tan-example (used to crash)

Percentage Accurate: 79.3% → 99.7%

Time: 30.4s

Alternatives: 8

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, 0.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

MATLAB

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

Derivation

1. Initial program 79.3%

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

   1. tan-sum 99.7%

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

   2. div-inv 99.6%

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

4. Applied egg-rr 99.6%

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

   1. associate-*r/ 99.7%

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

   2. *-rgt-identity 99.7%

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

6. Simplified 99.7%

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

   1. expm1-log1p-u 93.8%

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

   2. expm1-udef 93.8%

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

   3. log1p-udef 93.8%

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

   4. add-exp-log 99.7%

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

8. Applied egg-rr 99.7%

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

9. Final simplification 99.7%

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

Alternative 2: 89.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\tan a \leq -5 \cdot 10^{-9}:\\ \;\;\;\;x + \left(\tan \left(y + z\right) - \tan a\right)\\ \mathbf{elif}\;\tan a \leq 5 \cdot 10^{-23}:\\ \;\;\;\;x + \frac{\tan y + \tan z}{1 - \tan y \cdot \tan z}\\ \mathbf{else}:\\ \;\;\;\;x + \left(\frac{1}{\frac{\cos \left(y + z\right)}{\sin \left(y + z\right)}} - \tan a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (if (<= (tan a) -5e-9)
   (+ x (- (tan (+ y z)) (tan a)))
   (if (<= (tan a) 5e-23)
     (+ x (/ (+ (tan y) (tan z)) (- 1.0 (* (tan y) (tan z)))))
     (+ x (- (/ 1.0 (/ (cos (+ y z)) (sin (+ y z)))) (tan a))))))

C

double code(double x, double y, double z, double a) {
	double tmp;
	if (tan(a) <= -5e-9) {
		tmp = x + (tan((y + z)) - tan(a));
	} else if (tan(a) <= 5e-23) {
		tmp = x + ((tan(y) + tan(z)) / (1.0 - (tan(y) * tan(z))));
	} else {
		tmp = x + ((1.0 / (cos((y + z)) / sin((y + 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 (tan(a) <= (-5d-9)) then
        tmp = x + (tan((y + z)) - tan(a))
    else if (tan(a) <= 5d-23) then
        tmp = x + ((tan(y) + tan(z)) / (1.0d0 - (tan(y) * tan(z))))
    else
        tmp = x + ((1.0d0 / (cos((y + z)) / sin((y + 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 (Math.tan(a) <= -5e-9) {
		tmp = x + (Math.tan((y + z)) - Math.tan(a));
	} else if (Math.tan(a) <= 5e-23) {
		tmp = x + ((Math.tan(y) + Math.tan(z)) / (1.0 - (Math.tan(y) * Math.tan(z))));
	} else {
		tmp = x + ((1.0 / (Math.cos((y + z)) / Math.sin((y + z)))) - Math.tan(a));
	}
	return tmp;
}

Python

def code(x, y, z, a):
	tmp = 0
	if math.tan(a) <= -5e-9:
		tmp = x + (math.tan((y + z)) - math.tan(a))
	elif math.tan(a) <= 5e-23:
		tmp = x + ((math.tan(y) + math.tan(z)) / (1.0 - (math.tan(y) * math.tan(z))))
	else:
		tmp = x + ((1.0 / (math.cos((y + z)) / math.sin((y + z)))) - math.tan(a))
	return tmp

Julia

function code(x, y, z, a)
	tmp = 0.0
	if (tan(a) <= -5e-9)
		tmp = Float64(x + Float64(tan(Float64(y + z)) - tan(a)));
	elseif (tan(a) <= 5e-23)
		tmp = Float64(x + Float64(Float64(tan(y) + tan(z)) / Float64(1.0 - Float64(tan(y) * tan(z)))));
	else
		tmp = Float64(x + Float64(Float64(1.0 / Float64(cos(Float64(y + z)) / sin(Float64(y + z)))) - tan(a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, a)
	tmp = 0.0;
	if (tan(a) <= -5e-9)
		tmp = x + (tan((y + z)) - tan(a));
	elseif (tan(a) <= 5e-23)
		tmp = x + ((tan(y) + tan(z)) / (1.0 - (tan(y) * tan(z))));
	else
		tmp = x + ((1.0 / (cos((y + z)) / sin((y + z)))) - tan(a));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, a_] := If[LessEqual[N[Tan[a], $MachinePrecision], -5e-9], N[(x + N[(N[Tan[N[(y + z), $MachinePrecision]], $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Tan[a], $MachinePrecision], 5e-23], 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], N[(x + N[(N[(1.0 / N[(N[Cos[N[(y + z), $MachinePrecision]], $MachinePrecision] / N[Sin[N[(y + z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (tan.f64 a) < -5.0000000000000001e-9

