tan-example (used to crash)

Percentage Accurate: 80.1% → 99.7%

Time: 30.8s

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: 80.1% 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

\[\begin{array}{l} \\ x + \left(\frac{\tan y + \tan z}{1 - \sqrt[3]{{\left(\tan y \cdot \tan z\right)}^{3}}} - \tan a\right) \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (+
  x
  (-
   (/ (+ (tan y) (tan z)) (- 1.0 (cbrt (pow (* (tan y) (tan z)) 3.0))))
   (tan a))))

C

double code(double x, double y, double z, double a) {
	return x + (((tan(y) + tan(z)) / (1.0 - cbrt(pow((tan(y) * tan(z)), 3.0)))) - tan(a));
}

Java

public static double code(double x, double y, double z, double a) {
	return x + (((Math.tan(y) + Math.tan(z)) / (1.0 - Math.cbrt(Math.pow((Math.tan(y) * Math.tan(z)), 3.0)))) - Math.tan(a));
}

Julia

function code(x, y, z, a)
	return Float64(x + Float64(Float64(Float64(tan(y) + tan(z)) / Float64(1.0 - cbrt((Float64(tan(y) * tan(z)) ^ 3.0)))) - 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[Power[N[Power[N[(N[Tan[y], $MachinePrecision] * N[Tan[z], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 77.4%

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

   1. tan-sum 99.7%

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

   2. div-inv 99.7%

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

4. Applied egg-rr 99.7%

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

   1. associate-*r/ 99.7%

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

   2. *-rgt-identity 99.7%

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

6. Simplified 99.7%

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

   1. add-cbrt-cube 99.8%

      \[\leadsto x + \left(\frac{\tan y + \tan z}{1 - \color{blue}{\sqrt[3]{\left(\left(\tan y \cdot \tan z\right) \cdot \left(\tan y \cdot \tan z\right)\right) \cdot \left(\tan y \cdot \tan z\right)}}} - \tan a\right) \]

   2. pow3 99.8%

      \[\leadsto x + \left(\frac{\tan y + \tan z}{1 - \sqrt[3]{\color{blue}{{\left(\tan y \cdot \tan z\right)}^{3}}}} - \tan a\right) \]

8. Applied egg-rr 99.8%

   \[\leadsto x + \left(\frac{\tan y + \tan z}{1 - \color{blue}{\sqrt[3]{{\left(\tan y \cdot \tan z\right)}^{3}}}} - \tan a\right) \]

9. Final simplification 99.8%

   \[\leadsto x + \left(\frac{\tan y + \tan z}{1 - \sqrt[3]{{\left(\tan y \cdot \tan z\right)}^{3}}} - \tan a\right) \]
10. Add Preprocessing

Alternative 2: 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 77.4%

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

   1. tan-sum 99.7%

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

   2. div-inv 99.7%

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

4. Applied egg-rr 99.7%

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

   1. associate-*r/ 99.7%

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

   2. *-rgt-identity 99.7%

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

6. Simplified 99.7%

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

7. Final simplification 99.7%

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

Alternative 3: 60.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y + z \leq 4 \cdot 10^{-14}:\\ \;\;\;\;x + \left(\tan y - \tan a\right)\\ \mathbf{else}:\\ \;\;\;\;x + \tan \left(y + z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (if (<= (+ y z) 4e-14) (+ 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) <= 4e-14) {
		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) <= 4d-14) 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) <= 4e-14) {
		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) <= 4e-14:
		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) <= 4e-14)
		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) <= 4e-14)
		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], 4e-14], 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) < 4e-14

   1. Initial program 80.3%

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

   3. Taylor expanded in z around 0 67.3%

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

      1. +-commutative 67.3%

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

      2. associate--l+ 67.3%

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

   5. Simplified 67.3%

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

      1. tan-quot 67.3%

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

      2. tan-quot 67.4%

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

      3. associate-+r- 67.3%

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

   7. Applied egg-rr 67.3%

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

      1. +-commutative 67.3%

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

      2. associate--l+ 67.4%

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

   9. Simplified 67.4%

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

   if 4e-14 < (+.f64 y z)

   1. Initial program 72.1%

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

      1. expm1-log1p-u 66.8%

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

      2. +-commutative 66.8%

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

      3. associate-+l- 66.7%

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

   4. Applied egg-rr 66.7%

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

   5. Taylor expanded in a around 0 44.7%

      \[\leadsto \mathsf{expm1}\left(\color{blue}{\log \left(1 + \left(x + \frac{\sin \left(y + z\right)}{\cos \left(y + z\right)}\right)\right)}\right) \]
   6. Step-by-step derivation

