tan-example (used to crash)

Percentage Accurate: 79.2% → 99.7%

Time: 31.4s

Alternatives: 20

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: 79.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 82.4%

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

   1. +-commutative N/A

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

   2. tan-sum N/A

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

   3. tan-quot N/A

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

   4. frac-sub N/A

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

   5. div-inv N/A

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

   6. accelerator-lowering-fma.f64 N/A

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

4. Applied egg-rr 99.7%

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

   1. associate-*r/ N/A

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

   2. times-frac N/A

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

   3. accelerator-lowering-fma.f64 N/A

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

6. Applied egg-rr 99.7%

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

   1. sub-neg N/A

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

   2. +-commutative N/A

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

   3. *-commutative N/A

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

   4. distribute-rgt-neg-in N/A

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

   5. accelerator-lowering-fma.f64 N/A

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

   6. tan-lowering-tan.f64 N/A

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

   7. neg-lowering-neg.f64 N/A

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

   8. tan-lowering-tan.f64 99.7

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

8. Applied egg-rr 99.7%

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

Alternative 2: 99.7% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 82.4%

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

   1. +-commutative N/A

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

   2. tan-sum N/A

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

   3. tan-quot N/A

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

   4. frac-sub N/A

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

   5. div-inv N/A

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

   6. accelerator-lowering-fma.f64 N/A

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

4. Applied egg-rr 99.7%

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

   1. associate-*r/ N/A

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

   2. times-frac N/A

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

   3. accelerator-lowering-fma.f64 N/A

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

6. Applied egg-rr 99.7%

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

   1. associate-*l/ N/A

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

   2. associate-/l* N/A

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

   3. accelerator-lowering-fma.f64 N/A

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

8. Applied egg-rr 99.7%

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

9. Final simplification 99.7%

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

Alternative 3: 99.7% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 82.4%

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

   1. +-commutative N/A

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

   2. tan-sum N/A

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

   3. tan-quot N/A

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

   4. frac-sub N/A

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

   5. div-inv N/A

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

   6. accelerator-lowering-fma.f64 N/A

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

4. Applied egg-rr 99.7%

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

5. Final simplification 99.7%

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

Alternative 4: 99.7% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 82.4%

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

   1. sub-neg N/A

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

   2. +-commutative N/A

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

   3. tan-quot N/A

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

   4. distribute-neg-frac N/A

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

   5. tan-sum N/A

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

   6. frac-add N/A

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

   7. *-commutative N/A

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

   8. /-lowering-/.f64 N/A

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

4. Applied egg-rr 99.6%

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

5. Final simplification 99.6%

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

Alternative 5: 88.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x + \mathsf{fma}\left(\frac{1}{\cos \left(y + z\right)}, \sin \left(y + z\right), -\tan a\right)\\ \mathbf{if}\;\tan a \leq -0.05:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\tan a \leq 3 \cdot 10^{-13}:\\ \;\;\;\;x + \left(\frac{1}{\frac{1 - \tan y \cdot \tan z}{\tan y + \tan z}} - a\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (let* ((t_0 (+ x (fma (/ 1.0 (cos (+ y z))) (sin (+ y z)) (- (tan a))))))
   (if (<= (tan a) -0.05)
     t_0
     (if (<= (tan a) 3e-13)
       (+ x (- (/ 1.0 (/ (- 1.0 (* (tan y) (tan z))) (+ (tan y) (tan z)))) a))
       t_0))))

C

double code(double x, double y, double z, double a) {
	double t_0 = x + fma((1.0 / cos((y + z))), sin((y + z)), -tan(a));
	double tmp;
	if (tan(a) <= -0.05) {
		tmp = t_0;
	} else if (tan(a) <= 3e-13) {
		tmp = x + ((1.0 / ((1.0 - (tan(y) * tan(z))) / (tan(y) + tan(z)))) - a);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z, a)
	t_0 = Float64(x + fma(Float64(1.0 / cos(Float64(y + z))), sin(Float64(y + z)), Float64(-tan(a))))
	tmp = 0.0
	if (tan(a) <= -0.05)
		tmp = t_0;
	elseif (tan(a) <= 3e-13)
		tmp = Float64(x + Float64(Float64(1.0 / Float64(Float64(1.0 - Float64(tan(y) * tan(z))) / Float64(tan(y) + tan(z)))) - a));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, a_] := Block[{t$95$0 = N[(x + N[(N[(1.0 / N[Cos[N[(y + z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[N[(y + z), $MachinePrecision]], $MachinePrecision] + (-N[Tan[a], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Tan[a], $MachinePrecision], -0.05], t$95$0, If[LessEqual[N[Tan[a], $MachinePrecision], 3e-13], N[(x + N[(N[(1.0 / N[(N[(1.0 - N[(N[Tan[y], $MachinePrecision] * N[Tan[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Tan[y], $MachinePrecision] + N[Tan[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if (tan.f64 a) < -0.050000000000000003 or 2.99999999999999984e-13 < (tan.f64 a)

