tan-example (used to crash)

Percentage Accurate: 79.1% → 99.7%

Time: 23.8s

Alternatives: 12

Speedup: 1.0×

• Unsound rule application detected in e-graph. Results may not be sound.

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: 79.1% 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

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 83.1%

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

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

   3. fma-neg 99.7%

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

3. Applied egg-rr 99.7%

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

4. Final simplification 99.7%

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

Alternative 2: 89.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan \left(y + z\right)\\ \mathbf{if}\;\tan a \leq -0.002:\\ \;\;\;\;x + \left(t_0 - \tan a\right)\\ \mathbf{elif}\;\tan a \leq 0.0002:\\ \;\;\;\;x + \left(\left(\tan y + \tan z\right) \cdot \frac{1}{1 - \tan y \cdot \tan z} - a\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{t_0 + \left(x - \tan a\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (let* ((t_0 (tan (+ y z))))
   (if (<= (tan a) -0.002)
     (+ x (- t_0 (tan a)))
     (if (<= (tan a) 0.0002)
       (+ x (- (* (+ (tan y) (tan z)) (/ 1.0 (- 1.0 (* (tan y) (tan z))))) a))
       (log (exp (+ t_0 (- x (tan a)))))))))

C

double code(double x, double y, double z, double a) {
	double t_0 = tan((y + z));
	double tmp;
	if (tan(a) <= -0.002) {
		tmp = x + (t_0 - tan(a));
	} else if (tan(a) <= 0.0002) {
		tmp = x + (((tan(y) + tan(z)) * (1.0 / (1.0 - (tan(y) * tan(z))))) - a);
	} else {
		tmp = log(exp((t_0 + (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) :: t_0
    real(8) :: tmp
    t_0 = tan((y + z))
    if (tan(a) <= (-0.002d0)) then
        tmp = x + (t_0 - tan(a))
    else if (tan(a) <= 0.0002d0) then
        tmp = x + (((tan(y) + tan(z)) * (1.0d0 / (1.0d0 - (tan(y) * tan(z))))) - a)
    else
        tmp = log(exp((t_0 + (x - tan(a)))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double a) {
	double t_0 = Math.tan((y + z));
	double tmp;
	if (Math.tan(a) <= -0.002) {
		tmp = x + (t_0 - Math.tan(a));
	} else if (Math.tan(a) <= 0.0002) {
		tmp = x + (((Math.tan(y) + Math.tan(z)) * (1.0 / (1.0 - (Math.tan(y) * Math.tan(z))))) - a);
	} else {
		tmp = Math.log(Math.exp((t_0 + (x - Math.tan(a)))));
	}
	return tmp;
}

Python

def code(x, y, z, a):
	t_0 = math.tan((y + z))
	tmp = 0
	if math.tan(a) <= -0.002:
		tmp = x + (t_0 - math.tan(a))
	elif math.tan(a) <= 0.0002:
		tmp = x + (((math.tan(y) + math.tan(z)) * (1.0 / (1.0 - (math.tan(y) * math.tan(z))))) - a)
	else:
		tmp = math.log(math.exp((t_0 + (x - math.tan(a)))))
	return tmp

Julia

function code(x, y, z, a)
	t_0 = tan(Float64(y + z))
	tmp = 0.0
	if (tan(a) <= -0.002)
		tmp = Float64(x + Float64(t_0 - tan(a)));
	elseif (tan(a) <= 0.0002)
		tmp = Float64(x + Float64(Float64(Float64(tan(y) + tan(z)) * Float64(1.0 / Float64(1.0 - Float64(tan(y) * tan(z))))) - a));
	else
		tmp = log(exp(Float64(t_0 + Float64(x - tan(a)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, a)
	t_0 = tan((y + z));
	tmp = 0.0;
	if (tan(a) <= -0.002)
		tmp = x + (t_0 - tan(a));
	elseif (tan(a) <= 0.0002)
		tmp = x + (((tan(y) + tan(z)) * (1.0 / (1.0 - (tan(y) * tan(z))))) - a);
	else
		tmp = log(exp((t_0 + (x - tan(a)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, a_] := Block[{t$95$0 = N[Tan[N[(y + z), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[Tan[a], $MachinePrecision], -0.002], N[(x + N[(t$95$0 - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Tan[a], $MachinePrecision], 0.0002], N[(x + N[(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]), $MachinePrecision] - a), $MachinePrecision]), $MachinePrecision], N[Log[N[Exp[N[(t$95$0 + N[(x - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (tan.f64 a) < -2e-3

