tan-example (used to crash)

Percentage Accurate: 79.5% → 99.7%

Time: 47.8s

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: 79.5% 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

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

FPCore

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

C

double code(double x, double y, double z, double a) {
	return (((tan(y) + tan(z)) / (1.0 - (tan(y) * tan(z)))) - tan(a)) + 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 = (((tan(y) + tan(z)) / (1.0d0 - (tan(y) * tan(z)))) - tan(a)) + x
end function

Java

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

Python

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

Julia

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

MATLAB

function tmp = code(x, y, z, a)
	tmp = (((tan(y) + tan(z)) / (1.0 - (tan(y) * tan(z)))) - tan(a)) + x;
end

Wolfram

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

Derivation

1. Initial program 81.5%

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

   1. +-commutative 81.5%

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

   2. sub-neg 81.5%

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

   3. associate-+l+ 81.4%

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

   4. tan-sum 99.7%

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

   5. div-inv 99.7%

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

   6. fma-define 99.7%

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

   7. neg-mul-1 99.7%

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

   8. fma-define 99.7%

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

4. Applied egg-rr 99.7%

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

   1. fma-undefine 99.7%

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

   2. fma-undefine 99.7%

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

   3. neg-mul-1 99.7%

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

   4. associate-+r+ 99.8%

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

   5. sub-neg 99.8%

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

   6. associate-*r/ 99.8%

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

   7. *-rgt-identity 99.8%

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

6. Simplified 99.8%

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

Alternative 2: 89.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan y + \tan z\\ \mathbf{if}\;\tan a \leq -0.005 \lor \neg \left(\tan a \leq 2 \cdot 10^{-16}\right):\\ \;\;\;\;x + \left(\frac{1}{\frac{1}{t\_0}} - \tan a\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(\frac{t\_0}{1 - \tan y \cdot \tan z} - a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (let* ((t_0 (+ (tan y) (tan z))))
   (if (or (<= (tan a) -0.005) (not (<= (tan a) 2e-16)))
     (+ x (- (/ 1.0 (/ 1.0 t_0)) (tan a)))
     (+ x (- (/ t_0 (- 1.0 (* (tan y) (tan z)))) a)))))

C

double code(double x, double y, double z, double a) {
	double t_0 = tan(y) + tan(z);
	double tmp;
	if ((tan(a) <= -0.005) || !(tan(a) <= 2e-16)) {
		tmp = x + ((1.0 / (1.0 / t_0)) - tan(a));
	} else {
		tmp = x + ((t_0 / (1.0 - (tan(y) * tan(z)))) - a);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: a
    real(8) :: t_0
    real(8) :: tmp
    t_0 = tan(y) + tan(z)
    if ((tan(a) <= (-0.005d0)) .or. (.not. (tan(a) <= 2d-16))) then
        tmp = x + ((1.0d0 / (1.0d0 / t_0)) - tan(a))
    else
        tmp = x + ((t_0 / (1.0d0 - (tan(y) * tan(z)))) - a)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double a) {
	double t_0 = Math.tan(y) + Math.tan(z);
	double tmp;
	if ((Math.tan(a) <= -0.005) || !(Math.tan(a) <= 2e-16)) {
		tmp = x + ((1.0 / (1.0 / t_0)) - Math.tan(a));
	} else {
		tmp = x + ((t_0 / (1.0 - (Math.tan(y) * Math.tan(z)))) - a);
	}
	return tmp;
}

Python

def code(x, y, z, a):
	t_0 = math.tan(y) + math.tan(z)
	tmp = 0
	if (math.tan(a) <= -0.005) or not (math.tan(a) <= 2e-16):
		tmp = x + ((1.0 / (1.0 / t_0)) - math.tan(a))
	else:
		tmp = x + ((t_0 / (1.0 - (math.tan(y) * math.tan(z)))) - a)
	return tmp

Julia

function code(x, y, z, a)
	t_0 = Float64(tan(y) + tan(z))
	tmp = 0.0
	if ((tan(a) <= -0.005) || !(tan(a) <= 2e-16))
		tmp = Float64(x + Float64(Float64(1.0 / Float64(1.0 / t_0)) - tan(a)));
	else
		tmp = Float64(x + Float64(Float64(t_0 / Float64(1.0 - Float64(tan(y) * tan(z)))) - a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, a)
	t_0 = tan(y) + tan(z);
	tmp = 0.0;
	if ((tan(a) <= -0.005) || ~((tan(a) <= 2e-16)))
		tmp = x + ((1.0 / (1.0 / t_0)) - tan(a));
	else
		tmp = x + ((t_0 / (1.0 - (tan(y) * tan(z)))) - a);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, a_] := Block[{t$95$0 = N[(N[Tan[y], $MachinePrecision] + N[Tan[z], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[N[Tan[a], $MachinePrecision], -0.005], N[Not[LessEqual[N[Tan[a], $MachinePrecision], 2e-16]], $MachinePrecision]], N[(x + N[(N[(1.0 / N[(1.0 / t$95$0), $MachinePrecision]), $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[(t$95$0 / N[(1.0 - N[(N[Tan[y], $MachinePrecision] * N[Tan[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (tan.f64 a) < -0.0050000000000000001 or 2e-16 < (tan.f64 a)

