tan-example (used to crash)

Percentage Accurate: 79.4% → 99.7%

Time: 24.3s

Alternatives: 7

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 82.7%

   \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
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 2: 88.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\tan a \leq -0.05 \lor \neg \left(\tan a \leq 5 \cdot 10^{-15}\right):\\ \;\;\;\;x + \left(\tan \left(y + z\right) - \tan a\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x + \frac{\tan y + \tan z}{1 - \tan y \cdot \tan z}\right) - a\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (if (or (<= (tan a) -0.05) (not (<= (tan a) 5e-15)))
   (+ 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) <= -0.05) || !(tan(a) <= 5e-15)) {
		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) <= (-0.05d0)) .or. (.not. (tan(a) <= 5d-15))) 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) <= -0.05) || !(Math.tan(a) <= 5e-15)) {
		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) <= -0.05) or not (math.tan(a) <= 5e-15):
		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) <= -0.05) || !(tan(a) <= 5e-15))
		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) <= -0.05) || ~((tan(a) <= 5e-15)))
		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], -0.05], N[Not[LessEqual[N[Tan[a], $MachinePrecision], 5e-15]], $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) < -0.050000000000000003 or 4.99999999999999999e-15 < (tan.f64 a)

   1. Initial program 81.8%

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

   if -0.050000000000000003 < (tan.f64 a) < 4.99999999999999999e-15

   1. Initial program 83.7%

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

      1. associate-+r- 83.7%

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

   3. Simplified 83.7%

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

   4. Taylor expanded in a around 0 83.7%

      \[\leadsto \left(x + \tan \left(y + z\right)\right) - \color{blue}{a} \]
   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 90.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\tan a \leq -0.05 \lor \neg \left(\tan a \leq 5 \cdot 10^{-15}\right):\\ \;\;\;\;x + \left(\tan \left(y + z\right) - \tan a\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x + \frac{\tan y + \tan z}{1 - \tan y \cdot \tan z}\right) - a\\ \end{array} \]

Alternative 3: 69.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\tan a \leq -2 \cdot 10^{-26} \lor \neg \left(\tan a \leq 5 \cdot 10^{-19}\right):\\ \;\;\;\;x + \left(\tan z - \tan a\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x + \tan \left(y + z\right)\right) - a\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (if (or (<= (tan a) -2e-26) (not (<= (tan a) 5e-19)))
   (+ x (- (tan z) (tan a)))
   (- (+ x (tan (+ y z))) a)))

C

double code(double x, double y, double z, double a) {
	double tmp;
	if ((tan(a) <= -2e-26) || !(tan(a) <= 5e-19)) {
		tmp = x + (tan(z) - tan(a));
	} else {
		tmp = (x + tan((y + z))) - a;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: a
    real(8) :: tmp
    if ((tan(a) <= (-2d-26)) .or. (.not. (tan(a) <= 5d-19))) then
        tmp = x + (tan(z) - tan(a))
    else
        tmp = (x + tan((y + z))) - a
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double a) {
	double tmp;
	if ((Math.tan(a) <= -2e-26) || !(Math.tan(a) <= 5e-19)) {
		tmp = x + (Math.tan(z) - Math.tan(a));
	} else {
		tmp = (x + Math.tan((y + z))) - a;
	}
	return tmp;
}

Python

def code(x, y, z, a):
	tmp = 0
	if (math.tan(a) <= -2e-26) or not (math.tan(a) <= 5e-19):
		tmp = x + (math.tan(z) - math.tan(a))
	else:
		tmp = (x + math.tan((y + z))) - a
	return tmp

Julia

function code(x, y, z, a)
	tmp = 0.0
	if ((tan(a) <= -2e-26) || !(tan(a) <= 5e-19))
		tmp = Float64(x + Float64(tan(z) - tan(a)));
	else
		tmp = Float64(Float64(x + tan(Float64(y + z))) - a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, a)
	tmp = 0.0;
	if ((tan(a) <= -2e-26) || ~((tan(a) <= 5e-19)))
		tmp = x + (tan(z) - tan(a));
	else
		tmp = (x + tan((y + z))) - a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, a_] := If[Or[LessEqual[N[Tan[a], $MachinePrecision], -2e-26], N[Not[LessEqual[N[Tan[a], $MachinePrecision], 5e-19]], $MachinePrecision]], N[(x + N[(N[Tan[z], $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x + N[Tan[N[(y + z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (tan.f64 a) < -2.0000000000000001e-26 or 5.0000000000000004e-19 < (tan.f64 a)

   1. Initial program 81.7%

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

   2. Taylor expanded in y around 0 66.8%

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

      1. associate--l+ 66.8%

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

   4. Simplified 66.8%

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

      1. tan-quot 66.8%

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

      2. tan-quot 66.8%

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

      3. associate-+r- 66.9%

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

   6. Applied egg-rr 66.9%

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

      1. +-commutative 66.9%

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

      2. associate--l+ 66.9%

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

   8. Simplified 66.9%

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

   if -2.0000000000000001e-26 < (tan.f64 a) < 5.0000000000000004e-19

   1. Initial program 84.0%

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

      1. associate-+r- 84.0%

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

   3. Simplified 84.0%

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

   4. Taylor expanded in a around 0 84.0%

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

4. Final simplification 74.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\tan a \leq -2 \cdot 10^{-26} \lor \neg \left(\tan a \leq 5 \cdot 10^{-19}\right):\\ \;\;\;\;x + \left(\tan z - \tan a\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x + \tan \left(y + z\right)\right) - a\\ \end{array} \]

