tan-example (used to crash)

Percentage Accurate: 79.0% → 99.7%

Time: 34.4s

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: 79.0% 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

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.4%

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

   1. lift-fma.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-neg.f64 N/A

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

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

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

   5. lift-tan.f64 N/A

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

   6. lift-tan.f64 N/A

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

   7. lower--.f64 N/A

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

   8. lift-tan.f64 N/A

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

   9. lift-tan.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 99.8

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

6. Applied rewrites 99.8%

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

7. Final simplification 99.8%

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

Alternative 2: 89.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan y + \tan z\\ t_1 := \left(\frac{t\_0}{1} - \tan a\right) + x\\ \mathbf{if}\;a \leq -0.00136:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 0.00062:\\ \;\;\;\;\frac{t\_0}{-\mathsf{fma}\left(\tan y, \tan z, -1\right)} - \left(a - x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (let* ((t_0 (+ (tan y) (tan z))) (t_1 (+ (- (/ t_0 1.0) (tan a)) x)))
   (if (<= a -0.00136)
     t_1
     (if (<= a 0.00062)
       (- (/ t_0 (- (fma (tan y) (tan z) -1.0))) (- a x))
       t_1))))

C

double code(double x, double y, double z, double a) {
	double t_0 = tan(y) + tan(z);
	double t_1 = ((t_0 / 1.0) - tan(a)) + x;
	double tmp;
	if (a <= -0.00136) {
		tmp = t_1;
	} else if (a <= 0.00062) {
		tmp = (t_0 / -fma(tan(y), tan(z), -1.0)) - (a - x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, a)
	t_0 = Float64(tan(y) + tan(z))
	t_1 = Float64(Float64(Float64(t_0 / 1.0) - tan(a)) + x)
	tmp = 0.0
	if (a <= -0.00136)
		tmp = t_1;
	elseif (a <= 0.00062)
		tmp = Float64(Float64(t_0 / Float64(-fma(tan(y), tan(z), -1.0))) - Float64(a - x));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, a_] := Block[{t$95$0 = N[(N[Tan[y], $MachinePrecision] + N[Tan[z], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(t$95$0 / 1.0), $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[a, -0.00136], t$95$1, If[LessEqual[a, 0.00062], N[(N[(t$95$0 / (-N[(N[Tan[y], $MachinePrecision] * N[Tan[z], $MachinePrecision] + -1.0), $MachinePrecision])), $MachinePrecision] - N[(a - x), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if a < -0.00136 or 6.2e-4 < a

   1. Initial program 76.9%

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

      1. lift-tan.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. tan-sum N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 N/A

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

      7. lower-tan.f64 N/A

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

      8. lower-tan.f64 N/A

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

      9. sub-neg N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

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

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

      13. lower-fma.f64 N/A

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

      14. lower-neg.f64 N/A

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

      15. lower-tan.f64 N/A

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

      16. lower-tan.f64 99.7

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in z around 0

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

   7. Applied rewrites 77.3%

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

   if -0.00136 < a < 6.2e-4

   1. Initial program 79.8%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   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 79.8

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

   4. Applied rewrites 79.8%

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

   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 79.8

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

   7. Applied rewrites 79.8%

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

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

      7. lift-+.f64 N/A

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

      8. lift-tan.f64 N/A

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

      9. lift-tan.f64 N/A

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

      10. *-commutative N/A

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

      11. sub-neg N/A

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

      12. metadata-eval N/A

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

      13. distribute-neg-in N/A

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

      14. +-commutative N/A

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

      15. lift-fma.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lower-neg.f64 99.9

