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

Percentage Accurate: 100.0% → 100.0%

Time: 5.3s

Alternatives: 8

Speedup: 1.0×

• 21.1% 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. Add Preprocessing

Alternative 2: 60.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot \left(-x\right)\\ \mathbf{if}\;z \leq -6.2 \cdot 10^{+185}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq -2.15 \cdot 10^{+142}:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;z \leq -1.1 \cdot 10^{+40}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq -1.55 \cdot 10^{-81}:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;z \leq 1:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 6.7 \cdot 10^{+87} \lor \neg \left(z \leq 1.05 \cdot 10^{+157}\right):\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;y \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* z (- x))))
   (if (<= z -6.2e+185)
     t_0
     (if (<= z -2.15e+142)
       (* y z)
       (if (<= z -1.1e+40)
         t_0
         (if (<= z -1.55e-81)
           (* y z)
           (if (<= z 1.0)
             x
             (if (or (<= z 6.7e+87) (not (<= z 1.05e+157))) t_0 (* y z)))))))))

C

double code(double x, double y, double z) {
	double t_0 = z * -x;
	double tmp;
	if (z <= -6.2e+185) {
		tmp = t_0;
	} else if (z <= -2.15e+142) {
		tmp = y * z;
	} else if (z <= -1.1e+40) {
		tmp = t_0;
	} else if (z <= -1.55e-81) {
		tmp = y * z;
	} else if (z <= 1.0) {
		tmp = x;
	} else if ((z <= 6.7e+87) || !(z <= 1.05e+157)) {
		tmp = t_0;
	} 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) :: t_0
    real(8) :: tmp
    t_0 = z * -x
    if (z <= (-6.2d+185)) then
        tmp = t_0
    else if (z <= (-2.15d+142)) then
        tmp = y * z
    else if (z <= (-1.1d+40)) then
        tmp = t_0
    else if (z <= (-1.55d-81)) then
        tmp = y * z
    else if (z <= 1.0d0) then
        tmp = x
    else if ((z <= 6.7d+87) .or. (.not. (z <= 1.05d+157))) then
        tmp = t_0
    else
        tmp = y * z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = z * -x;
	double tmp;
	if (z <= -6.2e+185) {
		tmp = t_0;
	} else if (z <= -2.15e+142) {
		tmp = y * z;
	} else if (z <= -1.1e+40) {
		tmp = t_0;
	} else if (z <= -1.55e-81) {
		tmp = y * z;
	} else if (z <= 1.0) {
		tmp = x;
	} else if ((z <= 6.7e+87) || !(z <= 1.05e+157)) {
		tmp = t_0;
	} else {
		tmp = y * z;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = z * -x
	tmp = 0
	if z <= -6.2e+185:
		tmp = t_0
	elif z <= -2.15e+142:
		tmp = y * z
	elif z <= -1.1e+40:
		tmp = t_0
	elif z <= -1.55e-81:
		tmp = y * z
	elif z <= 1.0:
		tmp = x
	elif (z <= 6.7e+87) or not (z <= 1.05e+157):
		tmp = t_0
	else:
		tmp = y * z
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(z * Float64(-x))
	tmp = 0.0
	if (z <= -6.2e+185)
		tmp = t_0;
	elseif (z <= -2.15e+142)
		tmp = Float64(y * z);
	elseif (z <= -1.1e+40)
		tmp = t_0;
	elseif (z <= -1.55e-81)
		tmp = Float64(y * z);
	elseif (z <= 1.0)
		tmp = x;
	elseif ((z <= 6.7e+87) || !(z <= 1.05e+157))
		tmp = t_0;
	else
		tmp = Float64(y * z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = z * -x;
	tmp = 0.0;
	if (z <= -6.2e+185)
		tmp = t_0;
	elseif (z <= -2.15e+142)
		tmp = y * z;
	elseif (z <= -1.1e+40)
		tmp = t_0;
	elseif (z <= -1.55e-81)
		tmp = y * z;
	elseif (z <= 1.0)
		tmp = x;
	elseif ((z <= 6.7e+87) || ~((z <= 1.05e+157)))
		tmp = t_0;
	else
		tmp = y * z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(z * (-x)), $MachinePrecision]}, If[LessEqual[z, -6.2e+185], t$95$0, If[LessEqual[z, -2.15e+142], N[(y * z), $MachinePrecision], If[LessEqual[z, -1.1e+40], t$95$0, If[LessEqual[z, -1.55e-81], N[(y * z), $MachinePrecision], If[LessEqual[z, 1.0], x, If[Or[LessEqual[z, 6.7e+87], N[Not[LessEqual[z, 1.05e+157]], $MachinePrecision]], t$95$0, N[(y * z), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if z < -6.19999999999999999e185 or -2.15000000000000006e142 < z < -1.0999999999999999e40 or 1 < z < 6.7000000000000003e87 or 1.05e157 < z

