tan-example (used to crash)

Percentage Accurate: 79.8% → 99.7%

Time: 31.0s

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: 79.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 79.5%

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

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

   3. flip-+ N/A

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

   4. div-inv N/A

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

   5. lower-*.f64 N/A

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

   6. difference-of-squares N/A

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

   7. +-commutative N/A

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

   8. lift-+.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. lift-+.f64 N/A

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

   11. +-commutative N/A

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

   12. lower-+.f64 N/A

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

   13. lower--.f64 N/A

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

   14. lower-/.f64 N/A

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

   15. lower--.f64 43.4

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

4. Applied rewrites 43.4%

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

6. Applied rewrites 99.7%

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

7. Final simplification 99.7%

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

Alternative 2: 99.7% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 79.5%

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

   1. lift--.f64 N/A

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

   2. sub-neg N/A

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

   3. lift-tan.f64 N/A

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

   4. lift-+.f64 N/A

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

   5. tan-sum N/A

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

   6. frac-2neg N/A

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

   7. div-inv N/A

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

   8. lower-fma.f64 N/A

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

4. Applied rewrites 99.7%

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

   1. lift-fma.f64 N/A

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

   2. *-commutative N/A

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

6. Applied rewrites 99.7%

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

7. Final simplification 99.7%

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

Alternative 3: 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 79.5%

   \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
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 4: 90.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan z + \tan y\\ \mathbf{if}\;a \leq -0.0065:\\ \;\;\;\;\left(\tan \left(z + y\right) - \tan a\right) + x\\ \mathbf{elif}\;a \leq 0.0068:\\ \;\;\;\;\mathsf{fma}\left(t\_0, \frac{-1}{\mathsf{fma}\left(\tan y, \tan z, -1\right)}, \mathsf{fma}\left(\mathsf{fma}\left(a \cdot a, -0.3333333333333333, -1\right), a, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1, t\_0, -\tan a\right) + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (let* ((t_0 (+ (tan z) (tan y))))
   (if (<= a -0.0065)
     (+ (- (tan (+ z y)) (tan a)) x)
     (if (<= a 0.0068)
       (fma
        t_0
        (/ -1.0 (fma (tan y) (tan z) -1.0))
        (fma (fma (* a a) -0.3333333333333333 -1.0) a x))
       (+ (fma 1.0 t_0 (- (tan a))) x)))))

C

double code(double x, double y, double z, double a) {
	double t_0 = tan(z) + tan(y);
	double tmp;
	if (a <= -0.0065) {
		tmp = (tan((z + y)) - tan(a)) + x;
	} else if (a <= 0.0068) {
		tmp = fma(t_0, (-1.0 / fma(tan(y), tan(z), -1.0)), fma(fma((a * a), -0.3333333333333333, -1.0), a, x));
	} else {
		tmp = fma(1.0, t_0, -tan(a)) + x;
	}
	return tmp;
}

Julia

function code(x, y, z, a)
	t_0 = Float64(tan(z) + tan(y))
	tmp = 0.0
	if (a <= -0.0065)
		tmp = Float64(Float64(tan(Float64(z + y)) - tan(a)) + x);
	elseif (a <= 0.0068)
		tmp = fma(t_0, Float64(-1.0 / fma(tan(y), tan(z), -1.0)), fma(fma(Float64(a * a), -0.3333333333333333, -1.0), a, x));
	else
		tmp = Float64(fma(1.0, t_0, Float64(-tan(a))) + x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, a_] := Block[{t$95$0 = N[(N[Tan[z], $MachinePrecision] + N[Tan[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -0.0065], N[(N[(N[Tan[N[(z + y), $MachinePrecision]], $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[a, 0.0068], N[(t$95$0 * N[(-1.0 / N[(N[Tan[y], $MachinePrecision] * N[Tan[z], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(a * a), $MachinePrecision] * -0.3333333333333333 + -1.0), $MachinePrecision] * a + x), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 * t$95$0 + (-N[Tan[a], $MachinePrecision])), $MachinePrecision] + x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if a < -0.0064999999999999997

   1. Initial program 78.6%

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

   if -0.0064999999999999997 < a < 0.00679999999999999962

   1. Initial program 76.4%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. lift-tan.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. tan-sum N/A

