Diagrams.TwoD.Segment.Bernstein:evaluateBernstein from diagrams-lib-1.3.0.3

Percentage Accurate: 88.4% → 99.8%

Time: 6.8s

Alternatives: 8

Speedup: 1.0×

• 20.7% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x, y, z):
	return (x * ((y - z) + 1.0)) / z

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 88.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x, y, z):
	return (x * ((y - z) + 1.0)) / z

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1 \cdot 10^{-16} \lor \neg \left(z \leq 9.8 \cdot 10^{-23}\right):\\ \;\;\;\;x \cdot \frac{1 + \left(y - z\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(1 + y\right)}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -1e-16) (not (<= z 9.8e-23)))
   (* x (/ (+ 1.0 (- y z)) z))
   (/ (* x (+ 1.0 y)) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1e-16) || !(z <= 9.8e-23)) {
		tmp = x * ((1.0 + (y - z)) / z);
	} else {
		tmp = (x * (1.0 + y)) / z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z <= (-1d-16)) .or. (.not. (z <= 9.8d-23))) then
        tmp = x * ((1.0d0 + (y - z)) / z)
    else
        tmp = (x * (1.0d0 + y)) / z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1e-16) || !(z <= 9.8e-23)) {
		tmp = x * ((1.0 + (y - z)) / z);
	} else {
		tmp = (x * (1.0 + y)) / z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -1e-16) or not (z <= 9.8e-23):
		tmp = x * ((1.0 + (y - z)) / z)
	else:
		tmp = (x * (1.0 + y)) / z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -1e-16) || !(z <= 9.8e-23))
		tmp = Float64(x * Float64(Float64(1.0 + Float64(y - z)) / z));
	else
		tmp = Float64(Float64(x * Float64(1.0 + y)) / z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -1e-16) || ~((z <= 9.8e-23)))
		tmp = x * ((1.0 + (y - z)) / z);
	else
		tmp = (x * (1.0 + y)) / z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -1e-16], N[Not[LessEqual[z, 9.8e-23]], $MachinePrecision]], N[(x * N[(N[(1.0 + N[(y - z), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], N[(N[(x * N[(1.0 + y), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -9.9999999999999998e-17 or 9.7999999999999996e-23 < z

   1. Initial program 83.2%

      \[\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \]

   2. Taylor expanded in x around 0 83.2%

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

      1. associate--l+ 83.2%

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

      2. +-commutative 83.2%

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

      3. associate-*r/ 99.8%

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

      4. +-commutative 99.8%

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

   4. Simplified 99.8%

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

   if -9.9999999999999998e-17 < z < 9.7999999999999996e-23

   1. Initial program 99.8%

      \[\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \]

   2. Taylor expanded in z around 0 99.8%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1 \cdot 10^{-16} \lor \neg \left(z \leq 9.8 \cdot 10^{-23}\right):\\ \;\;\;\;x \cdot \frac{1 + \left(y - z\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(1 + y\right)}{z}\\ \end{array} \]

Alternative 2: 65.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{z} \cdot y\\ \mathbf{if}\;z \leq -1.05 \cdot 10^{+79}:\\ \;\;\;\;-x\\ \mathbf{elif}\;z \leq -1 \cdot 10^{-197}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 2.8 \cdot 10^{-291}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;z \leq 7.5 \cdot 10^{-82}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 4.5 \cdot 10^{-15}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;z \leq 2.7 \cdot 10^{+25}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;-x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (/ x z) y)))
   (if (<= z -1.05e+79)
     (- x)
     (if (<= z -1e-197)
       t_0
       (if (<= z 2.8e-291)
         (/ x z)
         (if (<= z 7.5e-82)
           t_0
           (if (<= z 4.5e-15) (/ x z) (if (<= z 2.7e+25) t_0 (- x)))))))))