      1. associate-+r+ 44.7%

         \[\leadsto \mathsf{expm1}\left(\log \color{blue}{\left(\left(1 + x\right) + \frac{\sin \left(y + z\right)}{\cos \left(y + z\right)}\right)}\right) \]

   7. Simplified 44.7%

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

      1. associate-+l+ 44.7%

         \[\leadsto \mathsf{expm1}\left(\log \color{blue}{\left(1 + \left(x + \frac{\sin \left(y + z\right)}{\cos \left(y + z\right)}\right)\right)}\right) \]

      2. log1p-define 44.7%

         \[\leadsto \mathsf{expm1}\left(\color{blue}{\mathsf{log1p}\left(x + \frac{\sin \left(y + z\right)}{\cos \left(y + z\right)}\right)}\right) \]

      3. expm1-log1p-u 45.4%

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

      4. quot-tan 45.4%

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

   9. Applied egg-rr 45.4%

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

4. Final simplification 59.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y + z \leq 4 \cdot 10^{-14}:\\ \;\;\;\;x + \left(\tan y - \tan a\right)\\ \mathbf{else}:\\ \;\;\;\;x + \tan \left(y + z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 60.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y + z \leq -1 \cdot 10^{-12}:\\ \;\;\;\;x + \left(\tan y - \tan a\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(\tan z - \tan a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (if (<= (+ y z) -1e-12) (+ x (- (tan y) (tan a))) (+ x (- (tan z) (tan a)))))

C

double code(double x, double y, double z, double a) {
	double tmp;
	if ((y + z) <= -1e-12) {
		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 ((y + z) <= (-1d-12)) 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 ((y + z) <= -1e-12) {
		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 (y + z) <= -1e-12:
		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 (Float64(y + z) <= -1e-12)
		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 ((y + z) <= -1e-12)
		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[N[(y + z), $MachinePrecision], -1e-12], 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 (+.f64 y z) < -9.9999999999999998e-13

   1. Initial program 68.9%

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

   3. Taylor expanded in z around 0 48.2%

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

      1. +-commutative 48.2%

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

      2. associate--l+ 48.2%

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

   5. Simplified 48.2%

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

      1. tan-quot 48.3%

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

      2. tan-quot 48.3%

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

      3. associate-+r- 48.3%

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

   7. Applied egg-rr 48.3%

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

      1. +-commutative 48.3%

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

      2. associate--l+ 48.3%

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

   9. Simplified 48.3%

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

   if -9.9999999999999998e-13 < (+.f64 y z)

   1. Initial program 83.3%

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

   3. Taylor expanded in y around 0 69.7%

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

      1. tan-quot 69.7%

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

      2. *-un-lft-identity 69.7%

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

   5. Applied egg-rr 69.7%

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

      1. *-lft-identity 69.7%

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

   7. Simplified 69.7%

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

4. Final simplification 61.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y + z \leq -1 \cdot 10^{-12}:\\ \;\;\;\;x + \left(\tan y - \tan a\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(\tan z - \tan a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 80.1% 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 77.4%

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

3. Final simplification 77.4%

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

Alternative 6: 60.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y + z \leq -1 \cdot 10^{-12} \lor \neg \left(y + z \leq 4 \cdot 10^{-14}\right):\\ \;\;\;\;x + \tan \left(y + z\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(z - \tan a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (if (or (<= (+ y z) -1e-12) (not (<= (+ y z) 4e-14)))
   (+ x (tan (+ y z)))
   (+ x (- z (tan a)))))

C

double code(double x, double y, double z, double a) {
	double tmp;
	if (((y + z) <= -1e-12) || !((y + z) <= 4e-14)) {
		tmp = x + tan((y + z));
	} else {
		tmp = x + (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) <= (-1d-12)) .or. (.not. ((y + z) <= 4d-14))) then
        tmp = x + tan((y + z))
    else
        tmp = x + (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) <= -1e-12) || !((y + z) <= 4e-14)) {
		tmp = x + Math.tan((y + z));
	} else {
		tmp = x + (z - Math.tan(a));
	}
	return tmp;
}

Python

def code(x, y, z, a):
	tmp = 0
	if ((y + z) <= -1e-12) or not ((y + z) <= 4e-14):
		tmp = x + math.tan((y + z))
	else:
		tmp = x + (z - math.tan(a))
	return tmp