   1. Initial program 79.1%

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

   if -2e-3 < (tan.f64 a) < 2.0000000000000001e-4

   1. Initial program 82.1%

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

   2. Taylor expanded in a around 0 82.1%

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

      1. tan-sum 99.7%

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

      2. div-inv 99.7%

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

   4. Applied egg-rr 99.7%

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

   if 2.0000000000000001e-4 < (tan.f64 a)

   1. Initial program 89.6%

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

      1. add-log-exp 89.7%

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

      2. associate-+r- 89.5%

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

      3. +-commutative 89.5%

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

      4. associate--l+ 89.7%

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

   3. Applied egg-rr 89.7%

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

4. Final simplification 91.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\tan a \leq -0.002:\\ \;\;\;\;x + \left(\tan \left(y + z\right) - \tan a\right)\\ \mathbf{elif}\;\tan a \leq 0.0002:\\ \;\;\;\;x + \left(\left(\tan y + \tan z\right) \cdot \frac{1}{1 - \tan y \cdot \tan z} - a\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{\tan \left(y + z\right) + \left(x - \tan a\right)}\right)\\ \end{array} \]

Alternative 3: 89.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan \left(y + z\right)\\ \mathbf{if}\;\tan a \leq -0.002:\\ \;\;\;\;x + \left(t_0 - \tan a\right)\\ \mathbf{elif}\;\tan a \leq 0.0002:\\ \;\;\;\;x + \left(\frac{\tan y + \tan z}{1 - \tan y \cdot \tan z} - a\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{t_0 + \left(x - \tan a\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (let* ((t_0 (tan (+ y z))))
   (if (<= (tan a) -0.002)
     (+ x (- t_0 (tan a)))
     (if (<= (tan a) 0.0002)
       (+ x (- (/ (+ (tan y) (tan z)) (- 1.0 (* (tan y) (tan z)))) a))
       (log (exp (+ t_0 (- x (tan a)))))))))

C

double code(double x, double y, double z, double a) {
	double t_0 = tan((y + z));
	double tmp;
	if (tan(a) <= -0.002) {
		tmp = x + (t_0 - tan(a));
	} else if (tan(a) <= 0.0002) {
		tmp = x + (((tan(y) + tan(z)) / (1.0 - (tan(y) * tan(z)))) - a);
	} else {
		tmp = log(exp((t_0 + (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) :: t_0
    real(8) :: tmp
    t_0 = tan((y + z))
    if (tan(a) <= (-0.002d0)) then
        tmp = x + (t_0 - tan(a))
    else if (tan(a) <= 0.0002d0) then
        tmp = x + (((tan(y) + tan(z)) / (1.0d0 - (tan(y) * tan(z)))) - a)
    else
        tmp = log(exp((t_0 + (x - tan(a)))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double a) {
	double t_0 = Math.tan((y + z));
	double tmp;
	if (Math.tan(a) <= -0.002) {
		tmp = x + (t_0 - Math.tan(a));
	} else if (Math.tan(a) <= 0.0002) {
		tmp = x + (((Math.tan(y) + Math.tan(z)) / (1.0 - (Math.tan(y) * Math.tan(z)))) - a);
	} else {
		tmp = Math.log(Math.exp((t_0 + (x - Math.tan(a)))));
	}
	return tmp;
}

Python

def code(x, y, z, a):
	t_0 = math.tan((y + z))
	tmp = 0
	if math.tan(a) <= -0.002:
		tmp = x + (t_0 - math.tan(a))
	elif math.tan(a) <= 0.0002:
		tmp = x + (((math.tan(y) + math.tan(z)) / (1.0 - (math.tan(y) * math.tan(z)))) - a)
	else:
		tmp = math.log(math.exp((t_0 + (x - math.tan(a)))))
	return tmp