   1. Initial program 81.5%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   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. clear-num 99.8%

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in y around 0 81.8%

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

   if -0.0050000000000000001 < (tan.f64 a) < 2e-16

   1. Initial program 81.5%

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

   3. Taylor expanded in a around 0 81.5%

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

      1. tan-sum 99.8%

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

      2. div-inv 99.8%

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

   5. Applied egg-rr 99.8%

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

      1. associate-*r/ 99.8%

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

      2. *-rgt-identity 99.8%

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

   7. Simplified 99.8%

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

4. Final simplification 90.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\tan a \leq -0.005 \lor \neg \left(\tan a \leq 2 \cdot 10^{-16}\right):\\ \;\;\;\;x + \left(\frac{1}{\frac{1}{\tan y + \tan z}} - \tan a\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(\frac{\tan y + \tan z}{1 - \tan y \cdot \tan z} - a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 79.9% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Python

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.5%

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

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

4. Applied egg-rr 99.7%

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

5. Taylor expanded in y around 0 81.9%

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

Alternative 4: 59.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y + z \leq -1 \cdot 10^{-16}:\\ \;\;\;\;x + \left(\tan y - \tan a\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(\tan z - \tan a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (if (<= (+ y z) -1e-16) (+ 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-16) {
		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-16)) 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-16) {
		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-16:
		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-16)
		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-16)
		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-16], 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-17

   1. Initial program 75.0%

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

   3. Taylor expanded in y around inf 46.0%

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

   if -9.9999999999999998e-17 < (+.f64 y z)

   1. Initial program 85.4%

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

   3. Taylor expanded in y around 0 71.1%

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

Alternative 5: 64.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 2.05 \cdot 10^{-7}:\\ \;\;\;\;x + \left(\tan y - \tan a\right)\\ \mathbf{else}:\\ \;\;\;\;x + \tan \left(y + z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (if (<= z 2.05e-7) (+ x (- (tan y) (tan a))) (+ x (tan (+ y z)))))

C

double code(double x, double y, double z, double a) {
	double tmp;
	if (z <= 2.05e-7) {
		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 (z <= 2.05d-7) 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 (z <= 2.05e-7) {
		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 z <= 2.05e-7:
		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 (z <= 2.05e-7)
		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 (z <= 2.05e-7)
		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[z, 2.05e-7], 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 z < 2.05e-7

   1. Initial program 86.9%

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

   3. Taylor expanded in y around inf 71.9%

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

   if 2.05e-7 < z

   1. Initial program 67.8%

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

      1. add-log-exp 67.8%

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

   4. Applied egg-rr 67.8%

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

      1. rem-log-exp 67.8%

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

      2. add-sqr-sqrt 35.4%

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

      3. sqrt-unprod 46.3%

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

      4. pow2 46.3%

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

   6. Applied egg-rr 46.3%

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

      1. unpow2 46.3%

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

      2. rem-sqrt-square 46.3%

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

      3. +-commutative 46.3%

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

   8. Simplified 46.3%

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

   9. Taylor expanded in a around 0 34.0%

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

       1. rem-square-sqrt 23.7%

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

       2. fabs-sqr 23.7%

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

       3. rem-square-sqrt 45.3%

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

   11. Simplified 45.3%

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

Alternative 6: 79.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.5%

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

Alternative 7: 60.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y + z \leq -0.001 \lor \neg \left(y + z \leq 5 \cdot 10^{-14}\right):\\ \;\;\;\;x + \tan \left(y + z\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(\left(y + z\right) - \tan a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (if (or (<= (+ y z) -0.001) (not (<= (+ y z) 5e-14)))
   (+ x (tan (+ y z)))
   (+ x (- (+ y z) (tan a)))))

