tan-example (used to crash)

Percentage Accurate: 79.1% → 99.6%

Time: 28.2s

Alternatives: 9

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 79.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-\tan z, \tan y, 1\right)\\ t_1 := \frac{1}{\tan a}\\ \frac{t\_1 \cdot \left(\tan y + \tan z\right) - t\_0}{t\_0 \cdot t\_1} + x \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (let* ((t_0 (fma (- (tan z)) (tan y) 1.0)) (t_1 (/ 1.0 (tan a))))
   (+ (/ (- (* t_1 (+ (tan y) (tan z))) t_0) (* t_0 t_1)) x)))

C

double code(double x, double y, double z, double a) {
	double t_0 = fma(-tan(z), tan(y), 1.0);
	double t_1 = 1.0 / tan(a);
	return (((t_1 * (tan(y) + tan(z))) - t_0) / (t_0 * t_1)) + x;
}

Julia

function code(x, y, z, a)
	t_0 = fma(Float64(-tan(z)), tan(y), 1.0)
	t_1 = Float64(1.0 / tan(a))
	return Float64(Float64(Float64(Float64(t_1 * Float64(tan(y) + tan(z))) - t_0) / Float64(t_0 * t_1)) + x)
end

Wolfram

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

Derivation

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--.f64 N/A

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

   2. lift-tan.f64 N/A

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

   3. lift-+.f64 N/A

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

   4. tan-sum N/A

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

   5. lift-tan.f64 N/A

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

   6. tan-quot N/A

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

   7. clear-num N/A

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

   8. frac-sub N/A

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

   9. lower-/.f64 N/A

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

4. Applied rewrites 99.8%

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

5. Final simplification 99.8%

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

Alternative 2: 99.7% accurate, 0.4× speedup

Math

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

FPCore

(FPCore (x y z a)
 :precision binary64
 (+ (- (/ (+ (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 (((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) + 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[Tan[y], $MachinePrecision] + N[Tan[z], $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 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. Final simplification 99.7%

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

Alternative 3: 87.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -0.011:\\ \;\;\;\;\left(\tan \left(\frac{1}{\frac{1}{y + z}}\right) - \tan a\right) + x\\ \mathbf{elif}\;a \leq 1.02 \cdot 10^{-7}:\\ \;\;\;\;\left(\frac{\tan y + \tan z}{\mathsf{fma}\left(-\tan z, \tan y, 1\right)} - \mathsf{fma}\left(a \cdot a, 0.3333333333333333, 1\right) \cdot a\right) + x\\ \mathbf{else}:\\ \;\;\;\;\left(\tan \left(y + z\right) - \tan a\right) + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (if (<= a -0.011)
   (+ (- (tan (/ 1.0 (/ 1.0 (+ y z)))) (tan a)) x)
   (if (<= a 1.02e-7)
     (+
      (-
       (/ (+ (tan y) (tan z)) (fma (- (tan z)) (tan y) 1.0))
       (* (fma (* a a) 0.3333333333333333 1.0) a))
      x)
     (+ (- (tan (+ y z)) (tan a)) x))))

C

double code(double x, double y, double z, double a) {
	double tmp;
	if (a <= -0.011) {
		tmp = (tan((1.0 / (1.0 / (y + z)))) - tan(a)) + x;
	} else if (a <= 1.02e-7) {
		tmp = (((tan(y) + tan(z)) / fma(-tan(z), tan(y), 1.0)) - (fma((a * a), 0.3333333333333333, 1.0) * a)) + x;
	} else {
		tmp = (tan((y + z)) - tan(a)) + x;
	}
	return tmp;
}

