tan-example (used to crash)

Percentage Accurate: 79.4% → 99.7%

Time: 24.6s

Alternatives: 9

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: 79.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 0.4× speedup

Math

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

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 77.9%

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

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

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 2: 89.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan y + \tan z\\ \mathbf{if}\;\tan a \leq -0.002 \lor \neg \left(\tan a \leq 5 \cdot 10^{-11}\right):\\ \;\;\;\;\mathsf{fma}\left(t_0, 1, x - \tan a\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(\frac{t_0}{1 - \tan y \cdot \tan z} - a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (let* ((t_0 (+ (tan y) (tan z))))
   (if (or (<= (tan a) -0.002) (not (<= (tan a) 5e-11)))
     (fma t_0 1.0 (- x (tan a)))
     (+ x (- (/ t_0 (- 1.0 (* (tan y) (tan z)))) a)))))

C

double code(double x, double y, double z, double a) {
	double t_0 = tan(y) + tan(z);
	double tmp;
	if ((tan(a) <= -0.002) || !(tan(a) <= 5e-11)) {
		tmp = fma(t_0, 1.0, (x - tan(a)));
	} else {
		tmp = x + ((t_0 / (1.0 - (tan(y) * tan(z)))) - a);
	}
	return tmp;
}

Julia

function code(x, y, z, a)
	t_0 = Float64(tan(y) + tan(z))
	tmp = 0.0
	if ((tan(a) <= -0.002) || !(tan(a) <= 5e-11))
		tmp = fma(t_0, 1.0, Float64(x - tan(a)));
	else
		tmp = Float64(x + Float64(Float64(t_0 / Float64(1.0 - Float64(tan(y) * tan(z)))) - a));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, a_] := Block[{t$95$0 = N[(N[Tan[y], $MachinePrecision] + N[Tan[z], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[N[Tan[a], $MachinePrecision], -0.002], N[Not[LessEqual[N[Tan[a], $MachinePrecision], 5e-11]], $MachinePrecision]], N[(t$95$0 * 1.0 + N[(x - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[(t$95$0 / N[(1.0 - N[(N[Tan[y], $MachinePrecision] * N[Tan[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (tan.f64 a) < -2e-3 or 5.00000000000000018e-11 < (tan.f64 a)

   1. Initial program 76.5%

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

      1. associate-+r- 76.4%

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

      2. +-commutative 76.4%

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

      3. associate--l+ 76.4%

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

      4. tan-sum 99.6%

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

      5. div-inv 99.6%

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

      6. fma-def 99.6%

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

   3. Applied egg-rr 99.6%

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

   4. Taylor expanded in y around 0 76.8%

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

   if -2e-3 < (tan.f64 a) < 5.00000000000000018e-11

   1. Initial program 79.1%

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

   2. Taylor expanded in a around 0 79.1%

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

      1. tan-sum 99.6%

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

      2. div-inv 99.6%

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

      3. fma-neg 99.6%

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

   4. Applied egg-rr 99.6%

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

      1. fma-udef 99.6%

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

      2. unsub-neg 99.6%

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

      3. associate-*r/ 99.6%

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

      4. *-rgt-identity 99.6%

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

   6. Simplified 99.6%

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

4. Final simplification 89.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\tan a \leq -0.002 \lor \neg \left(\tan a \leq 5 \cdot 10^{-11}\right):\\ \;\;\;\;\mathsf{fma}\left(\tan y + \tan z, 1, x - \tan a\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(\frac{\tan y + \tan z}{1 - \tan y \cdot \tan z} - a\right)\\ \end{array} \]

Alternative 3: 79.8% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 77.9%

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

   1. associate-+r- 77.9%

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

   2. +-commutative 77.9%

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

   3. associate--l+ 77.9%

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

   4. tan-sum 99.6%

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

   5. div-inv 99.6%

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

   6. fma-def 99.6%

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

3. Applied egg-rr 99.6%

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

4. Taylor expanded in y around 0 78.0%

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

5. Final simplification 78.0%

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

Alternative 4: 69.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.65 \cdot 10^{-16} \lor \neg \left(a \leq 7.5 \cdot 10^{-25}\right):\\ \;\;\;\;x + \left(\tan y - \tan a\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(\tan \left(y + z\right) - a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (if (or (<= a -1.65e-16) (not (<= a 7.5e-25)))
   (+ x (- (tan y) (tan a)))
   (+ x (- (tan (+ y z)) a))))

C

double code(double x, double y, double z, double a) {
	double tmp;
	if ((a <= -1.65e-16) || !(a <= 7.5e-25)) {
		tmp = x + (tan(y) - 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 <= (-1.65d-16)) .or. (.not. (a <= 7.5d-25))) then
        tmp = x + (tan(y) - 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 <= -1.65e-16) || !(a <= 7.5e-25)) {
		tmp = x + (Math.tan(y) - Math.tan(a));
	} else {
		tmp = x + (Math.tan((y + z)) - a);
	}
	return tmp;
}

