tan-example (used to crash)

Percentage Accurate: 79.7% → 99.7%

Time: 33.8s

Alternatives: 11

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 79.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

function tmp = code(x, y, z, a)
	tmp = x + (((tan(y) + tan(z)) / (1.0 - ((sin(y) * sin(z)) / (cos(y) * cos(z))))) - (sin(a) / cos(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[(N[Sin[y], $MachinePrecision] * N[Sin[z], $MachinePrecision]), $MachinePrecision] / N[(N[Cos[y], $MachinePrecision] * N[Cos[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Sin[a], $MachinePrecision] / N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 79.5%

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

   1. tan-sum 99.7%

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

   2. tan-quot 99.7%

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

   3. frac-sub 99.7%

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

4. Applied egg-rr 99.7%

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

   1. div-sub 99.7%

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

   2. times-frac 99.7%

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

   3. *-inverses 99.7%

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

   4. cancel-sign-sub-inv 99.7%

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

6. Simplified 99.7%

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

7. Taylor expanded in y around inf 99.7%

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

8. Final simplification 99.7%

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

Alternative 2: 99.7% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Python

def code(x, y, z, a):
	return x + (((math.tan(y) + math.tan(z)) / (1.0 - ((math.sin(y) * math.sin(z)) / (math.cos(y) * math.cos(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(Float64(sin(y) * sin(z)) / Float64(cos(y) * cos(z))))) - tan(a)))
end

MATLAB

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

Derivation

1. Initial program 79.5%

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

   1. tan-sum 99.7%

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

   2. div-inv 99.7%

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

4. Applied egg-rr 99.7%

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

   1. associate-*r/ 99.7%

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

   2. *-rgt-identity 99.7%

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

6. Simplified 99.7%

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

7. Taylor expanded in y around inf 99.7%

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

8. Final simplification 99.7%

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

Alternative 3: 88.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\tan a \leq -2 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(\sin \left(y + z\right), \frac{1}{\cos \left(y + z\right)}, x - \tan a\right)\\ \mathbf{elif}\;\tan a \leq 5 \cdot 10^{-82}:\\ \;\;\;\;x + \left(\frac{\tan y + \tan z}{1 - \tan y \cdot \tan z} - a\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(\tan \left(y + z\right) - \frac{\sin a}{\cos a}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (if (<= (tan a) -2e-7)
   (fma (sin (+ y z)) (/ 1.0 (cos (+ y z))) (- x (tan a)))
   (if (<= (tan a) 5e-82)
     (+ x (- (/ (+ (tan y) (tan z)) (- 1.0 (* (tan y) (tan z)))) a))
     (+ x (- (tan (+ y z)) (/ (sin a) (cos a)))))))

C

double code(double x, double y, double z, double a) {
	double tmp;
	if (tan(a) <= -2e-7) {
		tmp = fma(sin((y + z)), (1.0 / cos((y + z))), (x - tan(a)));
	} else if (tan(a) <= 5e-82) {
		tmp = x + (((tan(y) + tan(z)) / (1.0 - (tan(y) * tan(z)))) - a);
	} else {
		tmp = x + (tan((y + z)) - (sin(a) / cos(a)));
	}
	return tmp;
}

Julia

function code(x, y, z, a)
	tmp = 0.0
	if (tan(a) <= -2e-7)
		tmp = fma(sin(Float64(y + z)), Float64(1.0 / cos(Float64(y + z))), Float64(x - tan(a)));
	elseif (tan(a) <= 5e-82)
		tmp = Float64(x + Float64(Float64(Float64(tan(y) + tan(z)) / Float64(1.0 - Float64(tan(y) * tan(z)))) - a));
	else
		tmp = Float64(x + Float64(tan(Float64(y + z)) - Float64(sin(a) / cos(a))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, a_] := If[LessEqual[N[Tan[a], $MachinePrecision], -2e-7], N[(N[Sin[N[(y + z), $MachinePrecision]], $MachinePrecision] * N[(1.0 / N[Cos[N[(y + z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(x - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Tan[a], $MachinePrecision], 5e-82], 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], N[(x + N[(N[Tan[N[(y + z), $MachinePrecision]], $MachinePrecision] - N[(N[Sin[a], $MachinePrecision] / N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (tan.f64 a) < -1.9999999999999999e-7