Julia

function code(x, y, z, a)
	t_0 = tan(Float64(y + z))
	tmp = 0.0
	if (tan(a) <= -0.002)
		tmp = Float64(x + Float64(t_0 - tan(a)));
	elseif (tan(a) <= 0.0002)
		tmp = Float64(x + Float64(Float64(Float64(tan(y) + tan(z)) / Float64(1.0 - Float64(tan(y) * tan(z)))) - a));
	else
		tmp = log(exp(Float64(t_0 + Float64(x - tan(a)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, a)
	t_0 = tan((y + z));
	tmp = 0.0;
	if (tan(a) <= -0.002)
		tmp = x + (t_0 - tan(a));
	elseif (tan(a) <= 0.0002)
		tmp = x + (((tan(y) + tan(z)) / (1.0 - (tan(y) * tan(z)))) - a);
	else
		tmp = log(exp((t_0 + (x - tan(a)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, a_] := Block[{t$95$0 = N[Tan[N[(y + z), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[Tan[a], $MachinePrecision], -0.002], N[(x + N[(t$95$0 - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Tan[a], $MachinePrecision], 0.0002], 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[Log[N[Exp[N[(t$95$0 + N[(x - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (tan.f64 a) < -2e-3

   1. Initial program 79.1%

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

   if -2e-3 < (tan.f64 a) < 2.0000000000000001e-4

   1. Initial program 82.1%

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

   2. Taylor expanded in a around 0 82.1%

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

      1. tan-sum 99.7%

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

      2. div-inv 99.7%

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

   4. Applied egg-rr 99.7%

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

      1. associate-*r/ 99.7%

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

      2. *-rgt-identity 99.7%

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

   6. Simplified 99.7%

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

   if 2.0000000000000001e-4 < (tan.f64 a)

   1. Initial program 89.6%

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

      1. add-log-exp 89.7%

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

      2. associate-+r- 89.5%

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

      3. +-commutative 89.5%

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

      4. associate--l+ 89.7%

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

   3. Applied egg-rr 89.7%

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

4. Final simplification 91.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\tan a \leq -0.002:\\ \;\;\;\;x + \left(\tan \left(y + z\right) - \tan a\right)\\ \mathbf{elif}\;\tan a \leq 0.0002:\\ \;\;\;\;x + \left(\frac{\tan y + \tan z}{1 - \tan y \cdot \tan z} - a\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{\tan \left(y + z\right) + \left(x - \tan a\right)}\right)\\ \end{array} \]

Alternative 4: 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 83.1%

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

   1. tan-sum 49.0%

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

   2. div-inv 49.0%

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

3. Applied egg-rr 99.7%

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

   1. associate-*r/ 49.0%

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

   2. *-rgt-identity 49.0%

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

5. Simplified 99.7%

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

6. Final simplification 99.7%

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

Alternative 5: 60.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\tan a \leq -0.002 \lor \neg \left(\tan a \leq 2 \cdot 10^{-8}\right):\\ \;\;\;\;x - \tan a\\ \mathbf{else}:\\ \;\;\;\;x + \left(\tan \left(y + z\right) - a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (if (or (<= (tan a) -0.002) (not (<= (tan a) 2e-8)))
   (- x (tan a))
   (+ x (- (tan (+ y z)) a))))

C

double code(double x, double y, double z, double a) {
	double tmp;
	if ((tan(a) <= -0.002) || !(tan(a) <= 2e-8)) {
		tmp = x - tan(a);
	} else {
		tmp = x + (tan((y + 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 ((tan(a) <= (-0.002d0)) .or. (.not. (tan(a) <= 2d-8))) then
        tmp = x - tan(a)
    else
        tmp = x + (tan((y + z)) - a)
    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) <= -0.002) || !(Math.tan(a) <= 2e-8)) {
		tmp = x - Math.tan(a);
	} else {
		tmp = x + (Math.tan((y + z)) - a);
	}
	return tmp;
}