   1. Initial program 82.3%

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

      1. sub-neg N/A

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

      2. tan-quot N/A

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

      3. clear-num N/A

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

      4. associate-/r/ N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. /-lowering-/.f64 N/A

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

      7. cos-lowering-cos.f64 N/A

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

      8. +-lowering-+.f64 N/A

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

      9. sin-lowering-sin.f64 N/A

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

      10. +-lowering-+.f64 N/A

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

      11. neg-lowering-neg.f64 N/A

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

      12. tan-lowering-tan.f64 82.3

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

   4. Applied egg-rr 82.3%

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

   if -0.050000000000000003 < (tan.f64 a) < 2.99999999999999984e-13

   1. Initial program 82.5%

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

   3. Taylor expanded in a around 0

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

   5. Simplified 82.5%

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

      1. tan-sum N/A

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

      2. clear-num N/A

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

      3. /-lowering-/.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. --lowering--.f64 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. tan-lowering-tan.f64 N/A

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

      8. tan-lowering-tan.f64 N/A

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

      9. +-lowering-+.f64 N/A

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

      10. tan-lowering-tan.f64 N/A

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

      11. tan-lowering-tan.f64 99.8

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

   7. Applied egg-rr 99.8%

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

Alternative 6: 88.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x + \mathsf{fma}\left(\frac{1}{\cos \left(y + z\right)}, \sin \left(y + z\right), -\tan a\right)\\ \mathbf{if}\;\tan a \leq -0.05:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\tan a \leq 3 \cdot 10^{-13}:\\ \;\;\;\;\mathsf{fma}\left(\tan y + \tan z, \frac{1}{1 - \tan y \cdot \tan z}, x - a\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (let* ((t_0 (+ x (fma (/ 1.0 (cos (+ y z))) (sin (+ y z)) (- (tan a))))))
   (if (<= (tan a) -0.05)
     t_0
     (if (<= (tan a) 3e-13)
       (fma (+ (tan y) (tan z)) (/ 1.0 (- 1.0 (* (tan y) (tan z)))) (- x a))
       t_0))))

C

double code(double x, double y, double z, double a) {
	double t_0 = x + fma((1.0 / cos((y + z))), sin((y + z)), -tan(a));
	double tmp;
	if (tan(a) <= -0.05) {
		tmp = t_0;
	} else if (tan(a) <= 3e-13) {
		tmp = fma((tan(y) + tan(z)), (1.0 / (1.0 - (tan(y) * tan(z)))), (x - a));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z, a)
	t_0 = Float64(x + fma(Float64(1.0 / cos(Float64(y + z))), sin(Float64(y + z)), Float64(-tan(a))))
	tmp = 0.0
	if (tan(a) <= -0.05)
		tmp = t_0;
	elseif (tan(a) <= 3e-13)
		tmp = fma(Float64(tan(y) + tan(z)), Float64(1.0 / Float64(1.0 - Float64(tan(y) * tan(z)))), Float64(x - a));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, a_] := Block[{t$95$0 = N[(x + N[(N[(1.0 / N[Cos[N[(y + z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[N[(y + z), $MachinePrecision]], $MachinePrecision] + (-N[Tan[a], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Tan[a], $MachinePrecision], -0.05], t$95$0, If[LessEqual[N[Tan[a], $MachinePrecision], 3e-13], N[(N[(N[Tan[y], $MachinePrecision] + N[Tan[z], $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(1.0 - N[(N[Tan[y], $MachinePrecision] * N[Tan[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x - a), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if (tan.f64 a) < -0.050000000000000003 or 2.99999999999999984e-13 < (tan.f64 a)

   1. Initial program 82.3%

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

      1. sub-neg N/A

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

      2. tan-quot N/A

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

      3. clear-num N/A

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

      4. associate-/r/ N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. /-lowering-/.f64 N/A

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

      7. cos-lowering-cos.f64 N/A

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

      8. +-lowering-+.f64 N/A

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

      9. sin-lowering-sin.f64 N/A

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

      10. +-lowering-+.f64 N/A

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

      11. neg-lowering-neg.f64 N/A

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

      12. tan-lowering-tan.f64 82.3

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

   4. Applied egg-rr 82.3%

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

   if -0.050000000000000003 < (tan.f64 a) < 2.99999999999999984e-13

   1. Initial program 82.5%

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

   3. Taylor expanded in a around 0

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

   5. Simplified 82.5%

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

      1. +-commutative N/A

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

      2. sub-neg N/A

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

      3. associate-+l+ N/A

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

      4. tan-sum N/A

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

      5. div-inv N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. +-lowering-+.f64 N/A

