tan-example (used to crash)

Percentage Accurate: 79.3% → 99.7%

Time: 33.8s

Alternatives: 10

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 79.3% 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 81.8%

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

   1. tan-sum 99.8%

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

   2. div-inv 99.8%

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

   3. fma-neg 99.8%

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

4. Applied egg-rr 99.8%

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

5. Final simplification 99.8%

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

Alternative 2: 88.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\tan a \leq -0.005 \lor \neg \left(\tan a \leq 2 \cdot 10^{-11}\right):\\ \;\;\;\;x + \log \left(e^{\tan \left(y + z\right) - \tan a}\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{\tan y + \tan z}{1 - \tan y \cdot \tan z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (if (or (<= (tan a) -0.005) (not (<= (tan a) 2e-11)))
   (+ x (log (exp (- (tan (+ y z)) (tan a)))))
   (+ x (/ (+ (tan y) (tan z)) (- 1.0 (* (tan y) (tan z)))))))

C

double code(double x, double y, double z, double a) {
	double tmp;
	if ((tan(a) <= -0.005) || !(tan(a) <= 2e-11)) {
		tmp = x + log(exp((tan((y + z)) - tan(a))));
	} else {
		tmp = x + ((tan(y) + tan(z)) / (1.0 - (tan(y) * tan(z))));
	}
	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) <= (-0.005d0)) .or. (.not. (tan(a) <= 2d-11))) then
        tmp = x + log(exp((tan((y + z)) - tan(a))))
    else
        tmp = x + ((tan(y) + tan(z)) / (1.0d0 - (tan(y) * tan(z))))
    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) <= -0.005) || !(Math.tan(a) <= 2e-11)) {
		tmp = x + Math.log(Math.exp((Math.tan((y + z)) - Math.tan(a))));
	} else {
		tmp = x + ((Math.tan(y) + Math.tan(z)) / (1.0 - (Math.tan(y) * Math.tan(z))));
	}
	return tmp;
}

Python

def code(x, y, z, a):
	tmp = 0
	if (math.tan(a) <= -0.005) or not (math.tan(a) <= 2e-11):
		tmp = x + math.log(math.exp((math.tan((y + z)) - math.tan(a))))
	else:
		tmp = x + ((math.tan(y) + math.tan(z)) / (1.0 - (math.tan(y) * math.tan(z))))
	return tmp

Julia

function code(x, y, z, a)
	tmp = 0.0
	if ((tan(a) <= -0.005) || !(tan(a) <= 2e-11))
		tmp = Float64(x + log(exp(Float64(tan(Float64(y + z)) - tan(a)))));
	else
		tmp = Float64(x + Float64(Float64(tan(y) + tan(z)) / Float64(1.0 - Float64(tan(y) * tan(z)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, a)
	tmp = 0.0;
	if ((tan(a) <= -0.005) || ~((tan(a) <= 2e-11)))
		tmp = x + log(exp((tan((y + z)) - tan(a))));
	else
		tmp = x + ((tan(y) + tan(z)) / (1.0 - (tan(y) * tan(z))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, a_] := If[Or[LessEqual[N[Tan[a], $MachinePrecision], -0.005], N[Not[LessEqual[N[Tan[a], $MachinePrecision], 2e-11]], $MachinePrecision]], N[(x + N[Log[N[Exp[N[(N[Tan[N[(y + z), $MachinePrecision]], $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(x + 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]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (tan.f64 a) < -0.0050000000000000001 or 1.99999999999999988e-11 < (tan.f64 a)

