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: 61.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(-z\right)\\ \mathbf{if}\;z \leq -1.1 \cdot 10^{+71}:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;z \leq -2.1 \cdot 10^{+29}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq -7.5 \cdot 10^{-17}:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;z \leq 3.2 \cdot 10^{-78}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 2.3 \cdot 10^{+126}:\\ \;\;\;\;y \cdot z\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (- z))))
   (if (<= z -1.1e+71)
     (* y z)
     (if (<= z -2.1e+29)
       t_0
       (if (<= z -7.5e-17)
         (* y z)
         (if (<= z 3.2e-78) x (if (<= z 2.3e+126) (* y z) t_0)))))))

C

double code(double x, double y, double z) {
	double t_0 = x * -z;
	double tmp;
	if (z <= -1.1e+71) {
		tmp = y * z;
	} else if (z <= -2.1e+29) {
		tmp = t_0;
	} else if (z <= -7.5e-17) {
		tmp = y * z;
	} else if (z <= 3.2e-78) {
		tmp = x;
	} else if (z <= 2.3e+126) {
		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 <= (-1.1d+71)) then
        tmp = y * z
    else if (z <= (-2.1d+29)) then
        tmp = t_0
    else if (z <= (-7.5d-17)) then
        tmp = y * z
    else if (z <= 3.2d-78) then
        tmp = x
    else if (z <= 2.3d+126) 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 <= -1.1e+71) {
		tmp = y * z;
	} else if (z <= -2.1e+29) {
		tmp = t_0;
	} else if (z <= -7.5e-17) {
		tmp = y * z;
	} else if (z <= 3.2e-78) {
		tmp = x;
	} else if (z <= 2.3e+126) {
		tmp = y * z;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x * -z
	tmp = 0
	if z <= -1.1e+71:
		tmp = y * z
	elif z <= -2.1e+29:
		tmp = t_0
	elif z <= -7.5e-17:
		tmp = y * z
	elif z <= 3.2e-78:
		tmp = x
	elif z <= 2.3e+126:
		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 <= -1.1e+71)
		tmp = Float64(y * z);
	elseif (z <= -2.1e+29)
		tmp = t_0;
	elseif (z <= -7.5e-17)
		tmp = Float64(y * z);
	elseif (z <= 3.2e-78)
		tmp = x;
	elseif (z <= 2.3e+126)
		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 <= -1.1e+71)
		tmp = y * z;
	elseif (z <= -2.1e+29)
		tmp = t_0;
	elseif (z <= -7.5e-17)
		tmp = y * z;
	elseif (z <= 3.2e-78)
		tmp = x;
	elseif (z <= 2.3e+126)
		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, -1.1e+71], N[(y * z), $MachinePrecision], If[LessEqual[z, -2.1e+29], t$95$0, If[LessEqual[z, -7.5e-17], N[(y * z), $MachinePrecision], If[LessEqual[z, 3.2e-78], x, If[LessEqual[z, 2.3e+126], N[(y * z), $MachinePrecision], t$95$0]]]]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.09999999999999997e71 or -2.1000000000000002e29 < z < -7.49999999999999984e-17 or 3.2e-78 < z < 2.3000000000000001e126

   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 95.7%

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

   4. Applied egg-rr 95.7%

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

   5. Taylor expanded in x around -inf 95.7%

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

      1. +-commutative 95.7%

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

      2. *-commutative 95.7%

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

      3. mul-1-neg 95.7%

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

      4. unsub-neg 95.7%

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

      5. sub-neg 95.7%

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

      6. metadata-eval 95.7%

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

   7. Simplified 95.7%

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

   8. Taylor expanded in y around inf 65.5%

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

      1. *-commutative 65.5%

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

   10. Simplified 65.5%

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

   if -1.09999999999999997e71 < z < -2.1000000000000002e29 or 2.3000000000000001e126 < 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 95.1%