Python

def code(x, y, z, a):
	tmp = 0
	if (math.tan(a) <= -0.002) or not (math.tan(a) <= 2e-8):
		tmp = x - math.tan(a)
	else:
		tmp = x + (math.tan((y + z)) - a)
	return tmp

Julia

function code(x, y, z, a)
	tmp = 0.0
	if ((tan(a) <= -0.002) || !(tan(a) <= 2e-8))
		tmp = Float64(x - tan(a));
	else
		tmp = Float64(x + Float64(tan(Float64(y + z)) - a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, a)
	tmp = 0.0;
	if ((tan(a) <= -0.002) || ~((tan(a) <= 2e-8)))
		tmp = x - tan(a);
	else
		tmp = x + (tan((y + z)) - a);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, a_] := If[Or[LessEqual[N[Tan[a], $MachinePrecision], -0.002], N[Not[LessEqual[N[Tan[a], $MachinePrecision], 2e-8]], $MachinePrecision]], N[(x - N[Tan[a], $MachinePrecision]), $MachinePrecision], N[(x + N[(N[Tan[N[(y + z), $MachinePrecision]], $MachinePrecision] - a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (tan.f64 a) < -2e-3 or 2e-8 < (tan.f64 a)

   1. Initial program 83.4%

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

   2. Taylor expanded in y around 0 63.4%

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

      1. tan-quot 63.5%

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

      2. add-log-exp 63.5%

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

   4. Applied egg-rr 63.5%

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

   5. Taylor expanded in z around 0 44.9%

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

   if -2e-3 < (tan.f64 a) < 2e-8

   1. Initial program 82.8%

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

   2. Taylor expanded in a around 0 82.8%

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

4. Final simplification 62.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\tan a \leq -0.002 \lor \neg \left(\tan a \leq 2 \cdot 10^{-8}\right):\\ \;\;\;\;x - \tan a\\ \mathbf{else}:\\ \;\;\;\;x + \left(\tan \left(y + z\right) - a\right)\\ \end{array} \]

Alternative 6: 60.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\tan a \leq -0.002:\\ \;\;\;\;x + \left(\sin z - \tan a\right)\\ \mathbf{elif}\;\tan a \leq 2 \cdot 10^{-8}:\\ \;\;\;\;x + \left(\tan \left(y + z\right) - a\right)\\ \mathbf{else}:\\ \;\;\;\;x - \tan a\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (if (<= (tan a) -0.002)
   (+ x (- (sin z) (tan a)))
   (if (<= (tan a) 2e-8) (+ x (- (tan (+ y z)) a)) (- x (tan a)))))

C

double code(double x, double y, double z, double a) {
	double tmp;
	if (tan(a) <= -0.002) {
		tmp = x + (sin(z) - tan(a));
	} else if (tan(a) <= 2e-8) {
		tmp = x + (tan((y + z)) - a);
	} else {
		tmp = 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 (tan(a) <= (-0.002d0)) then
        tmp = x + (sin(z) - tan(a))
    else if (tan(a) <= 2d-8) then
        tmp = x + (tan((y + z)) - a)
    else
        tmp = 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 (Math.tan(a) <= -0.002) {
		tmp = x + (Math.sin(z) - Math.tan(a));
	} else if (Math.tan(a) <= 2e-8) {
		tmp = x + (Math.tan((y + z)) - a);
	} else {
		tmp = x - Math.tan(a);
	}
	return tmp;
}

Python

def code(x, y, z, a):
	tmp = 0
	if math.tan(a) <= -0.002:
		tmp = x + (math.sin(z) - math.tan(a))
	elif math.tan(a) <= 2e-8:
		tmp = x + (math.tan((y + z)) - a)
	else:
		tmp = x - math.tan(a)
	return tmp