   1. Initial program 78.0%

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

      1. +-commutative 78.0%

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

      2. associate-+l- 78.0%

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

      3. tan-quot 77.9%

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

      4. div-inv 78.0%

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

      5. fmm-def 78.2%

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

   4. Applied egg-rr 78.2%

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

   if -1.9999999999999999e-7 < (tan.f64 a) < 4.9999999999999998e-82

   1. Initial program 79.0%

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

      1. tan-sum 99.7%

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

      2. tan-quot 99.7%

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

      3. frac-sub 99.7%

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

   4. Applied egg-rr 99.7%

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

      1. div-sub 99.7%

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

      2. times-frac 99.7%

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

      3. *-inverses 99.7%

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

      4. cancel-sign-sub-inv 99.7%

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

   6. Simplified 99.7%

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

   7. Taylor expanded in a around 0 99.7%

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

      1. mul-1-neg 99.7%

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

   9. Simplified 99.7%

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

   if 4.9999999999999998e-82 < (tan.f64 a)

   1. Initial program 81.9%

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

      1. tan-sum 99.7%

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

      2. tan-quot 99.7%

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

      3. frac-sub 99.7%

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

   4. Applied egg-rr 99.7%

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

      1. div-sub 99.7%

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

      2. times-frac 99.7%

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

      3. *-inverses 99.7%

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

      4. cancel-sign-sub-inv 99.7%

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

   6. Simplified 99.7%

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

      1. tan-sum 82.0%

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

      2. *-un-lft-identity 82.0%

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

   8. Applied egg-rr 82.0%

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

      1. *-lft-identity 82.0%

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

   10. Simplified 82.0%

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

4. Final simplification 89.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\tan a \leq -2 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(\sin \left(y + z\right), \frac{1}{\cos \left(y + z\right)}, x - \tan a\right)\\ \mathbf{elif}\;\tan a \leq 5 \cdot 10^{-82}:\\ \;\;\;\;x + \left(\frac{\tan y + \tan z}{1 - \tan y \cdot \tan z} - a\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(\tan \left(y + z\right) - \frac{\sin a}{\cos a}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.7% accurate, 0.3× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z, double a) {
	return x + (((tan(y) + tan(z)) / (1.0 - (tan(y) * tan(z)))) - (sin(a) / cos(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)))) - (sin(a) / cos(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.sin(a) / Math.cos(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.sin(a) / math.cos(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)))) - Float64(sin(a) / cos(a))))
end

MATLAB

function tmp = code(x, y, z, a)
	tmp = x + (((tan(y) + tan(z)) / (1.0 - (tan(y) * tan(z)))) - (sin(a) / cos(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[(N[Sin[a], $MachinePrecision] / N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 79.5%

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

   1. tan-sum 99.7%

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

   2. tan-quot 99.7%

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

   3. frac-sub 99.7%

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

4. Applied egg-rr 99.7%

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

   1. div-sub 99.7%

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

   2. times-frac 99.7%

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

   3. *-inverses 99.7%

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

   4. cancel-sign-sub-inv 99.7%

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

6. Simplified 99.7%

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

7. Final simplification 99.7%

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

Alternative 5: 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 79.5%

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

   1. tan-sum 99.7%

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

   2. div-inv 99.7%

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

4. Applied egg-rr 99.7%

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

   1. associate-*r/ 99.7%

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

   2. *-rgt-identity 99.7%

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

6. Simplified 99.7%

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

7. Final simplification 99.7%

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

Alternative 6: 79.7% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.5%

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

   1. tan-sum 99.7%

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

   2. tan-quot 99.7%

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

   3. frac-sub 99.7%

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

4. Applied egg-rr 99.7%

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

   1. div-sub 99.7%

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

   2. times-frac 99.7%

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

   3. *-inverses 99.7%

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

   4. cancel-sign-sub-inv 99.7%

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

6. Simplified 99.7%

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

   1. tan-sum 79.5%

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

   2. *-un-lft-identity 79.5%

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

8. Applied egg-rr 79.5%

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

   1. *-lft-identity 79.5%

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

10. Simplified 79.5%

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

11. Final simplification 79.5%

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

Alternative 7: 60.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y + z \leq 2 \cdot 10^{-6}:\\ \;\;\;\;x + \left(\tan y - \tan a\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\frac{\tan \left(y + z\right)}{x} + 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (if (<= (+ y z) 2e-6)
   (+ x (- (tan y) (tan a)))
   (* x (+ (/ (tan (+ y z)) x) 1.0))))