   1. Initial program 82.0%

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

      1. add-log-exp 82.0%

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

   4. Applied egg-rr 82.0%

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

   if -0.0050000000000000001 < (tan.f64 a) < 1.99999999999999988e-11

   1. Initial program 81.7%

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

      1. +-commutative 81.7%

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

      2. associate-+l- 81.7%

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

   4. Applied egg-rr 81.7%

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

   5. Taylor expanded in a around 0 81.6%

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

      1. neg-mul-1 81.6%

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

   7. Simplified 81.6%

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

      1. sub-neg 81.6%

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

      2. add-sqr-sqrt 0.0%

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

      3. sqrt-unprod 2.6%

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

      4. sqr-neg 2.6%

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

      5. sqrt-unprod 2.6%

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

      6. add-sqr-sqrt 2.6%

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

      7. add-sqr-sqrt 0.0%

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

      8. sqrt-unprod 81.6%

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

      9. sqr-neg 81.6%

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

      10. sqrt-unprod 81.0%

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

      11. add-sqr-sqrt 81.6%

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

   9. Applied egg-rr 81.6%

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

       1. tan-sum 99.6%

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

       2. div-inv 99.6%

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

   11. Applied egg-rr 99.6%

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

       1. associate-*r/ 99.6%

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

       2. *-rgt-identity 99.6%

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

   13. Simplified 99.6%

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

4. Final simplification 90.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\tan a \leq -0.005 \lor \neg \left(\tan a \leq 2 \cdot 10^{-11}\right):\\ \;\;\;\;x + \log \left(e^{\tan \left(y + z\right) - \tan a}\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{\tan y + \tan z}{1 - \tan y \cdot \tan z}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 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 81.8%

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

   1. tan-sum 61.2%

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

   2. div-inv 61.2%

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

4. Applied egg-rr 99.8%

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

   1. associate-*r/ 61.2%

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

   2. *-rgt-identity 61.2%

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

6. Simplified 99.8%

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

7. Final simplification 99.8%

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

Alternative 4: 59.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan \left(y + z\right)\\ t_1 := x + t_0\\ \mathbf{if}\;t_0 \leq -0.02:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_0 \leq 0.015:\\ \;\;\;\;y + \left(x - \tan a\right)\\ \mathbf{else}:\\ \;\;\;\;\left|t_1\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (let* ((t_0 (tan (+ y z))) (t_1 (+ x t_0)))
   (if (<= t_0 -0.02) t_1 (if (<= t_0 0.015) (+ y (- x (tan a))) (fabs t_1)))))

C

double code(double x, double y, double z, double a) {
	double t_0 = tan((y + z));
	double t_1 = x + t_0;
	double tmp;
	if (t_0 <= -0.02) {
		tmp = t_1;
	} else if (t_0 <= 0.015) {
		tmp = y + (x - tan(a));
	} else {
		tmp = fabs(t_1);
	}
	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) :: t_1
    real(8) :: tmp
    t_0 = tan((y + z))
    t_1 = x + t_0
    if (t_0 <= (-0.02d0)) then
        tmp = t_1
    else if (t_0 <= 0.015d0) then
        tmp = y + (x - tan(a))
    else
        tmp = abs(t_1)
    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 t_1 = x + t_0;
	double tmp;
	if (t_0 <= -0.02) {
		tmp = t_1;
	} else if (t_0 <= 0.015) {
		tmp = y + (x - Math.tan(a));
	} else {
		tmp = Math.abs(t_1);
	}
	return tmp;
}

Python

def code(x, y, z, a):
	t_0 = math.tan((y + z))
	t_1 = x + t_0
	tmp = 0
	if t_0 <= -0.02:
		tmp = t_1
	elif t_0 <= 0.015:
		tmp = y + (x - math.tan(a))
	else:
		tmp = math.fabs(t_1)
	return tmp

Julia

function code(x, y, z, a)
	t_0 = tan(Float64(y + z))
	t_1 = Float64(x + t_0)
	tmp = 0.0
	if (t_0 <= -0.02)
		tmp = t_1;
	elseif (t_0 <= 0.015)
		tmp = Float64(y + Float64(x - tan(a)));
	else
		tmp = abs(t_1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, a)
	t_0 = tan((y + z));
	t_1 = x + t_0;
	tmp = 0.0;
	if (t_0 <= -0.02)
		tmp = t_1;
	elseif (t_0 <= 0.015)
		tmp = y + (x - tan(a));
	else
		tmp = abs(t_1);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, a_] := Block[{t$95$0 = N[Tan[N[(y + z), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(x + t$95$0), $MachinePrecision]}, If[LessEqual[t$95$0, -0.02], t$95$1, If[LessEqual[t$95$0, 0.015], N[(y + N[(x - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Abs[t$95$1], $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (tan.f64 (+.f64 y z)) < -0.0200000000000000004

