tan-example (used to crash)

Percentage Accurate: 79.1% → 99.7%

Time: 29.7s

Alternatives: 13

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 79.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - \tan y \cdot \tan z\\ x + \frac{\cos a \cdot \left(\tan y + \tan z\right) - \sin a \cdot t\_0}{\cos a \cdot t\_0} \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (let* ((t_0 (- 1.0 (* (tan y) (tan z)))))
   (+
    x
    (/ (- (* (cos a) (+ (tan y) (tan z))) (* (sin a) t_0)) (* (cos a) t_0)))))

C

double code(double x, double y, double z, double a) {
	double t_0 = 1.0 - (tan(y) * tan(z));
	return x + (((cos(a) * (tan(y) + tan(z))) - (sin(a) * t_0)) / (cos(a) * t_0));
}

Fortran

real(8) function code(x, y, z, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: a
    real(8) :: t_0
    t_0 = 1.0d0 - (tan(y) * tan(z))
    code = x + (((cos(a) * (tan(y) + tan(z))) - (sin(a) * t_0)) / (cos(a) * t_0))
end function

Java

public static double code(double x, double y, double z, double a) {
	double t_0 = 1.0 - (Math.tan(y) * Math.tan(z));
	return x + (((Math.cos(a) * (Math.tan(y) + Math.tan(z))) - (Math.sin(a) * t_0)) / (Math.cos(a) * t_0));
}

Python

def code(x, y, z, a):
	t_0 = 1.0 - (math.tan(y) * math.tan(z))
	return x + (((math.cos(a) * (math.tan(y) + math.tan(z))) - (math.sin(a) * t_0)) / (math.cos(a) * t_0))

Julia

function code(x, y, z, a)
	t_0 = Float64(1.0 - Float64(tan(y) * tan(z)))
	return Float64(x + Float64(Float64(Float64(cos(a) * Float64(tan(y) + tan(z))) - Float64(sin(a) * t_0)) / Float64(cos(a) * t_0)))
end

MATLAB

function tmp = code(x, y, z, a)
	t_0 = 1.0 - (tan(y) * tan(z));
	tmp = x + (((cos(a) * (tan(y) + tan(z))) - (sin(a) * t_0)) / (cos(a) * t_0));
end

Wolfram

code[x_, y_, z_, a_] := Block[{t$95$0 = N[(1.0 - N[(N[Tan[y], $MachinePrecision] * N[Tan[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(x + N[(N[(N[(N[Cos[a], $MachinePrecision] * N[(N[Tan[y], $MachinePrecision] + N[Tan[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Sin[a], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] / N[(N[Cos[a], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 78.9%

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

   1. lift--.f64 N/A

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

   2. lift-tan.f64 N/A

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

   3. lift-+.f64 N/A

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

   4. tan-sum N/A

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

   5. lift-tan.f64 N/A

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

   6. tan-quot N/A

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

   7. frac-sub N/A

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

   8. lower-/.f64 N/A

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

4. Applied rewrites 99.7%

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

Alternative 2: 99.7% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.9%

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

   1. lift-tan.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. tan-sum N/A

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

   4. lower-/.f64 N/A

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

   5. lower-+.f64 N/A

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

   6. lower-tan.f64 N/A

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

   7. lower-tan.f64 N/A

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

   8. lower--.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. lower-tan.f64 N/A

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

   11. lower-tan.f64 99.7

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

4. Applied rewrites 99.7%

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

Alternative 3: 88.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan y + \tan z\\ \mathbf{if}\;a \leq -2.1 \cdot 10^{-12}:\\ \;\;\;\;x + \left(\tan \left(y + z\right) - \tan a\right)\\ \mathbf{elif}\;a \leq 2.36 \cdot 10^{-63}:\\ \;\;\;\;x + \frac{t\_0}{1 - \tan y \cdot \tan z}\\ \mathbf{else}:\\ \;\;\;\;x + \left(t\_0 \cdot 1 - \tan a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (let* ((t_0 (+ (tan y) (tan z))))
   (if (<= a -2.1e-12)
     (+ x (- (tan (+ y z)) (tan a)))
     (if (<= a 2.36e-63)
       (+ x (/ t_0 (- 1.0 (* (tan y) (tan z)))))
       (+ x (- (* t_0 1.0) (tan a)))))))