C

double code(double x, double y, double z, double a) {
	double tmp;
	if (((y + z) <= -0.001) || !((y + z) <= 5e-14)) {
		tmp = x + tan((y + z));
	} else {
		tmp = x + ((y + 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) <= (-0.001d0)) .or. (.not. ((y + z) <= 5d-14))) then
        tmp = x + tan((y + z))
    else
        tmp = x + ((y + 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) <= -0.001) || !((y + z) <= 5e-14)) {
		tmp = x + Math.tan((y + z));
	} else {
		tmp = x + ((y + z) - Math.tan(a));
	}
	return tmp;
}

Python

def code(x, y, z, a):
	tmp = 0
	if ((y + z) <= -0.001) or not ((y + z) <= 5e-14):
		tmp = x + math.tan((y + z))
	else:
		tmp = x + ((y + z) - math.tan(a))
	return tmp

Julia

function code(x, y, z, a)
	tmp = 0.0
	if ((Float64(y + z) <= -0.001) || !(Float64(y + z) <= 5e-14))
		tmp = Float64(x + tan(Float64(y + z)));
	else
		tmp = Float64(x + Float64(Float64(y + z) - tan(a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, a)
	tmp = 0.0;
	if (((y + z) <= -0.001) || ~(((y + z) <= 5e-14)))
		tmp = x + tan((y + z));
	else
		tmp = x + ((y + z) - tan(a));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, a_] := If[Or[LessEqual[N[(y + z), $MachinePrecision], -0.001], N[Not[LessEqual[N[(y + z), $MachinePrecision], 5e-14]], $MachinePrecision]], N[(x + N[Tan[N[(y + z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(x + N[(N[(y + z), $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 y z) < -1e-3 or 5.0000000000000002e-14 < (+.f64 y z)

   1. Initial program 75.6%

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

      1. add-log-exp 75.6%

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

   4. Applied egg-rr 75.6%

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

      1. rem-log-exp 75.6%

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

      2. add-sqr-sqrt 38.6%

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

      3. sqrt-unprod 49.1%

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

      4. pow2 49.1%

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

   6. Applied egg-rr 49.1%

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

      1. unpow2 49.1%

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

      2. rem-sqrt-square 49.1%

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

      3. +-commutative 49.1%

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

   8. Simplified 49.1%

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

   9. Taylor expanded in a around 0 37.6%

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

       1. rem-square-sqrt 27.5%

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

       2. fabs-sqr 27.5%

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

       3. rem-square-sqrt 52.3%

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

   11. Simplified 52.3%

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

   if -1e-3 < (+.f64 y z) < 5.0000000000000002e-14

   1. Initial program 99.9%

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

      1. +-commutative 99.9%

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

      2. sub-neg 99.9%

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

      3. associate-+l+ 99.8%

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

      4. tan-sum 99.8%

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

      5. div-inv 99.8%

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

      6. fma-define 99.8%

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

      7. neg-mul-1 99.8%

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

      8. fma-define 99.8%

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

   4. Applied egg-rr 99.8%

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

      1. fma-undefine 99.8%

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

      2. fma-undefine 99.8%

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

      3. neg-mul-1 99.8%

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

      4. associate-+r+ 99.9%

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

      5. sub-neg 99.9%

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

      6. associate-*r/ 99.9%

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

      7. *-rgt-identity 99.9%

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

   6. Simplified 99.9%

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

   7. Taylor expanded in y around 0 99.9%

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

   8. Taylor expanded in y around 0 99.9%

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

   9. Taylor expanded in z around 0 99.9%

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

4. Final simplification 63.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y + z \leq -0.001 \lor \neg \left(y + z \leq 5 \cdot 10^{-14}\right):\\ \;\;\;\;x + \tan \left(y + z\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(\left(y + z\right) - \tan a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 50.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.5%

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

   1. add-log-exp 81.5%

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

4. Applied egg-rr 81.5%

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

   1. rem-log-exp 81.5%

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

   2. add-sqr-sqrt 42.5%

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

   3. sqrt-unprod 61.2%

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

   4. pow2 61.2%

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

6. Applied egg-rr 61.2%

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

   1. unpow2 61.2%

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

   2. rem-sqrt-square 61.2%

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

   3. +-commutative 61.2%

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

8. Simplified 61.2%

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

9. Taylor expanded in a around 0 42.0%

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

    1. rem-square-sqrt 27.9%

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

    2. fabs-sqr 27.9%

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

    3. rem-square-sqrt 53.4%

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

11. Simplified 53.4%

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

Alternative 9: 31.5% accurate, 207.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.5%

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

3. Taylor expanded in x around inf 30.9%

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

Reproduce

herbie shell --seed 2024157 
(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))))