Julia

function code(x, y, z, a)
	tmp = 0.0
	if (a <= -0.011)
		tmp = Float64(Float64(tan(Float64(1.0 / Float64(1.0 / Float64(y + z)))) - tan(a)) + x);
	elseif (a <= 1.02e-7)
		tmp = Float64(Float64(Float64(Float64(tan(y) + tan(z)) / fma(Float64(-tan(z)), tan(y), 1.0)) - Float64(fma(Float64(a * a), 0.3333333333333333, 1.0) * a)) + x);
	else
		tmp = Float64(Float64(tan(Float64(y + z)) - tan(a)) + x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, a_] := If[LessEqual[a, -0.011], N[(N[(N[Tan[N[(1.0 / N[(1.0 / N[(y + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[a, 1.02e-7], N[(N[(N[(N[(N[Tan[y], $MachinePrecision] + N[Tan[z], $MachinePrecision]), $MachinePrecision] / N[((-N[Tan[z], $MachinePrecision]) * N[Tan[y], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(a * a), $MachinePrecision] * 0.3333333333333333 + 1.0), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], N[(N[(N[Tan[N[(y + z), $MachinePrecision]], $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -0.010999999999999999

   1. Initial program 87.5%

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

      1. lift-+.f64 N/A

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

      2. flip-+ N/A

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

      3. clear-num N/A

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

      4. lower-/.f64 N/A

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

      5. clear-num N/A

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

      6. flip-+ N/A

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

      7. lift-+.f64 N/A

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

      8. lower-/.f64 87.7

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 87.7

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

   4. Applied rewrites 87.7%

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

   if -0.010999999999999999 < a < 1.02e-7

   1. Initial program 84.3%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   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 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in a around 0

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

      1. *-commutative N/A

         \[\leadsto x + \left(\frac{\tan z + \tan y}{\mathsf{fma}\left(-\tan z, \tan y, 1\right)} - \color{blue}{\left(1 + \frac{1}{3} \cdot {a}^{2}\right) \cdot a}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto x + \left(\frac{\tan z + \tan y}{\mathsf{fma}\left(-\tan z, \tan y, 1\right)} - \color{blue}{\left(1 + \frac{1}{3} \cdot {a}^{2}\right) \cdot a}\right) \]

      3. +-commutative N/A

         \[\leadsto x + \left(\frac{\tan z + \tan y}{\mathsf{fma}\left(-\tan z, \tan y, 1\right)} - \color{blue}{\left(\frac{1}{3} \cdot {a}^{2} + 1\right)} \cdot a\right) \]

      4. *-commutative N/A

         \[\leadsto x + \left(\frac{\tan z + \tan y}{\mathsf{fma}\left(-\tan z, \tan y, 1\right)} - \left(\color{blue}{{a}^{2} \cdot \frac{1}{3}} + 1\right) \cdot a\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto x + \left(\frac{\tan z + \tan y}{\mathsf{fma}\left(-\tan z, \tan y, 1\right)} - \color{blue}{\mathsf{fma}\left({a}^{2}, \frac{1}{3}, 1\right)} \cdot a\right) \]

      6. unpow2 N/A

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

      7. lower-*.f64 99.8

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

   7. Applied rewrites 99.8%

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

   if 1.02e-7 < a

   1. Initial program 74.7%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   2. Add Preprocessing
3. Recombined 3 regimes into one program.

4. Final simplification 89.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -0.011:\\ \;\;\;\;\left(\tan \left(\frac{1}{\frac{1}{y + z}}\right) - \tan a\right) + x\\ \mathbf{elif}\;a \leq 1.02 \cdot 10^{-7}:\\ \;\;\;\;\left(\frac{\tan y + \tan z}{\mathsf{fma}\left(-\tan z, \tan y, 1\right)} - \mathsf{fma}\left(a \cdot a, 0.3333333333333333, 1\right) \cdot a\right) + x\\ \mathbf{else}:\\ \;\;\;\;\left(\tan \left(y + z\right) - \tan a\right) + x\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 87.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.6 \cdot 10^{-10}:\\ \;\;\;\;\left(\tan \left(\frac{1}{\frac{1}{y + z}}\right) - \tan a\right) + x\\ \mathbf{elif}\;a \leq 1.02 \cdot 10^{-7}:\\ \;\;\;\;\frac{\tan y + \tan z}{\mathsf{fma}\left(-\tan z, \tan y, 1\right)} - \left(a - x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\tan \left(y + z\right) - \tan a\right) + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (if (<= a -1.6e-10)
   (+ (- (tan (/ 1.0 (/ 1.0 (+ y z)))) (tan a)) x)
   (if (<= a 1.02e-7)
     (- (/ (+ (tan y) (tan z)) (fma (- (tan z)) (tan y) 1.0)) (- a x))
     (+ (- (tan (+ y z)) (tan a)) x))))