C

double code(double x, double y, double z, double a) {
	double t_0 = tan(y) + tan(z);
	double tmp;
	if (a <= -2.1e-12) {
		tmp = x + (tan((y + z)) - tan(a));
	} else if (a <= 2.36e-63) {
		tmp = x + (t_0 / (1.0 - (tan(y) * tan(z))));
	} else {
		tmp = x + ((t_0 * 1.0) - tan(a));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: a
    real(8) :: t_0
    real(8) :: tmp
    t_0 = tan(y) + tan(z)
    if (a <= (-2.1d-12)) then
        tmp = x + (tan((y + z)) - tan(a))
    else if (a <= 2.36d-63) then
        tmp = x + (t_0 / (1.0d0 - (tan(y) * tan(z))))
    else
        tmp = x + ((t_0 * 1.0d0) - tan(a))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double a) {
	double t_0 = Math.tan(y) + Math.tan(z);
	double tmp;
	if (a <= -2.1e-12) {
		tmp = x + (Math.tan((y + z)) - Math.tan(a));
	} else if (a <= 2.36e-63) {
		tmp = x + (t_0 / (1.0 - (Math.tan(y) * Math.tan(z))));
	} else {
		tmp = x + ((t_0 * 1.0) - Math.tan(a));
	}
	return tmp;
}

Python

def code(x, y, z, a):
	t_0 = math.tan(y) + math.tan(z)
	tmp = 0
	if a <= -2.1e-12:
		tmp = x + (math.tan((y + z)) - math.tan(a))
	elif a <= 2.36e-63:
		tmp = x + (t_0 / (1.0 - (math.tan(y) * math.tan(z))))
	else:
		tmp = x + ((t_0 * 1.0) - math.tan(a))
	return tmp

Julia

function code(x, y, z, a)
	t_0 = Float64(tan(y) + tan(z))
	tmp = 0.0
	if (a <= -2.1e-12)
		tmp = Float64(x + Float64(tan(Float64(y + z)) - tan(a)));
	elseif (a <= 2.36e-63)
		tmp = Float64(x + Float64(t_0 / Float64(1.0 - Float64(tan(y) * tan(z)))));
	else
		tmp = Float64(x + Float64(Float64(t_0 * 1.0) - tan(a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, a)
	t_0 = tan(y) + tan(z);
	tmp = 0.0;
	if (a <= -2.1e-12)
		tmp = x + (tan((y + z)) - tan(a));
	elseif (a <= 2.36e-63)
		tmp = x + (t_0 / (1.0 - (tan(y) * tan(z))));
	else
		tmp = x + ((t_0 * 1.0) - tan(a));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, a_] := Block[{t$95$0 = N[(N[Tan[y], $MachinePrecision] + N[Tan[z], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -2.1e-12], N[(x + N[(N[Tan[N[(y + z), $MachinePrecision]], $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 2.36e-63], N[(x + N[(t$95$0 / N[(1.0 - N[(N[Tan[y], $MachinePrecision] * N[Tan[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[(t$95$0 * 1.0), $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if a < -2.09999999999999994e-12

   1. Initial program 77.5%

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

   if -2.09999999999999994e-12 < a < 2.35999999999999998e-63

   1. Initial program 82.1%

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

      1. lift-+.f64 N/A

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

      2. flip-+ N/A

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

      3. clear-num N/A

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

      4. lower-/.f64 N/A

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

      5. clear-num N/A

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

      6. flip-+ N/A

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

      7. lift-+.f64 N/A

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

      8. lower-/.f64 71.0

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

   4. Applied rewrites 71.0%

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

   5. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-+.f64 82.1

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

   7. Applied rewrites 82.1%

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

   9. Applied rewrites 99.8%

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

   if 2.35999999999999998e-63 < a

   1. Initial program 75.9%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. associate-+r- N/A

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

      4. +-commutative N/A

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

      5. associate--l+ N/A

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

      6. lift-tan.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. tan-sum N/A

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

      9. clear-num N/A

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

      10. associate-/r/ N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower--.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-tan.f64 N/A