Python

def code(x, y, z, a):
	tmp = 0
	if (a <= -1.65e-16) or not (a <= 7.5e-25):
		tmp = x + (math.tan(y) - 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 <= -1.65e-16) || !(a <= 7.5e-25))
		tmp = Float64(x + Float64(tan(y) - 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 <= -1.65e-16) || ~((a <= 7.5e-25)))
		tmp = x + (tan(y) - tan(a));
	else
		tmp = x + (tan((y + z)) - a);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, a_] := If[Or[LessEqual[a, -1.65e-16], N[Not[LessEqual[a, 7.5e-25]], $MachinePrecision]], N[(x + N[(N[Tan[y], $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 < -1.64999999999999994e-16 or 7.49999999999999989e-25 < a

   1. Initial program 75.3%

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

      1. add-sqr-sqrt 38.2%

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

      2. sqrt-unprod 59.2%

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

      3. pow2 59.2%

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

   3. Applied egg-rr 59.2%

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

   4. Taylor expanded in z around 0 48.9%

      \[\leadsto x + \left(\sqrt{{\color{blue}{\left(\frac{\sin y}{\cos y}\right)}}^{2}} - \tan a\right) \]
   5. Step-by-step derivation

      1. tan-quot 48.9%

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

      2. sqrt-pow1 57.0%

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

      3. metadata-eval 57.0%

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

      4. pow1 57.0%

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

      5. expm1-log1p-u 51.4%

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

      6. expm1-udef 51.4%

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

   6. Applied egg-rr 51.4%

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

      1. expm1-def 51.4%

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

      2. expm1-log1p 57.0%

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

   8. Simplified 57.0%

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

   if -1.64999999999999994e-16 < a < 7.49999999999999989e-25

   1. Initial program 80.5%

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

   2. Taylor expanded in a around 0 80.5%

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

4. Final simplification 69.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.65 \cdot 10^{-16} \lor \neg \left(a \leq 7.5 \cdot 10^{-25}\right):\\ \;\;\;\;x + \left(\tan y - \tan a\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(\tan \left(y + z\right) - a\right)\\ \end{array} \]

Alternative 5: 70.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 2.1 \cdot 10^{-8}:\\ \;\;\;\;x + \left(\tan y - \tan a\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(\tan z - \tan a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (if (<= z 2.1e-8) (+ x (- (tan y) (tan a))) (+ x (- (tan z) (tan a)))))

C

double code(double x, double y, double z, double a) {
	double tmp;
	if (z <= 2.1e-8) {
		tmp = x + (tan(y) - tan(a));
	} else {
		tmp = x + (tan(z) - tan(a));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z <= 2.1d-8) then
        tmp = x + (tan(y) - tan(a))
    else
        tmp = x + (tan(z) - tan(a))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double a) {
	double tmp;
	if (z <= 2.1e-8) {
		tmp = x + (Math.tan(y) - Math.tan(a));
	} else {
		tmp = x + (Math.tan(z) - Math.tan(a));
	}
	return tmp;
}

Python

def code(x, y, z, a):
	tmp = 0
	if z <= 2.1e-8:
		tmp = x + (math.tan(y) - math.tan(a))
	else:
		tmp = x + (math.tan(z) - math.tan(a))
	return tmp

Julia

function code(x, y, z, a)
	tmp = 0.0
	if (z <= 2.1e-8)
		tmp = Float64(x + Float64(tan(y) - tan(a)));
	else
		tmp = Float64(x + Float64(tan(z) - tan(a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, a)
	tmp = 0.0;
	if (z <= 2.1e-8)
		tmp = x + (tan(y) - tan(a));
	else
		tmp = x + (tan(z) - tan(a));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, a_] := If[LessEqual[z, 2.1e-8], N[(x + N[(N[Tan[y], $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[Tan[z], $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 2.09999999999999994e-8

   1. Initial program 86.1%

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

      1. add-sqr-sqrt 42.0%

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

      2. sqrt-unprod 68.2%

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

      3. pow2 68.2%

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

   3. Applied egg-rr 68.2%

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

   4. Taylor expanded in z around 0 61.0%

      \[\leadsto x + \left(\sqrt{{\color{blue}{\left(\frac{\sin y}{\cos y}\right)}}^{2}} - \tan a\right) \]
   5. Step-by-step derivation

      1. tan-quot 61.0%

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

      2. sqrt-pow1 72.3%

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

      3. metadata-eval 72.3%

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

      4. pow1 72.3%

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

      5. expm1-log1p-u 65.3%

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

      6. expm1-udef 65.3%

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

   6. Applied egg-rr 65.3%

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

      1. expm1-def 65.3%

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

      2. expm1-log1p 72.3%

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

   8. Simplified 72.3%

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

   if 2.09999999999999994e-8 < z

   1. Initial program 58.5%

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

      1. add-sqr-sqrt 35.7%

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

      2. sqrt-unprod 44.0%

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

      3. pow2 44.0%

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

   3. Applied egg-rr 44.0%

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

   4. Taylor expanded in y around 0 43.8%

      \[\leadsto x + \left(\sqrt{{\color{blue}{\left(\frac{\sin z}{\cos z}\right)}}^{2}} - \tan a\right) \]
   5. Step-by-step derivation