Julia

function code(x, y, z, a)
	tmp = 0.0
	if (tan(a) <= -0.002)
		tmp = Float64(x + Float64(sin(z) - tan(a)));
	elseif (tan(a) <= 2e-8)
		tmp = Float64(x + Float64(tan(Float64(y + z)) - a));
	else
		tmp = Float64(x - tan(a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, a)
	tmp = 0.0;
	if (tan(a) <= -0.002)
		tmp = x + (sin(z) - tan(a));
	elseif (tan(a) <= 2e-8)
		tmp = x + (tan((y + z)) - a);
	else
		tmp = x - tan(a);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, a_] := If[LessEqual[N[Tan[a], $MachinePrecision], -0.002], N[(x + N[(N[Sin[z], $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Tan[a], $MachinePrecision], 2e-8], N[(x + N[(N[Tan[N[(y + z), $MachinePrecision]], $MachinePrecision] - a), $MachinePrecision]), $MachinePrecision], N[(x - N[Tan[a], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (tan.f64 a) < -2e-3

   1. Initial program 79.1%

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

      1. tan-quot 79.0%

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

      2. clear-num 79.1%

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

   3. Applied egg-rr 79.1%

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

   4. Taylor expanded in z around 0 60.0%

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

      1. +-commutative 60.0%

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

      2. mul-1-neg 60.0%

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

      3. unsub-neg 60.0%

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

   6. Simplified 60.0%

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

   7. Taylor expanded in y around 0 42.0%

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

   if -2e-3 < (tan.f64 a) < 2e-8

   1. Initial program 82.8%

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

   2. Taylor expanded in a around 0 82.8%

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

   if 2e-8 < (tan.f64 a)

   1. Initial program 88.3%

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

   2. Taylor expanded in y around 0 66.3%

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

      1. tan-quot 66.4%

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

      2. add-log-exp 66.4%

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

   4. Applied egg-rr 66.4%

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

   5. Taylor expanded in z around 0 48.4%

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

4. Final simplification 62.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\tan a \leq -0.002:\\ \;\;\;\;x + \left(\sin z - \tan a\right)\\ \mathbf{elif}\;\tan a \leq 2 \cdot 10^{-8}:\\ \;\;\;\;x + \left(\tan \left(y + z\right) - a\right)\\ \mathbf{else}:\\ \;\;\;\;x - \tan a\\ \end{array} \]

Alternative 7: 59.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y + z \leq -0.0005:\\ \;\;\;\;x + \left(\tan a + \tan \left(y + z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(\tan z - \tan a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (if (<= (+ y z) -0.0005)
   (+ x (+ (tan a) (tan (+ y z))))
   (+ x (- (tan z) (tan a)))))

C

double code(double x, double y, double z, double a) {
	double tmp;
	if ((y + z) <= -0.0005) {
		tmp = x + (tan(a) + tan((y + z)));
	} 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 ((y + z) <= (-0.0005d0)) then
        tmp = x + (tan(a) + tan((y + z)))
    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 ((y + z) <= -0.0005) {
		tmp = x + (Math.tan(a) + Math.tan((y + z)));
	} else {
		tmp = x + (Math.tan(z) - Math.tan(a));
	}
	return tmp;
}

Python

def code(x, y, z, a):
	tmp = 0
	if (y + z) <= -0.0005:
		tmp = x + (math.tan(a) + math.tan((y + z)))
	else:
		tmp = x + (math.tan(z) - math.tan(a))
	return tmp

Julia

function code(x, y, z, a)
	tmp = 0.0
	if (Float64(y + z) <= -0.0005)
		tmp = Float64(x + Float64(tan(a) + tan(Float64(y + z))));
	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 ((y + z) <= -0.0005)
		tmp = x + (tan(a) + tan((y + z)));
	else
		tmp = x + (tan(z) - tan(a));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, a_] := If[LessEqual[N[(y + z), $MachinePrecision], -0.0005], N[(x + N[(N[Tan[a], $MachinePrecision] + N[Tan[N[(y + z), $MachinePrecision]], $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 (+.f64 y z) < -5.0000000000000001e-4