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

      6. frac-2neg N/A

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

      7. div-inv N/A

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

      8. lower-fma.f64 N/A

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

   4. Applied rewrites 99.8%

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

   6. Applied rewrites 99.8%

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

   7. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sub-neg N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 99.8

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

   9. Applied rewrites 99.8%

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

   if 0.00679999999999999962 < a

   1. Initial program 86.3%

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

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

      3. flip-+ N/A

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

      4. div-inv N/A

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

      5. lower-*.f64 N/A

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

      6. difference-of-squares N/A

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

      7. +-commutative N/A

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

      8. lift-+.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. lower--.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower--.f64 40.5

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

   4. Applied rewrites 40.5%

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

   6. Applied rewrites 99.6%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-/r/ N/A

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

      6. metadata-eval N/A

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

      7. lift-neg.f64 N/A

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

      8. frac-2neg N/A

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

      9. lift-/.f64 N/A

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

      10. lower-fma.f64 N/A

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

   8. Applied rewrites 99.6%

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

   9. Taylor expanded in z around 0

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

   11. Applied rewrites 87.1%

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

4. Final simplification 90.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -0.0065:\\ \;\;\;\;\left(\tan \left(z + y\right) - \tan a\right) + x\\ \mathbf{elif}\;a \leq 0.0068:\\ \;\;\;\;\mathsf{fma}\left(\tan z + \tan y, \frac{-1}{\mathsf{fma}\left(\tan y, \tan z, -1\right)}, \mathsf{fma}\left(\mathsf{fma}\left(a \cdot a, -0.3333333333333333, -1\right), a, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1, \tan z + \tan y, -\tan a\right) + x\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 89.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan z + \tan y\\ \mathbf{if}\;a \leq -3 \cdot 10^{-5}:\\ \;\;\;\;\left(\tan \left(z + y\right) - \tan a\right) + x\\ \mathbf{elif}\;a \leq 0.00023:\\ \;\;\;\;\frac{t\_0}{-\mathsf{fma}\left(\tan y, \tan z, -1\right)} - \left(a - x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1, t\_0, -\tan a\right) + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (let* ((t_0 (+ (tan z) (tan y))))
   (if (<= a -3e-5)
     (+ (- (tan (+ z y)) (tan a)) x)
     (if (<= a 0.00023)
       (- (/ t_0 (- (fma (tan y) (tan z) -1.0))) (- a x))
       (+ (fma 1.0 t_0 (- (tan a))) x)))))

C

double code(double x, double y, double z, double a) {
	double t_0 = tan(z) + tan(y);
	double tmp;
	if (a <= -3e-5) {
		tmp = (tan((z + y)) - tan(a)) + x;
	} else if (a <= 0.00023) {
		tmp = (t_0 / -fma(tan(y), tan(z), -1.0)) - (a - x);
	} else {
		tmp = fma(1.0, t_0, -tan(a)) + x;
	}
	return tmp;
}

Julia

function code(x, y, z, a)
	t_0 = Float64(tan(z) + tan(y))
	tmp = 0.0
	if (a <= -3e-5)
		tmp = Float64(Float64(tan(Float64(z + y)) - tan(a)) + x);
	elseif (a <= 0.00023)
		tmp = Float64(Float64(t_0 / Float64(-fma(tan(y), tan(z), -1.0))) - Float64(a - x));
	else
		tmp = Float64(fma(1.0, t_0, Float64(-tan(a))) + x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, a_] := Block[{t$95$0 = N[(N[Tan[z], $MachinePrecision] + N[Tan[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -3e-5], N[(N[(N[Tan[N[(z + y), $MachinePrecision]], $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[a, 0.00023], N[(N[(t$95$0 / (-N[(N[Tan[y], $MachinePrecision] * N[Tan[z], $MachinePrecision] + -1.0), $MachinePrecision])), $MachinePrecision] - N[(a - x), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 * t$95$0 + (-N[Tan[a], $MachinePrecision])), $MachinePrecision] + x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if a < -3.00000000000000008e-5