C

double code(double x, double y, double z) {
	double t_0 = (x / z) * y;
	double tmp;
	if (z <= -1.05e+79) {
		tmp = -x;
	} else if (z <= -1e-197) {
		tmp = t_0;
	} else if (z <= 2.8e-291) {
		tmp = x / z;
	} else if (z <= 7.5e-82) {
		tmp = t_0;
	} else if (z <= 4.5e-15) {
		tmp = x / z;
	} else if (z <= 2.7e+25) {
		tmp = t_0;
	} else {
		tmp = -x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x / z) * y
    if (z <= (-1.05d+79)) then
        tmp = -x
    else if (z <= (-1d-197)) then
        tmp = t_0
    else if (z <= 2.8d-291) then
        tmp = x / z
    else if (z <= 7.5d-82) then
        tmp = t_0
    else if (z <= 4.5d-15) then
        tmp = x / z
    else if (z <= 2.7d+25) then
        tmp = t_0
    else
        tmp = -x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (x / z) * y;
	double tmp;
	if (z <= -1.05e+79) {
		tmp = -x;
	} else if (z <= -1e-197) {
		tmp = t_0;
	} else if (z <= 2.8e-291) {
		tmp = x / z;
	} else if (z <= 7.5e-82) {
		tmp = t_0;
	} else if (z <= 4.5e-15) {
		tmp = x / z;
	} else if (z <= 2.7e+25) {
		tmp = t_0;
	} else {
		tmp = -x;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (x / z) * y
	tmp = 0
	if z <= -1.05e+79:
		tmp = -x
	elif z <= -1e-197:
		tmp = t_0
	elif z <= 2.8e-291:
		tmp = x / z
	elif z <= 7.5e-82:
		tmp = t_0
	elif z <= 4.5e-15:
		tmp = x / z
	elif z <= 2.7e+25:
		tmp = t_0
	else:
		tmp = -x
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(x / z) * y)
	tmp = 0.0
	if (z <= -1.05e+79)
		tmp = Float64(-x);
	elseif (z <= -1e-197)
		tmp = t_0;
	elseif (z <= 2.8e-291)
		tmp = Float64(x / z);
	elseif (z <= 7.5e-82)
		tmp = t_0;
	elseif (z <= 4.5e-15)
		tmp = Float64(x / z);
	elseif (z <= 2.7e+25)
		tmp = t_0;
	else
		tmp = Float64(-x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (x / z) * y;
	tmp = 0.0;
	if (z <= -1.05e+79)
		tmp = -x;
	elseif (z <= -1e-197)
		tmp = t_0;
	elseif (z <= 2.8e-291)
		tmp = x / z;
	elseif (z <= 7.5e-82)
		tmp = t_0;
	elseif (z <= 4.5e-15)
		tmp = x / z;
	elseif (z <= 2.7e+25)
		tmp = t_0;
	else
		tmp = -x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(x / z), $MachinePrecision] * y), $MachinePrecision]}, If[LessEqual[z, -1.05e+79], (-x), If[LessEqual[z, -1e-197], t$95$0, If[LessEqual[z, 2.8e-291], N[(x / z), $MachinePrecision], If[LessEqual[z, 7.5e-82], t$95$0, If[LessEqual[z, 4.5e-15], N[(x / z), $MachinePrecision], If[LessEqual[z, 2.7e+25], t$95$0, (-x)]]]]]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.05000000000000004e79 or 2.7e25 < z

   1. Initial program 76.1%

      \[\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \]

   2. Taylor expanded in z around inf 80.7%

      \[\leadsto \color{blue}{-1 \cdot x} \]
   3. Step-by-step derivation

      1. neg-mul-1 80.7%

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

   4. Simplified 80.7%

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

   if -1.05000000000000004e79 < z < -9.9999999999999999e-198 or 2.8e-291 < z < 7.4999999999999997e-82 or 4.4999999999999998e-15 < z < 2.7e25

   1. Initial program 99.8%

      \[\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \]

   2. Taylor expanded in y around inf 65.7%

      \[\leadsto \color{blue}{\frac{x \cdot y}{z}} \]
   3. Step-by-step derivation

      1. associate-/l* 57.8%

         \[\leadsto \color{blue}{\frac{x}{\frac{z}{y}}} \]

      2. associate-/r/ 67.2%

         \[\leadsto \color{blue}{\frac{x}{z} \cdot y} \]

   4. Simplified 67.2%

      \[\leadsto \color{blue}{\frac{x}{z} \cdot y} \]

   if -9.9999999999999999e-198 < z < 2.8e-291 or 7.4999999999999997e-82 < z < 4.4999999999999998e-15

   1. Initial program 99.9%

      \[\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \]

