tan-example (used to crash)

Percentage Accurate: 79.7% → 99.7%

Time: 33.7s

Alternatives: 10

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 79.7% 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 80.4%

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

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

   2. lift-fma.f64 N/A

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

   3. +-commutative N/A

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

   4. lift-neg.f64 N/A

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

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

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

   6. lift-tan.f64 N/A

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

   7. lift-tan.f64 N/A

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

   8. lift-+.f64 N/A

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

   9. lift-tan.f64 N/A

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

   10. lift-tan.f64 N/A

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

   11. tan-sum N/A

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

   12. lift-+.f64 N/A

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

   13. *-lft-identity N/A

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

   14. lift-+.f64 N/A

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

   15. distribute-rgt-in N/A

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

   16. tan-sum N/A

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

   17. lower-/.f64 N/A

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

6. Applied rewrites 99.8%

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

   1. lift-*.f64 N/A

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

   2. *-rgt-identity 99.8

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

   3. lift-*.f64 N/A

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

   4. *-rgt-identity 99.8

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

8. Applied rewrites 99.8%

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

   1. lift-*.f64 N/A

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

   2. *-rgt-identity 99.8

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

   3. lift-*.f64 N/A

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

   4. *-rgt-identity 99.8

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

10. Applied rewrites 99.8%

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

11. Final simplification 99.8%

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

Alternative 2: 89.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\tan \left(y + z\right) - \tan a\right) + x\\ \mathbf{if}\;\tan a \leq -2 \cdot 10^{-14}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\tan a \leq 2 \cdot 10^{-23}:\\ \;\;\;\;\frac{\tan y + \tan z}{\mathsf{fma}\left(-\tan y, \tan z, 1\right)} - \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (let* ((t_0 (+ (- (tan (+ y z)) (tan a)) x)))
   (if (<= (tan a) -2e-14)
     t_0
     (if (<= (tan a) 2e-23)
       (- (/ (+ (tan y) (tan z)) (fma (- (tan y)) (tan z) 1.0)) (- x))
       t_0))))

C

double code(double x, double y, double z, double a) {
	double t_0 = (tan((y + z)) - tan(a)) + x;
	double tmp;
	if (tan(a) <= -2e-14) {
		tmp = t_0;
	} else if (tan(a) <= 2e-23) {
		tmp = ((tan(y) + tan(z)) / fma(-tan(y), tan(z), 1.0)) - -x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z, a)
	t_0 = Float64(Float64(tan(Float64(y + z)) - tan(a)) + x)
	tmp = 0.0
	if (tan(a) <= -2e-14)
		tmp = t_0;
	elseif (tan(a) <= 2e-23)
		tmp = Float64(Float64(Float64(tan(y) + tan(z)) / fma(Float64(-tan(y)), tan(z), 1.0)) - Float64(-x));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, a_] := Block[{t$95$0 = N[(N[(N[Tan[N[(y + z), $MachinePrecision]], $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[N[Tan[a], $MachinePrecision], -2e-14], t$95$0, If[LessEqual[N[Tan[a], $MachinePrecision], 2e-23], N[(N[(N[(N[Tan[y], $MachinePrecision] + N[Tan[z], $MachinePrecision]), $MachinePrecision] / N[((-N[Tan[y], $MachinePrecision]) * N[Tan[z], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] - (-x)), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if (tan.f64 a) < -2e-14 or 1.99999999999999992e-23 < (tan.f64 a)

   1. Initial program 82.0%

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

   if -2e-14 < (tan.f64 a) < 1.99999999999999992e-23

   1. Initial program 77.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift--.f64 N/A

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

      4. associate-+l- N/A

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

      5. lower--.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. lower--.f64 77.9

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

   4. Applied rewrites 77.9%

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

   5. Taylor expanded in x around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 77.9

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

   7. Applied rewrites 77.9%

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

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

      5. *-rgt-identity N/A

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

      6. lift-*.f64 N/A

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

      7. lower-tan.f64 N/A

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

      8. *-rgt-identity N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift--.f64 N/A