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

      16. lower-tan.f64 N/A

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

      17. lower-+.f64 N/A

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

      18. lower-tan.f64 N/A

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

      19. lower-tan.f64 N/A

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

      20. lower--.f64 99.6

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in y around 0

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

   7. Applied rewrites 76.0%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift--.f64 N/A

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

      4. associate-+l- N/A

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

      5. lower--.f64 N/A

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

      6. lower--.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 76.1

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

   9. Applied rewrites 76.1%

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

4. Final simplification 86.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -2.1 \cdot 10^{-12}:\\ \;\;\;\;x + \left(\tan \left(y + z\right) - \tan a\right)\\ \mathbf{elif}\;a \leq 2.36 \cdot 10^{-63}:\\ \;\;\;\;x + \frac{\tan y + \tan z}{1 - \tan y \cdot \tan z}\\ \mathbf{else}:\\ \;\;\;\;x + \left(\left(\tan y + \tan z\right) \cdot 1 - \tan a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 59.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan \left(y + z\right)\\ t_1 := x + t\_0\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{-11}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 0.02:\\ \;\;\;\;x + \left(\mathsf{fma}\left(\mathsf{fma}\left(z, z \cdot \mathsf{fma}\left(z \cdot z, 0.05396825396825397, 0.13333333333333333\right), 0.3333333333333333\right), z \cdot \left(z \cdot z\right), z\right) - \tan a\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (let* ((t_0 (tan (+ y z))) (t_1 (+ x t_0)))
   (if (<= t_0 -1e-11)
     t_1
     (if (<= t_0 0.02)
       (+
        x
        (-
         (fma
          (fma
           z
           (* z (fma (* z z) 0.05396825396825397 0.13333333333333333))
           0.3333333333333333)
          (* z (* z z))
          z)
         (tan a)))
       t_1))))

C

double code(double x, double y, double z, double a) {
	double t_0 = tan((y + z));
	double t_1 = x + t_0;
	double tmp;
	if (t_0 <= -1e-11) {
		tmp = t_1;
	} else if (t_0 <= 0.02) {
		tmp = x + (fma(fma(z, (z * fma((z * z), 0.05396825396825397, 0.13333333333333333)), 0.3333333333333333), (z * (z * z)), z) - tan(a));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, a)
	t_0 = tan(Float64(y + z))
	t_1 = Float64(x + t_0)
	tmp = 0.0
	if (t_0 <= -1e-11)
		tmp = t_1;
	elseif (t_0 <= 0.02)
		tmp = Float64(x + Float64(fma(fma(z, Float64(z * fma(Float64(z * z), 0.05396825396825397, 0.13333333333333333)), 0.3333333333333333), Float64(z * Float64(z * z)), z) - tan(a)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, a_] := Block[{t$95$0 = N[Tan[N[(y + z), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(x + t$95$0), $MachinePrecision]}, If[LessEqual[t$95$0, -1e-11], t$95$1, If[LessEqual[t$95$0, 0.02], N[(x + N[(N[(N[(z * N[(z * N[(N[(z * z), $MachinePrecision] * 0.05396825396825397 + 0.13333333333333333), $MachinePrecision]), $MachinePrecision] + 0.3333333333333333), $MachinePrecision] * N[(z * N[(z * z), $MachinePrecision]), $MachinePrecision] + z), $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (tan.f64 (+.f64 y z)) < -9.99999999999999939e-12 or 0.0200000000000000004 < (tan.f64 (+.f64 y z))

