Diagrams.ThreeD.Shapes:frustum from diagrams-lib-1.3.0.3, B

Percentage Accurate: 100.0% → 100.0%

Time: 5.8s

Alternatives: 8

Speedup: 1.0×

• 20.5% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

double code(double x, double y, double z) {
	return x + ((y - x) * 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 - x) * z)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z) {
	return x + ((y - x) * 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 - x) * z)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(y - x, z, x\right) \end{array} \]

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

   \[x + \left(y - x\right) \cdot z \]
2. Step-by-step derivation

   1. +-commutative 100.0%

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

   2. fma-define 100.0%

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

3. Simplified 100.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, z, x\right)} \]
4. Add Preprocessing

5. Final simplification 100.0%

   \[\leadsto \mathsf{fma}\left(y - x, z, x\right) \]
6. Add Preprocessing

Alternative 2: 73.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -310000000000 \lor \neg \left(x \leq -3.25 \cdot 10^{-217} \lor \neg \left(x \leq -1.95 \cdot 10^{-245}\right) \land x \leq 2.15 \cdot 10^{-7}\right):\\ \;\;\;\;x \cdot \left(1 - z\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -310000000000.0)
         (not
          (or (<= x -3.25e-217) (and (not (<= x -1.95e-245)) (<= x 2.15e-7)))))
   (* x (- 1.0 z))
   (* y z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -310000000000.0) || !((x <= -3.25e-217) || (!(x <= -1.95e-245) && (x <= 2.15e-7)))) {
		tmp = x * (1.0 - z);
	} else {
		tmp = 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 ((x <= (-310000000000.0d0)) .or. (.not. (x <= (-3.25d-217)) .or. (.not. (x <= (-1.95d-245))) .and. (x <= 2.15d-7))) then
        tmp = x * (1.0d0 - z)
    else
        tmp = y * z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= -310000000000.0) || !((x <= -3.25e-217) || (!(x <= -1.95e-245) && (x <= 2.15e-7)))) {
		tmp = x * (1.0 - z);
	} else {
		tmp = y * z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -310000000000.0) or not ((x <= -3.25e-217) or (not (x <= -1.95e-245) and (x <= 2.15e-7))):
		tmp = x * (1.0 - z)
	else:
		tmp = y * z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -310000000000.0) || !((x <= -3.25e-217) || (!(x <= -1.95e-245) && (x <= 2.15e-7))))
		tmp = Float64(x * Float64(1.0 - z));
	else
		tmp = Float64(y * z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -310000000000.0) || ~(((x <= -3.25e-217) || (~((x <= -1.95e-245)) && (x <= 2.15e-7)))))
		tmp = x * (1.0 - z);
	else
		tmp = y * z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -310000000000.0], N[Not[Or[LessEqual[x, -3.25e-217], And[N[Not[LessEqual[x, -1.95e-245]], $MachinePrecision], LessEqual[x, 2.15e-7]]]], $MachinePrecision]], N[(x * N[(1.0 - z), $MachinePrecision]), $MachinePrecision], N[(y * z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -3.1e11 or -3.2499999999999998e-217 < x < -1.9499999999999999e-245 or 2.1500000000000001e-7 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 87.1%

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

      1. mul-1-neg 87.1%

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

      2. unsub-neg 87.1%

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

   5. Simplified 87.1%

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

   if -3.1e11 < x < -3.2499999999999998e-217 or -1.9499999999999999e-245 < x < 2.1500000000000001e-7

   1. Initial program 100.0%

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

   3. Taylor expanded in z around inf 99.9%

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

      1. associate--l+ 99.9%

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

      2. +-commutative 99.9%

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

      3. associate-+l- 99.9%

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

   5. Simplified 99.9%

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

   6. Taylor expanded in x around 0 73.6%

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

4. Final simplification 80.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -310000000000 \lor \neg \left(x \leq -3.25 \cdot 10^{-217} \lor \neg \left(x \leq -1.95 \cdot 10^{-245}\right) \land x \leq 2.15 \cdot 10^{-7}\right):\\ \;\;\;\;x \cdot \left(1 - z\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 61.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(-z\right)\\ \mathbf{if}\;z \leq -1:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 3.4 \cdot 10^{-36}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 1.35 \cdot 10^{+99} \lor \neg \left(z \leq 6 \cdot 10^{+244}\right):\\ \;\;\;\;y \cdot z\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (- z))))
   (if (<= z -1.0)
     t_0
     (if (<= z 3.4e-36)
       x
       (if (or (<= z 1.35e+99) (not (<= z 6e+244))) (* y z) t_0)))))