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. sub-neg 100.0%

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

      3. distribute-lft-in 94.3%

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

   4. Applied egg-rr 94.3%

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

      1. associate-+r+ 94.3%

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

      2. distribute-rgt-neg-out 94.3%

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

      3. unsub-neg 94.3%

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

      4. +-commutative 94.3%

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

   6. Applied egg-rr 94.3%

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

   7. Taylor expanded in z around inf 98.7%

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

   8. Taylor expanded in y around 0 68.3%

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

      1. mul-1-neg 68.3%

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

      2. distribute-rgt-neg-in 68.3%

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

   10. Simplified 68.3%

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

   if -6.19999999999999999e185 < z < -2.15000000000000006e142 or -1.0999999999999999e40 < z < -1.54999999999999994e-81 or 6.7000000000000003e87 < z < 1.05e157

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. sub-neg 100.0%

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

      3. distribute-lft-in 100.0%

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

   4. Applied egg-rr 100.0%

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

      1. associate-+r+ 100.0%

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

      2. distribute-rgt-neg-out 100.0%

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

      3. unsub-neg 100.0%

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

      4. +-commutative 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in y around inf 78.0%

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

      1. *-commutative 78.0%

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

   9. Simplified 78.0%

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

   if -1.54999999999999994e-81 < z < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in z around 0 73.4%

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

4. Final simplification 72.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -6.2 \cdot 10^{+185}:\\ \;\;\;\;z \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq -2.15 \cdot 10^{+142}:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;z \leq -1.1 \cdot 10^{+40}:\\ \;\;\;\;z \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq -1.55 \cdot 10^{-81}:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;z \leq 1:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 6.7 \cdot 10^{+87} \lor \neg \left(z \leq 1.05 \cdot 10^{+157}\right):\\ \;\;\;\;z \cdot \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -60000000000000 \lor \neg \left(z \leq 1\right):\\ \;\;\;\;\left(y - x\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -60000000000000.0) (not (<= z 1.0)))
   (* (- y x) z)
   (+ x (* y z))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -60000000000000.0) || !(z <= 1.0)) {
		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 <= (-60000000000000.0d0)) .or. (.not. (z <= 1.0d0))) 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 <= -60000000000000.0) || !(z <= 1.0)) {
		tmp = (y - x) * z;
	} else {
		tmp = x + (y * z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -60000000000000.0) or not (z <= 1.0):
		tmp = (y - x) * z
	else:
		tmp = x + (y * z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -60000000000000.0) || !(z <= 1.0))
		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 <= -60000000000000.0) || ~((z <= 1.0)))
		tmp = (y - x) * z;
	else
		tmp = x + (y * z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -60000000000000.0], N[Not[LessEqual[z, 1.0]], $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 < -6e13 or 1 < z

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. sub-neg 100.0%

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

      3. distribute-lft-in 95.6%

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

   4. Applied egg-rr 95.6%

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

      1. associate-+r+ 95.6%

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

      2. distribute-rgt-neg-out 95.6%

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

      3. unsub-neg 95.6%

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

      4. +-commutative 95.6%

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

   6. Applied egg-rr 95.6%

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

   7. Taylor expanded in z around inf 99.0%

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

   if -6e13 < z < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf 99.1%

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

      1. *-commutative 99.1%

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

   5. Simplified 99.1%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -60000000000000 \lor \neg \left(z \leq 1\right):\\ \;\;\;\;\left(y - x\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 84.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -5.5 \cdot 10^{-82} \lor \neg \left(z \leq 2.4 \cdot 10^{+14}\right):\\ \;\;\;\;\left(y - x\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 - z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -5.5e-82) (not (<= z 2.4e+14))) (* (- y x) z) (* x (- 1.0 z))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -5.5e-82) || !(z <= 2.4e+14)) {
		tmp = (y - x) * z;
	} else {
		tmp = x * (1.0 - 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 <= (-5.5d-82)) .or. (.not. (z <= 2.4d+14))) then
        tmp = (y - x) * z
    else
        tmp = x * (1.0d0 - z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -5.5e-82) || !(z <= 2.4e+14)) {
		tmp = (y - x) * z;
	} else {
		tmp = x * (1.0 - z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -5.5e-82) or not (z <= 2.4e+14):
		tmp = (y - x) * z
	else:
		tmp = x * (1.0 - z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -5.5e-82) || !(z <= 2.4e+14))
		tmp = Float64(Float64(y - x) * z);
	else
		tmp = Float64(x * Float64(1.0 - z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -5.5e-82) || ~((z <= 2.4e+14)))
		tmp = (y - x) * z;
	else
		tmp = x * (1.0 - z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -5.5e-82], N[Not[LessEqual[z, 2.4e+14]], $MachinePrecision]], N[(N[(y - x), $MachinePrecision] * z), $MachinePrecision], N[(x * N[(1.0 - z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -5.4999999999999998e-82 or 2.4e14 < z