   1. Initial program 72.4%

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

      1. lift-+.f64 N/A

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

      2. flip-+ N/A

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

      3. clear-num N/A

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

      4. lower-/.f64 N/A

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

      5. clear-num N/A

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

      6. flip-+ N/A

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

      7. lift-+.f64 N/A

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

      8. lower-/.f64 63.2

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

   4. Applied rewrites 63.2%

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

   5. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-+.f64 47.4

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

   7. Applied rewrites 47.4%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 47.4

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

   9. Applied rewrites 47.4%

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

   if -9.99999999999999939e-12 < (tan.f64 (+.f64 y z)) < 0.0200000000000000004

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-cos.f64 95.9

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

   5. Applied rewrites 95.9%

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

   6. Taylor expanded in z around 0

      \[\leadsto x + \left(z \cdot \color{blue}{\left(1 + {z}^{2} \cdot \left(\frac{1}{3} + {z}^{2} \cdot \left(\frac{2}{15} + \frac{17}{315} \cdot {z}^{2}\right)\right)\right)} - \tan a\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 95.9%

      \[\leadsto x + \left(\mathsf{fma}\left(\mathsf{fma}\left(z, z \cdot \mathsf{fma}\left(z \cdot z, 0.05396825396825397, 0.13333333333333333\right), 0.3333333333333333\right), \color{blue}{\left(z \cdot z\right) \cdot z}, z\right) - \tan a\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 59.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\tan \left(y + z\right) \leq -1 \cdot 10^{-11}:\\ \;\;\;\;x + \tan \left(y + z\right)\\ \mathbf{elif}\;\tan \left(y + z\right) \leq 0.02:\\ \;\;\;\;x + \left(\mathsf{fma}\left(\mathsf{fma}\left(z, z \cdot \mathsf{fma}\left(z \cdot z, 0.05396825396825397, 0.13333333333333333\right), 0.3333333333333333\right), z \cdot \left(z \cdot z\right), z\right) - \tan a\right)\\ \mathbf{else}:\\ \;\;\;\;x + \tan \left(y + z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 59.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan \left(y + z\right)\\ t_1 := x + t\_0\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{-11}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 0.02:\\ \;\;\;\;x + \left(\mathsf{fma}\left(\mathsf{fma}\left(z, z \cdot 0.13333333333333333, 0.3333333333333333\right), z \cdot \left(z \cdot z\right), z\right) - \tan a\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (let* ((t_0 (tan (+ y z))) (t_1 (+ x t_0)))
   (if (<= t_0 -1e-11)
     t_1
     (if (<= t_0 0.02)
       (+
        x
        (-
         (fma
          (fma z (* z 0.13333333333333333) 0.3333333333333333)
          (* z (* z z))
          z)
         (tan a)))
       t_1))))

C

double code(double x, double y, double z, double a) {
	double t_0 = tan((y + z));
	double t_1 = x + t_0;
	double tmp;
	if (t_0 <= -1e-11) {
		tmp = t_1;
	} else if (t_0 <= 0.02) {
		tmp = x + (fma(fma(z, (z * 0.13333333333333333), 0.3333333333333333), (z * (z * z)), z) - tan(a));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, a)
	t_0 = tan(Float64(y + z))
	t_1 = Float64(x + t_0)
	tmp = 0.0
	if (t_0 <= -1e-11)
		tmp = t_1;
	elseif (t_0 <= 0.02)
		tmp = Float64(x + Float64(fma(fma(z, Float64(z * 0.13333333333333333), 0.3333333333333333), Float64(z * Float64(z * z)), z) - tan(a)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, a_] := Block[{t$95$0 = N[Tan[N[(y + z), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(x + t$95$0), $MachinePrecision]}, If[LessEqual[t$95$0, -1e-11], t$95$1, If[LessEqual[t$95$0, 0.02], N[(x + N[(N[(N[(z * N[(z * 0.13333333333333333), $MachinePrecision] + 0.3333333333333333), $MachinePrecision] * N[(z * N[(z * z), $MachinePrecision]), $MachinePrecision] + z), $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (tan.f64 (+.f64 y z)) < -9.99999999999999939e-12 or 0.0200000000000000004 < (tan.f64 (+.f64 y z))

