tan-example (used to crash)

Percentage Accurate: 79.3% → 99.7%

Time: 29.1s

Alternatives: 10

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

\[\begin{array}{l} \\ \left(\frac{{\tan y}^{2} - {\tan z}^{2}}{\left(\tan y - \tan z\right) \cdot \mathsf{fma}\left(-\tan z, \tan y, 1\right)} - \tan a\right) + x \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (+
  (-
   (/
    (- (pow (tan y) 2.0) (pow (tan z) 2.0))
    (* (- (tan y) (tan z)) (fma (- (tan z)) (tan y) 1.0)))
   (tan a))
  x))

C

double code(double x, double y, double z, double a) {
	return (((pow(tan(y), 2.0) - pow(tan(z), 2.0)) / ((tan(y) - tan(z)) * fma(-tan(z), tan(y), 1.0))) - tan(a)) + x;
}

Julia

function code(x, y, z, a)
	return Float64(Float64(Float64(Float64((tan(y) ^ 2.0) - (tan(z) ^ 2.0)) / Float64(Float64(tan(y) - tan(z)) * fma(Float64(-tan(z)), tan(y), 1.0))) - tan(a)) + x)
end

Wolfram

code[x_, y_, z_, a_] := N[(N[(N[(N[(N[Power[N[Tan[y], $MachinePrecision], 2.0], $MachinePrecision] - N[Power[N[Tan[z], $MachinePrecision], 2.0], $MachinePrecision]), $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] - N[Tan[a], $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]

Derivation

1. Initial program 80.6%

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

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

   5. associate-/l/ N/A

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

   6. lower-/.f64 N/A

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

   7. lower--.f64 N/A

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

   8. pow2 N/A

      \[\leadsto x + \left(\frac{\color{blue}{{\tan y}^{2}} - \tan z \cdot \tan z}{\left(1 - \tan y \cdot \tan z\right) \cdot \left(\tan y - \tan z\right)} - \tan a\right) \]

   9. lower-pow.f64 N/A

      \[\leadsto x + \left(\frac{\color{blue}{{\tan y}^{2}} - \tan z \cdot \tan z}{\left(1 - \tan y \cdot \tan z\right) \cdot \left(\tan y - \tan z\right)} - \tan a\right) \]

   10. lower-tan.f64 N/A

       \[\leadsto x + \left(\frac{{\color{blue}{\tan y}}^{2} - \tan z \cdot \tan z}{\left(1 - \tan y \cdot \tan z\right) \cdot \left(\tan y - \tan z\right)} - \tan a\right) \]

   11. pow2 N/A

       \[\leadsto x + \left(\frac{{\tan y}^{2} - \color{blue}{{\tan z}^{2}}}{\left(1 - \tan y \cdot \tan z\right) \cdot \left(\tan y - \tan z\right)} - \tan a\right) \]

   12. lower-pow.f64 N/A

       \[\leadsto x + \left(\frac{{\tan y}^{2} - \color{blue}{{\tan z}^{2}}}{\left(1 - \tan y \cdot \tan z\right) \cdot \left(\tan y - \tan z\right)} - \tan a\right) \]

   13. lower-tan.f64 N/A

       \[\leadsto x + \left(\frac{{\tan y}^{2} - {\color{blue}{\tan z}}^{2}}{\left(1 - \tan y \cdot \tan z\right) \cdot \left(\tan y - \tan z\right)} - \tan a\right) \]

   14. lower-*.f64 N/A

       \[\leadsto x + \left(\frac{{\tan y}^{2} - {\tan z}^{2}}{\color{blue}{\left(1 - \tan y \cdot \tan z\right) \cdot \left(\tan y - \tan z\right)}} - \tan a\right) \]

4. Applied rewrites 99.7%

   \[\leadsto x + \left(\color{blue}{\frac{{\tan y}^{2} - {\tan z}^{2}}{\mathsf{fma}\left(-\tan z, \tan y, 1\right) \cdot \left(\tan y - \tan z\right)}} - \tan a\right) \]

5. Final simplification 99.7%

   \[\leadsto \left(\frac{{\tan y}^{2} - {\tan z}^{2}}{\left(\tan y - \tan z\right) \cdot \mathsf{fma}\left(-\tan z, \tan y, 1\right)} - \tan a\right) + x \]
6. Add Preprocessing

Alternative 2: 88.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan z + \tan y\\ t_1 := \left(\frac{t\_0}{1} - \tan a\right) + x\\ \mathbf{if}\;\tan a \leq -0.0004:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\tan a \leq 5 \cdot 10^{-69}:\\ \;\;\;\;\mathsf{fma}\left(t\_0, \frac{-1}{\mathsf{fma}\left(\tan y, \tan z, -1\right)}, -\left(a - x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (let* ((t_0 (+ (tan z) (tan y))) (t_1 (+ (- (/ t_0 1.0) (tan a)) x)))
   (if (<= (tan a) -0.0004)
     t_1
     (if (<= (tan a) 5e-69)
       (fma t_0 (/ -1.0 (fma (tan y) (tan z) -1.0)) (- (- a x)))
       t_1))))