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

      12. lower-/.f64 N/A

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

   9. Applied rewrites 99.7%

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

4. Final simplification 89.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\tan a \leq -2 \cdot 10^{-14}:\\ \;\;\;\;\left(\tan \left(y + z\right) - \tan a\right) + x\\ \mathbf{elif}\;\tan a \leq 2 \cdot 10^{-23}:\\ \;\;\;\;\frac{\tan y + \tan z}{\mathsf{fma}\left(-\tan y, \tan z, 1\right)} - \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\tan \left(y + z\right) - \tan a\right) + x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 88.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -2.2 \cdot 10^{-14}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{\sin \left(y + z\right)}{\cos \left(y + z\right)} - \frac{\sin a}{\cos a}}{x}, x, x\right)\\ \mathbf{elif}\;a \leq 1.8 \cdot 10^{-33}:\\ \;\;\;\;\frac{\tan y + \tan z}{\mathsf{fma}\left(-\tan y, \tan z, 1\right)} - \left(a - x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\tan \left(y + z\right) - \tan a\right) + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (if (<= a -2.2e-14)
   (fma (/ (- (/ (sin (+ y z)) (cos (+ y z))) (/ (sin a) (cos a))) x) x x)
   (if (<= a 1.8e-33)
     (- (/ (+ (tan y) (tan z)) (fma (- (tan y)) (tan z) 1.0)) (- a x))
     (+ (- (tan (+ y z)) (tan a)) x))))

C

double code(double x, double y, double z, double a) {
	double tmp;
	if (a <= -2.2e-14) {
		tmp = fma((((sin((y + z)) / cos((y + z))) - (sin(a) / cos(a))) / x), x, x);
	} else if (a <= 1.8e-33) {
		tmp = ((tan(y) + tan(z)) / fma(-tan(y), tan(z), 1.0)) - (a - x);
	} else {
		tmp = (tan((y + z)) - tan(a)) + x;
	}
	return tmp;
}

Julia

function code(x, y, z, a)
	tmp = 0.0
	if (a <= -2.2e-14)
		tmp = fma(Float64(Float64(Float64(sin(Float64(y + z)) / cos(Float64(y + z))) - Float64(sin(a) / cos(a))) / x), x, x);
	elseif (a <= 1.8e-33)
		tmp = Float64(Float64(Float64(tan(y) + tan(z)) / fma(Float64(-tan(y)), tan(z), 1.0)) - Float64(a - x));
	else
		tmp = Float64(Float64(tan(Float64(y + z)) - tan(a)) + x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, a_] := If[LessEqual[a, -2.2e-14], N[(N[(N[(N[(N[Sin[N[(y + z), $MachinePrecision]], $MachinePrecision] / N[Cos[N[(y + z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(N[Sin[a], $MachinePrecision] / N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] * x + x), $MachinePrecision], If[LessEqual[a, 1.8e-33], N[(N[(N[(N[Tan[y], $MachinePrecision] + N[Tan[z], $MachinePrecision]), $MachinePrecision] / N[((-N[Tan[y], $MachinePrecision]) * N[Tan[z], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] - N[(a - x), $MachinePrecision]), $MachinePrecision], N[(N[(N[Tan[N[(y + z), $MachinePrecision]], $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -2.2000000000000001e-14

   1. Initial program 80.5%

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

   3. Taylor expanded in x around inf

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. associate-/l/ N/A

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

      4. associate-/l/ N/A

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

      5. div-sub N/A

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

      6. distribute-rgt-in N/A

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

      7. *-lft-identity N/A

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

      8. lower-fma.f64 N/A

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

   5. Applied rewrites 80.5%

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

   if -2.2000000000000001e-14 < a < 1.80000000000000017e-33

   1. Initial program 77.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift--.f64 N/A

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

      4. associate-+l- N/A

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

      5. lower--.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. lower--.f64 77.3