   1. Initial program 72.4%

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

      1. lift-+.f64 N/A

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

      2. flip-+ N/A

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

      3. clear-num N/A

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

      4. lower-/.f64 N/A

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

      5. clear-num N/A

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

      6. flip-+ N/A

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

      7. lift-+.f64 N/A

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

      8. lower-/.f64 63.2

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

   4. Applied rewrites 63.2%

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

   5. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-+.f64 47.4

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

   7. Applied rewrites 47.4%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 47.4

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

   9. Applied rewrites 47.4%

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

   if -9.99999999999999939e-12 < (tan.f64 (+.f64 y z)) < 0.0200000000000000004

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-cos.f64 95.9

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

   5. Applied rewrites 95.9%

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

   6. Taylor expanded in z around 0

      \[\leadsto x + \left(z \cdot \color{blue}{\left(1 + {z}^{2} \cdot \left(\frac{1}{3} + \frac{2}{15} \cdot {z}^{2}\right)\right)} - \tan a\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 95.9%

      \[\leadsto x + \left(\mathsf{fma}\left(\mathsf{fma}\left(z, z \cdot 0.13333333333333333, 0.3333333333333333\right), \color{blue}{\left(z \cdot z\right) \cdot z}, z\right) - \tan a\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 59.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\tan \left(y + z\right) \leq -1 \cdot 10^{-11}:\\ \;\;\;\;x + \tan \left(y + z\right)\\ \mathbf{elif}\;\tan \left(y + z\right) \leq 0.02:\\ \;\;\;\;x + \left(\mathsf{fma}\left(\mathsf{fma}\left(z, z \cdot 0.13333333333333333, 0.3333333333333333\right), z \cdot \left(z \cdot z\right), z\right) - \tan a\right)\\ \mathbf{else}:\\ \;\;\;\;x + \tan \left(y + z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 59.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan \left(y + z\right)\\ t_1 := x + t\_0\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{-11}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 0.02:\\ \;\;\;\;x + \left(\mathsf{fma}\left(z, \left(z \cdot z\right) \cdot 0.3333333333333333, z\right) - \tan a\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (let* ((t_0 (tan (+ y z))) (t_1 (+ x t_0)))
   (if (<= t_0 -1e-11)
     t_1
     (if (<= t_0 0.02)
       (+ x (- (fma z (* (* z z) 0.3333333333333333) z) (tan a)))
       t_1))))

C

double code(double x, double y, double z, double a) {
	double t_0 = tan((y + z));
	double t_1 = x + t_0;
	double tmp;
	if (t_0 <= -1e-11) {
		tmp = t_1;
	} else if (t_0 <= 0.02) {
		tmp = x + (fma(z, ((z * z) * 0.3333333333333333), z) - tan(a));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, a)
	t_0 = tan(Float64(y + z))
	t_1 = Float64(x + t_0)
	tmp = 0.0
	if (t_0 <= -1e-11)
		tmp = t_1;
	elseif (t_0 <= 0.02)
		tmp = Float64(x + Float64(fma(z, Float64(Float64(z * z) * 0.3333333333333333), z) - tan(a)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, a_] := Block[{t$95$0 = N[Tan[N[(y + z), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(x + t$95$0), $MachinePrecision]}, If[LessEqual[t$95$0, -1e-11], t$95$1, If[LessEqual[t$95$0, 0.02], N[(x + N[(N[(z * N[(N[(z * z), $MachinePrecision] * 0.3333333333333333), $MachinePrecision] + z), $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (tan.f64 (+.f64 y z)) < -9.99999999999999939e-12 or 0.0200000000000000004 < (tan.f64 (+.f64 y z))