C

double code(double x, double y, double z, double a) {
	double tmp;
	if (a <= -1.6e-10) {
		tmp = (tan((1.0 / (1.0 / (y + z)))) - tan(a)) + x;
	} else if (a <= 1.02e-7) {
		tmp = ((tan(y) + tan(z)) / fma(-tan(z), tan(y), 1.0)) - (a - x);
	} else {
		tmp = (tan((y + z)) - tan(a)) + x;
	}
	return tmp;
}

Julia

function code(x, y, z, a)
	tmp = 0.0
	if (a <= -1.6e-10)
		tmp = Float64(Float64(tan(Float64(1.0 / Float64(1.0 / Float64(y + z)))) - tan(a)) + x);
	elseif (a <= 1.02e-7)
		tmp = Float64(Float64(Float64(tan(y) + tan(z)) / fma(Float64(-tan(z)), tan(y), 1.0)) - Float64(a - x));
	else
		tmp = Float64(Float64(tan(Float64(y + z)) - tan(a)) + x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, a_] := If[LessEqual[a, -1.6e-10], N[(N[(N[Tan[N[(1.0 / N[(1.0 / N[(y + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[a, 1.02e-7], N[(N[(N[(N[Tan[y], $MachinePrecision] + N[Tan[z], $MachinePrecision]), $MachinePrecision] / N[((-N[Tan[z], $MachinePrecision]) * N[Tan[y], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] - N[(a - x), $MachinePrecision]), $MachinePrecision], N[(N[(N[Tan[N[(y + z), $MachinePrecision]], $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -1.5999999999999999e-10

   1. Initial program 88.1%

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

      1. lift-+.f64 N/A

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

      2. flip-+ N/A

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

      3. clear-num N/A

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

      4. lower-/.f64 N/A

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

      5. clear-num N/A

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

      6. flip-+ N/A

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

      7. lift-+.f64 N/A

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

      8. lower-/.f64 88.2

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 88.2

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

   4. Applied rewrites 88.2%

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

   if -1.5999999999999999e-10 < a < 1.02e-7

   1. Initial program 83.8%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   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 83.8

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

   4. Applied rewrites 83.8%

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

      1. lift-tan.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. tan-sum N/A

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

      4. lift-tan.f64 N/A

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

      5. lift-tan.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-tan.f64 N/A

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

      8. lift-tan.f64 N/A

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

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

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

      10. lift-neg.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-fma.f64 N/A

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

      13. lift-/.f64 99.8

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 99.8

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

   6. Applied rewrites 99.8%

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

   7. Taylor expanded in a around 0

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

      1. lower--.f64 99.8

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

   9. Applied rewrites 99.8%

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

   if 1.02e-7 < a

   1. Initial program 74.7%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   2. Add Preprocessing
3. Recombined 3 regimes into one program.

4. Final simplification 89.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.6 \cdot 10^{-10}:\\ \;\;\;\;\left(\tan \left(\frac{1}{\frac{1}{y + z}}\right) - \tan a\right) + x\\ \mathbf{elif}\;a \leq 1.02 \cdot 10^{-7}:\\ \;\;\;\;\frac{\tan y + \tan z}{\mathsf{fma}\left(-\tan z, \tan y, 1\right)} - \left(a - x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\tan \left(y + z\right) - \tan a\right) + x\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 87.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.22 \cdot 10^{-11}:\\ \;\;\;\;\left(\tan \left(\frac{1}{\frac{1}{y + z}}\right) - \tan a\right) + x\\ \mathbf{elif}\;a \leq 5.2 \cdot 10^{-10}:\\ \;\;\;\;\frac{\tan y + \tan z}{\mathsf{fma}\left(-\tan z, \tan y, 1\right)} - \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\tan \left(y + z\right) - \tan a\right) + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (if (<= a -1.22e-11)
   (+ (- (tan (/ 1.0 (/ 1.0 (+ y z)))) (tan a)) x)
   (if (<= a 5.2e-10)
     (- (/ (+ (tan y) (tan z)) (fma (- (tan z)) (tan y) 1.0)) (- x))
     (+ (- (tan (+ y z)) (tan a)) x))))

