tan-example (used to crash)

Percentage Accurate: 79.3% → 99.7%

Time: 40.1s

Alternatives: 11

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, 0.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 77.2%

   \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
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. +-commutative N/A

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

   3. lift--.f64 N/A

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

   4. lift-tan.f64 N/A

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

   5. lift-+.f64 N/A

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

   6. tan-sum N/A

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

   7. lift-tan.f64 N/A

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

   8. tan-quot N/A

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

   9. frac-sub N/A

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

   10. div-inv N/A

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

   11. lower-fma.f64 N/A

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

4. Applied rewrites 99.7%

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

6. Applied rewrites 99.7%

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

   1. lift-pow.f64 N/A

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

   2. unpow-1 N/A

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

   3. lower-/.f64 99.7

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

8. Applied rewrites 99.7%

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

9. Final simplification 99.7%

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

Alternative 2: 99.7% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 77.2%

   \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
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. +-commutative N/A

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

   3. lift--.f64 N/A

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

   4. lift-tan.f64 N/A

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

   5. lift-+.f64 N/A

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

   6. tan-sum N/A

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

   7. lift-tan.f64 N/A

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

   8. tan-quot N/A

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

   9. frac-sub N/A

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

   10. div-inv N/A

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

   11. lower-fma.f64 N/A

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

4. Applied rewrites 99.7%

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

5. Final simplification 99.7%

   \[\leadsto \mathsf{fma}\left(\left(\tan y + \tan z\right) \cdot \cos a - \mathsf{fma}\left(-\tan z, \tan y, 1\right) \cdot \sin a, \frac{1}{\cos a \cdot \mathsf{fma}\left(-\tan z, \tan y, 1\right)}, x\right) \]
6. Add Preprocessing

Alternative 3: 89.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\tan \left(\frac{y}{y - z} \cdot y - \frac{z}{y - z} \cdot z\right) - \tan a\right) + x\\ \mathbf{if}\;\tan a \leq -2 \cdot 10^{-8}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\tan a \leq 2 \cdot 10^{-15}:\\ \;\;\;\;\frac{\tan y + \tan z}{\mathsf{fma}\left(\tan y, -\tan z, 1\right)} - \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (let* ((t_0
         (+ (- (tan (- (* (/ y (- y z)) y) (* (/ z (- y z)) z))) (tan a)) x)))
   (if (<= (tan a) -2e-8)
     t_0
     (if (<= (tan a) 2e-15)
       (- (/ (+ (tan y) (tan z)) (fma (tan y) (- (tan z)) 1.0)) (- x))
       t_0))))

C

double code(double x, double y, double z, double a) {
	double t_0 = (tan((((y / (y - z)) * y) - ((z / (y - z)) * z))) - tan(a)) + x;
	double tmp;
	if (tan(a) <= -2e-8) {
		tmp = t_0;
	} else if (tan(a) <= 2e-15) {
		tmp = ((tan(y) + tan(z)) / fma(tan(y), -tan(z), 1.0)) - -x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z, a)
	t_0 = Float64(Float64(tan(Float64(Float64(Float64(y / Float64(y - z)) * y) - Float64(Float64(z / Float64(y - z)) * z))) - tan(a)) + x)
	tmp = 0.0
	if (tan(a) <= -2e-8)
		tmp = t_0;
	elseif (tan(a) <= 2e-15)
		tmp = Float64(Float64(Float64(tan(y) + tan(z)) / fma(tan(y), Float64(-tan(z)), 1.0)) - Float64(-x));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, a_] := Block[{t$95$0 = N[(N[(N[Tan[N[(N[(N[(y / N[(y - z), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision] - N[(N[(z / N[(y - z), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[N[Tan[a], $MachinePrecision], -2e-8], t$95$0, If[LessEqual[N[Tan[a], $MachinePrecision], 2e-15], N[(N[(N[(N[Tan[y], $MachinePrecision] + N[Tan[z], $MachinePrecision]), $MachinePrecision] / N[(N[Tan[y], $MachinePrecision] * (-N[Tan[z], $MachinePrecision]) + 1.0), $MachinePrecision]), $MachinePrecision] - (-x)), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if (tan.f64 a) < -2e-8 or 2.0000000000000002e-15 < (tan.f64 a)

   1. Initial program 77.1%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   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. div-sub N/A

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

      4. lower--.f64 N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. associate-/l* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 77.2

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

   4. Applied rewrites 77.2%

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

   if -2e-8 < (tan.f64 a) < 2.0000000000000002e-15

   1. Initial program 77.2%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   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. +-commutative N/A