   2. Taylor expanded in z around 0 99.8%

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

   3. Taylor expanded in y around 0 80.6%

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

4. Final simplification 74.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.05 \cdot 10^{+79}:\\ \;\;\;\;-x\\ \mathbf{elif}\;z \leq -1 \cdot 10^{-197}:\\ \;\;\;\;\frac{x}{z} \cdot y\\ \mathbf{elif}\;z \leq 2.8 \cdot 10^{-291}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;z \leq 7.5 \cdot 10^{-82}:\\ \;\;\;\;\frac{x}{z} \cdot y\\ \mathbf{elif}\;z \leq 4.5 \cdot 10^{-15}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;z \leq 2.7 \cdot 10^{+25}:\\ \;\;\;\;\frac{x}{z} \cdot y\\ \mathbf{else}:\\ \;\;\;\;-x\\ \end{array} \]

Alternative 3: 84.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.75 \cdot 10^{+43} \lor \neg \left(y \leq 750000000000\right) \land \left(y \leq 1.12 \cdot 10^{+52} \lor \neg \left(y \leq 3.05 \cdot 10^{+106}\right)\right):\\ \;\;\;\;\frac{x}{z} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} - x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -1.75e+43)
         (and (not (<= y 750000000000.0))
              (or (<= y 1.12e+52) (not (<= y 3.05e+106)))))
   (* (/ x z) y)
   (- (/ x z) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -1.75e+43) || (!(y <= 750000000000.0) && ((y <= 1.12e+52) || !(y <= 3.05e+106)))) {
		tmp = (x / z) * y;
	} else {
		tmp = (x / z) - x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((y <= (-1.75d+43)) .or. (.not. (y <= 750000000000.0d0)) .and. (y <= 1.12d+52) .or. (.not. (y <= 3.05d+106))) then
        tmp = (x / z) * y
    else
        tmp = (x / z) - x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((y <= -1.75e+43) || (!(y <= 750000000000.0) && ((y <= 1.12e+52) || !(y <= 3.05e+106)))) {
		tmp = (x / z) * y;
	} else {
		tmp = (x / z) - x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -1.75e+43) or (not (y <= 750000000000.0) and ((y <= 1.12e+52) or not (y <= 3.05e+106))):
		tmp = (x / z) * y
	else:
		tmp = (x / z) - x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -1.75e+43) || (!(y <= 750000000000.0) && ((y <= 1.12e+52) || !(y <= 3.05e+106))))
		tmp = Float64(Float64(x / z) * y);
	else
		tmp = Float64(Float64(x / z) - x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -1.75e+43) || (~((y <= 750000000000.0)) && ((y <= 1.12e+52) || ~((y <= 3.05e+106)))))
		tmp = (x / z) * y;
	else
		tmp = (x / z) - x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -1.75e+43], And[N[Not[LessEqual[y, 750000000000.0]], $MachinePrecision], Or[LessEqual[y, 1.12e+52], N[Not[LessEqual[y, 3.05e+106]], $MachinePrecision]]]], N[(N[(x / z), $MachinePrecision] * y), $MachinePrecision], N[(N[(x / z), $MachinePrecision] - x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.7500000000000001e43 or 7.5e11 < y < 1.12000000000000002e52 or 3.05e106 < y

   1. Initial program 92.4%

      \[\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \]

   2. Taylor expanded in y around inf 83.9%

      \[\leadsto \color{blue}{\frac{x \cdot y}{z}} \]
   3. Step-by-step derivation

      1. associate-/l* 79.1%

         \[\leadsto \color{blue}{\frac{x}{\frac{z}{y}}} \]

      2. associate-/r/ 86.9%

         \[\leadsto \color{blue}{\frac{x}{z} \cdot y} \]

   4. Simplified 86.9%

      \[\leadsto \color{blue}{\frac{x}{z} \cdot y} \]

   if -1.7500000000000001e43 < y < 7.5e11 or 1.12000000000000002e52 < y < 3.05e106

   1. Initial program 89.2%

      \[\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \]

   2. Taylor expanded in z around 0 100.0%

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

      1. neg-mul-1 100.0%

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

      2. +-commutative 100.0%

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

      3. unsub-neg 100.0%

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

      4. associate-/l* 99.6%

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

      5. associate-/r/ 100.0%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in y around 0 96.5%

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

4. Final simplification 92.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.75 \cdot 10^{+43} \lor \neg \left(y \leq 750000000000\right) \land \left(y \leq 1.12 \cdot 10^{+52} \lor \neg \left(y \leq 3.05 \cdot 10^{+106}\right)\right):\\ \;\;\;\;\frac{x}{z} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} - x\\ \end{array} \]