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

   4. Applied rewrites 77.3%

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

   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 77.3

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

   7. Applied rewrites 77.3%

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

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

      6. lift-*.f64 N/A

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

      7. *-rgt-identity N/A

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

      8. lift-*.f64 N/A

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

      9. lift-tan.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. lift-tan.f64 N/A

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

      12. lift-tan.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lift--.f64 N/A

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

      15. lift-/.f64 99.8

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

      16. lift-+.f64 N/A

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

      17. +-commutative N/A

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

      18. lower-+.f64 99.8

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

      19. lift-*.f64 N/A

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

      20. *-rgt-identity 99.8

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

      21. lift-*.f64 N/A

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

      22. *-rgt-identity 99.8

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

   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 1.80000000000000017e-33 < a

   1. Initial program 84.3%

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

4. Final simplification 89.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -2.2 \cdot 10^{-14}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{\sin \left(y + z\right)}{\cos \left(y + z\right)} - \frac{\sin a}{\cos a}}{x}, x, x\right)\\ \mathbf{elif}\;a \leq 1.8 \cdot 10^{-33}:\\ \;\;\;\;\frac{\tan y + \tan z}{\mathsf{fma}\left(-\tan y, \tan z, 1\right)} - \left(a - x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\tan \left(y + z\right) - \tan a\right) + x\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 88.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\tan \left(y + z\right) - \tan a\right) + x\\ \mathbf{if}\;a \leq -2.2 \cdot 10^{-14}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;a \leq 1.8 \cdot 10^{-33}:\\ \;\;\;\;\frac{\tan y + \tan z}{\mathsf{fma}\left(-\tan y, \tan z, 1\right)} - \left(a - x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (let* ((t_0 (+ (- (tan (+ y z)) (tan a)) x)))
   (if (<= a -2.2e-14)
     t_0
     (if (<= a 1.8e-33)
       (- (/ (+ (tan y) (tan z)) (fma (- (tan y)) (tan z) 1.0)) (- a x))
       t_0))))

C

double code(double x, double y, double z, double a) {
	double t_0 = (tan((y + z)) - tan(a)) + x;
	double tmp;
	if (a <= -2.2e-14) {
		tmp = t_0;
	} else if (a <= 1.8e-33) {
		tmp = ((tan(y) + tan(z)) / fma(-tan(y), tan(z), 1.0)) - (a - x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z, a)
	t_0 = Float64(Float64(tan(Float64(y + z)) - tan(a)) + x)
	tmp = 0.0
	if (a <= -2.2e-14)
		tmp = t_0;
	elseif (a <= 1.8e-33)
		tmp = Float64(Float64(Float64(tan(y) + tan(z)) / fma(Float64(-tan(y)), tan(z), 1.0)) - Float64(a - x));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, a_] := Block[{t$95$0 = N[(N[(N[Tan[N[(y + z), $MachinePrecision]], $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[a, -2.2e-14], t$95$0, If[LessEqual[a, 1.8e-33], N[(N[(N[(N[Tan[y], $MachinePrecision] + N[Tan[z], $MachinePrecision]), $MachinePrecision] / N[((-N[Tan[y], $MachinePrecision]) * N[Tan[z], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] - N[(a - x), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if a < -2.2000000000000001e-14 or 1.80000000000000017e-33 < a

   1. Initial program 82.3%

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

   if -2.2000000000000001e-14 < a < 1.80000000000000017e-33

   1. Initial program 77.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift--.f64 N/A

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

      4. associate-+l- N/A

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

      5. lower--.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. lower--.f64 77.3

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

   4. Applied rewrites 77.3%

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

   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 77.3

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

   7. Applied rewrites 77.3%

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

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

      6. lift-*.f64 N/A

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

      7. *-rgt-identity N/A

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

      8. lift-*.f64 N/A

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

      9. lift-tan.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. lift-tan.f64 N/A