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

      8. tan-lowering-tan.f64 N/A

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

      9. tan-lowering-tan.f64 N/A

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

      10. /-lowering-/.f64 N/A

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

      11. --lowering--.f64 N/A

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

      12. *-lowering-*.f64 N/A

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

      13. tan-lowering-tan.f64 N/A

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

      14. tan-lowering-tan.f64 N/A

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

      15. +-lowering-+.f64 N/A

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

      16. neg-lowering-neg.f64 99.8

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

   7. Applied egg-rr 99.8%

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

4. Final simplification 90.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\tan a \leq -0.05:\\ \;\;\;\;x + \mathsf{fma}\left(\frac{1}{\cos \left(y + z\right)}, \sin \left(y + z\right), -\tan a\right)\\ \mathbf{elif}\;\tan a \leq 3 \cdot 10^{-13}:\\ \;\;\;\;\mathsf{fma}\left(\tan y + \tan z, \frac{1}{1 - \tan y \cdot \tan z}, x - a\right)\\ \mathbf{else}:\\ \;\;\;\;x + \mathsf{fma}\left(\frac{1}{\cos \left(y + z\right)}, \sin \left(y + z\right), -\tan a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 88.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x + \mathsf{fma}\left(\frac{1}{\cos \left(y + z\right)}, \sin \left(y + z\right), -\tan a\right)\\ \mathbf{if}\;\tan a \leq -0.05:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\tan a \leq 3 \cdot 10^{-13}:\\ \;\;\;\;x + \left(\frac{\tan y + \tan z}{1 - \tan y \cdot \tan z} - a\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (let* ((t_0 (+ x (fma (/ 1.0 (cos (+ y z))) (sin (+ y z)) (- (tan a))))))
   (if (<= (tan a) -0.05)
     t_0
     (if (<= (tan a) 3e-13)
       (+ x (- (/ (+ (tan y) (tan z)) (- 1.0 (* (tan y) (tan z)))) a))
       t_0))))

C

double code(double x, double y, double z, double a) {
	double t_0 = x + fma((1.0 / cos((y + z))), sin((y + z)), -tan(a));
	double tmp;
	if (tan(a) <= -0.05) {
		tmp = t_0;
	} else if (tan(a) <= 3e-13) {
		tmp = x + (((tan(y) + tan(z)) / (1.0 - (tan(y) * tan(z)))) - a);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z, a)
	t_0 = Float64(x + fma(Float64(1.0 / cos(Float64(y + z))), sin(Float64(y + z)), Float64(-tan(a))))
	tmp = 0.0
	if (tan(a) <= -0.05)
		tmp = t_0;
	elseif (tan(a) <= 3e-13)
		tmp = Float64(x + Float64(Float64(Float64(tan(y) + tan(z)) / Float64(1.0 - Float64(tan(y) * tan(z)))) - a));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, a_] := Block[{t$95$0 = N[(x + N[(N[(1.0 / N[Cos[N[(y + z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[N[(y + z), $MachinePrecision]], $MachinePrecision] + (-N[Tan[a], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Tan[a], $MachinePrecision], -0.05], t$95$0, If[LessEqual[N[Tan[a], $MachinePrecision], 3e-13], N[(x + 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] - a), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if (tan.f64 a) < -0.050000000000000003 or 2.99999999999999984e-13 < (tan.f64 a)

   1. Initial program 82.3%

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

      1. sub-neg N/A

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

      2. tan-quot N/A

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

      3. clear-num N/A

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

      4. associate-/r/ N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. /-lowering-/.f64 N/A

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

      7. cos-lowering-cos.f64 N/A

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

      8. +-lowering-+.f64 N/A

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

      9. sin-lowering-sin.f64 N/A

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

      10. +-lowering-+.f64 N/A

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

      11. neg-lowering-neg.f64 N/A

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

      12. tan-lowering-tan.f64 82.3

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

   4. Applied egg-rr 82.3%

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

   if -0.050000000000000003 < (tan.f64 a) < 2.99999999999999984e-13

   1. Initial program 82.5%

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

   3. Taylor expanded in a around 0

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

   5. Simplified 82.5%

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

      1. tan-sum N/A

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

      2. /-lowering-/.f64 N/A

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

      3. +-lowering-+.f64 N/A

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

      4. tan-lowering-tan.f64 N/A

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

      5. tan-lowering-tan.f64 N/A

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

      6. --lowering--.f64 N/A

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

      7. *-lowering-*.f64 N/A

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

      8. tan-lowering-tan.f64 N/A

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

      9. tan-lowering-tan.f64 99.8

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

   7. Applied egg-rr 99.8%

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

Alternative 8: 99.7% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 82.4%

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

   1. flip-+ N/A

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

   2. div-inv N/A

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

   3. difference-of-squares N/A

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

   4. associate-*l* N/A

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

   5. *-lowering-*.f64 N/A

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

   6. +-lowering-+.f64 N/A

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

   7. *-lowering-*.f64 N/A

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

   8. --lowering--.f64 N/A

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

   9. /-lowering-/.f64 N/A

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

   10. --lowering--.f64 76.4

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

4. Applied egg-rr 76.4%

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

   1. rgt-mult-inverse N/A

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

   2. *-rgt-identity 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. clear-num N/A