   1. Initial program 73.0%

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

      1. +-commutative 73.0%

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

      2. associate-+l- 72.9%

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

   4. Applied egg-rr 72.9%

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

   5. Taylor expanded in a around 0 45.2%

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

      1. neg-mul-1 45.2%

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

   7. Simplified 45.2%

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

      1. sub-neg 45.2%

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

      2. add-sqr-sqrt 0.0%

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

      3. sqrt-unprod 3.1%

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

      4. sqr-neg 3.1%

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

      5. sqrt-unprod 3.1%

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

      6. add-sqr-sqrt 3.1%

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

      7. add-sqr-sqrt 0.0%

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

      8. sqrt-unprod 45.2%

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

      9. sqr-neg 45.2%

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

      10. sqrt-unprod 45.0%

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

      11. add-sqr-sqrt 45.2%

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

   9. Applied egg-rr 45.2%

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

   if -0.0200000000000000004 < (tan.f64 (+.f64 y z)) < 0.014999999999999999

   1. Initial program 99.8%

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

      1. +-commutative 99.8%

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

      2. associate-+l- 99.7%

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

   4. Applied egg-rr 99.7%

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

   5. Taylor expanded in z around 0 98.0%

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

   6. Taylor expanded in y around 0 96.7%

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

   if 0.014999999999999999 < (tan.f64 (+.f64 y z))

   1. Initial program 77.1%

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

      1. +-commutative 77.1%

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

      2. associate-+l- 77.0%

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

   4. Applied egg-rr 77.0%

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

   5. Taylor expanded in a around 0 51.7%

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

      1. neg-mul-1 51.7%

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

   7. Simplified 51.7%

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

      1. add-sqr-sqrt 51.5%

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

      2. sqrt-unprod 51.7%

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

      3. pow2 51.7%

         \[\leadsto \sqrt{\color{blue}{{\left(\tan \left(y + z\right) - \left(-x\right)\right)}^{2}}} \]

      4. sub-neg 51.7%

         \[\leadsto \sqrt{{\color{blue}{\left(\tan \left(y + z\right) + \left(-\left(-x\right)\right)\right)}}^{2}} \]

      5. add-sqr-sqrt 0.0%

         \[\leadsto \sqrt{{\left(\tan \left(y + z\right) + \left(-\color{blue}{\sqrt{-x} \cdot \sqrt{-x}}\right)\right)}^{2}} \]

      6. sqrt-unprod 21.6%

         \[\leadsto \sqrt{{\left(\tan \left(y + z\right) + \left(-\color{blue}{\sqrt{\left(-x\right) \cdot \left(-x\right)}}\right)\right)}^{2}} \]

      7. sqr-neg 21.6%

         \[\leadsto \sqrt{{\left(\tan \left(y + z\right) + \left(-\sqrt{\color{blue}{x \cdot x}}\right)\right)}^{2}} \]

      8. sqrt-unprod 21.6%

         \[\leadsto \sqrt{{\left(\tan \left(y + z\right) + \left(-\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right)\right)}^{2}} \]

      9. add-sqr-sqrt 21.6%

         \[\leadsto \sqrt{{\left(\tan \left(y + z\right) + \left(-\color{blue}{x}\right)\right)}^{2}} \]

      10. add-sqr-sqrt 0.0%

          \[\leadsto \sqrt{{\left(\tan \left(y + z\right) + \color{blue}{\sqrt{-x} \cdot \sqrt{-x}}\right)}^{2}} \]

      11. sqrt-unprod 51.7%

          \[\leadsto \sqrt{{\left(\tan \left(y + z\right) + \color{blue}{\sqrt{\left(-x\right) \cdot \left(-x\right)}}\right)}^{2}} \]

      12. sqr-neg 51.7%

          \[\leadsto \sqrt{{\left(\tan \left(y + z\right) + \sqrt{\color{blue}{x \cdot x}}\right)}^{2}} \]

      13. sqrt-unprod 51.4%

          \[\leadsto \sqrt{{\left(\tan \left(y + z\right) + \color{blue}{\sqrt{x} \cdot \sqrt{x}}\right)}^{2}} \]

      14. add-sqr-sqrt 51.7%

          \[\leadsto \sqrt{{\left(\tan \left(y + z\right) + \color{blue}{x}\right)}^{2}} \]