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

      12. lift-tan.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lift--.f64 N/A

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

      15. lift-/.f64 99.8

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

      16. lift-+.f64 N/A

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

      17. +-commutative N/A

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

      18. lower-+.f64 99.8

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

      19. lift-*.f64 N/A

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

      20. *-rgt-identity 99.8

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

      21. lift-*.f64 N/A

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

      22. *-rgt-identity 99.8

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

   9. Applied rewrites 99.8%

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

4. Final simplification 89.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -2.2 \cdot 10^{-14}:\\ \;\;\;\;\left(\tan \left(y + z\right) - \tan a\right) + x\\ \mathbf{elif}\;a \leq 1.8 \cdot 10^{-33}:\\ \;\;\;\;\frac{\tan y + \tan z}{\mathsf{fma}\left(-\tan y, \tan z, 1\right)} - \left(a - x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\tan \left(y + z\right) - \tan a\right) + x\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 64.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2900:\\ \;\;\;\;\tan y - \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\tan z - \tan a\right) + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (if (<= y -2900.0) (- (tan y) (- x)) (+ (- (tan z) (tan a)) x)))

C

double code(double x, double y, double z, double a) {
	double tmp;
	if (y <= -2900.0) {
		tmp = tan(y) - -x;
	} else {
		tmp = (tan(z) - tan(a)) + x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: a
    real(8) :: tmp
    if (y <= (-2900.0d0)) then
        tmp = tan(y) - -x
    else
        tmp = (tan(z) - tan(a)) + x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double a) {
	double tmp;
	if (y <= -2900.0) {
		tmp = Math.tan(y) - -x;
	} else {
		tmp = (Math.tan(z) - Math.tan(a)) + x;
	}
	return tmp;
}

Python

def code(x, y, z, a):
	tmp = 0
	if y <= -2900.0:
		tmp = math.tan(y) - -x
	else:
		tmp = (math.tan(z) - math.tan(a)) + x
	return tmp

Julia

function code(x, y, z, a)
	tmp = 0.0
	if (y <= -2900.0)
		tmp = Float64(tan(y) - Float64(-x));
	else
		tmp = Float64(Float64(tan(z) - tan(a)) + x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, a)
	tmp = 0.0;
	if (y <= -2900.0)
		tmp = tan(y) - -x;
	else
		tmp = (tan(z) - tan(a)) + x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, a_] := If[LessEqual[y, -2900.0], N[(N[Tan[y], $MachinePrecision] - (-x)), $MachinePrecision], N[(N[(N[Tan[z], $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -2900

   1. Initial program 59.5%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   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 59.4

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

   4. Applied rewrites 59.4%

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

   5. Taylor expanded in x around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 38.6

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

   7. Applied rewrites 38.6%

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

   8. Taylor expanded in z around 0

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

      1. lower-/.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-cos.f64 38.9

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

   10. Applied rewrites 38.9%

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

   12. Applied rewrites 38.9%

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

   if -2900 < y

   1. Initial program 86.9%

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

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

   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 y around 0

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

      1. lower-/.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-cos.f64 70.4

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

   7. Applied rewrites 70.4%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 70.4

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

   9. Applied rewrites 70.4%

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

Alternative 6: 79.7% 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 80.4%

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

3. Final simplification 80.4%

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

Alternative 7: 49.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y + z \leq -100000:\\ \;\;\;\;\tan y - \left(-x\right)\\ \mathbf{elif}\;y + z \leq 5 \cdot 10^{-32}:\\ \;\;\;\;\left(\mathsf{fma}\left(z \cdot z, 0.3333333333333333, 1\right) \cdot z - \tan a\right) + x\\ \mathbf{else}:\\ \;\;\;\;\tan \left(\mathsf{fma}\left(\frac{y}{z}, z, z\right)\right) - \left(-x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (if (<= (+ y z) -100000.0)
   (- (tan y) (- x))
   (if (<= (+ y z) 5e-32)
     (+ (- (* (fma (* z z) 0.3333333333333333 1.0) z) (tan a)) x)
     (- (tan (fma (/ y z) z z)) (- x)))))