   1. Initial program 85.4%

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

   if -5.0000000000000001e-9 < (tan.f64 a) < 5.0000000000000002e-23

   1. Initial program 77.8%

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

      1. +-commutative 77.8%

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

      2. associate-+l- 77.8%

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

   4. Applied egg-rr 77.8%

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

   5. Taylor expanded in a around 0 77.8%

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

      1. neg-mul-1 77.8%

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

   7. Simplified 77.8%

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

      1. tan-sum 99.7%

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

      2. div-inv 99.7%

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

      3. fma-neg 99.7%

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

   9. Applied egg-rr 99.7%

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

       1. fma-udef 99.7%

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

       2. associate-*r/ 99.7%

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

       3. *-rgt-identity 99.7%

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

       4. remove-double-neg 99.7%

          \[\leadsto \frac{\tan y + \tan z}{1 - \tan y \cdot \tan z} + \color{blue}{x} \]

   11. Simplified 99.7%

       \[\leadsto \color{blue}{\frac{\tan y + \tan z}{1 - \tan y \cdot \tan z} + x} \]

   if 5.0000000000000002e-23 < (tan.f64 a)

   1. Initial program 75.6%

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

      1. tan-quot 75.6%

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

      2. clear-num 75.6%

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

   4. Applied egg-rr 75.6%

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

4. Final simplification 89.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\tan a \leq -5 \cdot 10^{-9}:\\ \;\;\;\;x + \left(\tan \left(y + z\right) - \tan a\right)\\ \mathbf{elif}\;\tan a \leq 5 \cdot 10^{-23}:\\ \;\;\;\;x + \frac{\tan y + \tan z}{1 - \tan y \cdot \tan z}\\ \mathbf{else}:\\ \;\;\;\;x + \left(\frac{1}{\frac{\cos \left(y + z\right)}{\sin \left(y + z\right)}} - \tan a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.7% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.3%

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

   1. tan-sum 99.7%

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

   2. div-inv 99.6%

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

4. Applied egg-rr 99.6%

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

   1. associate-*r/ 99.7%

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

   2. *-rgt-identity 99.7%

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

6. Simplified 99.7%

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

7. Final simplification 99.7%

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

Alternative 4: 59.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y + z \leq -0.5:\\ \;\;\;\;x + \tan \left(y + z\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(\tan z - \tan a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (if (<= (+ y z) -0.5) (+ x (tan (+ y z))) (+ x (- (tan z) (tan a)))))

C

double code(double x, double y, double z, double a) {
	double tmp;
	if ((y + z) <= -0.5) {
		tmp = x + 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 + z) <= (-0.5d0)) then
        tmp = x + 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 + z) <= -0.5) {
		tmp = x + 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 + z) <= -0.5:
		tmp = x + 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 (Float64(y + z) <= -0.5)
		tmp = Float64(x + 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 + z) <= -0.5)
		tmp = x + tan((y + z));
	else
		tmp = x + (tan(z) - tan(a));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, a_] := If[LessEqual[N[(y + z), $MachinePrecision], -0.5], N[(x + N[Tan[N[(y + z), $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 (+.f64 y z) < -0.5

   1. Initial program 74.5%

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

      1. +-commutative 74.5%

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

      2. associate-+l- 74.5%

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

   4. Applied egg-rr 74.5%

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

   5. Taylor expanded in a around 0 43.6%

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

      1. neg-mul-1 43.6%

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

   7. Simplified 43.6%

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

      1. sub-neg 43.6%

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

   9. Applied egg-rr 43.6%

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

       1. remove-double-neg 43.6%

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

       2. +-commutative 43.6%

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

   11. Simplified 43.6%

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

   if -0.5 < (+.f64 y z)