   9. Applied egg-rr 51.7%

      \[\leadsto \color{blue}{\sqrt{{\left(\tan \left(y + z\right) + x\right)}^{2}}} \]
   10. Step-by-step derivation

       1. unpow2 51.7%

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

       2. rem-sqrt-square 51.7%

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

       3. +-commutative 51.7%

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

   11. Simplified 51.7%

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

4. Final simplification 61.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\tan \left(y + z\right) \leq -0.02:\\ \;\;\;\;x + \tan \left(y + z\right)\\ \mathbf{elif}\;\tan \left(y + z\right) \leq 0.015:\\ \;\;\;\;y + \left(x - \tan a\right)\\ \mathbf{else}:\\ \;\;\;\;\left|x + \tan \left(y + z\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 79.2% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.8%

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

   1. add-log-exp 81.8%

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

4. Applied egg-rr 81.8%

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

5. Final simplification 81.8%

   \[\leadsto x + \log \left(e^{\tan \left(y + z\right) - \tan a}\right) \]
6. Add Preprocessing

Alternative 6: 59.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan \left(y + z\right)\\ \mathbf{if}\;t_0 \leq -0.02 \lor \neg \left(t_0 \leq 0.015\right):\\ \;\;\;\;x + t_0\\ \mathbf{else}:\\ \;\;\;\;y + \left(x - \tan a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (let* ((t_0 (tan (+ y z))))
   (if (or (<= t_0 -0.02) (not (<= t_0 0.015)))
     (+ x t_0)
     (+ y (- x (tan a))))))

C

double code(double x, double y, double z, double a) {
	double t_0 = tan((y + z));
	double tmp;
	if ((t_0 <= -0.02) || !(t_0 <= 0.015)) {
		tmp = x + t_0;
	} else {
		tmp = y + (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 ((t_0 <= (-0.02d0)) .or. (.not. (t_0 <= 0.015d0))) then
        tmp = x + t_0
    else
        tmp = y + (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 ((t_0 <= -0.02) || !(t_0 <= 0.015)) {
		tmp = x + t_0;
	} else {
		tmp = y + (x - Math.tan(a));
	}
	return tmp;
}

Python

def code(x, y, z, a):
	t_0 = math.tan((y + z))
	tmp = 0
	if (t_0 <= -0.02) or not (t_0 <= 0.015):
		tmp = x + t_0
	else:
		tmp = y + (x - math.tan(a))
	return tmp

Julia

function code(x, y, z, a)
	t_0 = tan(Float64(y + z))
	tmp = 0.0
	if ((t_0 <= -0.02) || !(t_0 <= 0.015))
		tmp = Float64(x + t_0);
	else
		tmp = Float64(y + 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 ((t_0 <= -0.02) || ~((t_0 <= 0.015)))
		tmp = x + t_0;
	else
		tmp = y + (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[Or[LessEqual[t$95$0, -0.02], N[Not[LessEqual[t$95$0, 0.015]], $MachinePrecision]], N[(x + t$95$0), $MachinePrecision], N[(y + N[(x - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (tan.f64 (+.f64 y z)) < -0.0200000000000000004 or 0.014999999999999999 < (tan.f64 (+.f64 y z))