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

   5. /-lowering-/.f64 N/A

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

   6. /-lowering-/.f64 N/A

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

   7. --lowering--.f64 N/A

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

   8. *-lowering-*.f64 N/A

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

   9. tan-lowering-tan.f64 N/A

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

   10. tan-lowering-tan.f64 N/A

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

   11. +-lowering-+.f64 N/A

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

   12. tan-lowering-tan.f64 N/A

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

   13. tan-lowering-tan.f64 99.6

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

6. Applied egg-rr 99.6%

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

Alternative 9: 99.7% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 82.4%

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

   1. tan-sum N/A

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

   2. /-lowering-/.f64 N/A

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

   3. +-lowering-+.f64 N/A

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

   4. tan-lowering-tan.f64 N/A

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

   5. tan-lowering-tan.f64 N/A

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

   6. --lowering--.f64 N/A

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

   7. *-lowering-*.f64 N/A

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

   8. tan-lowering-tan.f64 N/A

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

   9. tan-lowering-tan.f64 99.6

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

4. Applied egg-rr 99.6%

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

Alternative 10: 89.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(y + z\right)\\ t_1 := \cos \left(y + z\right)\\ \mathbf{if}\;a \leq -0.00011:\\ \;\;\;\;x + \mathsf{fma}\left(\frac{1}{t\_1}, t\_0, -\tan a\right)\\ \mathbf{elif}\;a \leq 3.7 \cdot 10^{-13}:\\ \;\;\;\;x + \left(\frac{1}{\frac{1 - \tan y \cdot \tan z}{\tan y + \tan z}} - a\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{t\_0}{x \cdot t\_1} - \frac{\sin a}{\cos a \cdot x}, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (let* ((t_0 (sin (+ y z))) (t_1 (cos (+ y z))))
   (if (<= a -0.00011)
     (+ x (fma (/ 1.0 t_1) t_0 (- (tan a))))
     (if (<= a 3.7e-13)
       (+ x (- (/ 1.0 (/ (- 1.0 (* (tan y) (tan z))) (+ (tan y) (tan z)))) a))
       (fma x (- (/ t_0 (* x t_1)) (/ (sin a) (* (cos a) x))) x)))))

C

double code(double x, double y, double z, double a) {
	double t_0 = sin((y + z));
	double t_1 = cos((y + z));
	double tmp;
	if (a <= -0.00011) {
		tmp = x + fma((1.0 / t_1), t_0, -tan(a));
	} else if (a <= 3.7e-13) {
		tmp = x + ((1.0 / ((1.0 - (tan(y) * tan(z))) / (tan(y) + tan(z)))) - a);
	} else {
		tmp = fma(x, ((t_0 / (x * t_1)) - (sin(a) / (cos(a) * x))), x);
	}
	return tmp;
}

Julia

function code(x, y, z, a)
	t_0 = sin(Float64(y + z))
	t_1 = cos(Float64(y + z))
	tmp = 0.0
	if (a <= -0.00011)
		tmp = Float64(x + fma(Float64(1.0 / t_1), t_0, Float64(-tan(a))));
	elseif (a <= 3.7e-13)
		tmp = Float64(x + Float64(Float64(1.0 / Float64(Float64(1.0 - Float64(tan(y) * tan(z))) / Float64(tan(y) + tan(z)))) - a));
	else
		tmp = fma(x, Float64(Float64(t_0 / Float64(x * t_1)) - Float64(sin(a) / Float64(cos(a) * x))), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, a_] := Block[{t$95$0 = N[Sin[N[(y + z), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(y + z), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[a, -0.00011], N[(x + N[(N[(1.0 / t$95$1), $MachinePrecision] * t$95$0 + (-N[Tan[a], $MachinePrecision])), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 3.7e-13], N[(x + N[(N[(1.0 / N[(N[(1.0 - N[(N[Tan[y], $MachinePrecision] * N[Tan[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Tan[y], $MachinePrecision] + N[Tan[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision]), $MachinePrecision], N[(x * N[(N[(t$95$0 / N[(x * t$95$1), $MachinePrecision]), $MachinePrecision] - N[(N[Sin[a], $MachinePrecision] / N[(N[Cos[a], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if a < -1.10000000000000004e-4

   1. Initial program 82.6%

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

      1. sub-neg N/A

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

      2. tan-quot N/A

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

      3. clear-num N/A

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

      4. associate-/r/ N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. /-lowering-/.f64 N/A

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

      7. cos-lowering-cos.f64 N/A

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

      8. +-lowering-+.f64 N/A

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

      9. sin-lowering-sin.f64 N/A

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

      10. +-lowering-+.f64 N/A

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

      11. neg-lowering-neg.f64 N/A

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

      12. tan-lowering-tan.f64 82.6

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

   4. Applied egg-rr 82.6%

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

   if -1.10000000000000004e-4 < a < 3.69999999999999989e-13

   1. Initial program 82.5%

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

   3. Taylor expanded in a around 0

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

   5. Simplified 82.5%

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

      1. tan-sum N/A

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

      2. clear-num N/A

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

      3. /-lowering-/.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. --lowering--.f64 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. tan-lowering-tan.f64 N/A