C

double code(double x, double y, double z, double a) {
	double tmp;
	if ((y + z) <= -100000.0) {
		tmp = tan(y) - -x;
	} else if ((y + z) <= 5e-32) {
		tmp = ((fma((z * z), 0.3333333333333333, 1.0) * z) - tan(a)) + x;
	} else {
		tmp = tan(fma((y / z), z, z)) - -x;
	}
	return tmp;
}

Julia

function code(x, y, z, a)
	tmp = 0.0
	if (Float64(y + z) <= -100000.0)
		tmp = Float64(tan(y) - Float64(-x));
	elseif (Float64(y + z) <= 5e-32)
		tmp = Float64(Float64(Float64(fma(Float64(z * z), 0.3333333333333333, 1.0) * z) - tan(a)) + x);
	else
		tmp = Float64(tan(fma(Float64(y / z), z, z)) - Float64(-x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, a_] := If[LessEqual[N[(y + z), $MachinePrecision], -100000.0], N[(N[Tan[y], $MachinePrecision] - (-x)), $MachinePrecision], If[LessEqual[N[(y + z), $MachinePrecision], 5e-32], N[(N[(N[(N[(N[(z * z), $MachinePrecision] * 0.3333333333333333 + 1.0), $MachinePrecision] * z), $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], N[(N[Tan[N[(N[(y / z), $MachinePrecision] * z + z), $MachinePrecision]], $MachinePrecision] - (-x)), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 y z) < -1e5

   1. Initial program 72.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift--.f64 N/A

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

      4. associate-+l- N/A

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

      5. lower--.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. lower--.f64 72.8

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

   4. Applied rewrites 72.8%

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

   5. Taylor expanded in x around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 42.8

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

   7. Applied rewrites 42.8%

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

   8. Taylor expanded in z around 0

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

      1. lower-/.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-cos.f64 34.1

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

   10. Applied rewrites 34.1%

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

   12. Applied rewrites 34.1%

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

   if -1e5 < (+.f64 y z) < 5e-32

   1. Initial program 99.9%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   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 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-cos.f64 98.2

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

   7. Applied rewrites 98.2%

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

   8. Taylor expanded in z around 0

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

   10. Applied rewrites 98.2%

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

   if 5e-32 < (+.f64 y z)

   1. Initial program 76.5%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   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 76.3

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

   4. Applied rewrites 76.3%

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

   5. Taylor expanded in x around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 48.1

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

   7. Applied rewrites 48.1%

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

   8. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-lft-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 36.8

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

   10. Applied rewrites 36.8%

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

4. Final simplification 49.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y + z \leq -100000:\\ \;\;\;\;\tan y - \left(-x\right)\\ \mathbf{elif}\;y + z \leq 5 \cdot 10^{-32}:\\ \;\;\;\;\left(\mathsf{fma}\left(z \cdot z, 0.3333333333333333, 1\right) \cdot z - \tan a\right) + x\\ \mathbf{else}:\\ \;\;\;\;\tan \left(\mathsf{fma}\left(\frac{y}{z}, z, z\right)\right) - \left(-x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 53.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y + z \leq -100000:\\ \;\;\;\;\tan y - \left(-x\right)\\ \mathbf{elif}\;y + z \leq 5 \cdot 10^{-32}:\\ \;\;\;\;\left(\mathsf{fma}\left(z \cdot z, 0.3333333333333333, 1\right) \cdot z - \tan a\right) + x\\ \mathbf{else}:\\ \;\;\;\;\tan \left(y + z\right) - \left(-x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (if (<= (+ y z) -100000.0)
   (- (tan y) (- x))
   (if (<= (+ y z) 5e-32)
     (+ (- (* (fma (* z z) 0.3333333333333333 1.0) z) (tan a)) x)
     (- (tan (+ y z)) (- x)))))