   1. Initial program 72.4%

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

      1. lift-+.f64 N/A

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

      2. flip-+ N/A

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

      3. clear-num N/A

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

      4. lower-/.f64 N/A

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

      5. clear-num N/A

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

      6. flip-+ N/A

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

      7. lift-+.f64 N/A

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

      8. lower-/.f64 63.2

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

   4. Applied rewrites 63.2%

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

   5. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-+.f64 47.4

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

   7. Applied rewrites 47.4%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 47.4

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

   9. Applied rewrites 47.4%

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

   if -9.99999999999999939e-12 < (tan.f64 (+.f64 y z)) < 0.0200000000000000004

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-cos.f64 95.9

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

   5. Applied rewrites 95.9%

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

   6. Taylor expanded in z around 0

      \[\leadsto x + \left(z \cdot \color{blue}{\left(1 + \frac{1}{3} \cdot {z}^{2}\right)} - \tan a\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 95.9%

      \[\leadsto x + \left(\mathsf{fma}\left(z, \color{blue}{0.3333333333333333 \cdot \left(z \cdot z\right)}, z\right) - \tan a\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 59.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\tan \left(y + z\right) \leq -1 \cdot 10^{-11}:\\ \;\;\;\;x + \tan \left(y + z\right)\\ \mathbf{elif}\;\tan \left(y + z\right) \leq 0.02:\\ \;\;\;\;x + \left(\mathsf{fma}\left(z, \left(z \cdot z\right) \cdot 0.3333333333333333, z\right) - \tan a\right)\\ \mathbf{else}:\\ \;\;\;\;x + \tan \left(y + z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 79.5% accurate, 0.7× speedup

Math

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

FPCore

(FPCore (x y z a)
 :precision binary64
 (+ x (- (* (+ (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) - 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(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(a));
}

Python

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

MATLAB

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

Derivation

1. Initial program 78.9%

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

   1. lift-+.f64 N/A

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

   2. lift--.f64 N/A

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

   3. associate-+r- N/A

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

   4. +-commutative N/A

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

   5. associate--l+ N/A

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

   6. lift-tan.f64 N/A

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

   7. lift-+.f64 N/A

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

   8. tan-sum N/A

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

   9. clear-num N/A

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

   10. associate-/r/ N/A

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

   11. lower-fma.f64 N/A

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

   12. lower-/.f64 N/A

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

   13. lower--.f64 N/A

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

   14. lower-*.f64 N/A

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

   15. lower-tan.f64 N/A

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

   16. lower-tan.f64 N/A

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

   17. lower-+.f64 N/A

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

   18. lower-tan.f64 N/A

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

   19. lower-tan.f64 N/A

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

   20. lower--.f64 99.6

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

4. Applied rewrites 99.6%

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

5. Taylor expanded in y around 0

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

7. Applied rewrites 79.1%

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

   1. lift-fma.f64 N/A

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

   2. +-commutative N/A

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

   3. lift--.f64 N/A

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

   4. associate-+l- N/A

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

   5. lower--.f64 N/A

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

   6. lower--.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 79.2

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

9. Applied rewrites 79.2%

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

10. Final simplification 79.2%

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

Alternative 8: 79.4% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 78.9%

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

   1. lift-+.f64 N/A

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

   2. lift--.f64 N/A

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

   3. associate-+r- N/A

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

   4. +-commutative N/A

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

   5. associate--l+ N/A

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

   6. lift-tan.f64 N/A

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

   7. lift-+.f64 N/A

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

   8. tan-sum N/A

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

   9. clear-num N/A

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

   10. associate-/r/ N/A

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

   11. lower-fma.f64 N/A

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

   12. lower-/.f64 N/A

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

   13. lower--.f64 N/A

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

   14. lower-*.f64 N/A

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

   15. lower-tan.f64 N/A

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

   16. lower-tan.f64 N/A

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

   17. lower-+.f64 N/A

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

   18. lower-tan.f64 N/A

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

   19. lower-tan.f64 N/A

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

   20. lower--.f64 99.6

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

4. Applied rewrites 99.6%

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

5. Taylor expanded in y around 0

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

7. Applied rewrites 79.1%

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

   1. lift-fma.f64 N/A

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

   2. lift--.f64 N/A

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

   3. associate-+r- N/A

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

   4. lower--.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f64 79.1

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

9. Applied rewrites 79.1%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\tan y + \tan z, 1, x\right) - \tan a} \]
10. Add Preprocessing