   1. Initial program 74.8%

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

      1. +-commutative 74.8%

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

      2. associate-+l- 74.7%

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

   4. Applied egg-rr 74.7%

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

   5. Taylor expanded in a around 0 48.1%

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

      1. neg-mul-1 48.1%

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

   7. Simplified 48.1%

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

      1. sub-neg 48.1%

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

      2. add-sqr-sqrt 0.0%

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

      3. sqrt-unprod 3.4%

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

      4. sqr-neg 3.4%

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

      5. sqrt-unprod 3.4%

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

      6. add-sqr-sqrt 3.4%

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

      7. add-sqr-sqrt 0.0%

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

      8. sqrt-unprod 48.1%

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

      9. sqr-neg 48.1%

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

      10. sqrt-unprod 47.9%

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

      11. add-sqr-sqrt 48.1%

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

   9. Applied egg-rr 48.1%

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

   if -0.0200000000000000004 < (tan.f64 (+.f64 y z)) < 0.014999999999999999

   1. Initial program 99.8%

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

      1. +-commutative 99.8%

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

      2. associate-+l- 99.7%

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

   4. Applied egg-rr 99.7%

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

   5. Taylor expanded in z around 0 98.0%

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

   6. Taylor expanded in y around 0 96.7%

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

4. Final simplification 61.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\tan \left(y + z\right) \leq -0.02 \lor \neg \left(\tan \left(y + z\right) \leq 0.015\right):\\ \;\;\;\;x + \tan \left(y + z\right)\\ \mathbf{else}:\\ \;\;\;\;y + \left(x - \tan a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 59.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y + z \leq 0.002:\\ \;\;\;\;x + \left(\tan y - \tan a\right)\\ \mathbf{else}:\\ \;\;\;\;x + \tan \left(y + z\right)\\ \end{array} \end{array} \]

FPCore

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

C

double code(double x, double y, double z, double a) {
	double tmp;
	if ((y + z) <= 0.002) {
		tmp = x + (tan(y) - tan(a));
	} else {
		tmp = x + tan((y + z));
	}
	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) <= 0.002d0) then
        tmp = x + (tan(y) - tan(a))
    else
        tmp = x + tan((y + z))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double a) {
	double tmp;
	if ((y + z) <= 0.002) {
		tmp = x + (Math.tan(y) - Math.tan(a));
	} else {
		tmp = x + Math.tan((y + z));
	}
	return tmp;
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (+.f64 y z) < 2e-3

   1. Initial program 87.3%

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

      1. +-commutative 87.3%

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

      2. associate-+l- 87.2%

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

   4. Applied egg-rr 87.2%

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

   5. Taylor expanded in z around 0 70.4%

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

      1. tan-quot 70.4%

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

      2. associate--r- 70.5%

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

   7. Applied egg-rr 70.5%

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

   if 2e-3 < (+.f64 y z)

   1. Initial program 71.7%

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

      1. +-commutative 71.7%

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

      2. associate-+l- 71.5%

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

   4. Applied egg-rr 71.5%

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

   5. Taylor expanded in a around 0 45.5%

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

      1. neg-mul-1 45.5%

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

   7. Simplified 45.5%

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

      1. sub-neg 45.5%

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

      2. add-sqr-sqrt 0.0%

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

      3. sqrt-unprod 3.6%

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

      4. sqr-neg 3.6%

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

      5. sqrt-unprod 3.6%

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

      6. add-sqr-sqrt 3.6%

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

      7. add-sqr-sqrt 0.0%

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

      8. sqrt-unprod 45.5%

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

      9. sqr-neg 45.5%

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

      10. sqrt-unprod 45.3%

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

      11. add-sqr-sqrt 45.5%

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

   9. Applied egg-rr 45.5%

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

4. Final simplification 61.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y + z \leq 0.002:\\ \;\;\;\;x + \left(\tan y - \tan a\right)\\ \mathbf{else}:\\ \;\;\;\;x + \tan \left(y + z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 79.3% 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 81.8%

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

3. Final simplification 81.8%

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

Alternative 9: 50.2% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

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))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.8%

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

   1. +-commutative 81.8%

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

   2. associate-+l- 81.8%

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

4. Applied egg-rr 81.8%

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

5. Taylor expanded in a around 0 52.0%

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

   1. neg-mul-1 52.0%

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

7. Simplified 52.0%

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

   1. sub-neg 52.0%

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

   2. add-sqr-sqrt 0.0%

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

   3. sqrt-unprod 3.1%

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

   4. sqr-neg 3.1%

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

   5. sqrt-unprod 3.1%

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

   6. add-sqr-sqrt 3.1%

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

   7. add-sqr-sqrt 0.0%

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

   8. sqrt-unprod 52.0%

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

   9. sqr-neg 52.0%

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

   10. sqrt-unprod 51.7%

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

   11. add-sqr-sqrt 52.0%

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

9. Applied egg-rr 52.0%

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

10. Final simplification 52.0%

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

Alternative 10: 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 81.8%

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

3. Taylor expanded in x around inf 33.7%

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

4. Final simplification 33.7%

   \[\leadsto x \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024021 
(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))))