Alternative 4: 84.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.75 \cdot 10^{+43}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{elif}\;y \leq 3.2 \cdot 10^{+17} \lor \neg \left(y \leq 6 \cdot 10^{+53}\right) \land y \leq 3.8 \cdot 10^{+106}:\\ \;\;\;\;\frac{x}{z} - x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -1.75e+43)
   (/ (* x y) z)
   (if (or (<= y 3.2e+17) (and (not (<= y 6e+53)) (<= y 3.8e+106)))
     (- (/ x z) x)
     (* (/ x z) y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.75e+43) {
		tmp = (x * y) / z;
	} else if ((y <= 3.2e+17) || (!(y <= 6e+53) && (y <= 3.8e+106))) {
		tmp = (x / z) - x;
	} else {
		tmp = (x / z) * y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= (-1.75d+43)) then
        tmp = (x * y) / z
    else if ((y <= 3.2d+17) .or. (.not. (y <= 6d+53)) .and. (y <= 3.8d+106)) then
        tmp = (x / z) - x
    else
        tmp = (x / z) * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.75e+43) {
		tmp = (x * y) / z;
	} else if ((y <= 3.2e+17) || (!(y <= 6e+53) && (y <= 3.8e+106))) {
		tmp = (x / z) - x;
	} else {
		tmp = (x / z) * y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -1.75e+43:
		tmp = (x * y) / z
	elif (y <= 3.2e+17) or (not (y <= 6e+53) and (y <= 3.8e+106)):
		tmp = (x / z) - x
	else:
		tmp = (x / z) * y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -1.75e+43)
		tmp = Float64(Float64(x * y) / z);
	elseif ((y <= 3.2e+17) || (!(y <= 6e+53) && (y <= 3.8e+106)))
		tmp = Float64(Float64(x / z) - x);
	else
		tmp = Float64(Float64(x / z) * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -1.75e+43)
		tmp = (x * y) / z;
	elseif ((y <= 3.2e+17) || (~((y <= 6e+53)) && (y <= 3.8e+106)))
		tmp = (x / z) - x;
	else
		tmp = (x / z) * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -1.75e+43], N[(N[(x * y), $MachinePrecision] / z), $MachinePrecision], If[Or[LessEqual[y, 3.2e+17], And[N[Not[LessEqual[y, 6e+53]], $MachinePrecision], LessEqual[y, 3.8e+106]]], N[(N[(x / z), $MachinePrecision] - x), $MachinePrecision], N[(N[(x / z), $MachinePrecision] * y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.7500000000000001e43

   1. Initial program 96.1%

      \[\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \]

   2. Taylor expanded in y around inf 83.3%

      \[\leadsto \color{blue}{\frac{x \cdot y}{z}} \]

   if -1.7500000000000001e43 < y < 3.2e17 or 5.99999999999999996e53 < y < 3.7999999999999998e106

   1. Initial program 89.2%

      \[\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \]

   2. Taylor expanded in z around 0 100.0%

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

      1. neg-mul-1 100.0%

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

      2. +-commutative 100.0%

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

      3. unsub-neg 100.0%

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

      4. associate-/l* 99.6%

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

      5. associate-/r/ 100.0%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in y around 0 96.5%

      \[\leadsto \color{blue}{\frac{x}{z}} - x \]

   if 3.2e17 < y < 5.99999999999999996e53 or 3.7999999999999998e106 < y

   1. Initial program 89.0%

      \[\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \]

   2. Taylor expanded in y around inf 84.5%

      \[\leadsto \color{blue}{\frac{x \cdot y}{z}} \]
   3. Step-by-step derivation

      1. associate-/l* 79.8%

         \[\leadsto \color{blue}{\frac{x}{\frac{z}{y}}} \]

      2. associate-/r/ 90.6%

         \[\leadsto \color{blue}{\frac{x}{z} \cdot y} \]

   4. Simplified 90.6%

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

4. Final simplification 92.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.75 \cdot 10^{+43}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{elif}\;y \leq 3.2 \cdot 10^{+17} \lor \neg \left(y \leq 6 \cdot 10^{+53}\right) \land y \leq 3.8 \cdot 10^{+106}:\\ \;\;\;\;\frac{x}{z} - x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot y\\ \end{array} \]