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

   9. Applied rewrites 99.9%

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

4. Final simplification 89.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -0.00136:\\ \;\;\;\;\left(\frac{\tan y + \tan z}{1} - \tan a\right) + x\\ \mathbf{elif}\;a \leq 0.00062:\\ \;\;\;\;\frac{\tan y + \tan z}{-\mathsf{fma}\left(\tan y, \tan z, -1\right)} - \left(a - x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\tan y + \tan z}{1} - \tan a\right) + x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 88.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan y + \tan z\\ t_1 := \left(\frac{t\_0}{1} - \tan a\right) + x\\ \mathbf{if}\;a \leq -1.9 \cdot 10^{-12}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 8.6 \cdot 10^{-9}:\\ \;\;\;\;\frac{t\_0}{-\mathsf{fma}\left(\tan y, \tan z, -1\right)} - \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (let* ((t_0 (+ (tan y) (tan z))) (t_1 (+ (- (/ t_0 1.0) (tan a)) x)))
   (if (<= a -1.9e-12)
     t_1
     (if (<= a 8.6e-9) (- (/ t_0 (- (fma (tan y) (tan z) -1.0))) (- x)) t_1))))

C

double code(double x, double y, double z, double a) {
	double t_0 = tan(y) + tan(z);
	double t_1 = ((t_0 / 1.0) - tan(a)) + x;
	double tmp;
	if (a <= -1.9e-12) {
		tmp = t_1;
	} else if (a <= 8.6e-9) {
		tmp = (t_0 / -fma(tan(y), tan(z), -1.0)) - -x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, a)
	t_0 = Float64(tan(y) + tan(z))
	t_1 = Float64(Float64(Float64(t_0 / 1.0) - tan(a)) + x)
	tmp = 0.0
	if (a <= -1.9e-12)
		tmp = t_1;
	elseif (a <= 8.6e-9)
		tmp = Float64(Float64(t_0 / Float64(-fma(tan(y), tan(z), -1.0))) - Float64(-x));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, a_] := Block[{t$95$0 = N[(N[Tan[y], $MachinePrecision] + N[Tan[z], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(t$95$0 / 1.0), $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[a, -1.9e-12], t$95$1, If[LessEqual[a, 8.6e-9], N[(N[(t$95$0 / (-N[(N[Tan[y], $MachinePrecision] * N[Tan[z], $MachinePrecision] + -1.0), $MachinePrecision])), $MachinePrecision] - (-x)), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if a < -1.89999999999999998e-12 or 8.59999999999999925e-9 < a

   1. Initial program 77.2%

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

      1. lift-tan.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. tan-sum N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 N/A

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

      7. lower-tan.f64 N/A

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

      8. lower-tan.f64 N/A

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

      9. sub-neg N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

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

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

      13. lower-fma.f64 N/A

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

      14. lower-neg.f64 N/A

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

      15. lower-tan.f64 N/A

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

      16. lower-tan.f64 99.7

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in z around 0

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

   7. Applied rewrites 77.7%

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

   if -1.89999999999999998e-12 < a < 8.59999999999999925e-9

   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 79.5

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

   4. Applied rewrites 79.5%

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

   5. Taylor expanded in a around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 79.5

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

   7. Applied rewrites 79.5%

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

      1. lift-tan.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. tan-sum N/A

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

      4. lift-tan.f64 N/A

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

      5. lift-tan.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-tan.f64 N/A

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

      8. lift-tan.f64 N/A

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

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

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

      10. lift-neg.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-fma.f64 N/A

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

      13. lower-/.f64 99.7

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 99.7

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

      17. remove-double-neg N/A

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

      18. lift-neg.f64 N/A

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

      19. lower-neg.f64 99.7

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

      20. lift-neg.f64 N/A

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

   9. Applied rewrites 99.7%

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

4. Final simplification 89.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.9 \cdot 10^{-12}:\\ \;\;\;\;\left(\frac{\tan y + \tan z}{1} - \tan a\right) + x\\ \mathbf{elif}\;a \leq 8.6 \cdot 10^{-9}:\\ \;\;\;\;\frac{\tan y + \tan z}{-\mathsf{fma}\left(\tan y, \tan z, -1\right)} - \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\tan y + \tan z}{1} - \tan a\right) + x\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 79.4% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Derivation

1. Initial program 78.4%

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

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

7. Applied rewrites 78.6%

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

8. Final simplification 78.6%

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

Alternative 5: 79.3% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 78.4%

   \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
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. lift--.f64 N/A

