tan-example (used to crash)

Percentage Accurate: 79.9% → 99.7%

Time: 22.2s

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: 79.9% 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

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: a
    code = x + (((tan(y) + tan(z)) / (1.0d0 - (sin(y) * (tan(z) / cos(y))))) - tan(a))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Derivation

1. Initial program 79.6%

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

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

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/ 99.8%

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

   2. *-rgt-identity 99.8%

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

6. Simplified 99.8%

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

   1. *-commutative 99.8%

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

   2. tan-quot 99.8%

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

   3. associate-*r/ 99.8%

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

8. Applied egg-rr 99.8%

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

   1. *-commutative 99.8%

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

   2. associate-/l* 99.8%

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

10. Simplified 99.8%

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

Alternative 2: 88.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\tan a \leq -2 \cdot 10^{-13} \lor \neg \left(\tan a \leq 5 \cdot 10^{-41}\right):\\ \;\;\;\;x + \left(\tan \left(y + z\right) - \tan a\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{\tan y + \tan z}{1 - \tan y \cdot \tan z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (if (or (<= (tan a) -2e-13) (not (<= (tan a) 5e-41)))
   (+ x (- (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) <= -2e-13) || !(tan(a) <= 5e-41)) {
		tmp = x + (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) <= (-2d-13)) .or. (.not. (tan(a) <= 5d-41))) then
        tmp = x + (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) <= -2e-13) || !(Math.tan(a) <= 5e-41)) {
		tmp = x + (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) <= -2e-13) or not (math.tan(a) <= 5e-41):
		tmp = x + (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) <= -2e-13) || !(tan(a) <= 5e-41))
		tmp = Float64(x + 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) <= -2e-13) || ~((tan(a) <= 5e-41)))
		tmp = x + (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], -2e-13], N[Not[LessEqual[N[Tan[a], $MachinePrecision], 5e-41]], $MachinePrecision]], N[(x + N[(N[Tan[N[(y + z), $MachinePrecision]], $MachinePrecision] - N[Tan[a], $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) < -2.0000000000000001e-13 or 4.9999999999999996e-41 < (tan.f64 a)

   1. Initial program 82.4%

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

   if -2.0000000000000001e-13 < (tan.f64 a) < 4.9999999999999996e-41

   1. Initial program 77.0%

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

      1. add-sqr-sqrt 47.3%

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

      2. sqrt-unprod 65.3%

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

      3. pow2 65.3%

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

   4. Applied egg-rr 65.3%

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

   5. Taylor expanded in a around 0 65.3%

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

      1. sqrt-div 65.3%

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

      2. sqrt-pow1 65.7%

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

      3. metadata-eval 65.7%

         \[\leadsto x + \frac{{\sin \left(y + z\right)}^{\color{blue}{1}}}{\sqrt{{\cos \left(y + z\right)}^{2}}} \]

      4. pow1 65.7%

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

      5. sqrt-pow1 77.0%

         \[\leadsto x + \frac{\sin \left(y + z\right)}{\color{blue}{{\cos \left(y + z\right)}^{\left(\frac{2}{2}\right)}}} \]

      6. metadata-eval 77.0%

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

      7. pow1 77.0%

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

      8. tan-quot 77.0%

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

      9. tan-sum 99.8%

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

      10. +-commutative 99.8%

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

   7. Applied egg-rr 99.8%

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

      1. *-commutative 99.8%

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

   9. Simplified 99.8%

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

4. Final simplification 91.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\tan a \leq -2 \cdot 10^{-13} \lor \neg \left(\tan a \leq 5 \cdot 10^{-41}\right):\\ \;\;\;\;x + \left(\tan \left(y + z\right) - \tan a\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{\tan y + \tan z}{1 - \tan y \cdot \tan z}\\ \end{array} \]
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 79.6%

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

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

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/ 99.8%

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

   2. *-rgt-identity 99.8%

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

6. Simplified 99.8%

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

Alternative 4: 61.2% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x y z a)
 :precision binary64
 (if (<= (+ y z) -5e-14) (+ 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) <= -5e-14) {
		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) <= (-5d-14)) 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) <= -5e-14) {
		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) <= -5e-14:
		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) <= -5e-14)
		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) <= -5e-14)
		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], -5e-14], 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) < -5.0000000000000002e-14

   1. Initial program 72.7%

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

   3. Taylor expanded in z around 0 45.7%

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

      1. tan-quot 45.7%

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

      2. tan-quot 45.7%

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

      3. associate--l+ 45.7%

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

   5. Applied egg-rr 45.7%

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

   if -5.0000000000000002e-14 < (+.f64 y z)

   1. Initial program 83.7%

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

   3. Taylor expanded in y around 0 70.2%

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

      1. +-commutative 70.2%

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

      2. tan-quot 70.2%

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

      3. *-un-lft-identity 70.2%

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

      4. fma-define 70.2%

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

   5. Applied egg-rr 70.2%

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

      1. fma-undefine 70.2%

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

      2. *-lft-identity 70.2%

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

   7. Simplified 70.2%

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

4. Final simplification 61.0%

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

Alternative 5: 60.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y + z \leq 10^{-19}:\\ \;\;\;\;x + \left(\tan y - \tan a\right)\\ \mathbf{else}:\\ \;\;\;\;x + \tan \left(y + z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (if (<= (+ y z) 1e-19) (+ 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) <= 1e-19) {
		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) <= 1d-19) 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) <= 1e-19) {
		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) <= 1e-19:
		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) <= 1e-19)
		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) <= 1e-19)
		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], 1e-19], 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) < 9.9999999999999998e-20