Alternative 9: 59.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y + z \leq -1 \cdot 10^{-11}:\\ \;\;\;\;x + \tan \left(y + z\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(\tan z - \tan a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (if (<= (+ y z) -1e-11) (+ 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) <= -1e-11) {
		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) <= (-1d-11)) 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) <= -1e-11) {
		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) <= -1e-11:
		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) <= -1e-11)
		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) <= -1e-11)
		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], -1e-11], 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) < -9.99999999999999939e-12

   1. Initial program 73.0%

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

      1. lift-+.f64 N/A

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

      2. flip-+ N/A

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

      3. clear-num N/A

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

      4. lower-/.f64 N/A

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

      5. clear-num N/A

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

      6. flip-+ N/A

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

      7. lift-+.f64 N/A

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

      8. lower-/.f64 61.5

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

   4. Applied rewrites 61.5%

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

   5. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-+.f64 45.3

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

   7. Applied rewrites 45.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 45.3

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

   9. Applied rewrites 45.3%

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

   if -9.99999999999999939e-12 < (+.f64 y z)

   1. Initial program 82.2%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-cos.f64 64.0

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

   5. Applied rewrites 64.0%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 64.0

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

   7. Applied rewrites 64.0%

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

4. Final simplification 57.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y + z \leq -1 \cdot 10^{-11}:\\ \;\;\;\;x + \tan \left(y + z\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(\tan z - \tan a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 79.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.9%

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

Alternative 11: 50.2% 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 78.9%

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

   1. lift-+.f64 N/A

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

   2. flip-+ N/A

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

   3. clear-num N/A

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

   4. lower-/.f64 N/A

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

   5. clear-num N/A

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

   6. flip-+ N/A

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

   7. lift-+.f64 N/A

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

   8. lower-/.f64 72.0

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

4. Applied rewrites 72.0%

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

5. Taylor expanded in a around 0

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

   1. lower-/.f64 N/A

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

   2. lower-sin.f64 N/A

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

   3. lower-+.f64 N/A

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

   4. lower-cos.f64 N/A

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

   5. lower-+.f64 51.5

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

7. Applied rewrites 51.5%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lower-+.f64 51.5

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

9. Applied rewrites 51.5%

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

10. Final simplification 51.5%

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

Alternative 12: 31.7% accurate, 9.1× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\frac{1}{x}} \end{array} \]

FPCore

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

C

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

Java

public static double code(double x, double y, double z, double a) {
	return 1.0 / (1.0 / x);
}

Python

def code(x, y, z, a):
	return 1.0 / (1.0 / x)

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, a_] := N[(1.0 / N[(1.0 / x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 78.9%

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

   1. lift-+.f64 N/A

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

   2. flip3-+ N/A

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

   3. clear-num N/A

      \[\leadsto \color{blue}{\frac{1}{\frac{x \cdot x + \left(\left(\tan \left(y + z\right) - \tan a\right) \cdot \left(\tan \left(y + z\right) - \tan a\right) - x \cdot \left(\tan \left(y + z\right) - \tan a\right)\right)}{{x}^{3} + {\left(\tan \left(y + z\right) - \tan a\right)}^{3}}}} \]

   4. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{1}{\frac{x \cdot x + \left(\left(\tan \left(y + z\right) - \tan a\right) \cdot \left(\tan \left(y + z\right) - \tan a\right) - x \cdot \left(\tan \left(y + z\right) - \tan a\right)\right)}{{x}^{3} + {\left(\tan \left(y + z\right) - \tan a\right)}^{3}}}} \]