Julia

function code(x, y, z, a)
	tmp = 0.0
	if ((Float64(y + z) <= -1e-12) || !(Float64(y + z) <= 4e-14))
		tmp = Float64(x + tan(Float64(y + z)));
	else
		tmp = Float64(x + Float64(z - tan(a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, a)
	tmp = 0.0;
	if (((y + z) <= -1e-12) || ~(((y + z) <= 4e-14)))
		tmp = x + tan((y + z));
	else
		tmp = x + (z - tan(a));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, a_] := If[Or[LessEqual[N[(y + z), $MachinePrecision], -1e-12], N[Not[LessEqual[N[(y + z), $MachinePrecision], 4e-14]], $MachinePrecision]], N[(x + N[Tan[N[(y + z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(x + N[(z - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 y z) < -9.9999999999999998e-13 or 4e-14 < (+.f64 y z)

   1. Initial program 70.4%

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

      1. expm1-log1p-u 64.9%

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

      2. +-commutative 64.9%

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

      3. associate-+l- 64.9%

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

   4. Applied egg-rr 64.9%

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

   5. Taylor expanded in a around 0 44.8%

      \[\leadsto \mathsf{expm1}\left(\color{blue}{\log \left(1 + \left(x + \frac{\sin \left(y + z\right)}{\cos \left(y + z\right)}\right)\right)}\right) \]
   6. Step-by-step derivation

      1. associate-+r+ 44.8%

         \[\leadsto \mathsf{expm1}\left(\log \color{blue}{\left(\left(1 + x\right) + \frac{\sin \left(y + z\right)}{\cos \left(y + z\right)}\right)}\right) \]

   7. Simplified 44.8%

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

      1. associate-+l+ 44.8%

         \[\leadsto \mathsf{expm1}\left(\log \color{blue}{\left(1 + \left(x + \frac{\sin \left(y + z\right)}{\cos \left(y + z\right)}\right)\right)}\right) \]

      2. log1p-define 44.8%

         \[\leadsto \mathsf{expm1}\left(\color{blue}{\mathsf{log1p}\left(x + \frac{\sin \left(y + z\right)}{\cos \left(y + z\right)}\right)}\right) \]

      3. expm1-log1p-u 46.2%

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

      4. quot-tan 46.2%

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

   9. Applied egg-rr 46.2%

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

   if -9.9999999999999998e-13 < (+.f64 y z) < 4e-14

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0 99.9%

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

   4. Taylor expanded in z around 0 99.9%

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

4. Final simplification 59.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y + z \leq -1 \cdot 10^{-12} \lor \neg \left(y + z \leq 4 \cdot 10^{-14}\right):\\ \;\;\;\;x + \tan \left(y + z\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(z - \tan a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 51.3% 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 77.4%

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

   1. expm1-log1p-u 73.0%

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

   2. +-commutative 73.0%

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

   3. associate-+l- 73.0%

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

4. Applied egg-rr 73.0%

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

5. Taylor expanded in a around 0 47.1%

   \[\leadsto \mathsf{expm1}\left(\color{blue}{\log \left(1 + \left(x + \frac{\sin \left(y + z\right)}{\cos \left(y + z\right)}\right)\right)}\right) \]
6. Step-by-step derivation

   1. associate-+r+ 47.1%

      \[\leadsto \mathsf{expm1}\left(\log \color{blue}{\left(\left(1 + x\right) + \frac{\sin \left(y + z\right)}{\cos \left(y + z\right)}\right)}\right) \]

7. Simplified 47.1%

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

   1. associate-+l+ 47.1%

      \[\leadsto \mathsf{expm1}\left(\log \color{blue}{\left(1 + \left(x + \frac{\sin \left(y + z\right)}{\cos \left(y + z\right)}\right)\right)}\right) \]

   2. log1p-define 47.1%

      \[\leadsto \mathsf{expm1}\left(\color{blue}{\mathsf{log1p}\left(x + \frac{\sin \left(y + z\right)}{\cos \left(y + z\right)}\right)}\right) \]

   3. expm1-log1p-u 48.2%

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

   4. quot-tan 48.2%

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

9. Applied egg-rr 48.2%

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

10. Final simplification 48.2%

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

Alternative 8: 2.7% accurate, 207.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.4%

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

   1. expm1-log1p-u 73.0%

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

   2. +-commutative 73.0%

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

   3. associate-+l- 73.0%

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

4. Applied egg-rr 73.0%

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

5. Taylor expanded in x around inf 21.6%

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

   1. mul-1-neg 21.6%

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

   2. log-rec 21.6%

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

   3. remove-double-neg 21.6%

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

7. Simplified 21.6%

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

8. Taylor expanded in x around 0 2.5%

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

9. Final simplification 2.5%

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

Alternative 9: 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 77.4%

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

3. Taylor expanded in x around inf 30.5%

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

4. Final simplification 30.5%

   \[\leadsto x \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024048 
(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))))