Alternative 5: 94.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -15000000 \lor \neg \left(y \leq 7.3 \cdot 10^{-9}\right):\\ \;\;\;\;\frac{x \cdot y}{z} - x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} - x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -15000000.0) (not (<= y 7.3e-9)))
   (- (/ (* x y) z) x)
   (- (/ x z) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -15000000.0) || !(y <= 7.3e-9)) {
		tmp = ((x * y) / z) - x;
	} else {
		tmp = (x / z) - x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((y <= (-15000000.0d0)) .or. (.not. (y <= 7.3d-9))) then
        tmp = ((x * y) / z) - x
    else
        tmp = (x / z) - x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((y <= -15000000.0) || !(y <= 7.3e-9)) {
		tmp = ((x * y) / z) - x;
	} else {
		tmp = (x / z) - x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -15000000.0) or not (y <= 7.3e-9):
		tmp = ((x * y) / z) - x
	else:
		tmp = (x / z) - x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -15000000.0) || !(y <= 7.3e-9))
		tmp = Float64(Float64(Float64(x * y) / z) - x);
	else
		tmp = Float64(Float64(x / z) - x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -15000000.0) || ~((y <= 7.3e-9)))
		tmp = ((x * y) / z) - x;
	else
		tmp = (x / z) - x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -15000000.0], N[Not[LessEqual[y, 7.3e-9]], $MachinePrecision]], N[(N[(N[(x * y), $MachinePrecision] / z), $MachinePrecision] - x), $MachinePrecision], N[(N[(x / z), $MachinePrecision] - x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.5e7 or 7.30000000000000002e-9 < y

   1. Initial program 91.5%

      \[\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \]

   2. Taylor expanded in z around 0 93.6%

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

      1. neg-mul-1 93.6%

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

      2. +-commutative 93.6%

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

      3. unsub-neg 93.6%

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

      4. associate-/l* 91.3%

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

      5. associate-/r/ 98.2%

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

   4. Simplified 98.2%

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

   5. Taylor expanded in y around inf 93.3%

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

   if -1.5e7 < y < 7.30000000000000002e-9

   1. Initial program 89.6%

      \[\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \]

   2. Taylor expanded in z around 0 100.0%

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

      1. neg-mul-1 100.0%

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

      2. +-commutative 100.0%

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

      3. unsub-neg 100.0%

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

      4. associate-/l* 100.0%

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

      5. associate-/r/ 100.0%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in y around 0 99.9%

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

4. Final simplification 96.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -15000000 \lor \neg \left(y \leq 7.3 \cdot 10^{-9}\right):\\ \;\;\;\;\frac{x \cdot y}{z} - x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} - x\\ \end{array} \]

Alternative 6: 97.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x, y, z):
	return ((x / z) * (1.0 + y)) - x

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 90.6%

   \[\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \]

2. Taylor expanded in z around 0 96.5%

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

   1. neg-mul-1 96.5%

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

   2. +-commutative 96.5%

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

   3. unsub-neg 96.5%

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

   4. associate-/l* 95.3%

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

   5. associate-/r/ 99.0%

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

4. Simplified 99.0%

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

5. Final simplification 99.0%

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

Alternative 7: 64.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.9:\\ \;\;\;\;-x\\ \mathbf{elif}\;z \leq 1.15:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;-x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -2.9) (- x) (if (<= z 1.15) (/ x z) (- x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -2.9) {
		tmp = -x;
	} else if (z <= 1.15) {
		tmp = x / z;
	} else {
		tmp = -x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= (-2.9d0)) then
        tmp = -x
    else if (z <= 1.15d0) then
        tmp = x / z
    else
        tmp = -x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -2.9) {
		tmp = -x;
	} else if (z <= 1.15) {
		tmp = x / z;
	} else {
		tmp = -x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -2.9:
		tmp = -x
	elif z <= 1.15:
		tmp = x / z
	else:
		tmp = -x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -2.9)
		tmp = Float64(-x);
	elseif (z <= 1.15)
		tmp = Float64(x / z);
	else
		tmp = Float64(-x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -2.9)
		tmp = -x;
	elseif (z <= 1.15)
		tmp = x / z;
	else
		tmp = -x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -2.9], (-x), If[LessEqual[z, 1.15], N[(x / z), $MachinePrecision], (-x)]]

Derivation

1. Split input into 2 regimes

2. if z < -2.89999999999999991 or 1.1499999999999999 < z

   1. Initial program 81.9%

      \[\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \]

   2. Taylor expanded in z around inf 71.0%

      \[\leadsto \color{blue}{-1 \cdot x} \]
   3. Step-by-step derivation

      1. neg-mul-1 71.0%

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

   4. Simplified 71.0%

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

   if -2.89999999999999991 < z < 1.1499999999999999

   1. Initial program 99.8%

      \[\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \]

   2. Taylor expanded in z around 0 99.6%

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

   3. Taylor expanded in y around 0 54.0%

      \[\leadsto \frac{\color{blue}{x}}{z} \]
3. Recombined 2 regimes into one program.

