tan-example (used to crash)

Percentage Accurate: 79.0% → 99.7%

Time: 39.9s

Alternatives: 10

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 79.0% 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

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

FPCore

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

C

double code(double x, double y, double z, double a) {
	return x + (((tan(y) + tan(z)) / (1.0 - ((1.0 + (tan(y) * tan(z))) + -1.0))) - 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 - ((1.0d0 + (tan(y) * tan(z))) + (-1.0d0)))) - 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 - ((1.0 + (Math.tan(y) * Math.tan(z))) + -1.0))) - Math.tan(a));
}

Python

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

Julia

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

MATLAB

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

Derivation

1. Initial program 76.5%

   \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
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. expm1-log1p-u 91.9%

      \[\leadsto x + \left(\frac{\tan y + \tan z}{1 - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\tan y \cdot \tan z\right)\right)}} - \tan a\right) \]

   2. expm1-udef 91.9%

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

   3. log1p-udef 91.9%

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

   4. add-exp-log 99.7%

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

   5. +-commutative 99.7%

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

8. Applied egg-rr 99.7%

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

9. Final simplification 99.7%

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

Alternative 2: 99.7% accurate, 0.4× speedup

Math

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

FPCore

(FPCore (x y z a)
 :precision binary64
 (+ x (- (* (+ (tan y) (tan z)) (/ 1.0 (- 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 / (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 / (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 / (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 / (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(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 / (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[(1.0 - N[(N[Tan[y], $MachinePrecision] * N[Tan[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 76.5%

   \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
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. Final simplification 99.7%

   \[\leadsto x + \left(\left(\tan y + \tan z\right) \cdot \frac{1}{1 - \tan y \cdot \tan z} - \tan a\right) \]
6. 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 76.5%

   \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
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 4: 60.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y + z \leq -2000000000000:\\ \;\;\;\;x + \tan \left(y + z\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(\tan z - \tan a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (if (<= (+ y z) -2000000000000.0)
   (+ x (tan (+ y z)))
   (+ x (- (tan z) (tan a)))))

C

double code(double x, double y, double z, double a) {
	double tmp;
	if ((y + z) <= -2000000000000.0) {
		tmp = x + tan((y + z));
	} else {
		tmp = x + (tan(z) - tan(a));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: a
    real(8) :: tmp
    if ((y + z) <= (-2000000000000.0d0)) then
        tmp = x + tan((y + z))
    else
        tmp = x + (tan(z) - tan(a))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double a) {
	double tmp;
	if ((y + z) <= -2000000000000.0) {
		tmp = x + Math.tan((y + z));
	} else {
		tmp = x + (Math.tan(z) - Math.tan(a));
	}
	return tmp;
}

Python

def code(x, y, z, a):
	tmp = 0
	if (y + z) <= -2000000000000.0:
		tmp = x + math.tan((y + z))
	else:
		tmp = x + (math.tan(z) - math.tan(a))
	return tmp

Julia

function code(x, y, z, a)
	tmp = 0.0
	if (Float64(y + z) <= -2000000000000.0)
		tmp = Float64(x + tan(Float64(y + z)));
	else
		tmp = Float64(x + Float64(tan(z) - tan(a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, a)
	tmp = 0.0;
	if ((y + z) <= -2000000000000.0)
		tmp = x + tan((y + z));
	else
		tmp = x + (tan(z) - tan(a));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, a_] := If[LessEqual[N[(y + z), $MachinePrecision], -2000000000000.0], N[(x + N[Tan[N[(y + z), $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) < -2e12

   1. Initial program 67.3%

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

      1. add-cube-cbrt 66.3%

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

      2. pow3 66.3%

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

      3. +-commutative 66.3%

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

      4. associate-+l- 66.3%

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

   4. Applied egg-rr 66.3%

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

   5. Taylor expanded in a around 0 42.7%

      \[\leadsto {\color{blue}{\left({\left(x + \frac{\sin \left(y + z\right)}{\cos \left(y + z\right)}\right)}^{0.3333333333333333}\right)}}^{3} \]
   6. Step-by-step derivation

      1. unpow1/3 46.8%

         \[\leadsto {\color{blue}{\left(\sqrt[3]{x + \frac{\sin \left(y + z\right)}{\cos \left(y + z\right)}}\right)}}^{3} \]

   7. Simplified 46.8%

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

      1. rem-cube-cbrt 47.4%

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

      2. tan-quot 47.5%

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

      3. +-commutative 47.5%

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

   9. Applied egg-rr 47.5%

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

   if -2e12 < (+.f64 y z)

   1. Initial program 80.4%

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

   3. Taylor expanded in y around 0 66.1%

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

      1. tan-quot 66.1%

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

      2. expm1-log1p-u 60.5%

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

      3. expm1-udef 60.5%

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

   5. Applied egg-rr 60.5%

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

      1. expm1-def 60.5%

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

      2. expm1-log1p 66.1%

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

   7. Simplified 66.1%

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

4. Final simplification 60.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y + z \leq -2000000000000:\\ \;\;\;\;x + \tan \left(y + z\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(\tan z - \tan a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 64.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3200000000:\\ \;\;\;\;x + \frac{\sin y}{\cos y}\\ \mathbf{else}:\\ \;\;\;\;x + \left(\tan z - \tan a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (if (<= y -3200000000.0) (+ x (/ (sin y) (cos y))) (+ x (- (tan z) (tan a)))))