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

      8. tan-lowering-tan.f64 N/A

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

      9. +-lowering-+.f64 N/A

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

      10. tan-lowering-tan.f64 N/A

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

      11. tan-lowering-tan.f64 99.8

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

   7. Applied egg-rr 99.8%

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

   if 3.69999999999999989e-13 < a

   1. Initial program 82.0%

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

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

      3. *-rgt-identity N/A

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

      4. +-commutative N/A

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

      5. associate-/l/ N/A

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

      6. associate-/l/ N/A

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

      7. div-sub N/A

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

      8. accelerator-lowering-fma.f64 N/A

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

   5. Simplified 82.1%

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

4. Final simplification 91.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -0.00011:\\ \;\;\;\;x + \mathsf{fma}\left(\frac{1}{\cos \left(y + z\right)}, \sin \left(y + z\right), -\tan a\right)\\ \mathbf{elif}\;a \leq 3.7 \cdot 10^{-13}:\\ \;\;\;\;x + \left(\frac{1}{\frac{1 - \tan y \cdot \tan z}{\tan y + \tan z}} - a\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{\sin \left(y + z\right)}{x \cdot \cos \left(y + z\right)} - \frac{\sin a}{\cos a \cdot x}, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 68.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x + \left(\tan y - \tan a\right)\\ \mathbf{if}\;\tan a \leq -2 \cdot 10^{-32}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\tan a \leq 6.5 \cdot 10^{-57}:\\ \;\;\;\;x + \left(\tan \left(y + z\right) - a\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (let* ((t_0 (+ x (- (tan y) (tan a)))))
   (if (<= (tan a) -2e-32)
     t_0
     (if (<= (tan a) 6.5e-57) (+ x (- (tan (+ y z)) a)) t_0))))

C

double code(double x, double y, double z, double a) {
	double t_0 = x + (tan(y) - tan(a));
	double tmp;
	if (tan(a) <= -2e-32) {
		tmp = t_0;
	} else if (tan(a) <= 6.5e-57) {
		tmp = x + (tan((y + z)) - a);
	} else {
		tmp = t_0;
	}
	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) :: t_0
    real(8) :: tmp
    t_0 = x + (tan(y) - tan(a))
    if (tan(a) <= (-2d-32)) then
        tmp = t_0
    else if (tan(a) <= 6.5d-57) then
        tmp = x + (tan((y + z)) - a)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double a) {
	double t_0 = x + (Math.tan(y) - Math.tan(a));
	double tmp;
	if (Math.tan(a) <= -2e-32) {
		tmp = t_0;
	} else if (Math.tan(a) <= 6.5e-57) {
		tmp = x + (Math.tan((y + z)) - a);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z, a):
	t_0 = x + (math.tan(y) - math.tan(a))
	tmp = 0
	if math.tan(a) <= -2e-32:
		tmp = t_0
	elif math.tan(a) <= 6.5e-57:
		tmp = x + (math.tan((y + z)) - a)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z, a)
	t_0 = Float64(x + Float64(tan(y) - tan(a)))
	tmp = 0.0
	if (tan(a) <= -2e-32)
		tmp = t_0;
	elseif (tan(a) <= 6.5e-57)
		tmp = Float64(x + Float64(tan(Float64(y + z)) - a));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, a)
	t_0 = x + (tan(y) - tan(a));
	tmp = 0.0;
	if (tan(a) <= -2e-32)
		tmp = t_0;
	elseif (tan(a) <= 6.5e-57)
		tmp = x + (tan((y + z)) - a);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, a_] := Block[{t$95$0 = N[(x + N[(N[Tan[y], $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Tan[a], $MachinePrecision], -2e-32], t$95$0, If[LessEqual[N[Tan[a], $MachinePrecision], 6.5e-57], N[(x + N[(N[Tan[N[(y + z), $MachinePrecision]], $MachinePrecision] - a), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if (tan.f64 a) < -2.00000000000000011e-32 or 6.49999999999999992e-57 < (tan.f64 a)