   1. Initial program 84.3%

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

   3. Taylor expanded in z around 0 68.7%

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

      1. tan-quot 68.7%

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

      2. tan-quot 68.7%

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

      3. associate--l+ 68.8%

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

   5. Applied egg-rr 68.8%

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

   if 9.9999999999999998e-20 < (+.f64 y z)

   1. Initial program 70.8%

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

      1. add-sqr-sqrt 45.4%

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

      2. sqrt-unprod 52.6%

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

      3. pow2 52.6%

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

   4. Applied egg-rr 52.6%

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

   5. Taylor expanded in a around 0 45.2%

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

      1. sqrt-div 45.2%

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

      2. sqrt-pow1 40.8%

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

      3. metadata-eval 40.8%

         \[\leadsto x + \frac{{\sin \left(y + z\right)}^{\color{blue}{1}}}{\sqrt{{\cos \left(y + z\right)}^{2}}} \]

      4. pow1 40.8%

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

      5. sqrt-pow1 53.0%

         \[\leadsto x + \frac{\sin \left(y + z\right)}{\color{blue}{{\cos \left(y + z\right)}^{\left(\frac{2}{2}\right)}}} \]

      6. metadata-eval 53.0%

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

      7. pow1 53.0%

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

      8. tan-quot 53.1%

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

      9. *-un-lft-identity 53.1%

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

   7. Applied egg-rr 53.1%

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

      1. *-lft-identity 53.1%

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

   9. Simplified 53.1%

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

Alternative 6: 79.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.6%

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

Alternative 7: 59.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y + z \leq -0.2 \lor \neg \left(y + z \leq 0.002\right):\\ \;\;\;\;x + \tan \left(y + z\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(z - \tan a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (if (or (<= (+ y z) -0.2) (not (<= (+ y z) 0.002)))
   (+ x (tan (+ y z)))
   (+ x (- z (tan a)))))

C

double code(double x, double y, double z, double a) {
	double tmp;
	if (((y + z) <= -0.2) || !((y + z) <= 0.002)) {
		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) <= (-0.2d0)) .or. (.not. ((y + z) <= 0.002d0))) 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) <= -0.2) || !((y + z) <= 0.002)) {
		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) <= -0.2) or not ((y + z) <= 0.002):
		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) <= -0.2) || !(Float64(y + z) <= 0.002))
		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) <= -0.2) || ~(((y + z) <= 0.002)))
		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], -0.2], N[Not[LessEqual[N[(y + z), $MachinePrecision], 0.002]], $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) < -0.20000000000000001 or 2e-3 < (+.f64 y z)

   1. Initial program 70.5%

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

      1. add-sqr-sqrt 39.8%

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

      2. sqrt-unprod 48.5%

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

      3. pow2 48.5%

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

   4. Applied egg-rr 48.5%

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

   5. Taylor expanded in a around 0 37.5%

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

      1. sqrt-div 37.5%

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

      2. sqrt-pow1 37.0%

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

      3. metadata-eval 37.0%

         \[\leadsto x + \frac{{\sin \left(y + z\right)}^{\color{blue}{1}}}{\sqrt{{\cos \left(y + z\right)}^{2}}} \]

      4. pow1 37.0%

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

      5. sqrt-pow1 46.8%

         \[\leadsto x + \frac{\sin \left(y + z\right)}{\color{blue}{{\cos \left(y + z\right)}^{\left(\frac{2}{2}\right)}}} \]

      6. metadata-eval 46.8%

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

      7. pow1 46.8%

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

      8. tan-quot 46.8%

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

      9. *-un-lft-identity 46.8%

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

   7. Applied egg-rr 46.8%

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

      1. *-lft-identity 46.8%

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

   9. Simplified 46.8%

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

   if -0.20000000000000001 < (+.f64 y z) < 2e-3

   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 98.3%

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

   4. Taylor expanded in z around 0 98.1%

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

4. Final simplification 62.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y + z \leq -0.2 \lor \neg \left(y + z \leq 0.002\right):\\ \;\;\;\;x + \tan \left(y + z\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(z - \tan a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 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 79.6%

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

   1. add-sqr-sqrt 45.8%

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

   2. sqrt-unprod 59.9%

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

   3. pow2 59.9%

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

4. Applied egg-rr 59.9%

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

5. Taylor expanded in a around 0 45.6%

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

   1. sqrt-div 45.6%

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

   2. sqrt-pow1 45.3%

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

   3. metadata-eval 45.3%

      \[\leadsto x + \frac{{\sin \left(y + z\right)}^{\color{blue}{1}}}{\sqrt{{\cos \left(y + z\right)}^{2}}} \]

   4. pow1 45.3%

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

   5. sqrt-pow1 52.1%

      \[\leadsto x + \frac{\sin \left(y + z\right)}{\color{blue}{{\cos \left(y + z\right)}^{\left(\frac{2}{2}\right)}}} \]

   6. metadata-eval 52.1%

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

   7. pow1 52.1%

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

   8. tan-quot 52.1%

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

   9. *-un-lft-identity 52.1%

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

7. Applied egg-rr 52.1%

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

   1. *-lft-identity 52.1%

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

9. Simplified 52.1%

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

Alternative 9: 31.8% accurate, 207.0× speedup

Math

\[\begin{array}{l} \\ x \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, a_] := x

Derivation

1. Initial program 79.6%

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

3. Taylor expanded in x around inf 35.3%

   \[\leadsto \color{blue}{x} \]
4. Add Preprocessing

Reproduce

herbie shell --seed 2024091 
(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))))