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

      3. lift--.f64 N/A

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

      4. associate-+l- N/A

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

      5. lower--.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. lower--.f64 77.2

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

   4. Applied rewrites 77.2%

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

   5. Taylor expanded in x around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 77.1

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

   7. Applied rewrites 77.1%

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

      1. lift-tan.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. tan-sum N/A

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

      4. lift-tan.f64 N/A

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

      5. lift-tan.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. lift-tan.f64 N/A

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

      9. lift-tan.f64 N/A

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

      10. cancel-sign-sub-inv N/A

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

      11. lift-neg.f64 N/A

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

      12. +-commutative N/A

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

      13. lift-fma.f64 N/A

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

      14. lower-/.f64 99.6

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

      15. lift-fma.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-fma.f64 99.6

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

   9. Applied rewrites 99.6%

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

4. Final simplification 87.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\tan a \leq -2 \cdot 10^{-8}:\\ \;\;\;\;\left(\tan \left(\frac{y}{y - z} \cdot y - \frac{z}{y - z} \cdot z\right) - \tan a\right) + x\\ \mathbf{elif}\;\tan a \leq 2 \cdot 10^{-15}:\\ \;\;\;\;\frac{\tan y + \tan z}{\mathsf{fma}\left(\tan y, -\tan z, 1\right)} - \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\tan \left(\frac{y}{y - z} \cdot y - \frac{z}{y - z} \cdot z\right) - \tan a\right) + x\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.7% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 77.2%

   \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
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. +-commutative N/A

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

   6. lower-+.f64 N/A

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

   7. lower-tan.f64 N/A

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

   8. lower-tan.f64 N/A

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

   9. sub-neg N/A

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

   10. +-commutative N/A

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

   11. *-commutative N/A

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

   12. distribute-lft-neg-in N/A

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

   13. lower-fma.f64 N/A

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

   14. lower-neg.f64 N/A

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

   15. lower-tan.f64 N/A

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

   16. lower-tan.f64 99.7

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

4. Applied rewrites 99.7%

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

5. Final simplification 99.7%

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

Alternative 5: 79.3% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.2%

   \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
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. div-sub N/A

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

   4. lower--.f64 N/A

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

   5. associate-/l* N/A

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

   6. lower-*.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. lower--.f64 N/A

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

   9. associate-/l* N/A

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

   10. lower-*.f64 N/A

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

   11. lower-/.f64 N/A

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

   12. lower--.f64 77.2

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

4. Applied rewrites 77.2%

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

5. Final simplification 77.2%

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

Alternative 6: 60.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y + z \leq -1000000:\\ \;\;\;\;\tan \left(y + z\right) - \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\tan z - \tan a\right) + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (if (<= (+ y z) -1000000.0)
   (- (tan (+ y z)) (- x))
   (+ (- (tan z) (tan a)) x)))

C

double code(double x, double y, double z, double a) {
	double tmp;
	if ((y + z) <= -1000000.0) {
		tmp = tan((y + z)) - -x;
	} else {
		tmp = (tan(z) - tan(a)) + x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: a
    real(8) :: tmp
    if ((y + z) <= (-1000000.0d0)) then
        tmp = tan((y + z)) - -x
    else
        tmp = (tan(z) - tan(a)) + x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double a) {
	double tmp;
	if ((y + z) <= -1000000.0) {
		tmp = Math.tan((y + z)) - -x;
	} else {
		tmp = (Math.tan(z) - Math.tan(a)) + x;
	}
	return tmp;
}

Python

def code(x, y, z, a):
	tmp = 0
	if (y + z) <= -1000000.0:
		tmp = math.tan((y + z)) - -x
	else:
		tmp = (math.tan(z) - math.tan(a)) + x
	return tmp

Julia

function code(x, y, z, a)
	tmp = 0.0
	if (Float64(y + z) <= -1000000.0)
		tmp = Float64(tan(Float64(y + z)) - Float64(-x));
	else
		tmp = Float64(Float64(tan(z) - tan(a)) + x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, a)
	tmp = 0.0;
	if ((y + z) <= -1000000.0)
		tmp = tan((y + z)) - -x;
	else
		tmp = (tan(z) - tan(a)) + x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, a_] := If[LessEqual[N[(y + z), $MachinePrecision], -1000000.0], N[(N[Tan[N[(y + z), $MachinePrecision]], $MachinePrecision] - (-x)), $MachinePrecision], N[(N[(N[Tan[z], $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 y z) < -1e6