C

double code(double x, double y, double z, double a) {
	double tmp;
	if ((y + z) <= 2e-6) {
		tmp = x + (tan(y) - tan(a));
	} else {
		tmp = x * ((tan((y + z)) / x) + 1.0);
	}
	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) :: tmp
    if ((y + z) <= 2d-6) then
        tmp = x + (tan(y) - tan(a))
    else
        tmp = x * ((tan((y + z)) / x) + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double a) {
	double tmp;
	if ((y + z) <= 2e-6) {
		tmp = x + (Math.tan(y) - Math.tan(a));
	} else {
		tmp = x * ((Math.tan((y + z)) / x) + 1.0);
	}
	return tmp;
}

Python

def code(x, y, z, a):
	tmp = 0
	if (y + z) <= 2e-6:
		tmp = x + (math.tan(y) - math.tan(a))
	else:
		tmp = x * ((math.tan((y + z)) / x) + 1.0)
	return tmp

Julia

function code(x, y, z, a)
	tmp = 0.0
	if (Float64(y + z) <= 2e-6)
		tmp = Float64(x + Float64(tan(y) - tan(a)));
	else
		tmp = Float64(x * Float64(Float64(tan(Float64(y + z)) / x) + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, a)
	tmp = 0.0;
	if ((y + z) <= 2e-6)
		tmp = x + (tan(y) - tan(a));
	else
		tmp = x * ((tan((y + z)) / x) + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, a_] := If[LessEqual[N[(y + z), $MachinePrecision], 2e-6], N[(x + N[(N[Tan[y], $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(N[(N[Tan[N[(y + z), $MachinePrecision]], $MachinePrecision] / x), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 y z) < 1.99999999999999991e-6

   1. Initial program 85.1%

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

   3. Taylor expanded in z around 0 72.0%

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

      1. +-commutative 72.0%

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

      2. associate--l+ 72.0%

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

   5. Simplified 72.0%

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

      1. tan-quot 72.0%

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

      2. tan-quot 72.0%

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

      3. associate-+r- 72.0%

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

   7. Applied egg-rr 72.0%

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

      1. +-commutative 72.0%

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

      2. associate--l+ 72.0%

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

   9. Simplified 72.0%

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

   if 1.99999999999999991e-6 < (+.f64 y z)

   1. Initial program 69.3%

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

   3. Taylor expanded in x around inf 69.0%

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

      1. associate--l+ 69.0%

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

      2. +-commutative 69.0%

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

      3. *-commutative 69.0%

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

      4. +-commutative 69.0%

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

      5. associate-/r* 69.1%

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

   5. Simplified 69.1%

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

   6. Taylor expanded in a around 0 45.0%

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

      1. associate-/r* 45.0%

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

      2. remove-double-neg 45.0%

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

      3. mul-1-neg 45.0%

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

      4. sub-neg 45.0%

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

      5. remove-double-neg 45.0%

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

      6. mul-1-neg 45.0%

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

      7. sub-neg 45.0%

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

      8. sub-neg 45.0%

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

      9. mul-1-neg 45.0%

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

      10. remove-double-neg 45.0%

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

      11. +-commutative 45.0%

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

      12. remove-double-neg 45.0%

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

      13. mul-1-neg 45.0%

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

      14. sub-neg 45.0%

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

      15. sub-neg 45.0%

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

      16. mul-1-neg 45.0%

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

   8. Simplified 45.0%

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

      1. *-un-lft-identity 45.0%

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

      2. associate-/r* 45.0%

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

      3. quot-tan 45.1%

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

   10. Applied egg-rr 45.1%

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

       1. *-lft-identity 45.1%

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

       2. +-commutative 45.1%

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

   12. Simplified 45.1%

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

4. Final simplification 62.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y + z \leq 2 \cdot 10^{-6}:\\ \;\;\;\;x + \left(\tan y - \tan a\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\frac{\tan \left(y + z\right)}{x} + 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 79.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.5%

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

3. Final simplification 79.5%

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

Alternative 9: 51.0% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

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)) / x) + 1.0d0)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.5%