   1. Initial program 81.9%

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

   3. Taylor expanded in y around inf

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

   5. Simplified 64.3%

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

   if -2.00000000000000011e-32 < (tan.f64 a) < 6.49999999999999992e-57

   1. Initial program 83.0%

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

   3. Taylor expanded in a around 0

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

   5. Simplified 83.0%

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

Alternative 12: 79.1% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 82.4%

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

   1. sub-neg N/A

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

   2. tan-quot N/A

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

   3. clear-num N/A

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

   4. associate-/r/ N/A

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

   5. accelerator-lowering-fma.f64 N/A

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

   6. /-lowering-/.f64 N/A

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

   7. cos-lowering-cos.f64 N/A

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

   8. +-lowering-+.f64 N/A

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

   9. sin-lowering-sin.f64 N/A

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

   10. +-lowering-+.f64 N/A

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

   11. neg-lowering-neg.f64 N/A

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

   12. tan-lowering-tan.f64 82.4

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

4. Applied egg-rr 82.4%

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

Alternative 13: 79.1% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 82.4%

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

   1. sub-neg N/A

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

   2. associate-+r+ N/A

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

   3. +-commutative N/A

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

   4. tan-quot N/A

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

   5. distribute-neg-frac2 N/A

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

   6. div-inv N/A

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

   7. accelerator-lowering-fma.f64 N/A

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

   8. sin-lowering-sin.f64 N/A

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

   9. /-lowering-/.f64 N/A

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

   10. neg-lowering-neg.f64 N/A

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

   11. cos-lowering-cos.f64 N/A

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

   12. +-lowering-+.f64 N/A

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

   13. tan-lowering-tan.f64 N/A

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

   14. +-lowering-+.f64 82.4

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

4. Applied egg-rr 82.4%

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

5. Final simplification 82.4%

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

Alternative 14: 50.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\tan a \leq -0.05:\\ \;\;\;\;x\\ \mathbf{elif}\;\tan a \leq 5 \cdot 10^{-12}:\\ \;\;\;\;x + \left(\tan \left(y + z\right) - a\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (if (<= (tan a) -0.05) x (if (<= (tan a) 5e-12) (+ x (- (tan (+ y z)) a)) x)))

C

double code(double x, double y, double z, double a) {
	double tmp;
	if (tan(a) <= -0.05) {
		tmp = x;
	} else if (tan(a) <= 5e-12) {
		tmp = x + (tan((y + z)) - a);
	} else {
		tmp = 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 (tan(a) <= (-0.05d0)) then
        tmp = x
    else if (tan(a) <= 5d-12) then
        tmp = x + (tan((y + z)) - a)
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double a) {
	double tmp;
	if (Math.tan(a) <= -0.05) {
		tmp = x;
	} else if (Math.tan(a) <= 5e-12) {
		tmp = x + (Math.tan((y + z)) - a);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, a):
	tmp = 0
	if math.tan(a) <= -0.05:
		tmp = x
	elif math.tan(a) <= 5e-12:
		tmp = x + (math.tan((y + z)) - a)
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, a)
	tmp = 0.0
	if (tan(a) <= -0.05)
		tmp = x;
	elseif (tan(a) <= 5e-12)
		tmp = Float64(x + Float64(tan(Float64(y + z)) - a));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, a)
	tmp = 0.0;
	if (tan(a) <= -0.05)
		tmp = x;
	elseif (tan(a) <= 5e-12)
		tmp = x + (tan((y + z)) - a);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, a_] := If[LessEqual[N[Tan[a], $MachinePrecision], -0.05], x, If[LessEqual[N[Tan[a], $MachinePrecision], 5e-12], N[(x + N[(N[Tan[N[(y + z), $MachinePrecision]], $MachinePrecision] - a), $MachinePrecision]), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if (tan.f64 a) < -0.050000000000000003 or 4.9999999999999997e-12 < (tan.f64 a)

   1. Initial program 82.0%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x} \]
   4. Step-by-step derivation

   5. Simplified 22.6%

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

   if -0.050000000000000003 < (tan.f64 a) < 4.9999999999999997e-12

   1. Initial program 82.8%

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

   3. Taylor expanded in a around 0

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

   5. Simplified 82.8%

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

Alternative 15: 40.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\tan a \leq -0.05:\\ \;\;\;\;x\\ \mathbf{elif}\;\tan a \leq 5 \cdot 10^{-12}:\\ \;\;\;\;x + \left(\tan y - a\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (if (<= (tan a) -0.05) x (if (<= (tan a) 5e-12) (+ x (- (tan y) a)) x)))

C

double code(double x, double y, double z, double a) {
	double tmp;
	if (tan(a) <= -0.05) {
		tmp = x;
	} else if (tan(a) <= 5e-12) {
		tmp = x + (tan(y) - a);
	} else {
		tmp = 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 (tan(a) <= (-0.05d0)) then
        tmp = x
    else if (tan(a) <= 5d-12) then
        tmp = x + (tan(y) - a)
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double a) {
	double tmp;
	if (Math.tan(a) <= -0.05) {
		tmp = x;
	} else if (Math.tan(a) <= 5e-12) {
		tmp = x + (Math.tan(y) - a);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, a):
	tmp = 0
	if math.tan(a) <= -0.05:
		tmp = x
	elif math.tan(a) <= 5e-12:
		tmp = x + (math.tan(y) - a)
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, a)
	tmp = 0.0
	if (tan(a) <= -0.05)
		tmp = x;
	elseif (tan(a) <= 5e-12)
		tmp = Float64(x + Float64(tan(y) - a));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, a)
	tmp = 0.0;
	if (tan(a) <= -0.05)
		tmp = x;
	elseif (tan(a) <= 5e-12)
		tmp = x + (tan(y) - a);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, a_] := If[LessEqual[N[Tan[a], $MachinePrecision], -0.05], x, If[LessEqual[N[Tan[a], $MachinePrecision], 5e-12], N[(x + N[(N[Tan[y], $MachinePrecision] - a), $MachinePrecision]), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if (tan.f64 a) < -0.050000000000000003 or 4.9999999999999997e-12 < (tan.f64 a)

