tan-example (used to crash)

Percentage Accurate: 79.3% → 99.7%

Time: 26.6s

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: 79.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Python

def code(x, y, z, a):
	return x + (((math.tan(y) + math.tan(z)) / (1.0 - ((math.sin(z) / math.cos(y)) * (math.sin(y) / math.cos(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(Float64(sin(z) / cos(y)) * Float64(sin(y) / cos(z))))) - tan(a)))
end

MATLAB

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

Derivation

1. Initial program 78.7%

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

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

3. Applied egg-rr 99.7%

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

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

5. Simplified 99.7%

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

6. Taylor expanded in y around inf 99.7%

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

   1. *-commutative 99.7%

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

   2. times-frac 99.7%

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

8. Simplified 99.7%

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

9. Final simplification 99.7%

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

Alternative 2: 89.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\tan a \leq -0.01 \lor \neg \left(\tan a \leq 4 \cdot 10^{-20}\right):\\ \;\;\;\;x + \left(\tan \left(y + z\right) - \tan a\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x + \frac{\tan y + \tan z}{1 - \tan y \cdot \tan z}\right) - a\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (if (or (<= (tan a) -0.01) (not (<= (tan a) 4e-20)))
   (+ x (- (tan (+ y z)) (tan a)))
   (- (+ x (/ (+ (tan y) (tan z)) (- 1.0 (* (tan y) (tan z))))) a)))

C

double code(double x, double y, double z, double a) {
	double tmp;
	if ((tan(a) <= -0.01) || !(tan(a) <= 4e-20)) {
		tmp = x + (tan((y + z)) - tan(a));
	} else {
		tmp = (x + ((tan(y) + tan(z)) / (1.0 - (tan(y) * tan(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 ((tan(a) <= (-0.01d0)) .or. (.not. (tan(a) <= 4d-20))) then
        tmp = x + (tan((y + z)) - tan(a))
    else
        tmp = (x + ((tan(y) + tan(z)) / (1.0d0 - (tan(y) * tan(z))))) - a
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double a) {
	double tmp;
	if ((Math.tan(a) <= -0.01) || !(Math.tan(a) <= 4e-20)) {
		tmp = x + (Math.tan((y + z)) - Math.tan(a));
	} else {
		tmp = (x + ((Math.tan(y) + Math.tan(z)) / (1.0 - (Math.tan(y) * Math.tan(z))))) - a;
	}
	return tmp;
}

Python

def code(x, y, z, a):
	tmp = 0
	if (math.tan(a) <= -0.01) or not (math.tan(a) <= 4e-20):
		tmp = x + (math.tan((y + z)) - math.tan(a))
	else:
		tmp = (x + ((math.tan(y) + math.tan(z)) / (1.0 - (math.tan(y) * math.tan(z))))) - a
	return tmp

Julia

function code(x, y, z, a)
	tmp = 0.0
	if ((tan(a) <= -0.01) || !(tan(a) <= 4e-20))
		tmp = Float64(x + Float64(tan(Float64(y + z)) - tan(a)));
	else
		tmp = Float64(Float64(x + Float64(Float64(tan(y) + tan(z)) / Float64(1.0 - Float64(tan(y) * tan(z))))) - a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, a)
	tmp = 0.0;
	if ((tan(a) <= -0.01) || ~((tan(a) <= 4e-20)))
		tmp = x + (tan((y + z)) - tan(a));
	else
		tmp = (x + ((tan(y) + tan(z)) / (1.0 - (tan(y) * tan(z))))) - a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, a_] := If[Or[LessEqual[N[Tan[a], $MachinePrecision], -0.01], N[Not[LessEqual[N[Tan[a], $MachinePrecision], 4e-20]], $MachinePrecision]], N[(x + N[(N[Tan[N[(y + z), $MachinePrecision]], $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x + 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]), $MachinePrecision] - a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (tan.f64 a) < -0.0100000000000000002 or 3.99999999999999978e-20 < (tan.f64 a)

   1. Initial program 77.7%

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

   if -0.0100000000000000002 < (tan.f64 a) < 3.99999999999999978e-20

   1. Initial program 79.7%

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

      1. associate-+r- 79.7%

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

   3. Simplified 79.7%

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

   4. Taylor expanded in a around 0 79.7%

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

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

   6. Applied egg-rr 99.7%

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

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

   8. Simplified 99.7%

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

4. Final simplification 88.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\tan a \leq -0.01 \lor \neg \left(\tan a \leq 4 \cdot 10^{-20}\right):\\ \;\;\;\;x + \left(\tan \left(y + z\right) - \tan a\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x + \frac{\tan y + \tan z}{1 - \tan y \cdot \tan z}\right) - a\\ \end{array} \]

Alternative 3: 99.7% accurate, 0.4× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z, double a) {
	return x + ((1.0 / ((1.0 - (tan(y) * tan(z))) / (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 + ((1.0d0 / ((1.0d0 - (tan(y) * tan(z))) / (tan(y) + tan(z)))) - tan(a))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.7%

   \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
2. 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. clear-num 99.7%

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

3. Applied egg-rr 99.7%

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

4. Final simplification 99.7%

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

Alternative 4: 99.7% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.7%

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

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

3. Applied egg-rr 99.7%

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

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

5. Simplified 99.7%

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

6. Final simplification 99.7%

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

Alternative 5: 49.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\tan a \leq -0.01:\\ \;\;\;\;x\\ \mathbf{elif}\;\tan a \leq 10^{-42}:\\ \;\;\;\;\left(x + \tan \left(y + z\right)\right) - a\\ \mathbf{else}:\\ \;\;\;\;x - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (if (<= (tan a) -0.01)
   x
   (if (<= (tan a) 1e-42) (- (+ x (tan (+ y z))) a) (- x 1.0))))