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

3. Taylor expanded in x around inf 79.2%

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

   1. associate--l+ 79.2%

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

   2. +-commutative 79.2%

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

   3. *-commutative 79.2%

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

   4. +-commutative 79.2%

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

   5. associate-/r* 79.3%

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

5. Simplified 79.3%

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

6. Taylor expanded in a around 0 51.2%

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

   1. associate-/r* 51.2%

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

   2. remove-double-neg 51.2%

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

   3. mul-1-neg 51.2%

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

   4. sub-neg 51.2%

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

   5. remove-double-neg 51.2%

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

   6. mul-1-neg 51.2%

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

   7. sub-neg 51.2%

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

   8. sub-neg 51.2%

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

   9. mul-1-neg 51.2%

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

   10. remove-double-neg 51.2%

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

   11. +-commutative 51.2%

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

   12. remove-double-neg 51.2%

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

   13. mul-1-neg 51.2%

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

   14. sub-neg 51.2%

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

   15. sub-neg 51.2%

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

   16. mul-1-neg 51.2%

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

8. Simplified 51.2%

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

   1. *-un-lft-identity 51.2%

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

   2. associate-/r* 51.2%

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

   3. quot-tan 51.3%

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

10. Applied egg-rr 51.3%

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

    1. *-lft-identity 51.3%

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

    2. +-commutative 51.3%

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

12. Simplified 51.3%

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

13. Final simplification 51.3%

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

Alternative 10: 2.7% accurate, 207.0× speedup

Math

\[\begin{array}{l} \\ -1 \end{array} \]

FPCore

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

C

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

Fortran

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

Java

public static double code(double x, double y, double z, double a) {
	return -1.0;
}

Python

def code(x, y, z, a):
	return -1.0

Julia

function code(x, y, z, a)
	return -1.0
end

MATLAB

function tmp = code(x, y, z, a)
	tmp = -1.0;
end

Wolfram

code[x_, y_, z_, a_] := -1.0

Derivation

1. Initial program 79.5%

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

   1. expm1-log1p-u 76.2%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x + \left(\tan \left(y + z\right) - \tan a\right)\right)\right)} \]

   2. expm1-undefine 76.2%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(x + \left(\tan \left(y + z\right) - \tan a\right)\right)} - 1} \]

   3. +-commutative 76.2%

      \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\left(\tan \left(y + z\right) - \tan a\right) + x}\right)} - 1 \]

   4. associate-+l- 76.2%

      \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\tan \left(y + z\right) - \left(\tan a - x\right)}\right)} - 1 \]

4. Applied egg-rr 76.2%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\tan \left(y + z\right) - \left(\tan a - x\right)\right)} - 1} \]
5. Step-by-step derivation

   1. expm1-define 76.1%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\tan \left(y + z\right) - \left(\tan a - x\right)\right)\right)} \]

   2. associate--r- 76.2%

      \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\left(\tan \left(y + z\right) - \tan a\right) + x}\right)\right) \]

   3. +-commutative 76.2%

      \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\left(\tan \color{blue}{\left(z + y\right)} - \tan a\right) + x\right)\right) \]

6. Simplified 76.2%

   \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\tan \left(z + y\right) - \tan a\right) + x\right)\right)} \]

7. Taylor expanded in x around inf 22.1%

   \[\leadsto \mathsf{expm1}\left(\color{blue}{-1 \cdot \log \left(\frac{1}{x}\right)}\right) \]
8. Step-by-step derivation

   1. mul-1-neg 22.1%

      \[\leadsto \mathsf{expm1}\left(\color{blue}{-\log \left(\frac{1}{x}\right)}\right) \]

   2. log-rec 22.1%

      \[\leadsto \mathsf{expm1}\left(-\color{blue}{\left(-\log x\right)}\right) \]

   3. remove-double-neg 22.1%

      \[\leadsto \mathsf{expm1}\left(\color{blue}{\log x}\right) \]

9. Simplified 22.1%

   \[\leadsto \mathsf{expm1}\left(\color{blue}{\log x}\right) \]

10. Taylor expanded in x around 0 2.7%

    \[\leadsto \color{blue}{-1} \]

11. Final simplification 2.7%

    \[\leadsto -1 \]
12. Add Preprocessing

Alternative 11: 31.9% accurate, 207.0× 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 79.5%

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

3. Taylor expanded in x around inf 33.2%

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

4. Final simplification 33.2%

   \[\leadsto x \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024112 
(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))))