   1. Initial program 82.0%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x} \]
   4. Step-by-step derivation

   5. Simplified 22.6%

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

   if -0.050000000000000003 < (tan.f64 a) < 4.9999999999999997e-12

   1. Initial program 82.8%

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

   3. Taylor expanded in a around 0

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

   5. Simplified 82.8%

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

   6. Taylor expanded in y around inf

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

   8. Simplified 66.2%

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

Alternative 16: 69.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 4.5 \cdot 10^{-18}:\\ \;\;\;\;x + \left(\tan y - \tan a\right)\\ \mathbf{else}:\\ \;\;\;\;\tan z + \left(x - \tan a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (if (<= z 4.5e-18) (+ x (- (tan y) (tan a))) (+ (tan z) (- x (tan a)))))

C

double code(double x, double y, double z, double a) {
	double tmp;
	if (z <= 4.5e-18) {
		tmp = x + (tan(y) - tan(a));
	} else {
		tmp = tan(z) + (x - tan(a));
	}
	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 (z <= 4.5d-18) then
        tmp = x + (tan(y) - tan(a))
    else
        tmp = tan(z) + (x - tan(a))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double a) {
	double tmp;
	if (z <= 4.5e-18) {
		tmp = x + (Math.tan(y) - Math.tan(a));
	} else {
		tmp = Math.tan(z) + (x - Math.tan(a));
	}
	return tmp;
}

Python

def code(x, y, z, a):
	tmp = 0
	if z <= 4.5e-18:
		tmp = x + (math.tan(y) - math.tan(a))
	else:
		tmp = math.tan(z) + (x - math.tan(a))
	return tmp

Julia

function code(x, y, z, a)
	tmp = 0.0
	if (z <= 4.5e-18)
		tmp = Float64(x + Float64(tan(y) - tan(a)));
	else
		tmp = Float64(tan(z) + Float64(x - tan(a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, a)
	tmp = 0.0;
	if (z <= 4.5e-18)
		tmp = x + (tan(y) - tan(a));
	else
		tmp = tan(z) + (x - tan(a));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, a_] := If[LessEqual[z, 4.5e-18], N[(x + N[(N[Tan[y], $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Tan[z], $MachinePrecision] + N[(x - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 4.49999999999999994e-18

   1. Initial program 87.9%

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

   3. Taylor expanded in y around inf

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

   5. Simplified 79.1%

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

   if 4.49999999999999994e-18 < z

   1. Initial program 65.9%

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

   3. Taylor expanded in y around 0

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

      1. --lowering--.f64 N/A

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

      2. /-lowering-/.f64 N/A

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

      3. sin-lowering-sin.f64 N/A

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

      4. cos-lowering-cos.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

      6. sin-lowering-sin.f64 N/A

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

      7. cos-lowering-cos.f64 66.0

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

   5. Simplified 66.0%

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

      1. +-commutative N/A

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

      2. tan-quot N/A

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

      3. tan-quot N/A

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

      4. associate-+l- N/A

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

      5. --lowering--.f64 N/A

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

      6. tan-lowering-tan.f64 N/A

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

      7. --lowering--.f64 N/A

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

      8. tan-lowering-tan.f64 65.9

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

   7. Applied egg-rr 65.9%

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

4. Final simplification 75.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 4.5 \cdot 10^{-18}:\\ \;\;\;\;x + \left(\tan y - \tan a\right)\\ \mathbf{else}:\\ \;\;\;\;\tan z + \left(x - \tan a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 69.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 4.5 \cdot 10^{-18}:\\ \;\;\;\;x + \left(\tan y - \tan a\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(\tan z - \tan a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (if (<= z 4.5e-18) (+ x (- (tan y) (tan a))) (+ x (- (tan z) (tan a)))))

C

double code(double x, double y, double z, double a) {
	double tmp;
	if (z <= 4.5e-18) {
		tmp = x + (tan(y) - tan(a));
	} else {
		tmp = x + (tan(z) - tan(a));
	}
	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 (z <= 4.5d-18) then
        tmp = x + (tan(y) - tan(a))
    else
        tmp = x + (tan(z) - tan(a))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double a) {
	double tmp;
	if (z <= 4.5e-18) {
		tmp = x + (Math.tan(y) - Math.tan(a));
	} else {
		tmp = x + (Math.tan(z) - Math.tan(a));
	}
	return tmp;
}