C

double code(double x, double y, double z, double a) {
	double tmp;
	if (y <= -3200000000.0) {
		tmp = x + (sin(y) / cos(y));
	} 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 <= (-3200000000.0d0)) then
        tmp = x + (sin(y) / cos(y))
    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 <= -3200000000.0) {
		tmp = x + (Math.sin(y) / Math.cos(y));
	} else {
		tmp = x + (Math.tan(z) - Math.tan(a));
	}
	return tmp;
}

Python

def code(x, y, z, a):
	tmp = 0
	if y <= -3200000000.0:
		tmp = x + (math.sin(y) / math.cos(y))
	else:
		tmp = x + (math.tan(z) - math.tan(a))
	return tmp

Julia

function code(x, y, z, a)
	tmp = 0.0
	if (y <= -3200000000.0)
		tmp = Float64(x + Float64(sin(y) / cos(y)));
	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 <= -3200000000.0)
		tmp = x + (sin(y) / cos(y));
	else
		tmp = x + (tan(z) - tan(a));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, a_] := If[LessEqual[y, -3200000000.0], N[(x + N[(N[Sin[y], $MachinePrecision] / N[Cos[y], $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 y < -3.2e9

   1. Initial program 51.7%

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

      1. add-cube-cbrt 50.9%

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

      2. pow3 50.9%

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

      3. +-commutative 50.9%

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

      4. associate-+l- 50.9%

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

   4. Applied egg-rr 50.9%

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

   5. Taylor expanded in a around 0 38.7%

      \[\leadsto {\color{blue}{\left({\left(x + \frac{\sin \left(y + z\right)}{\cos \left(y + z\right)}\right)}^{0.3333333333333333}\right)}}^{3} \]
   6. Step-by-step derivation

      1. unpow1/3 40.6%

         \[\leadsto {\color{blue}{\left(\sqrt[3]{x + \frac{\sin \left(y + z\right)}{\cos \left(y + z\right)}}\right)}}^{3} \]

   7. Simplified 40.6%

      \[\leadsto {\color{blue}{\left(\sqrt[3]{x + \frac{\sin \left(y + z\right)}{\cos \left(y + z\right)}}\right)}}^{3} \]

   8. Taylor expanded in z around 0 40.8%

      \[\leadsto \color{blue}{{1}^{0.3333333333333333} \cdot \left(x + \frac{\sin y}{\cos y}\right)} \]
   9. Step-by-step derivation

      1. pow-base-1 40.8%

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

      2. *-lft-identity 40.8%

         \[\leadsto \color{blue}{x + \frac{\sin y}{\cos y}} \]

   10. Simplified 40.8%

       \[\leadsto \color{blue}{x + \frac{\sin y}{\cos y}} \]

   if -3.2e9 < y

   1. Initial program 83.9%

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

   3. Taylor expanded in y around 0 70.8%

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

      1. tan-quot 70.8%

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

      2. expm1-log1p-u 59.8%

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

      3. expm1-udef 59.8%

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

   5. Applied egg-rr 59.8%

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

      1. expm1-def 59.8%

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

      2. expm1-log1p 70.8%

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

   7. Simplified 70.8%

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

4. Final simplification 63.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3200000000:\\ \;\;\;\;x + \frac{\sin y}{\cos y}\\ \mathbf{else}:\\ \;\;\;\;x + \left(\tan z - \tan a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 79.0% 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 76.5%

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

3. Final simplification 76.5%

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

Alternative 7: 58.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y + z \leq -2000000000000 \lor \neg \left(y + z \leq 4 \cdot 10^{-48}\right):\\ \;\;\;\;x + \tan \left(y + z\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(z - \tan a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (if (or (<= (+ y z) -2000000000000.0) (not (<= (+ y z) 4e-48)))
   (+ x (tan (+ y z)))
   (+ x (- z (tan a)))))

C

double code(double x, double y, double z, double a) {
	double tmp;
	if (((y + z) <= -2000000000000.0) || !((y + z) <= 4e-48)) {
		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) <= (-2000000000000.0d0)) .or. (.not. ((y + z) <= 4d-48))) 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) <= -2000000000000.0) || !((y + z) <= 4e-48)) {
		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) <= -2000000000000.0) or not ((y + z) <= 4e-48):
		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) <= -2000000000000.0) || !(Float64(y + z) <= 4e-48))
		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) <= -2000000000000.0) || ~(((y + z) <= 4e-48)))
		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], -2000000000000.0], N[Not[LessEqual[N[(y + z), $MachinePrecision], 4e-48]], $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) < -2e12 or 3.9999999999999999e-48 < (+.f64 y z)