4. Final simplification 62.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.9:\\ \;\;\;\;-x\\ \mathbf{elif}\;z \leq 1.15:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;-x\\ \end{array} \]

Alternative 8: 38.3% accurate, 4.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := (-x)

Derivation

1. Initial program 90.6%

   \[\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \]

2. Taylor expanded in z around inf 37.8%

   \[\leadsto \color{blue}{-1 \cdot x} \]
3. Step-by-step derivation

   1. neg-mul-1 37.8%

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

4. Simplified 37.8%

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

5. Final simplification 37.8%

   \[\leadsto -x \]

Developer target: 99.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(1 + y\right) \cdot \frac{x}{z} - x\\ \mathbf{if}\;x < -2.71483106713436 \cdot 10^{-162}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x < 3.874108816439546 \cdot 10^{-197}:\\ \;\;\;\;\left(x \cdot \left(\left(y - z\right) + 1\right)\right) \cdot \frac{1}{z}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- (* (+ 1.0 y) (/ x z)) x)))
   (if (< x -2.71483106713436e-162)
     t_0
     (if (< x 3.874108816439546e-197)
       (* (* x (+ (- y z) 1.0)) (/ 1.0 z))
       t_0))))

C

double code(double x, double y, double z) {
	double t_0 = ((1.0 + y) * (x / z)) - x;
	double tmp;
	if (x < -2.71483106713436e-162) {
		tmp = t_0;
	} else if (x < 3.874108816439546e-197) {
		tmp = (x * ((y - z) + 1.0)) * (1.0 / z);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((1.0d0 + y) * (x / z)) - x
    if (x < (-2.71483106713436d-162)) then
        tmp = t_0
    else if (x < 3.874108816439546d-197) then
        tmp = (x * ((y - z) + 1.0d0)) * (1.0d0 / z)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = ((1.0 + y) * (x / z)) - x;
	double tmp;
	if (x < -2.71483106713436e-162) {
		tmp = t_0;
	} else if (x < 3.874108816439546e-197) {
		tmp = (x * ((y - z) + 1.0)) * (1.0 / z);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = ((1.0 + y) * (x / z)) - x
	tmp = 0
	if x < -2.71483106713436e-162:
		tmp = t_0
	elif x < 3.874108816439546e-197:
		tmp = (x * ((y - z) + 1.0)) * (1.0 / z)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(Float64(1.0 + y) * Float64(x / z)) - x)
	tmp = 0.0
	if (x < -2.71483106713436e-162)
		tmp = t_0;
	elseif (x < 3.874108816439546e-197)
		tmp = Float64(Float64(x * Float64(Float64(y - z) + 1.0)) * Float64(1.0 / z));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = ((1.0 + y) * (x / z)) - x;
	tmp = 0.0;
	if (x < -2.71483106713436e-162)
		tmp = t_0;
	elseif (x < 3.874108816439546e-197)
		tmp = (x * ((y - z) + 1.0)) * (1.0 / z);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[(1.0 + y), $MachinePrecision] * N[(x / z), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]}, If[Less[x, -2.71483106713436e-162], t$95$0, If[Less[x, 3.874108816439546e-197], N[(N[(x * N[(N[(y - z), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * N[(1.0 / z), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Reproduce

herbie shell --seed 2023271 
(FPCore (x y z)
  :name "Diagrams.TwoD.Segment.Bernstein:evaluateBernstein from diagrams-lib-1.3.0.3"
  :precision binary64

  :herbie-target
  (if (< x -2.71483106713436e-162) (- (* (+ 1.0 y) (/ x z)) x) (if (< x 3.874108816439546e-197) (* (* x (+ (- y z) 1.0)) (/ 1.0 z)) (- (* (+ 1.0 y) (/ x z)) x)))

  (/ (* x (+ (- y z) 1.0)) z))