Python

def code(x, y, z, a):
	tmp = 0
	if z <= 4.5e-18:
		tmp = x + (math.tan(y) - math.tan(a))
	else:
		tmp = x + (math.tan(z) - math.tan(a))
	return tmp

Julia

function code(x, y, z, a)
	tmp = 0.0
	if (z <= 4.5e-18)
		tmp = Float64(x + Float64(tan(y) - tan(a)));
	else
		tmp = Float64(x + Float64(tan(z) - tan(a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, a)
	tmp = 0.0;
	if (z <= 4.5e-18)
		tmp = x + (tan(y) - tan(a));
	else
		tmp = x + (tan(z) - tan(a));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, a_] := If[LessEqual[z, 4.5e-18], N[(x + N[(N[Tan[y], $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[Tan[z], $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 4.49999999999999994e-18

   1. Initial program 87.9%

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

   3. Taylor expanded in y around inf

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

   5. Simplified 79.1%

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

   if 4.49999999999999994e-18 < z

   1. Initial program 65.9%

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

   3. Taylor expanded in y around 0

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

   5. Simplified 65.9%

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

Alternative 18: 79.2% 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]

Derivation

1. Initial program 82.4%

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

Alternative 19: 34.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y + z \leq -1 \cdot 10^{-13}:\\ \;\;\;\;x + \left(\tan y - a\right)\\ \mathbf{elif}\;y + z \leq 20000000:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x + \left(\tan z - a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (if (<= (+ y z) -1e-13)
   (+ x (- (tan y) a))
   (if (<= (+ y z) 20000000.0) x (+ x (- (tan z) a)))))

C

double code(double x, double y, double z, double a) {
	double tmp;
	if ((y + z) <= -1e-13) {
		tmp = x + (tan(y) - a);
	} else if ((y + z) <= 20000000.0) {
		tmp = x;
	} else {
		tmp = x + (tan(z) - a);
	}
	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) <= (-1d-13)) then
        tmp = x + (tan(y) - a)
    else if ((y + z) <= 20000000.0d0) then
        tmp = x
    else
        tmp = x + (tan(z) - a)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double a) {
	double tmp;
	if ((y + z) <= -1e-13) {
		tmp = x + (Math.tan(y) - a);
	} else if ((y + z) <= 20000000.0) {
		tmp = x;
	} else {
		tmp = x + (Math.tan(z) - a);
	}
	return tmp;
}

Python

def code(x, y, z, a):
	tmp = 0
	if (y + z) <= -1e-13:
		tmp = x + (math.tan(y) - a)
	elif (y + z) <= 20000000.0:
		tmp = x
	else:
		tmp = x + (math.tan(z) - a)
	return tmp

Julia

function code(x, y, z, a)
	tmp = 0.0
	if (Float64(y + z) <= -1e-13)
		tmp = Float64(x + Float64(tan(y) - a));
	elseif (Float64(y + z) <= 20000000.0)
		tmp = x;
	else
		tmp = Float64(x + Float64(tan(z) - a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, a)
	tmp = 0.0;
	if ((y + z) <= -1e-13)
		tmp = x + (tan(y) - a);
	elseif ((y + z) <= 20000000.0)
		tmp = x;
	else
		tmp = x + (tan(z) - a);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, a_] := If[LessEqual[N[(y + z), $MachinePrecision], -1e-13], N[(x + N[(N[Tan[y], $MachinePrecision] - a), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(y + z), $MachinePrecision], 20000000.0], x, N[(x + N[(N[Tan[z], $MachinePrecision] - a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 y z) < -1e-13

   1. Initial program 78.3%

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

   3. Taylor expanded in a around 0

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

   5. Simplified 46.1%

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

   6. Taylor expanded in y around inf

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

   8. Simplified 37.2%

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

   if -1e-13 < (+.f64 y z) < 2e7

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x} \]
   4. Step-by-step derivation

   5. Simplified 58.8%

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

   if 2e7 < (+.f64 y z)

   1. Initial program 73.5%

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

   3. Taylor expanded in a around 0

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

   5. Simplified 36.1%

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

   6. Taylor expanded in y around 0

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

   8. Simplified 27.3%

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

Alternative 20: 31.6% accurate, 210.0× speedup

Math

\[\begin{array}{l} \\ x \end{array} \]

FPCore

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

C

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

Java

public static double code(double x, double y, double z, double a) {
	return x;
}

Python

def code(x, y, z, a):
	return x

Julia

function code(x, y, z, a)
	return x
end

MATLAB

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

Wolfram

code[x_, y_, z_, a_] := x

Derivation

1. Initial program 82.4%

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

3. Taylor expanded in x around inf

   \[\leadsto \color{blue}{x} \]
4. Step-by-step derivation

5. Simplified 33.1%

   \[\leadsto \color{blue}{x} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024199 
(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))))