C

double code(double x, double y, double z, double a) {
	double t_0 = tan(z) + tan(y);
	double t_1 = ((t_0 / 1.0) - tan(a)) + x;
	double tmp;
	if (tan(a) <= -0.0004) {
		tmp = t_1;
	} else if (tan(a) <= 5e-69) {
		tmp = fma(t_0, (-1.0 / fma(tan(y), tan(z), -1.0)), -(a - x));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, a)
	t_0 = Float64(tan(z) + tan(y))
	t_1 = Float64(Float64(Float64(t_0 / 1.0) - tan(a)) + x)
	tmp = 0.0
	if (tan(a) <= -0.0004)
		tmp = t_1;
	elseif (tan(a) <= 5e-69)
		tmp = fma(t_0, Float64(-1.0 / fma(tan(y), tan(z), -1.0)), Float64(-Float64(a - x)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, a_] := Block[{t$95$0 = N[(N[Tan[z], $MachinePrecision] + N[Tan[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(t$95$0 / 1.0), $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[N[Tan[a], $MachinePrecision], -0.0004], t$95$1, If[LessEqual[N[Tan[a], $MachinePrecision], 5e-69], N[(t$95$0 * N[(-1.0 / N[(N[Tan[y], $MachinePrecision] * N[Tan[z], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] + (-N[(a - x), $MachinePrecision])), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (tan.f64 a) < -4.00000000000000019e-4 or 5.00000000000000033e-69 < (tan.f64 a)

   1. Initial program 81.9%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   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. +-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. Taylor expanded in z around 0

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

   7. Applied rewrites 82.4%

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

   if -4.00000000000000019e-4 < (tan.f64 a) < 5.00000000000000033e-69

   1. Initial program 78.7%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   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 78.7

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

   4. Applied rewrites 78.7%

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

   5. Taylor expanded in a around 0

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

      1. lower--.f64 78.7

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

   7. Applied rewrites 78.7%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

   9. Applied rewrites 99.7%

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

4. Final simplification 89.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\tan a \leq -0.0004:\\ \;\;\;\;\left(\frac{\tan z + \tan y}{1} - \tan a\right) + x\\ \mathbf{elif}\;\tan a \leq 5 \cdot 10^{-69}:\\ \;\;\;\;\mathsf{fma}\left(\tan z + \tan y, \frac{-1}{\mathsf{fma}\left(\tan y, \tan z, -1\right)}, -\left(a - x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\tan z + \tan y}{1} - \tan a\right) + x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 88.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan z + \tan y\\ t_1 := \left(\frac{t\_0}{1} - \tan a\right) + x\\ \mathbf{if}\;\tan a \leq -2 \cdot 10^{-11}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\tan a \leq 5 \cdot 10^{-69}:\\ \;\;\;\;\frac{t\_0}{\mathsf{fma}\left(-\tan z, \tan y, 1\right)} - \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (let* ((t_0 (+ (tan z) (tan y))) (t_1 (+ (- (/ t_0 1.0) (tan a)) x)))
   (if (<= (tan a) -2e-11)
     t_1
     (if (<= (tan a) 5e-69)
       (- (/ t_0 (fma (- (tan z)) (tan y) 1.0)) (- x))
       t_1))))

C

double code(double x, double y, double z, double a) {
	double t_0 = tan(z) + tan(y);
	double t_1 = ((t_0 / 1.0) - tan(a)) + x;
	double tmp;
	if (tan(a) <= -2e-11) {
		tmp = t_1;
	} else if (tan(a) <= 5e-69) {
		tmp = (t_0 / fma(-tan(z), tan(y), 1.0)) - -x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, a)
	t_0 = Float64(tan(z) + tan(y))
	t_1 = Float64(Float64(Float64(t_0 / 1.0) - tan(a)) + x)
	tmp = 0.0
	if (tan(a) <= -2e-11)
		tmp = t_1;
	elseif (tan(a) <= 5e-69)
		tmp = Float64(Float64(t_0 / fma(Float64(-tan(z)), tan(y), 1.0)) - Float64(-x));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, a_] := Block[{t$95$0 = N[(N[Tan[z], $MachinePrecision] + N[Tan[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(t$95$0 / 1.0), $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[N[Tan[a], $MachinePrecision], -2e-11], t$95$1, If[LessEqual[N[Tan[a], $MachinePrecision], 5e-69], N[(N[(t$95$0 / N[((-N[Tan[z], $MachinePrecision]) * N[Tan[y], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] - (-x)), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (tan.f64 a) < -1.99999999999999988e-11 or 5.00000000000000033e-69 < (tan.f64 a)

   1. Initial program 82.3%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   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. Taylor expanded in z around 0