   5. clear-num N/A

      \[\leadsto \frac{1}{\color{blue}{\frac{1}{\frac{{x}^{3} + {\left(\tan \left(y + z\right) - \tan a\right)}^{3}}{x \cdot x + \left(\left(\tan \left(y + z\right) - \tan a\right) \cdot \left(\tan \left(y + z\right) - \tan a\right) - x \cdot \left(\tan \left(y + z\right) - \tan a\right)\right)}}}} \]

   6. flip3-+ N/A

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

   7. lift-+.f64 N/A

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

   8. lower-/.f64 78.8

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

4. Applied rewrites 78.7%

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

5. Taylor expanded in x around inf

   \[\leadsto \frac{1}{\color{blue}{\frac{1}{x}}} \]
6. Step-by-step derivation

   1. lower-/.f64 32.7

      \[\leadsto \frac{1}{\color{blue}{\frac{1}{x}}} \]

7. Applied rewrites 32.7%

   \[\leadsto \frac{1}{\color{blue}{\frac{1}{x}}} \]
8. Add Preprocessing

Alternative 13: 2.7% accurate, 9.1× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\frac{-1}{x}} \end{array} \]

FPCore

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

C

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

Java

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

Python

def code(x, y, z, a):
	return 1.0 / (-1.0 / x)

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, a_] := N[(1.0 / N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 78.9%

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

   1. lift-+.f64 N/A

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

   2. flip3-+ N/A

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

   3. clear-num N/A

      \[\leadsto \color{blue}{\frac{1}{\frac{x \cdot x + \left(\left(\tan \left(y + z\right) - \tan a\right) \cdot \left(\tan \left(y + z\right) - \tan a\right) - x \cdot \left(\tan \left(y + z\right) - \tan a\right)\right)}{{x}^{3} + {\left(\tan \left(y + z\right) - \tan a\right)}^{3}}}} \]

   4. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{1}{\frac{x \cdot x + \left(\left(\tan \left(y + z\right) - \tan a\right) \cdot \left(\tan \left(y + z\right) - \tan a\right) - x \cdot \left(\tan \left(y + z\right) - \tan a\right)\right)}{{x}^{3} + {\left(\tan \left(y + z\right) - \tan a\right)}^{3}}}} \]

   5. clear-num N/A

      \[\leadsto \frac{1}{\color{blue}{\frac{1}{\frac{{x}^{3} + {\left(\tan \left(y + z\right) - \tan a\right)}^{3}}{x \cdot x + \left(\left(\tan \left(y + z\right) - \tan a\right) \cdot \left(\tan \left(y + z\right) - \tan a\right) - x \cdot \left(\tan \left(y + z\right) - \tan a\right)\right)}}}} \]

   6. flip3-+ N/A

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

   7. lift-+.f64 N/A

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

   8. lower-/.f64 78.8

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

4. Applied rewrites 78.7%

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

5. Taylor expanded in x around inf

   \[\leadsto \frac{1}{\color{blue}{\frac{1}{x}}} \]
6. Step-by-step derivation

   1. lower-/.f64 32.7

      \[\leadsto \frac{1}{\color{blue}{\frac{1}{x}}} \]

7. Applied rewrites 32.7%

   \[\leadsto \frac{1}{\color{blue}{\frac{1}{x}}} \]
8. Step-by-step derivation

9. Applied rewrites 32.7%

   \[\leadsto \frac{1}{{\left(x \cdot x\right)}^{\color{blue}{-0.5}}} \]

10. Taylor expanded in x around -inf

    \[\leadsto \frac{1}{\frac{-1}{\color{blue}{x}}} \]
11. Step-by-step derivation

12. Applied rewrites 2.5%

    \[\leadsto \frac{1}{\frac{-1}{\color{blue}{x}}} \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2024233 
(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))))