C

double code(double x, double y, double z, double a) {
	double tmp;
	if (a <= -1.22e-11) {
		tmp = (tan((1.0 / (1.0 / (y + z)))) - tan(a)) + x;
	} else if (a <= 5.2e-10) {
		tmp = ((tan(y) + tan(z)) / fma(-tan(z), tan(y), 1.0)) - -x;
	} else {
		tmp = (tan((y + z)) - tan(a)) + x;
	}
	return tmp;
}

Julia

function code(x, y, z, a)
	tmp = 0.0
	if (a <= -1.22e-11)
		tmp = Float64(Float64(tan(Float64(1.0 / Float64(1.0 / Float64(y + z)))) - tan(a)) + x);
	elseif (a <= 5.2e-10)
		tmp = Float64(Float64(Float64(tan(y) + tan(z)) / fma(Float64(-tan(z)), tan(y), 1.0)) - Float64(-x));
	else
		tmp = Float64(Float64(tan(Float64(y + z)) - tan(a)) + x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, a_] := If[LessEqual[a, -1.22e-11], N[(N[(N[Tan[N[(1.0 / N[(1.0 / N[(y + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[a, 5.2e-10], N[(N[(N[(N[Tan[y], $MachinePrecision] + N[Tan[z], $MachinePrecision]), $MachinePrecision] / N[((-N[Tan[z], $MachinePrecision]) * N[Tan[y], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] - (-x)), $MachinePrecision], N[(N[(N[Tan[N[(y + z), $MachinePrecision]], $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -1.2200000000000001e-11

   1. Initial program 88.4%

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

      1. lift-+.f64 N/A

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

      2. flip-+ N/A

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

      3. clear-num N/A

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

      4. lower-/.f64 N/A

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

      5. clear-num N/A

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

      6. flip-+ N/A

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

      7. lift-+.f64 N/A

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

      8. lower-/.f64 88.5

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 88.5

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

   4. Applied rewrites 88.5%

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

   if -1.2200000000000001e-11 < a < 5.19999999999999962e-10

   1. Initial program 83.5%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   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 83.5

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

   4. Applied rewrites 83.5%

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

      1. lift-tan.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. tan-sum N/A

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

      4. lift-tan.f64 N/A

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

      5. lift-tan.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-tan.f64 N/A

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

      8. lift-tan.f64 N/A

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

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

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

      10. lift-neg.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-fma.f64 N/A

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

      13. lift-/.f64 99.8

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 99.8

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

   6. Applied rewrites 99.8%

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

   7. Taylor expanded in a around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 99.5

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

   9. Applied rewrites 99.5%

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

   if 5.19999999999999962e-10 < a

   1. Initial program 74.7%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   2. Add Preprocessing
3. Recombined 3 regimes into one program.

4. Final simplification 89.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.22 \cdot 10^{-11}:\\ \;\;\;\;\left(\tan \left(\frac{1}{\frac{1}{y + z}}\right) - \tan a\right) + x\\ \mathbf{elif}\;a \leq 5.2 \cdot 10^{-10}:\\ \;\;\;\;\frac{\tan y + \tan z}{\mathsf{fma}\left(-\tan z, \tan y, 1\right)} - \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\tan \left(y + z\right) - \tan a\right) + x\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 79.0% 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 82.3%

   \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
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 82.3

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

4. Applied rewrites 82.3%

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

5. Final simplification 82.3%

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

Alternative 7: 79.1% 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 82.3%

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

3. Final simplification 82.3%

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

Alternative 8: 50.6% 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 82.3%

   \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
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 82.2

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

4. Applied rewrites 82.2%

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

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 50.6

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

7. Applied rewrites 50.6%

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

8. Final simplification 50.6%

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

Alternative 9: 32.3% 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 82.3%

   \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
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 82.1

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

4. Applied rewrites 82.0%

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

5. Taylor expanded in x around inf

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

   1. lower-/.f64 30.3

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

7. Applied rewrites 30.3%

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

Reproduce

herbie shell --seed 2024250 
(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))))