   1. Initial program 78.6%

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

   if -3.00000000000000008e-5 < a < 2.3000000000000001e-4

   1. Initial program 76.4%

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

      1. lift-+.f64 N/A

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

      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 76.4

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

   4. Applied rewrites 76.4%

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

   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 76.4

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

   7. Applied rewrites 76.4%

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

      1. lift-tan.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. tan-sum N/A

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

      4. lift-tan.f64 N/A

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

      5. lift-tan.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-tan.f64 N/A

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

      8. lift-tan.f64 N/A

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

      9. sub-neg N/A

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

      10. metadata-eval N/A

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

      11. distribute-neg-in N/A

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

      12. +-commutative N/A

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

      13. lift-fma.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lift-+.f64 N/A

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

      16. +-commutative N/A

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

      17. lift-+.f64 N/A

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

      18. lower-neg.f64 99.8

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

      19. lift-fma.f64 N/A

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

      20. *-commutative N/A

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

      21. lower-fma.f64 99.8

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

   9. Applied rewrites 99.8%

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

   if 2.3000000000000001e-4 < a

   1. Initial program 86.3%

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

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

      3. flip-+ N/A

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

      4. div-inv N/A

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

      5. lower-*.f64 N/A

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

      6. difference-of-squares N/A

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

      7. +-commutative N/A

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

      8. lift-+.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. lower--.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower--.f64 40.5

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

   4. Applied rewrites 40.5%

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

   6. Applied rewrites 99.6%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-/r/ N/A

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

      6. metadata-eval N/A

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

      7. lift-neg.f64 N/A

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

      8. frac-2neg N/A

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

      9. lift-/.f64 N/A

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

      10. lower-fma.f64 N/A

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

   8. Applied rewrites 99.6%

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

   9. Taylor expanded in z around 0

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

   11. Applied rewrites 87.1%

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

4. Final simplification 90.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -3 \cdot 10^{-5}:\\ \;\;\;\;\left(\tan \left(z + y\right) - \tan a\right) + x\\ \mathbf{elif}\;a \leq 0.00023:\\ \;\;\;\;\frac{\tan z + \tan y}{-\mathsf{fma}\left(\tan y, \tan z, -1\right)} - \left(a - x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1, \tan z + \tan y, -\tan a\right) + x\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 89.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan z + \tan y\\ \mathbf{if}\;a \leq -6.6 \cdot 10^{-10}:\\ \;\;\;\;\left(\tan \left(z + y\right) - \tan a\right) + x\\ \mathbf{elif}\;a \leq 5.5 \cdot 10^{-5}:\\ \;\;\;\;\frac{t\_0}{-\mathsf{fma}\left(\tan z, \tan y, -1\right)} - \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1, t\_0, -\tan a\right) + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (let* ((t_0 (+ (tan z) (tan y))))
   (if (<= a -6.6e-10)
     (+ (- (tan (+ z y)) (tan a)) x)
     (if (<= a 5.5e-5)
       (- (/ t_0 (- (fma (tan z) (tan y) -1.0))) (- x))
       (+ (fma 1.0 t_0 (- (tan a))) x)))))

C

double code(double x, double y, double z, double a) {
	double t_0 = tan(z) + tan(y);
	double tmp;
	if (a <= -6.6e-10) {
		tmp = (tan((z + y)) - tan(a)) + x;
	} else if (a <= 5.5e-5) {
		tmp = (t_0 / -fma(tan(z), tan(y), -1.0)) - -x;
	} else {
		tmp = fma(1.0, t_0, -tan(a)) + x;
	}
	return tmp;
}

Julia

function code(x, y, z, a)
	t_0 = Float64(tan(z) + tan(y))
	tmp = 0.0
	if (a <= -6.6e-10)
		tmp = Float64(Float64(tan(Float64(z + y)) - tan(a)) + x);
	elseif (a <= 5.5e-5)
		tmp = Float64(Float64(t_0 / Float64(-fma(tan(z), tan(y), -1.0))) - Float64(-x));
	else
		tmp = Float64(fma(1.0, t_0, Float64(-tan(a))) + x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, a_] := Block[{t$95$0 = N[(N[Tan[z], $MachinePrecision] + N[Tan[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -6.6e-10], N[(N[(N[Tan[N[(z + y), $MachinePrecision]], $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[a, 5.5e-5], N[(N[(t$95$0 / (-N[(N[Tan[z], $MachinePrecision] * N[Tan[y], $MachinePrecision] + -1.0), $MachinePrecision])), $MachinePrecision] - (-x)), $MachinePrecision], N[(N[(1.0 * t$95$0 + (-N[Tan[a], $MachinePrecision])), $MachinePrecision] + x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if a < -6.6e-10