   1. Initial program 72.8%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   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. +-commutative N/A

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

      3. lift--.f64 N/A

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

      4. associate-+l- N/A

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

      5. lower--.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. lower--.f64 72.6

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

   4. Applied rewrites 72.6%

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

   5. Taylor expanded in x around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 48.7

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

   7. Applied rewrites 48.7%

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

   if -1e6 < (+.f64 y z)

   1. Initial program 79.5%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   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 63.6

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

   5. Applied rewrites 63.6%

      \[\leadsto x + \left(\color{blue}{\frac{\sin z}{\cos z}} - \tan a\right) \]
   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 63.6

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

   7. Applied rewrites 63.7%

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

4. Final simplification 58.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y + z \leq -1000000:\\ \;\;\;\;\tan \left(y + z\right) - \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\tan z - \tan a\right) + x\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 79.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.2%

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

3. Final simplification 77.2%

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

Alternative 8: 55.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y + z \leq -1000000:\\ \;\;\;\;\tan \left(y + z\right) - \left(-x\right)\\ \mathbf{elif}\;y + z \leq 10^{-22}:\\ \;\;\;\;\left(\mathsf{fma}\left(z \cdot z, 0.3333333333333333, 1\right) \cdot z - \tan a\right) + x\\ \mathbf{else}:\\ \;\;\;\;\tan \left(\mathsf{fma}\left(\frac{y}{z}, z, z\right)\right) - \left(-x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (if (<= (+ y z) -1000000.0)
   (- (tan (+ y z)) (- x))
   (if (<= (+ y z) 1e-22)
     (+ (- (* (fma (* z z) 0.3333333333333333 1.0) z) (tan a)) x)
     (- (tan (fma (/ y z) z z)) (- x)))))

C

double code(double x, double y, double z, double a) {
	double tmp;
	if ((y + z) <= -1000000.0) {
		tmp = tan((y + z)) - -x;
	} else if ((y + z) <= 1e-22) {
		tmp = ((fma((z * z), 0.3333333333333333, 1.0) * z) - tan(a)) + x;
	} else {
		tmp = tan(fma((y / z), z, z)) - -x;
	}
	return tmp;
}

Julia

function code(x, y, z, a)
	tmp = 0.0
	if (Float64(y + z) <= -1000000.0)
		tmp = Float64(tan(Float64(y + z)) - Float64(-x));
	elseif (Float64(y + z) <= 1e-22)
		tmp = Float64(Float64(Float64(fma(Float64(z * z), 0.3333333333333333, 1.0) * z) - tan(a)) + x);
	else
		tmp = Float64(tan(fma(Float64(y / z), z, z)) - Float64(-x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, a_] := If[LessEqual[N[(y + z), $MachinePrecision], -1000000.0], N[(N[Tan[N[(y + z), $MachinePrecision]], $MachinePrecision] - (-x)), $MachinePrecision], If[LessEqual[N[(y + z), $MachinePrecision], 1e-22], N[(N[(N[(N[(N[(z * z), $MachinePrecision] * 0.3333333333333333 + 1.0), $MachinePrecision] * z), $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], N[(N[Tan[N[(N[(y / z), $MachinePrecision] * z + z), $MachinePrecision]], $MachinePrecision] - (-x)), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 y z) < -1e6

   1. Initial program 72.8%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   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. +-commutative N/A

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

      3. lift--.f64 N/A

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

      4. associate-+l- N/A

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

      5. lower--.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. lower--.f64 72.6

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

   4. Applied rewrites 72.6%

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

   5. Taylor expanded in x around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 48.7

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

   7. Applied rewrites 48.7%

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

   if -1e6 < (+.f64 y z) < 1e-22

   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 98.6

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

   5. Applied rewrites 98.6%

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

   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 97.6%

      \[\leadsto x + \left(\mathsf{fma}\left(z \cdot z, 0.3333333333333333, 1\right) \cdot \color{blue}{z} - \tan a\right) \]

   if 1e-22 < (+.f64 y z)

   1. Initial program 68.6%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   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. +-commutative N/A

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

      3. lift--.f64 N/A

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

      4. associate-+l- N/A

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

      5. lower--.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. lower--.f64 68.5

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

   4. Applied rewrites 68.5%

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

   5. Taylor expanded in x around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 45.0

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

   7. Applied rewrites 45.0%

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

   8. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-lft-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 29.4