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

   4. Applied egg-rr 95.1%

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

   5. Taylor expanded in x around -inf 95.1%

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

      1. +-commutative 95.1%

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

      2. *-commutative 95.1%

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

      3. mul-1-neg 95.1%

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

      4. unsub-neg 95.1%

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

      5. sub-neg 95.1%

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

      6. metadata-eval 95.1%

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

   7. Simplified 95.1%

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

   8. Taylor expanded in z around inf 100.0%

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

   9. Taylor expanded in y around 0 66.6%

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

       1. associate-*r* 66.6%

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

       2. neg-mul-1 66.6%

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

       3. *-commutative 66.6%

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

   11. Simplified 66.6%

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

   if -7.49999999999999984e-17 < z < 3.2e-78

   1. Initial program 100.0%

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

   3. Taylor expanded in z around 0 80.4%

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

4. Final simplification 72.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.1 \cdot 10^{+71}:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;z \leq -2.1 \cdot 10^{+29}:\\ \;\;\;\;x \cdot \left(-z\right)\\ \mathbf{elif}\;z \leq -7.5 \cdot 10^{-17}:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;z \leq 3.2 \cdot 10^{-78}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 2.3 \cdot 10^{+126}:\\ \;\;\;\;y \cdot z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(-z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 75.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.5 \cdot 10^{-45} \lor \neg \left(x \leq 1.8 \cdot 10^{-85}\right):\\ \;\;\;\;x \cdot \left(1 - z\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -4.5e-45) (not (<= x 1.8e-85))) (* x (- 1.0 z)) (* y z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -4.5e-45) || !(x <= 1.8e-85)) {
		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 <= (-4.5d-45)) .or. (.not. (x <= 1.8d-85))) 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 <= -4.5e-45) || !(x <= 1.8e-85)) {
		tmp = x * (1.0 - z);
	} else {
		tmp = y * z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -4.5e-45) or not (x <= 1.8e-85):
		tmp = x * (1.0 - z)
	else:
		tmp = y * z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -4.5e-45) || !(x <= 1.8e-85))
		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 <= -4.5e-45) || ~((x <= 1.8e-85)))
		tmp = x * (1.0 - z);
	else
		tmp = y * z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -4.5e-45], N[Not[LessEqual[x, 1.8e-85]], $MachinePrecision]], N[(x * N[(1.0 - z), $MachinePrecision]), $MachinePrecision], N[(y * z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -4.4999999999999999e-45 or 1.7999999999999999e-85 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 82.6%

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

      1. mul-1-neg 82.6%

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

      2. unsub-neg 82.6%

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

   5. Simplified 82.6%

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

   if -4.4999999999999999e-45 < x < 1.7999999999999999e-85

   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. Taylor expanded in x around -inf 100.0%

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

      1. +-commutative 100.0%

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

      2. *-commutative 100.0%

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

      3. mul-1-neg 100.0%

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

      4. unsub-neg 100.0%

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

      5. sub-neg 100.0%

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

      6. metadata-eval 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in y around inf 71.2%

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

      1. *-commutative 71.2%

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

   10. Simplified 71.2%

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

4. Final simplification 78.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.5 \cdot 10^{-45} \lor \neg \left(x \leq 1.8 \cdot 10^{-85}\right):\\ \;\;\;\;x \cdot \left(1 - z\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 84.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.02 \cdot 10^{-16} \lor \neg \left(z \leq 1.12 \cdot 10^{-79}\right):\\ \;\;\;\;\left(y - x\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 - z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -1.02e-16) (not (<= z 1.12e-79)))
   (* (- y x) z)
   (* x (- 1.0 z))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1.02e-16) || !(z <= 1.12e-79)) {
		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 <= (-1.02d-16)) .or. (.not. (z <= 1.12d-79))) 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 <= -1.02e-16) || !(z <= 1.12e-79)) {
		tmp = (y - x) * z;
	} else {
		tmp = x * (1.0 - z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -1.02e-16) or not (z <= 1.12e-79):
		tmp = (y - x) * z
	else:
		tmp = x * (1.0 - z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -1.02e-16) || !(z <= 1.12e-79))
		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 <= -1.02e-16) || ~((z <= 1.12e-79)))
		tmp = (y - x) * z;
	else
		tmp = x * (1.0 - z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -1.02e-16], N[Not[LessEqual[z, 1.12e-79]], $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 < -1.0200000000000001e-16 or 1.11999999999999996e-79 < 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. Taylor expanded in x around -inf 95.5%