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

   7. Applied rewrites 82.8%

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

   if -1.99999999999999988e-11 < (tan.f64 a) < 5.00000000000000033e-69

   1. Initial program 78.1%

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

      1. lift-+.f64 N/A

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

      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 78.1

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

   4. Applied rewrites 78.1%

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

   5. Taylor expanded in a around 0

      \[\leadsto \tan \left(z + y\right) - \color{blue}{-1 \cdot x} \]
   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 78.1

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

   7. Applied rewrites 78.1%

      \[\leadsto \tan \left(z + y\right) - \color{blue}{\left(-x\right)} \]
   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. lift-+.f64 N/A

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

      7. lift-tan.f64 N/A

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

      8. lift-tan.f64 N/A

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

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

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

      10. lift-neg.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-fma.f64 N/A

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

      13. lift-/.f64 99.7

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 99.7

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

   9. Applied rewrites 99.7%

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

4. Final simplification 89.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\tan a \leq -2 \cdot 10^{-11}:\\ \;\;\;\;\left(\frac{\tan z + \tan y}{1} - \tan a\right) + x\\ \mathbf{elif}\;\tan a \leq 5 \cdot 10^{-69}:\\ \;\;\;\;\frac{\tan z + \tan y}{\mathsf{fma}\left(-\tan z, \tan y, 1\right)} - \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\tan z + \tan y}{1} - \tan a\right) + x\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.7% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 80.6%

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

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

Alternative 5: 79.7% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 80.6%

   \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
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. Taylor expanded in z around 0

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

7. Applied rewrites 80.9%

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

8. Final simplification 80.9%

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

Alternative 6: 79.2% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 80.6%

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

   1. lift-+.f64 N/A

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

   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. sub-neg N/A

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

   5. associate-/l* N/A

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

   6. lower-fma.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. lower--.f64 N/A

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

   9. lower-neg.f64 N/A

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

   10. associate-/l* N/A

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

   11. lower-*.f64 N/A

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

   12. lower-/.f64 N/A

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

   13. lower--.f64 80.6

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

4. Applied rewrites 80.6%

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

5. Final simplification 80.6%

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

Alternative 7: 79.3% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 80.6%

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

3. Taylor expanded in x around inf

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

   1. associate--l+ N/A

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

   2. +-commutative N/A

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

   3. associate-/l/ N/A

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

   4. associate-/l/ N/A

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

   5. div-sub N/A

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

   6. distribute-rgt-in N/A

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

   7. *-lft-identity N/A

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

   8. lower-fma.f64 N/A

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

5. Applied rewrites 80.5%

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

7. Applied rewrites 80.6%

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

Alternative 8: 64.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4.3 \cdot 10^{-9}:\\ \;\;\;\;\tan \left(z + y\right) - \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\tan z - \tan a\right) + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (if (<= y -4.3e-9) (- (tan (+ z y)) (- x)) (+ (- (tan z) (tan a)) x)))

C

double code(double x, double y, double z, double a) {
	double tmp;
	if (y <= -4.3e-9) {
		tmp = tan((z + y)) - -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 <= (-4.3d-9)) then
        tmp = tan((z + y)) - -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 <= -4.3e-9) {
		tmp = Math.tan((z + y)) - -x;
	} else {
		tmp = (Math.tan(z) - Math.tan(a)) + x;
	}
	return tmp;
}

Python

def code(x, y, z, a):
	tmp = 0
	if y <= -4.3e-9:
		tmp = math.tan((z + y)) - -x
	else:
		tmp = (math.tan(z) - math.tan(a)) + x
	return tmp

Julia

function code(x, y, z, a)
	tmp = 0.0
	if (y <= -4.3e-9)
		tmp = Float64(tan(Float64(z + y)) - 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 <= -4.3e-9)
		tmp = tan((z + y)) - -x;
	else
		tmp = (tan(z) - tan(a)) + x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, a_] := If[LessEqual[y, -4.3e-9], N[(N[Tan[N[(z + y), $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 y < -4.29999999999999963e-9

   1. Initial program 63.0%

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

      1. lift-+.f64 N/A

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

      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 63.0

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

   4. Applied rewrites 63.0%

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

   5. Taylor expanded in a around 0

      \[\leadsto \tan \left(z + y\right) - \color{blue}{-1 \cdot x} \]
   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 38.5

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

   7. Applied rewrites 38.5%

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

   if -4.29999999999999963e-9 < y

   1. Initial program 86.8%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   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 74.6

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

   5. Applied rewrites 74.6%

      \[\leadsto x + \left(\color{blue}{\frac{\sin z}{\cos z}} - \tan a\right) \]
   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 74.6

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

   7. Applied rewrites 74.7%

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

Alternative 9: 79.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 80.6%

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

3. Final simplification 80.6%

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

Alternative 10: 51.1% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 80.6%

   \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
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 80.5

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

4. Applied rewrites 80.5%

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

5. Taylor expanded in a around 0

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

Reproduce

herbie shell --seed 2024244 
(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))))