   1. Initial program 78.6%

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

   if -6.6e-10 < a < 5.5000000000000002e-5

   1. Initial program 76.4%

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

      1. lift-+.f64 N/A

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

      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 76.4

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

   4. Applied rewrites 76.4%

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

   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 76.3

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

   7. Applied rewrites 76.3%

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

      1. lift-tan.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. tan-sum N/A

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

      4. lift-tan.f64 N/A

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

      5. lift-tan.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. lift-tan.f64 N/A

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

      9. lift-tan.f64 N/A

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

      10. *-commutative N/A

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

      11. sub-neg N/A

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

      12. metadata-eval N/A

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

      13. distribute-neg-in N/A

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

      14. +-commutative N/A

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

      15. lift-fma.f64 N/A

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

      16. lift-neg.f64 N/A

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

      17. lower-/.f64 99.3

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 99.3

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

      21. lift-fma.f64 N/A

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

      22. *-commutative N/A

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

   9. Applied rewrites 99.3%

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

   if 5.5000000000000002e-5 < a

   1. Initial program 86.3%

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

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

      3. flip-+ N/A

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

      4. div-inv N/A

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

      5. lower-*.f64 N/A

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

      6. difference-of-squares N/A

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

      7. +-commutative N/A

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

      8. lift-+.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. lower--.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower--.f64 40.5

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

   4. Applied rewrites 40.5%

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

   6. Applied rewrites 99.6%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-/r/ N/A

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

      6. metadata-eval N/A

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

      7. lift-neg.f64 N/A

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

      8. frac-2neg N/A

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

      9. lift-/.f64 N/A

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

      10. lower-fma.f64 N/A

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

   8. Applied rewrites 99.6%

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

   9. Taylor expanded in z around 0

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

   11. Applied rewrites 87.1%

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

4. Final simplification 90.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -6.6 \cdot 10^{-10}:\\ \;\;\;\;\left(\tan \left(z + y\right) - \tan a\right) + x\\ \mathbf{elif}\;a \leq 5.5 \cdot 10^{-5}:\\ \;\;\;\;\frac{\tan z + \tan y}{-\mathsf{fma}\left(\tan z, \tan y, -1\right)} - \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1, \tan z + \tan y, -\tan a\right) + x\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 80.2% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 79.5%

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

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

   3. flip-+ N/A

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

   4. div-inv N/A

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

   5. lower-*.f64 N/A

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

   6. difference-of-squares N/A

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

   7. +-commutative N/A

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

   8. lift-+.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. lift-+.f64 N/A

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

   11. +-commutative N/A

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

   12. lower-+.f64 N/A

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

   13. lower--.f64 N/A

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

   14. lower-/.f64 N/A

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

   15. lower--.f64 43.4

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

4. Applied rewrites 43.4%

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

6. Applied rewrites 99.7%

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

   1. lift--.f64 N/A

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

   2. sub-neg N/A

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

   3. lift-/.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. associate-/r/ N/A

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

   6. metadata-eval N/A

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

   7. lift-neg.f64 N/A

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

   8. frac-2neg N/A

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

   9. lift-/.f64 N/A

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

   10. lower-fma.f64 N/A

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

8. Applied rewrites 99.7%

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

9. Taylor expanded in z around 0

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

11. Applied rewrites 79.8%

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

12. Final simplification 79.8%

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

Alternative 8: 79.8% 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 79.5%

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

3. Final simplification 79.5%

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

Alternative 9: 51.0% 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 79.5%

   \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
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 79.4

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

4. Applied rewrites 79.4%

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

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 48.0

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

7. Applied rewrites 48.0%

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

Reproduce

herbie shell --seed 2024268 
(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))))