C

double code(double x, double y, double z) {
	double t_0 = x * -z;
	double tmp;
	if (z <= -1.0) {
		tmp = t_0;
	} else if (z <= 3.4e-36) {
		tmp = x;
	} else if ((z <= 1.35e+99) || !(z <= 6e+244)) {
		tmp = y * 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 = x * -z
    if (z <= (-1.0d0)) then
        tmp = t_0
    else if (z <= 3.4d-36) then
        tmp = x
    else if ((z <= 1.35d+99) .or. (.not. (z <= 6d+244))) then
        tmp = y * 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 = x * -z;
	double tmp;
	if (z <= -1.0) {
		tmp = t_0;
	} else if (z <= 3.4e-36) {
		tmp = x;
	} else if ((z <= 1.35e+99) || !(z <= 6e+244)) {
		tmp = y * z;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x * -z
	tmp = 0
	if z <= -1.0:
		tmp = t_0
	elif z <= 3.4e-36:
		tmp = x
	elif (z <= 1.35e+99) or not (z <= 6e+244):
		tmp = y * z
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x * Float64(-z))
	tmp = 0.0
	if (z <= -1.0)
		tmp = t_0;
	elseif (z <= 3.4e-36)
		tmp = x;
	elseif ((z <= 1.35e+99) || !(z <= 6e+244))
		tmp = Float64(y * z);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = x * -z;
	tmp = 0.0;
	if (z <= -1.0)
		tmp = t_0;
	elseif (z <= 3.4e-36)
		tmp = x;
	elseif ((z <= 1.35e+99) || ~((z <= 6e+244)))
		tmp = y * z;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x * (-z)), $MachinePrecision]}, If[LessEqual[z, -1.0], t$95$0, If[LessEqual[z, 3.4e-36], x, If[Or[LessEqual[z, 1.35e+99], N[Not[LessEqual[z, 6e+244]], $MachinePrecision]], N[(y * z), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if z < -1 or 1.34999999999999994e99 < z < 5.9999999999999995e244

   1. Initial program 100.0%

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

   3. Taylor expanded in z around inf 99.9%

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

      1. associate--l+ 99.9%

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

      2. +-commutative 99.9%

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

      3. associate-+l- 99.9%

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

   5. Simplified 99.9%

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

   6. Taylor expanded in z around inf 98.9%

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

      1. distribute-lft-out-- 98.9%

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

   8. Applied egg-rr 98.9%

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

   9. Taylor expanded in y around 0 58.6%

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

       1. associate-*r* 58.6%

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

       2. neg-mul-1 58.6%

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

       3. *-commutative 58.6%

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

   11. Simplified 58.6%

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

   if -1 < z < 3.4000000000000003e-36

   1. Initial program 100.0%

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

   3. Taylor expanded in z around 0 72.2%

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

   if 3.4000000000000003e-36 < z < 1.34999999999999994e99 or 5.9999999999999995e244 < z

   1. Initial program 100.0%

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

   3. Taylor expanded in z around inf 98.3%

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

      1. associate--l+ 98.3%

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

      2. +-commutative 98.3%

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

      3. associate-+l- 98.3%

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

   5. Simplified 98.3%

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

   6. Taylor expanded in x around 0 75.3%

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

4. Final simplification 68.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1:\\ \;\;\;\;x \cdot \left(-z\right)\\ \mathbf{elif}\;z \leq 3.4 \cdot 10^{-36}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 1.35 \cdot 10^{+99} \lor \neg \left(z \leq 6 \cdot 10^{+244}\right):\\ \;\;\;\;y \cdot z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(-z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 80.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.95 \cdot 10^{+45} \lor \neg \left(x \leq 1.16 \cdot 10^{+32}\right):\\ \;\;\;\;x \cdot \left(1 - z\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y - x\right) \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -1.95e+45) (not (<= x 1.16e+32)))
   (* x (- 1.0 z))
   (* (- y x) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -1.95e+45) || !(x <= 1.16e+32)) {
		tmp = x * (1.0 - z);
	} else {
		tmp = (y - x) * 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 ((x <= (-1.95d+45)) .or. (.not. (x <= 1.16d+32))) then
        tmp = x * (1.0d0 - z)
    else
        tmp = (y - x) * z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= -1.95e+45) || !(x <= 1.16e+32)) {
		tmp = x * (1.0 - z);
	} else {
		tmp = (y - x) * z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -1.95e+45) or not (x <= 1.16e+32):
		tmp = x * (1.0 - z)
	else:
		tmp = (y - x) * z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -1.95e+45) || !(x <= 1.16e+32))
		tmp = Float64(x * Float64(1.0 - z));
	else
		tmp = Float64(Float64(y - x) * z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -1.95e+45) || ~((x <= 1.16e+32)))
		tmp = x * (1.0 - z);
	else
		tmp = (y - x) * z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -1.95e+45], N[Not[LessEqual[x, 1.16e+32]], $MachinePrecision]], N[(x * N[(1.0 - z), $MachinePrecision]), $MachinePrecision], N[(N[(y - x), $MachinePrecision] * z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.95e45 or 1.16e32 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 93.3%