   1. Initial program 73.8%

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

      1. sub-neg 73.8%

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

   3. Applied egg-rr 73.8%

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

      1. +-commutative 73.8%

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

      2. rem-square-sqrt 36.7%

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

      3. fabs-sqr 36.7%

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

      4. rem-square-sqrt 58.6%

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

      5. fabs-neg 58.6%

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

      6. rem-square-sqrt 21.8%

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

      7. fabs-sqr 21.8%

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

      8. rem-square-sqrt 45.8%

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

   5. Simplified 45.8%

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

   if -5.0000000000000001e-4 < (+.f64 y z)

   1. Initial program 88.5%

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

   2. Taylor expanded in y around 0 71.0%

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

      1. tan-quot 71.0%

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

      2. expm1-log1p-u 65.4%

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

      3. expm1-udef 65.4%

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

   4. Applied egg-rr 65.4%

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

      1. expm1-def 65.4%

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

      2. expm1-log1p 71.0%

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

   6. Simplified 71.0%

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

4. Final simplification 61.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y + z \leq -0.0005:\\ \;\;\;\;x + \left(\tan a + \tan \left(y + z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(\tan z - \tan a\right)\\ \end{array} \]

Alternative 8: 69.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -0.0008 \lor \neg \left(a \leq 7.2\right):\\ \;\;\;\;x + \left(\tan z - \tan a\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(\tan \left(y + z\right) - a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (if (or (<= a -0.0008) (not (<= a 7.2)))
   (+ x (- (tan z) (tan a)))
   (+ x (- (tan (+ y z)) a))))

C

double code(double x, double y, double z, double a) {
	double tmp;
	if ((a <= -0.0008) || !(a <= 7.2)) {
		tmp = x + (tan(z) - tan(a));
	} else {
		tmp = x + (tan((y + 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 ((a <= (-0.0008d0)) .or. (.not. (a <= 7.2d0))) then
        tmp = x + (tan(z) - tan(a))
    else
        tmp = x + (tan((y + z)) - a)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double a) {
	double tmp;
	if ((a <= -0.0008) || !(a <= 7.2)) {
		tmp = x + (Math.tan(z) - Math.tan(a));
	} else {
		tmp = x + (Math.tan((y + z)) - a);
	}
	return tmp;
}

Python

def code(x, y, z, a):
	tmp = 0
	if (a <= -0.0008) or not (a <= 7.2):
		tmp = x + (math.tan(z) - math.tan(a))
	else:
		tmp = x + (math.tan((y + z)) - a)
	return tmp

Julia

function code(x, y, z, a)
	tmp = 0.0
	if ((a <= -0.0008) || !(a <= 7.2))
		tmp = Float64(x + Float64(tan(z) - tan(a)));
	else
		tmp = Float64(x + Float64(tan(Float64(y + z)) - a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, a)
	tmp = 0.0;
	if ((a <= -0.0008) || ~((a <= 7.2)))
		tmp = x + (tan(z) - tan(a));
	else
		tmp = x + (tan((y + z)) - a);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, a_] := If[Or[LessEqual[a, -0.0008], N[Not[LessEqual[a, 7.2]], $MachinePrecision]], N[(x + N[(N[Tan[z], $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[Tan[N[(y + z), $MachinePrecision]], $MachinePrecision] - a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -8.00000000000000038e-4 or 7.20000000000000018 < a

   1. Initial program 83.8%

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

   2. Taylor expanded in y around 0 64.3%

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

      1. tan-quot 64.4%

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

      2. expm1-log1p-u 57.6%

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

      3. expm1-udef 57.6%

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

   4. Applied egg-rr 57.6%

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

      1. expm1-def 57.6%

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

      2. expm1-log1p 64.4%

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

   6. Simplified 64.4%

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

   if -8.00000000000000038e-4 < a < 7.20000000000000018

   1. Initial program 82.4%

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

   2. Taylor expanded in a around 0 81.4%

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

4. Final simplification 72.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -0.0008 \lor \neg \left(a \leq 7.2\right):\\ \;\;\;\;x + \left(\tan z - \tan a\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(\tan \left(y + z\right) - a\right)\\ \end{array} \]