C

double code(double x, double y, double z, double a) {
	double tmp;
	if (tan(a) <= -0.01) {
		tmp = x;
	} else if (tan(a) <= 1e-42) {
		tmp = (x + tan((y + z))) - a;
	} else {
		tmp = x - 1.0;
	}
	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 (tan(a) <= (-0.01d0)) then
        tmp = x
    else if (tan(a) <= 1d-42) then
        tmp = (x + tan((y + z))) - a
    else
        tmp = x - 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double a) {
	double tmp;
	if (Math.tan(a) <= -0.01) {
		tmp = x;
	} else if (Math.tan(a) <= 1e-42) {
		tmp = (x + Math.tan((y + z))) - a;
	} else {
		tmp = x - 1.0;
	}
	return tmp;
}

Python

def code(x, y, z, a):
	tmp = 0
	if math.tan(a) <= -0.01:
		tmp = x
	elif math.tan(a) <= 1e-42:
		tmp = (x + math.tan((y + z))) - a
	else:
		tmp = x - 1.0
	return tmp

Julia

function code(x, y, z, a)
	tmp = 0.0
	if (tan(a) <= -0.01)
		tmp = x;
	elseif (tan(a) <= 1e-42)
		tmp = Float64(Float64(x + tan(Float64(y + z))) - a);
	else
		tmp = Float64(x - 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, a)
	tmp = 0.0;
	if (tan(a) <= -0.01)
		tmp = x;
	elseif (tan(a) <= 1e-42)
		tmp = (x + tan((y + z))) - a;
	else
		tmp = x - 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, a_] := If[LessEqual[N[Tan[a], $MachinePrecision], -0.01], x, If[LessEqual[N[Tan[a], $MachinePrecision], 1e-42], N[(N[(x + N[Tan[N[(y + z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision], N[(x - 1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (tan.f64 a) < -0.0100000000000000002

   1. Initial program 77.7%

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

   2. Taylor expanded in x around inf 23.9%

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

   if -0.0100000000000000002 < (tan.f64 a) < 1.00000000000000004e-42

   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%

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

   3. Simplified 80.6%

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

   4. Taylor expanded in a around 0 80.6%

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

   if 1.00000000000000004e-42 < (tan.f64 a)

   1. Initial program 75.9%

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

      1. associate-+r- 75.8%

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

      2. expm1-log1p-u 62.8%

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

      3. +-commutative 62.8%

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

      4. associate--l+ 62.8%

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

   3. Applied egg-rr 62.8%

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

   4. Taylor expanded in x around inf 20.9%

      \[\leadsto \mathsf{expm1}\left(\color{blue}{-1 \cdot \log \left(\frac{1}{x}\right)}\right) \]
   5. Step-by-step derivation

      1. mul-1-neg 20.9%

         \[\leadsto \mathsf{expm1}\left(\color{blue}{-\log \left(\frac{1}{x}\right)}\right) \]

      2. log-rec 20.9%

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

      3. remove-double-neg 20.9%

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

   6. Simplified 20.9%

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

   7. Taylor expanded in x around 0 20.9%

      \[\leadsto \color{blue}{x - 1} \]
3. Recombined 3 regimes into one program.

4. Final simplification 50.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\tan a \leq -0.01:\\ \;\;\;\;x\\ \mathbf{elif}\;\tan a \leq 10^{-42}:\\ \;\;\;\;\left(x + \tan \left(y + z\right)\right) - a\\ \mathbf{else}:\\ \;\;\;\;x - 1\\ \end{array} \]

Alternative 6: 79.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z, double a) {
	return x + ((1.0 / (1.0 / 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 + ((1.0d0 / (1.0d0 / tan((y + z)))) - tan(a))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.7%

   \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
2. 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. clear-num 99.7%

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

3. Applied egg-rr 99.7%

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

   1. expm1-log1p-u 65.3%

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

   2. expm1-udef 65.2%

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

   3. clear-num 65.3%

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

   4. tan-sum 50.7%

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

5. Applied egg-rr 50.7%

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

   1. expm1-def 50.8%

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

   2. expm1-log1p 78.7%

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

   3. +-commutative 78.7%

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

7. Simplified 78.7%

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

8. Final simplification 78.7%

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

Alternative 7: 50.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.7%

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

   1. +-commutative 78.7%

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

   2. associate-+l- 78.7%

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

3. Applied egg-rr 78.7%

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

   1. associate-+l- 78.7%

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

   2. +-commutative 78.7%

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

   3. sub-neg 78.7%

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

   4. associate-+r+ 78.7%

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

   5. rem-square-sqrt 41.3%

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

   6. fabs-sqr 41.3%

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

   7. rem-square-sqrt 65.1%

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

   8. fabs-neg 65.1%

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

   9. rem-square-sqrt 23.8%

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

   10. fabs-sqr 23.8%

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

   11. rem-square-sqrt 49.8%

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

   12. associate-+l+ 49.8%

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

   13. +-commutative 49.8%

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

5. Simplified 49.8%

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

6. Final simplification 49.8%

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

Alternative 8: 79.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.7%

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

2. Final simplification 78.7%

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

Alternative 9: 32.1% 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 78.7%

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

2. Taylor expanded in x around inf 31.9%

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

3. Final simplification 31.9%

   \[\leadsto x \]

Reproduce

herbie shell --seed 2023224 
(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))))