Alternative 4: 64.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -0.00295:\\ \;\;\;\;x + \left(\tan a + \tan \left(y + z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(\tan z - \tan a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (if (<= y -0.00295)
   (+ x (+ (tan a) (tan (+ y z))))
   (+ x (- (tan z) (tan a)))))

C

double code(double x, double y, double z, double a) {
	double tmp;
	if (y <= -0.00295) {
		tmp = x + (tan(a) + tan((y + z)));
	} else {
		tmp = x + (tan(z) - tan(a));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: a
    real(8) :: tmp
    if (y <= (-0.00295d0)) then
        tmp = x + (tan(a) + tan((y + z)))
    else
        tmp = x + (tan(z) - tan(a))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double a) {
	double tmp;
	if (y <= -0.00295) {
		tmp = x + (Math.tan(a) + Math.tan((y + z)));
	} else {
		tmp = x + (Math.tan(z) - Math.tan(a));
	}
	return tmp;
}

Python

def code(x, y, z, a):
	tmp = 0
	if y <= -0.00295:
		tmp = x + (math.tan(a) + math.tan((y + z)))
	else:
		tmp = x + (math.tan(z) - math.tan(a))
	return tmp

Julia

function code(x, y, z, a)
	tmp = 0.0
	if (y <= -0.00295)
		tmp = Float64(x + Float64(tan(a) + tan(Float64(y + z))));
	else
		tmp = Float64(x + Float64(tan(z) - tan(a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, a)
	tmp = 0.0;
	if (y <= -0.00295)
		tmp = x + (tan(a) + tan((y + z)));
	else
		tmp = x + (tan(z) - tan(a));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, a_] := If[LessEqual[y, -0.00295], N[(x + N[(N[Tan[a], $MachinePrecision] + N[Tan[N[(y + z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[Tan[z], $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -0.00294999999999999993

   1. Initial program 59.6%

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

      1. sub-neg 59.6%

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

   3. Applied egg-rr 59.6%

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

      1. +-commutative 59.6%

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

      2. rem-square-sqrt 24.4%

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

      3. fabs-sqr 24.4%

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

      4. rem-square-sqrt 47.0%

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

      5. fabs-neg 47.0%

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

      6. rem-square-sqrt 22.6%

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

      7. fabs-sqr 22.6%

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

      8. rem-square-sqrt 36.9%

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

   5. Simplified 36.9%

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

   if -0.00294999999999999993 < y

   1. Initial program 89.2%

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

   2. Taylor expanded in y around 0 76.4%

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

      1. associate--l+ 76.4%

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

   4. Simplified 76.4%

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

      1. tan-quot 76.4%

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

      2. tan-quot 76.4%

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

      3. associate-+r- 76.4%

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

   6. Applied egg-rr 76.4%

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

      1. +-commutative 76.4%

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

      2. associate--l+ 76.5%

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

   8. Simplified 76.5%

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

4. Final simplification 67.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.00295:\\ \;\;\;\;x + \left(\tan a + \tan \left(y + z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(\tan z - \tan a\right)\\ \end{array} \]

Alternative 5: 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 82.7%

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

2. Final simplification 82.7%

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

Alternative 6: 50.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.45:\\ \;\;\;\;x\\ \mathbf{elif}\;a \leq 1.6:\\ \;\;\;\;\left(x + \tan \left(y + z\right)\right) - a\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (if (<= a -1.45) x (if (<= a 1.6) (- (+ x (tan (+ y z))) a) x)))

C

double code(double x, double y, double z, double a) {
	double tmp;
	if (a <= -1.45) {
		tmp = x;
	} else if (a <= 1.6) {
		tmp = (x + tan((y + z))) - a;
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: a
    real(8) :: tmp
    if (a <= (-1.45d0)) then
        tmp = x
    else if (a <= 1.6d0) then
        tmp = (x + tan((y + z))) - a
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double a) {
	double tmp;
	if (a <= -1.45) {
		tmp = x;
	} else if (a <= 1.6) {
		tmp = (x + Math.tan((y + z))) - a;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, a):
	tmp = 0
	if a <= -1.45:
		tmp = x
	elif a <= 1.6:
		tmp = (x + math.tan((y + z))) - a
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, a)
	tmp = 0.0
	if (a <= -1.45)
		tmp = x;
	elseif (a <= 1.6)
		tmp = Float64(Float64(x + tan(Float64(y + z))) - a);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, a)
	tmp = 0.0;
	if (a <= -1.45)
		tmp = x;
	elseif (a <= 1.6)
		tmp = (x + tan((y + z))) - a;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, a_] := If[LessEqual[a, -1.45], x, If[LessEqual[a, 1.6], N[(N[(x + N[Tan[N[(y + z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if a < -1.44999999999999996 or 1.6000000000000001 < a

   1. Initial program 81.5%

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

   2. Taylor expanded in x around inf 22.8%

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

   if -1.44999999999999996 < a < 1.6000000000000001

   1. Initial program 83.9%

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

      1. associate-+r- 83.9%

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

   3. Simplified 83.9%

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

   4. Taylor expanded in a around 0 83.4%

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

4. Final simplification 52.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.45:\\ \;\;\;\;x\\ \mathbf{elif}\;a \leq 1.6:\\ \;\;\;\;\left(x + \tan \left(y + z\right)\right) - a\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 7: 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 82.7%

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

2. Taylor expanded in x around inf 30.1%

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

3. Final simplification 30.1%

   \[\leadsto x \]

Reproduce

herbie shell --seed 2023240 
(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))))