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. sub-neg 100.0%

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

      3. distribute-lft-in 95.8%

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

   4. Applied egg-rr 95.8%

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

      1. associate-+r+ 95.8%

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

      2. distribute-rgt-neg-out 95.8%

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

      3. unsub-neg 95.8%

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

      4. +-commutative 95.8%

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

   6. Applied egg-rr 95.8%

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

   7. Taylor expanded in z around inf 97.5%

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

   if -5.4999999999999998e-82 < z < 2.4e14

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 74.5%

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

      1. mul-1-neg 74.5%

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

      2. unsub-neg 74.5%

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

   5. Simplified 74.5%

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

4. Final simplification 87.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5.5 \cdot 10^{-82} \lor \neg \left(z \leq 2.4 \cdot 10^{+14}\right):\\ \;\;\;\;\left(y - x\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 - z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 75.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.1 \cdot 10^{-92} \lor \neg \left(x \leq 3.6 \cdot 10^{-88}\right):\\ \;\;\;\;x \cdot \left(1 - z\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -5.1e-92) (not (<= x 3.6e-88))) (* x (- 1.0 z)) (* y z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -5.1e-92) || !(x <= 3.6e-88)) {
		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 <= (-5.1d-92)) .or. (.not. (x <= 3.6d-88))) 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 <= -5.1e-92) || !(x <= 3.6e-88)) {
		tmp = x * (1.0 - z);
	} else {
		tmp = y * z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -5.1e-92) or not (x <= 3.6e-88):
		tmp = x * (1.0 - z)
	else:
		tmp = y * z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -5.1e-92) || !(x <= 3.6e-88))
		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 <= -5.1e-92) || ~((x <= 3.6e-88)))
		tmp = x * (1.0 - z);
	else
		tmp = y * z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -5.1e-92], N[Not[LessEqual[x, 3.6e-88]], $MachinePrecision]], N[(x * N[(1.0 - z), $MachinePrecision]), $MachinePrecision], N[(y * z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -5.09999999999999972e-92 or 3.5999999999999999e-88 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 80.6%

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

      1. mul-1-neg 80.6%

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

      2. unsub-neg 80.6%

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

   5. Simplified 80.6%

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

   if -5.09999999999999972e-92 < x < 3.5999999999999999e-88

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. sub-neg 100.0%

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

      3. distribute-lft-in 100.0%

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

   4. Applied egg-rr 100.0%

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

      1. associate-+r+ 100.0%

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

      2. distribute-rgt-neg-out 100.0%

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

      3. unsub-neg 100.0%

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

      4. +-commutative 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in y around inf 74.3%

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

      1. *-commutative 74.3%

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

   9. Simplified 74.3%

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

4. Final simplification 78.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.1 \cdot 10^{-92} \lor \neg \left(x \leq 3.6 \cdot 10^{-88}\right):\\ \;\;\;\;x \cdot \left(1 - z\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 62.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.65 \cdot 10^{-81} \lor \neg \left(z \leq 1.2 \cdot 10^{-7}\right):\\ \;\;\;\;y \cdot z\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -1.65e-81) (not (<= z 1.2e-7))) (* y z) x))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1.65e-81) || !(z <= 1.2e-7)) {
		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 <= (-1.65d-81)) .or. (.not. (z <= 1.2d-7))) 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 <= -1.65e-81) || !(z <= 1.2e-7)) {
		tmp = y * z;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -1.65e-81) or not (z <= 1.2e-7):
		tmp = y * z
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -1.65e-81) || !(z <= 1.2e-7))
		tmp = Float64(y * z);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -1.65e-81) || ~((z <= 1.2e-7)))
		tmp = y * z;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -1.65e-81], N[Not[LessEqual[z, 1.2e-7]], $MachinePrecision]], N[(y * z), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if z < -1.64999999999999994e-81 or 1.19999999999999989e-7 < z

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. sub-neg 100.0%

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

      3. distribute-lft-in 96.0%

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

   4. Applied egg-rr 96.0%

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

      1. associate-+r+ 96.0%

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

      2. distribute-rgt-neg-out 96.0%

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

      3. unsub-neg 96.0%

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

      4. +-commutative 96.0%

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

   6. Applied egg-rr 96.0%

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

   7. Taylor expanded in y around inf 51.0%

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

      1. *-commutative 51.0%

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

   9. Simplified 51.0%

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

   if -1.64999999999999994e-81 < z < 1.19999999999999989e-7

   1. Initial program 100.0%

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

   3. Taylor expanded in z around 0 73.4%

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

4. Final simplification 60.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.65 \cdot 10^{-81} \lor \neg \left(z \leq 1.2 \cdot 10^{-7}\right):\\ \;\;\;\;y \cdot z\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
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. 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 33.6%

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

Reproduce

herbie shell --seed 2024096 
(FPCore (x y z)
  :name "Diagrams.ThreeD.Shapes:frustum from diagrams-lib-1.3.0.3, B"
  :precision binary64
  (+ x (* (- y x) z)))