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

   10. Applied rewrites 29.4%

       \[\leadsto \tan \color{blue}{\left(\mathsf{fma}\left(\frac{y}{z}, z, z\right)\right)} - \left(-x\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 51.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y + z \leq -1000000:\\ \;\;\;\;\tan \left(y + z\right) - \left(-x\right)\\ \mathbf{elif}\;y + z \leq 10^{-22}:\\ \;\;\;\;\left(\mathsf{fma}\left(z \cdot z, 0.3333333333333333, 1\right) \cdot z - \tan a\right) + x\\ \mathbf{else}:\\ \;\;\;\;\tan \left(\mathsf{fma}\left(\frac{y}{z}, z, z\right)\right) - \left(-x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 58.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan \left(y + z\right) - \left(-x\right)\\ \mathbf{if}\;y + z \leq -1000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y + z \leq 10^{-22}:\\ \;\;\;\;\left(\mathsf{fma}\left(z \cdot z, 0.3333333333333333, 1\right) \cdot z - \tan a\right) + x\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (let* ((t_0 (- (tan (+ y z)) (- x))))
   (if (<= (+ y z) -1000000.0)
     t_0
     (if (<= (+ y z) 1e-22)
       (+ (- (* (fma (* z z) 0.3333333333333333 1.0) z) (tan a)) x)
       t_0))))

C

double code(double x, double y, double z, double a) {
	double t_0 = tan((y + z)) - -x;
	double tmp;
	if ((y + z) <= -1000000.0) {
		tmp = t_0;
	} else if ((y + z) <= 1e-22) {
		tmp = ((fma((z * z), 0.3333333333333333, 1.0) * z) - tan(a)) + x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z, a)
	t_0 = Float64(tan(Float64(y + z)) - Float64(-x))
	tmp = 0.0
	if (Float64(y + z) <= -1000000.0)
		tmp = t_0;
	elseif (Float64(y + z) <= 1e-22)
		tmp = Float64(Float64(Float64(fma(Float64(z * z), 0.3333333333333333, 1.0) * z) - tan(a)) + x);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, a_] := Block[{t$95$0 = N[(N[Tan[N[(y + z), $MachinePrecision]], $MachinePrecision] - (-x)), $MachinePrecision]}, If[LessEqual[N[(y + z), $MachinePrecision], -1000000.0], t$95$0, If[LessEqual[N[(y + z), $MachinePrecision], 1e-22], N[(N[(N[(N[(N[(z * z), $MachinePrecision] * 0.3333333333333333 + 1.0), $MachinePrecision] * z), $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 y z) < -1e6 or 1e-22 < (+.f64 y z)

   1. Initial program 70.5%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   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. +-commutative N/A

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

      3. lift--.f64 N/A

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

      4. associate-+l- N/A

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

      5. lower--.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. lower--.f64 70.3

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

   4. Applied rewrites 70.3%

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

   5. Taylor expanded in x around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 46.7

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

   7. Applied rewrites 46.7%

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

   if -1e6 < (+.f64 y z) < 1e-22

   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 98.6

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

   5. Applied rewrites 98.6%

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

   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 97.6%

      \[\leadsto x + \left(\mathsf{fma}\left(z \cdot z, 0.3333333333333333, 1\right) \cdot \color{blue}{z} - \tan a\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 58.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y + z \leq -1000000:\\ \;\;\;\;\tan \left(y + z\right) - \left(-x\right)\\ \mathbf{elif}\;y + z \leq 10^{-22}:\\ \;\;\;\;\left(\mathsf{fma}\left(z \cdot z, 0.3333333333333333, 1\right) \cdot z - \tan a\right) + x\\ \mathbf{else}:\\ \;\;\;\;\tan \left(y + z\right) - \left(-x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 49.9% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.2%

   \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
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. +-commutative N/A

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

   3. lift--.f64 N/A

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

   4. associate-+l- N/A

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

   5. lower--.f64 N/A

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

   6. lift-+.f64 N/A

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

   7. +-commutative N/A

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

   8. lower-+.f64 N/A

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

   9. lower--.f64 77.0

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

4. Applied rewrites 77.0%

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

5. Taylor expanded in x around inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 48.7

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

7. Applied rewrites 48.7%

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

8. Final simplification 48.7%

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

Alternative 11: 31.9% 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 77.2%

   \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
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 77.1

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

4. Applied rewrites 77.0%

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

5. Taylor expanded in x around inf

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

   1. lower-/.f64 30.6

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

7. Applied rewrites 30.6%

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

Reproduce

herbie shell --seed 2024304 
(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))))