   1. Initial program 82.1%

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

   3. Taylor expanded in y around 0 65.3%

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

      1. +-commutative 65.3%

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

   5. Simplified 65.3%

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

      1. tan-quot 65.3%

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

      2. tan-quot 65.3%

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

      3. associate--l+ 65.3%

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

   7. Applied egg-rr 65.3%

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

      1. associate-+r- 65.3%

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

      2. +-commutative 65.3%

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

      3. associate--l+ 65.4%

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

   9. Simplified 65.4%

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

4. Final simplification 57.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y + z \leq -0.5:\\ \;\;\;\;x + \tan \left(y + z\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(\tan z - \tan a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 54.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y + z \leq -0.5:\\ \;\;\;\;x + \frac{\sin y}{\cos y}\\ \mathbf{else}:\\ \;\;\;\;x + \left(\tan z - \tan a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (if (<= (+ y z) -0.5) (+ x (/ (sin y) (cos y))) (+ x (- (tan z) (tan a)))))

C

double code(double x, double y, double z, double a) {
	double tmp;
	if ((y + z) <= -0.5) {
		tmp = x + (sin(y) / cos(y));
	} 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 + z) <= (-0.5d0)) then
        tmp = x + (sin(y) / cos(y))
    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 + z) <= -0.5) {
		tmp = x + (Math.sin(y) / Math.cos(y));
	} else {
		tmp = x + (Math.tan(z) - Math.tan(a));
	}
	return tmp;
}

Python

def code(x, y, z, a):
	tmp = 0
	if (y + z) <= -0.5:
		tmp = x + (math.sin(y) / math.cos(y))
	else:
		tmp = x + (math.tan(z) - math.tan(a))
	return tmp

Julia

function code(x, y, z, a)
	tmp = 0.0
	if (Float64(y + z) <= -0.5)
		tmp = Float64(x + Float64(sin(y) / cos(y)));
	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 + z) <= -0.5)
		tmp = x + (sin(y) / cos(y));
	else
		tmp = x + (tan(z) - tan(a));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, a_] := If[LessEqual[N[(y + z), $MachinePrecision], -0.5], N[(x + N[(N[Sin[y], $MachinePrecision] / N[Cos[y], $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 (+.f64 y z) < -0.5

   1. Initial program 74.5%

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

      1. +-commutative 74.5%

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

      2. associate-+l- 74.5%

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

   4. Applied egg-rr 74.5%

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

   5. Taylor expanded in a around 0 43.6%

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

      1. neg-mul-1 43.6%

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

   7. Simplified 43.6%

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

   8. Taylor expanded in z around 0 32.0%

      \[\leadsto \color{blue}{x + \frac{\sin y}{\cos y}} \]
   9. Step-by-step derivation

      1. +-commutative 32.0%

         \[\leadsto \color{blue}{\frac{\sin y}{\cos y} + x} \]

   10. Simplified 32.0%

       \[\leadsto \color{blue}{\frac{\sin y}{\cos y} + x} \]

   if -0.5 < (+.f64 y z)

   1. Initial program 82.1%

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

   3. Taylor expanded in y around 0 65.3%

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

      1. +-commutative 65.3%

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

   5. Simplified 65.3%

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

      1. tan-quot 65.3%

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

      2. tan-quot 65.3%

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

      3. associate--l+ 65.3%

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

   7. Applied egg-rr 65.3%

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

      1. associate-+r- 65.3%

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

      2. +-commutative 65.3%

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

      3. associate--l+ 65.4%

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

   9. Simplified 65.4%

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

4. Final simplification 53.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y + z \leq -0.5:\\ \;\;\;\;x + \frac{\sin y}{\cos y}\\ \mathbf{else}:\\ \;\;\;\;x + \left(\tan z - \tan a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 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 79.3%

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

3. Final simplification 79.3%

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

Alternative 7: 50.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.3%

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

   1. +-commutative 79.3%

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

   2. associate-+l- 79.2%

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

4. Applied egg-rr 79.2%

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

5. Taylor expanded in a around 0 47.9%

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

   1. neg-mul-1 47.9%

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

7. Simplified 47.9%

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

   1. sub-neg 47.9%

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

9. Applied egg-rr 47.9%

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

    1. remove-double-neg 47.9%

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

    2. +-commutative 47.9%

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

11. Simplified 47.9%

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

12. Final simplification 47.9%

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

Alternative 8: 31.6% 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 79.3%

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

3. Taylor expanded in x around inf 31.0%

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

4. Final simplification 31.0%

   \[\leadsto x \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024010 
(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))))