C

double code(double x, double y, double z, double a) {
	double tmp;
	if ((y + z) <= -100000.0) {
		tmp = tan(y) - -x;
	} else if ((y + z) <= 5e-32) {
		tmp = ((fma((z * z), 0.3333333333333333, 1.0) * z) - tan(a)) + x;
	} else {
		tmp = tan((y + z)) - -x;
	}
	return tmp;
}

Julia

function code(x, y, z, a)
	tmp = 0.0
	if (Float64(y + z) <= -100000.0)
		tmp = Float64(tan(y) - Float64(-x));
	elseif (Float64(y + z) <= 5e-32)
		tmp = Float64(Float64(Float64(fma(Float64(z * z), 0.3333333333333333, 1.0) * z) - tan(a)) + x);
	else
		tmp = Float64(tan(Float64(y + z)) - Float64(-x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, a_] := If[LessEqual[N[(y + z), $MachinePrecision], -100000.0], N[(N[Tan[y], $MachinePrecision] - (-x)), $MachinePrecision], If[LessEqual[N[(y + z), $MachinePrecision], 5e-32], N[(N[(N[(N[(N[(z * z), $MachinePrecision] * 0.3333333333333333 + 1.0), $MachinePrecision] * z), $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], N[(N[Tan[N[(y + z), $MachinePrecision]], $MachinePrecision] - (-x)), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 y z) < -1e5

   1. Initial program 72.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift--.f64 N/A

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

      4. associate-+l- N/A

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

      5. lower--.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. lower--.f64 72.8

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

   4. Applied rewrites 72.8%

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

   5. Taylor expanded in x around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 42.8

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

   7. Applied rewrites 42.8%

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

   8. Taylor expanded in z around 0

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

      1. lower-/.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-cos.f64 34.1

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

   10. Applied rewrites 34.1%

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

   12. Applied rewrites 34.1%

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

   if -1e5 < (+.f64 y z) < 5e-32

   1. Initial program 99.9%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   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 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-cos.f64 98.2

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

   7. Applied rewrites 98.2%

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

   8. Taylor expanded in z around 0

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

   10. Applied rewrites 98.2%

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

   if 5e-32 < (+.f64 y z)

   1. Initial program 76.5%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   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 76.3

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

   4. Applied rewrites 76.3%

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

   5. Taylor expanded in x around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 48.1

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

   7. Applied rewrites 48.1%

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

4. Final simplification 54.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y + z \leq -100000:\\ \;\;\;\;\tan y - \left(-x\right)\\ \mathbf{elif}\;y + z \leq 5 \cdot 10^{-32}:\\ \;\;\;\;\left(\mathsf{fma}\left(z \cdot z, 0.3333333333333333, 1\right) \cdot z - \tan a\right) + x\\ \mathbf{else}:\\ \;\;\;\;\tan \left(y + z\right) - \left(-x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 49.9% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 80.4%

   \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
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 80.2

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

4. Applied rewrites 80.2%

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

5. Taylor expanded in x around inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 46.1

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

7. Applied rewrites 46.1%

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

8. Final simplification 46.1%

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

Alternative 10: 40.8% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 80.4%

   \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
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 80.2

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

4. Applied rewrites 80.2%

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

5. Taylor expanded in x around inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 46.1

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

7. Applied rewrites 46.1%

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

8. Taylor expanded in z around 0

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

   1. lower-/.f64 N/A

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

   2. lower-sin.f64 N/A

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

   3. lower-cos.f64 37.2

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

10. Applied rewrites 37.2%

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

12. Applied rewrites 37.2%

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

Reproduce

herbie shell --seed 2024296 
(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))))