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

      1. mul-1-neg 93.3%

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

      2. unsub-neg 93.3%

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

   5. Simplified 93.3%

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

   if -1.95e45 < x < 1.16e32

   1. Initial program 99.9%

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

   3. Taylor expanded in z around inf 99.9%

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

      1. associate--l+ 99.9%

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

      2. +-commutative 99.9%

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

      3. associate-+l- 99.9%

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

   5. Simplified 99.9%

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

   6. Taylor expanded in z around inf 80.2%

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

4. Final simplification 85.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.95 \cdot 10^{+45} \lor \neg \left(x \leq 1.16 \cdot 10^{+32}\right):\\ \;\;\;\;x \cdot \left(1 - z\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y - x\right) \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1 \lor \neg \left(z \leq 5.8 \cdot 10^{-5}\right):\\ \;\;\;\;\left(y - x\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -1.0) (not (<= z 5.8e-5))) (* (- y x) z) (+ x (* y z))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1.0) || !(z <= 5.8e-5)) {
		tmp = (y - x) * z;
	} else {
		tmp = x + (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 <= (-1.0d0)) .or. (.not. (z <= 5.8d-5))) then
        tmp = (y - x) * z
    else
        tmp = x + (y * z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1.0) || !(z <= 5.8e-5)) {
		tmp = (y - x) * z;
	} else {
		tmp = x + (y * z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -1.0) or not (z <= 5.8e-5):
		tmp = (y - x) * z
	else:
		tmp = x + (y * z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -1.0) || !(z <= 5.8e-5))
		tmp = Float64(Float64(y - x) * z);
	else
		tmp = Float64(x + Float64(y * z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -1.0) || ~((z <= 5.8e-5)))
		tmp = (y - x) * z;
	else
		tmp = x + (y * z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -1.0], N[Not[LessEqual[z, 5.8e-5]], $MachinePrecision]], N[(N[(y - x), $MachinePrecision] * z), $MachinePrecision], N[(x + N[(y * z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1 or 5.8e-5 < z

   1. Initial program 100.0%

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

   3. Taylor expanded in z around inf 100.0%

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

      1. associate--l+ 100.0%

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

      2. +-commutative 100.0%

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

      3. associate-+l- 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in z around inf 99.2%

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

   if -1 < z < 5.8e-5

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf 99.6%

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

      1. *-commutative 99.6%

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

   5. Simplified 99.6%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1 \lor \neg \left(z \leq 5.8 \cdot 10^{-5}\right):\\ \;\;\;\;\left(y - x\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 61.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.6 \cdot 10^{-18} \lor \neg \left(z \leq 3.8 \cdot 10^{-37}\right):\\ \;\;\;\;y \cdot z\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -2.6e-18) (not (<= z 3.8e-37))) (* y z) x))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -2.6e-18) || !(z <= 3.8e-37)) {
		tmp = y * 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 <= (-2.6d-18)) .or. (.not. (z <= 3.8d-37))) then
        tmp = y * 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 <= -2.6e-18) || !(z <= 3.8e-37)) {
		tmp = y * z;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -2.6e-18) or not (z <= 3.8e-37):
		tmp = y * z
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -2.6e-18) || !(z <= 3.8e-37))
		tmp = Float64(y * z);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -2.6e-18) || ~((z <= 3.8e-37)))
		tmp = y * z;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -2.6e-18], N[Not[LessEqual[z, 3.8e-37]], $MachinePrecision]], N[(y * z), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if z < -2.6e-18 or 3.8000000000000004e-37 < z

   1. Initial program 99.9%

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

   3. Taylor expanded in z around inf 99.3%

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

      1. associate--l+ 99.3%

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

      2. +-commutative 99.3%

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

      3. associate-+l- 99.3%

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

   5. Simplified 99.3%

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

   6. Taylor expanded in x around 0 55.9%

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

   if -2.6e-18 < z < 3.8000000000000004e-37

   1. Initial program 100.0%

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

   3. Taylor expanded in z around 0 74.6%

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

4. Final simplification 64.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.6 \cdot 10^{-18} \lor \neg \left(z \leq 3.8 \cdot 10^{-37}\right):\\ \;\;\;\;y \cdot z\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z) {
	return x + ((y - x) * 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 - x) * z)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Final simplification 100.0%

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

Alternative 8: 36.4% accurate, 7.0× 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 x
end

MATLAB

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

Wolfram

code[x_, y_, z_] := x

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in z around 0 34.9%

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

4. Final simplification 34.9%

   \[\leadsto x \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024067 
(FPCore (x y z)
  :name "Diagrams.ThreeD.Shapes:frustum from diagrams-lib-1.3.0.3, B"
  :precision binary64
  (+ x (* (- y x) z)))