Alternative 9: 79.1% 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 83.1%

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

2. Final simplification 83.1%

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

Alternative 10: 55.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -0.047 \lor \neg \left(a \leq 82000000\right):\\ \;\;\;\;x + \left(z - \tan a\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(\tan \left(y + z\right) - a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (if (or (<= a -0.047) (not (<= a 82000000.0)))
   (+ x (- z (tan a)))
   (+ x (- (tan (+ y z)) a))))

C

double code(double x, double y, double z, double a) {
	double tmp;
	if ((a <= -0.047) || !(a <= 82000000.0)) {
		tmp = x + (z - tan(a));
	} else {
		tmp = x + (tan((y + 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 ((a <= (-0.047d0)) .or. (.not. (a <= 82000000.0d0))) then
        tmp = x + (z - tan(a))
    else
        tmp = x + (tan((y + z)) - a)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double a) {
	double tmp;
	if ((a <= -0.047) || !(a <= 82000000.0)) {
		tmp = x + (z - Math.tan(a));
	} else {
		tmp = x + (Math.tan((y + z)) - a);
	}
	return tmp;
}

Python

def code(x, y, z, a):
	tmp = 0
	if (a <= -0.047) or not (a <= 82000000.0):
		tmp = x + (z - math.tan(a))
	else:
		tmp = x + (math.tan((y + z)) - a)
	return tmp

Julia

function code(x, y, z, a)
	tmp = 0.0
	if ((a <= -0.047) || !(a <= 82000000.0))
		tmp = Float64(x + Float64(z - tan(a)));
	else
		tmp = Float64(x + Float64(tan(Float64(y + z)) - a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, a)
	tmp = 0.0;
	if ((a <= -0.047) || ~((a <= 82000000.0)))
		tmp = x + (z - tan(a));
	else
		tmp = x + (tan((y + z)) - a);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, a_] := If[Or[LessEqual[a, -0.047], N[Not[LessEqual[a, 82000000.0]], $MachinePrecision]], N[(x + N[(z - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[Tan[N[(y + z), $MachinePrecision]], $MachinePrecision] - a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -0.047 or 8.2e7 < a

   1. Initial program 84.2%

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

   2. Taylor expanded in y around 0 64.8%

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

   3. Taylor expanded in z around 0 36.5%

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

   if -0.047 < a < 8.2e7

   1. Initial program 81.9%

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

   2. Taylor expanded in a around 0 80.7%

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

4. Final simplification 57.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -0.047 \lor \neg \left(a \leq 82000000\right):\\ \;\;\;\;x + \left(z - \tan a\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(\tan \left(y + z\right) - a\right)\\ \end{array} \]

Alternative 11: 36.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 1.46:\\ \;\;\;\;x + \left(z - \tan a\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a) :precision binary64 (if (<= z 1.46) (+ x (- z (tan a))) x))

C

double code(double x, double y, double z, double a) {
	double tmp;
	if (z <= 1.46) {
		tmp = x + (z - tan(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 (z <= 1.46d0) then
        tmp = x + (z - tan(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 (z <= 1.46) {
		tmp = x + (z - Math.tan(a));
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, a):
	tmp = 0
	if z <= 1.46:
		tmp = x + (z - math.tan(a))
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, a)
	tmp = 0.0
	if (z <= 1.46)
		tmp = Float64(x + Float64(z - tan(a)));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, a)
	tmp = 0.0;
	if (z <= 1.46)
		tmp = x + (z - tan(a));
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, a_] := If[LessEqual[z, 1.46], N[(x + N[(z - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if z < 1.46

   1. Initial program 88.2%

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

   2. Taylor expanded in y around 0 60.6%

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

   3. Taylor expanded in z around 0 43.1%

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

   if 1.46 < z

   1. Initial program 67.5%

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

   2. Taylor expanded in x around inf 22.6%

      \[\leadsto \color{blue}{x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 38.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1.46:\\ \;\;\;\;x + \left(z - \tan a\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 12: 32.0% 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 83.1%

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

2. Taylor expanded in x around inf 30.3%

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

3. Final simplification 30.3%

   \[\leadsto x \]

Reproduce

herbie shell --seed 2023172 
(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))))