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

      1. +-commutative 95.5%

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

      2. *-commutative 95.5%

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

      3. mul-1-neg 95.5%

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

      4. unsub-neg 95.5%

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

      5. sub-neg 95.5%

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

      6. metadata-eval 95.5%

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

   7. Simplified 95.5%

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

   8. Taylor expanded in z around inf 93.5%

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

   if -1.0200000000000001e-16 < z < 1.11999999999999996e-79

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 80.4%

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

      1. mul-1-neg 80.4%

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

      2. unsub-neg 80.4%

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

   5. Simplified 80.4%

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

4. Final simplification 87.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.02 \cdot 10^{-16} \lor \neg \left(z \leq 1.12 \cdot 10^{-79}\right):\\ \;\;\;\;\left(y - x\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 - z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.9% accurate, 0.5× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1.0) || !(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 <= (-1.0d0)) .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 <= -1.0) || !(z <= 1.0)) {
		tmp = (y - x) * z;
	} else {
		tmp = x + (y * z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -1.0) 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 <= -1.0) || !(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 <= -1.0) || ~((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, -1.0], 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 < -1 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 94.7%

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

   4. Applied egg-rr 94.7%

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

   5. Taylor expanded in x around -inf 94.7%

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

      1. +-commutative 94.7%

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

      2. *-commutative 94.7%

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

      3. mul-1-neg 94.7%

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

      4. unsub-neg 94.7%

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

      5. sub-neg 94.7%

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

      6. metadata-eval 94.7%

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

   7. Simplified 94.7%

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

   8. Taylor expanded in z around inf 99.2%

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

   if -1 < z < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf 99.3%

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

      1. *-commutative 99.3%

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

   5. Simplified 99.3%

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

4. Final simplification 99.2%

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

Alternative 6: 60.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.5 \cdot 10^{-16} \lor \neg \left(z \leq 9.6 \cdot 10^{-80}\right):\\ \;\;\;\;y \cdot z\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -1.5e-16) (not (<= z 9.6e-80))) (* y z) x))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1.5e-16) || !(z <= 9.6e-80)) {
		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 <= (-1.5d-16)) .or. (.not. (z <= 9.6d-80))) 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 <= -1.5e-16) || !(z <= 9.6e-80)) {
		tmp = y * z;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -1.5e-16) or not (z <= 9.6e-80):
		tmp = y * z
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -1.5e-16) || !(z <= 9.6e-80))
		tmp = Float64(y * z);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -1.5e-16) || ~((z <= 9.6e-80)))
		tmp = y * z;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -1.5e-16], N[Not[LessEqual[z, 9.6e-80]], $MachinePrecision]], N[(y * z), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if z < -1.49999999999999997e-16 or 9.5999999999999996e-80 < 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. Taylor expanded in x around -inf 95.5%

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

      1. +-commutative 95.5%

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

      2. *-commutative 95.5%

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

      3. mul-1-neg 95.5%

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

      4. unsub-neg 95.5%

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

      5. sub-neg 95.5%

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

      6. metadata-eval 95.5%

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

   7. Simplified 95.5%

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

   8. Taylor expanded in y around inf 58.7%

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

      1. *-commutative 58.7%

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

   10. Simplified 58.7%

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

   if -1.49999999999999997e-16 < z < 9.5999999999999996e-80

   1. Initial program 100.0%

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

   3. Taylor expanded in z around 0 80.4%

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

4. Final simplification 69.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.5 \cdot 10^{-16} \lor \neg \left(z \leq 9.6 \cdot 10^{-80}\right):\\ \;\;\;\;y \cdot z\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
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: 36.7% 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 42.2%

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

4. Final simplification 42.2%

   \[\leadsto x \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024034 
(FPCore (x y z)
  :name "Diagrams.ThreeD.Shapes:frustum from diagrams-lib-1.3.0.3, B"
  :precision binary64
  (+ x (* (- y x) z)))