      1. tan-quot 43.8%

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

      2. sqrt-pow1 58.4%

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

      3. metadata-eval 58.4%

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

      4. pow1 58.4%

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

      5. expm1-log1p-u 49.8%

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

      6. expm1-udef 49.8%

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

   6. Applied egg-rr 49.8%

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

      1. expm1-def 49.8%

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

      2. expm1-log1p 58.4%

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

   8. Simplified 58.4%

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

4. Final simplification 68.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 2.1 \cdot 10^{-8}:\\ \;\;\;\;x + \left(\tan y - \tan a\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(\tan z - \tan a\right)\\ \end{array} \]

Alternative 6: 79.4% 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 77.9%

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

2. Final simplification 77.9%

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

Alternative 7: 60.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -0.45 \lor \neg \left(a \leq 4.1 \cdot 10^{-11}\right):\\ \;\;\;\;x - \tan a\\ \mathbf{else}:\\ \;\;\;\;x + \left(\tan \left(y + z\right) - a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (if (or (<= a -0.45) (not (<= a 4.1e-11)))
   (- x (tan a))
   (+ x (- (tan (+ y z)) a))))

C

double code(double x, double y, double z, double a) {
	double tmp;
	if ((a <= -0.45) || !(a <= 4.1e-11)) {
		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 ((a <= (-0.45d0)) .or. (.not. (a <= 4.1d-11))) 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 ((a <= -0.45) || !(a <= 4.1e-11)) {
		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 (a <= -0.45) or not (a <= 4.1e-11):
		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 ((a <= -0.45) || !(a <= 4.1e-11))
		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 ((a <= -0.45) || ~((a <= 4.1e-11)))
		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[a, -0.45], N[Not[LessEqual[a, 4.1e-11]], $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 a < -0.450000000000000011 or 4.1000000000000001e-11 < a

   1. Initial program 75.7%

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

      1. associate-+r- 75.6%

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

   3. Simplified 75.6%

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

      1. add-exp-log 72.5%

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

      2. +-commutative 72.5%

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

   5. Applied egg-rr 72.5%

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

   6. Taylor expanded in x around inf 41.8%

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

      1. mul-1-neg 41.8%

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

      2. log-rec 41.8%

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

      3. remove-double-neg 41.8%

         \[\leadsto e^{\color{blue}{\log x}} - \tan a \]

   8. Simplified 41.8%

      \[\leadsto e^{\color{blue}{\log x}} - \tan a \]
   9. Step-by-step derivation

      1. add-exp-log 42.0%

         \[\leadsto \color{blue}{x} - \tan a \]

      2. sub-neg 42.0%

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

   10. Applied egg-rr 42.0%

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

       1. sub-neg 42.0%

          \[\leadsto \color{blue}{x - \tan a} \]

   12. Simplified 42.0%

       \[\leadsto \color{blue}{x - \tan a} \]

   if -0.450000000000000011 < a < 4.1000000000000001e-11

   1. Initial program 79.8%

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

   2. Taylor expanded in a around 0 79.1%

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

4. Final simplification 62.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -0.45 \lor \neg \left(a \leq 4.1 \cdot 10^{-11}\right):\\ \;\;\;\;x - \tan a\\ \mathbf{else}:\\ \;\;\;\;x + \left(\tan \left(y + z\right) - a\right)\\ \end{array} \]

Alternative 8: 41.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ x - \tan a \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.9%

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

   1. associate-+r- 77.9%

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

3. Simplified 77.9%

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

   1. add-exp-log 73.8%

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

   2. +-commutative 73.8%

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

5. Applied egg-rr 73.8%

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

6. Taylor expanded in x around inf 41.6%

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

   1. mul-1-neg 41.6%

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

   2. log-rec 41.6%

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

   3. remove-double-neg 41.6%

      \[\leadsto e^{\color{blue}{\log x}} - \tan a \]

8. Simplified 41.6%

   \[\leadsto e^{\color{blue}{\log x}} - \tan a \]
9. Step-by-step derivation

   1. add-exp-log 41.8%

      \[\leadsto \color{blue}{x} - \tan a \]

   2. sub-neg 41.8%

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

10. Applied egg-rr 41.8%

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

    1. sub-neg 41.8%

       \[\leadsto \color{blue}{x - \tan a} \]

12. Simplified 41.8%

    \[\leadsto \color{blue}{x - \tan a} \]

13. Final simplification 41.8%

    \[\leadsto x - \tan a \]

Alternative 9: 31.7% 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 77.9%

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

2. Taylor expanded in x around inf 33.1%

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

3. Final simplification 33.1%

   \[\leadsto x \]

Reproduce

herbie shell --seed 2023228 
(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))))