   1. Initial program 68.5%

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

      1. add-cube-cbrt 67.5%

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

      2. pow3 67.5%

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

      3. +-commutative 67.5%

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

      4. associate-+l- 67.5%

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

   4. Applied egg-rr 67.5%

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

   5. Taylor expanded in a around 0 42.8%

      \[\leadsto {\color{blue}{\left({\left(x + \frac{\sin \left(y + z\right)}{\cos \left(y + z\right)}\right)}^{0.3333333333333333}\right)}}^{3} \]
   6. Step-by-step derivation

      1. unpow1/3 45.2%

         \[\leadsto {\color{blue}{\left(\sqrt[3]{x + \frac{\sin \left(y + z\right)}{\cos \left(y + z\right)}}\right)}}^{3} \]

   7. Simplified 45.2%

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

      1. rem-cube-cbrt 45.8%

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

      2. tan-quot 45.8%

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

      3. +-commutative 45.8%

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

   9. Applied egg-rr 45.8%

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

   if -2e12 < (+.f64 y z) < 3.9999999999999999e-48

   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 99.4%

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

   4. Taylor expanded in z around 0 97.6%

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

4. Final simplification 58.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y + z \leq -2000000000000 \lor \neg \left(y + z \leq 4 \cdot 10^{-48}\right):\\ \;\;\;\;x + \tan \left(y + z\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(z - \tan a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 41.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.7:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 1.6:\\ \;\;\;\;x + \left(z - \tan a\right)\\ \mathbf{else}:\\ \;\;\;\;x + \sin z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (if (<= z -1.7) x (if (<= z 1.6) (+ x (- z (tan a))) (+ x (sin z)))))

C

double code(double x, double y, double z, double a) {
	double tmp;
	if (z <= -1.7) {
		tmp = x;
	} else if (z <= 1.6) {
		tmp = x + (z - tan(a));
	} else {
		tmp = x + sin(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 (z <= (-1.7d0)) then
        tmp = x
    else if (z <= 1.6d0) then
        tmp = x + (z - tan(a))
    else
        tmp = x + sin(z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double a) {
	double tmp;
	if (z <= -1.7) {
		tmp = x;
	} else if (z <= 1.6) {
		tmp = x + (z - Math.tan(a));
	} else {
		tmp = x + Math.sin(z);
	}
	return tmp;
}

Python

def code(x, y, z, a):
	tmp = 0
	if z <= -1.7:
		tmp = x
	elif z <= 1.6:
		tmp = x + (z - math.tan(a))
	else:
		tmp = x + math.sin(z)
	return tmp

Julia

function code(x, y, z, a)
	tmp = 0.0
	if (z <= -1.7)
		tmp = x;
	elseif (z <= 1.6)
		tmp = Float64(x + Float64(z - tan(a)));
	else
		tmp = Float64(x + sin(z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, a)
	tmp = 0.0;
	if (z <= -1.7)
		tmp = x;
	elseif (z <= 1.6)
		tmp = x + (z - tan(a));
	else
		tmp = x + sin(z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, a_] := If[LessEqual[z, -1.7], x, If[LessEqual[z, 1.6], N[(x + N[(z - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[Sin[z], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.69999999999999996

   1. Initial program 51.4%

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

   3. Taylor expanded in x around inf 22.1%

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

   if -1.69999999999999996 < z < 1.6000000000000001

   1. Initial program 99.3%

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

   3. Taylor expanded in y around 0 65.7%

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

   4. Taylor expanded in z around 0 65.1%

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

   if 1.6000000000000001 < z

   1. Initial program 54.3%

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

   3. Taylor expanded in y around 0 55.0%

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

   4. Taylor expanded in z around 0 24.0%

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

   5. Taylor expanded in a around 0 23.2%

      \[\leadsto \color{blue}{x + \sin z} \]
   6. Step-by-step derivation

      1. +-commutative 23.2%

         \[\leadsto \color{blue}{\sin z + x} \]

   7. Simplified 23.2%

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

4. Final simplification 44.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.7:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 1.6:\\ \;\;\;\;x + \left(z - \tan a\right)\\ \mathbf{else}:\\ \;\;\;\;x + \sin z\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 31.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ x + \sin z \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 76.5%

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

3. Taylor expanded in y around 0 59.8%

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

4. Taylor expanded in z around 0 44.8%

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

5. Taylor expanded in a around 0 34.7%

   \[\leadsto \color{blue}{x + \sin z} \]
6. Step-by-step derivation

   1. +-commutative 34.7%

      \[\leadsto \color{blue}{\sin z + x} \]

7. Simplified 34.7%

   \[\leadsto \color{blue}{\sin z + x} \]

8. Final simplification 34.7%

   \[\leadsto x + \sin z \]
9. Add Preprocessing

Alternative 10: 31.2% 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 76.5%

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

3. Taylor expanded in x around inf 34.5%

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

4. Final simplification 34.5%

   \[\leadsto x \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024027 
(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))))