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

   3. associate-+r- N/A

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

   4. +-commutative N/A

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

   5. associate--l+ N/A

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

   6. lift-tan.f64 N/A

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

   7. lift-+.f64 N/A

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

   8. tan-sum N/A

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

   9. frac-2neg N/A

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

   10. div-inv N/A

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

   11. lower-fma.f64 N/A

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

4. Applied rewrites 99.7%

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

5. Taylor expanded in z around 0

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

7. Applied rewrites 78.6%

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

8. Final simplification 78.6%

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

Alternative 6: 59.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y + z \leq 5 \cdot 10^{-7}:\\ \;\;\;\;\left(\tan y - \tan a\right) + x\\ \mathbf{else}:\\ \;\;\;\;\tan \left(y + z\right) - \left(-x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (if (<= (+ y z) 5e-7) (+ (- (tan y) (tan a)) x) (- (tan (+ y z)) (- x))))

C

double code(double x, double y, double z, double a) {
	double tmp;
	if ((y + z) <= 5e-7) {
		tmp = (tan(y) - tan(a)) + x;
	} else {
		tmp = tan((y + z)) - -x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: a
    real(8) :: tmp
    if ((y + z) <= 5d-7) then
        tmp = (tan(y) - tan(a)) + x
    else
        tmp = tan((y + z)) - -x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double a) {
	double tmp;
	if ((y + z) <= 5e-7) {
		tmp = (Math.tan(y) - Math.tan(a)) + x;
	} else {
		tmp = Math.tan((y + z)) - -x;
	}
	return tmp;
}

Python

def code(x, y, z, a):
	tmp = 0
	if (y + z) <= 5e-7:
		tmp = (math.tan(y) - math.tan(a)) + x
	else:
		tmp = math.tan((y + z)) - -x
	return tmp

Julia

function code(x, y, z, a)
	tmp = 0.0
	if (Float64(y + z) <= 5e-7)
		tmp = Float64(Float64(tan(y) - tan(a)) + x);
	else
		tmp = Float64(tan(Float64(y + z)) - Float64(-x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, a)
	tmp = 0.0;
	if ((y + z) <= 5e-7)
		tmp = (tan(y) - tan(a)) + x;
	else
		tmp = tan((y + z)) - -x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, a_] := If[LessEqual[N[(y + z), $MachinePrecision], 5e-7], N[(N[(N[Tan[y], $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], N[(N[Tan[N[(y + z), $MachinePrecision]], $MachinePrecision] - (-x)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 y z) < 4.99999999999999977e-7

   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. Taylor expanded in z around 0

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

      1. lower-/.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-cos.f64 69.3

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

   7. Applied rewrites 69.3%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. associate--r- N/A

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

      4. lower-+.f64 N/A

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

      5. lower--.f64 69.3

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

   9. Applied rewrites 69.3%

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

   if 4.99999999999999977e-7 < (+.f64 y z)

   1. Initial program 70.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift--.f64 N/A

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

      4. associate-+l- N/A

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

      5. lower--.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. lower--.f64 70.5

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

   4. Applied rewrites 70.5%

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

   5. Taylor expanded in a around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 44.3

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

   7. Applied rewrites 44.3%

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

4. Final simplification 59.4%

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

Alternative 7: 79.0% 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 78.4%

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

3. Final simplification 78.4%

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

Alternative 8: 49.8% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.4%

   \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
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 78.4

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

4. Applied rewrites 78.4%

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

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 52.9

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

7. Applied rewrites 52.9%

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

   1. lift-+.f64 N/A

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

   2. flip-+ N/A

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

   3. lift--.f64 N/A

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

   4. difference-of-squares N/A

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

   5. lift-+.f64 N/A

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

   6. lift--.f64 N/A

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

   7. *-commutative N/A

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

   8. associate-/l* N/A

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

   9. lower-*.f64 N/A

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

   10. lower-/.f64 52.9

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

   11. lift-+.f64 N/A

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

   12. +-commutative N/A

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

   13. lift-+.f64 52.9

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

9. Applied rewrites 52.9%

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

10. Final simplification 52.9%

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

Alternative 9: 49.8% 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 78.4%

   \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
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 78.4

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

4. Applied rewrites 78.4%

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

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 52.9

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

7. Applied rewrites 52.9%

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

8. Final simplification 52.9%

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

Reproduce

herbie shell --seed 2024249 
(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))))