\ \;\;\;\;x + \left(\tan y - \tan a\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(\tan z - \tan a\right)\\ \end{array} \]

Alternative 6: 79.4% 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 79.2%

   \[x + \left(\tan \left(y + z\right) - \tan a\right) \]

2. Final simplification 79.2%

   \[\leadsto x + \left(\tan \left(y + z\right) - \tan a\right) \]

Alternative 7: 60.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -8.6 \cdot 10^{-14} \lor \neg \left(a \leq 0.0235\right):\\ \;\;\;\;x - \tan a\\ \mathbf{else}:\\ \;\;\;\;x + \left(\tan \left(y + z\right) - a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (if (or (<= a -8.6e-14) (not (<= a 0.0235)))
   (- x (tan a))
   (+ x (- (tan (+ y z)) a))))

C

double code(double x, double y, double z, double a) {
	double tmp;
	if ((a <= -8.6e-14) || !(a <= 0.0235)) {
		tmp = x - 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 <= (-8.6d-14)) .or. (.not. (a <= 0.0235d0))) then
        tmp = x - 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 <= -8.6e-14) || !(a <= 0.0235)) {
		tmp = x - Math.tan(a);
	} else {
		tmp = x + (Math.tan((y + z)) - a);
	}
	return tmp;
}

Python

def code(x, y, z, a):
	tmp = 0
	if (a <= -8.6e-14) or not (a <= 0.0235):
		tmp = x - 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 <= -8.6e-14) || !(a <= 0.0235))
		tmp = Float64(x - 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 <= -8.6e-14) || ~((a <= 0.0235)))
		tmp = x - tan(a);
	else
		tmp = x + (tan((y + z)) - a);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, a_] := If[Or[LessEqual[a, -8.6e-14], N[Not[LessEqual[a, 0.0235]], $MachinePrecision]], N[(x - N[Tan[a], $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 < -8.59999999999999996e-14 or 0.0235 < a

   1. Initial program 78.1%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]

   2. Taylor expanded in y around 0 60.9%

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

      1. add-cbrt-cube 60.7%

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

      2. tan-quot 60.7%

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

      3. tan-quot 60.7%

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

      4. associate--l+ 60.7%

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

      5. tan-quot 60.7%

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

      6. tan-quot 60.7%

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

      7. associate--l+ 60.7%

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

      8. tan-quot 60.7%

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

      9. tan-quot 60.7%

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

   4. Applied egg-rr 60.7%

      \[\leadsto \color{blue}{\sqrt[3]{\left(\left(\tan z + \left(x - \tan a\right)\right) \cdot \left(\tan z + \left(x - \tan a\right)\right)\right) \cdot \left(\tan z + \left(x - \tan a\right)\right)}} \]
   5. Step-by-step derivation

      1. associate-*l* 60.7%

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

      2. cube-unmult 60.7%

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

      3. associate-+r- 60.8%

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

      4. +-commutative 60.8%

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

      5. associate--l+ 60.8%

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

   6. Simplified 60.8%

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

   7. Taylor expanded in z around 0 46.2%

      \[\leadsto \sqrt[3]{\color{blue}{{\left(x - \frac{\sin a}{\cos a}\right)}^{3}}} \]
   8. Step-by-step derivation

      1. rem-cbrt-cube 46.3%

         \[\leadsto \color{blue}{x - \frac{\sin a}{\cos a}} \]

      2. tan-quot 46.4%

         \[\leadsto x - \color{blue}{\tan a} \]

      3. sub-neg 46.4%

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

   9. Applied egg-rr 46.4%

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

       1. sub-neg 46.4%

          \[\leadsto \color{blue}{x - \tan a} \]

   11. Simplified 46.4%

       \[\leadsto \color{blue}{x - \tan a} \]

   if -8.59999999999999996e-14 < a < 0.0235

   1. Initial program 80.2%

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

      1. associate-+r- 80.2%

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

      2. +-commutative 80.2%

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

      3. associate--l+ 80.2%

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

   3. Simplified 80.2%

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

   4. Taylor expanded in a around 0 79.8%

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

      1. +-commutative 79.8%

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

      2. mul-1-neg 79.8%

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

      3. unsub-neg 79.8%

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

   6. Simplified 79.8%

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

      1. associate-+r- 79.8%

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

   8. Applied egg-rr 79.8%

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

      1. +-commutative 79.8%

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

      2. associate--l+ 79.8%

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

      3. +-commutative 79.8%

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

   10. Simplified 79.8%

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

4. Final simplification 63.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -8.6 \cdot 10^{-14} \lor \neg \left(a \leq 0.0235\right):\\ \;\;\;\;x - \tan a\\ \mathbf{else}:\\ \;\;\;\;x + \left(\tan \left(y + z\right) - a\right)\\ \end{array} \]

Alternative 8: 41.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ x - \tan a \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.2%

   \[x + \left(\tan \left(y + z\right) - \tan a\right) \]

2. Taylor expanded in y around 0 60.2%

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

   1. add-cbrt-cube 60.0%

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

   2. tan-quot 60.0%

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

   3. tan-quot 60.0%

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

   4. associate--l+ 60.0%

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

   5. tan-quot 60.0%

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

   6. tan-quot 60.0%

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

   7. associate--l+ 60.0%

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

   8. tan-quot 60.0%

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

   9. tan-quot 60.0%

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

4. Applied egg-rr 60.0%

   \[\leadsto \color{blue}{\sqrt[3]{\left(\left(\tan z + \left(x - \tan a\right)\right) \cdot \left(\tan z + \left(x - \tan a\right)\right)\right) \cdot \left(\tan z + \left(x - \tan a\right)\right)}} \]
5. Step-by-step derivation

   1. associate-*l* 60.0%

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

   2. cube-unmult 60.0%

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

   3. associate-+r- 60.0%

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

   4. +-commutative 60.0%

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

   5. associate--l+ 60.0%

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

6. Simplified 60.0%

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

7. Taylor expanded in z around 0 42.7%

   \[\leadsto \sqrt[3]{\color{blue}{{\left(x - \frac{\sin a}{\cos a}\right)}^{3}}} \]
8. Step-by-step derivation

   1. rem-cbrt-cube 42.9%

      \[\leadsto \color{blue}{x - \frac{\sin a}{\cos a}} \]

   2. tan-quot 42.9%

      \[\leadsto x - \color{blue}{\tan a} \]

   3. sub-neg 42.9%

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

9. Applied egg-rr 42.9%

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

    1. sub-neg 42.9%

       \[\leadsto \color{blue}{x - \tan a} \]

11. Simplified 42.9%

    \[\leadsto \color{blue}{x - \tan a} \]

12. Final simplification 42.9%

    \[\leadsto x - \tan a \]

Alternative 9: 31.7% 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 79.2%

   \[x + \left(\tan \left(y + z\right) - \tan a\right) \]

2. Taylor expanded in x around inf 30.8%

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

3. Final simplification 30.8%

   \[\leadsto x \]

Reproduce

herbie shell --seed 2023279 
(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))))