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

Percentage Accurate: 100.0% → 100.0%

Time: 5.1s

Alternatives: 8

Speedup: 1.0×

• 20.8% 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-def 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: 61.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(-z\right)\\ \mathbf{if}\;z \leq -3.1 \cdot 10^{+171}:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;z \leq -5.2 \cdot 10^{+26}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq -9.5 \cdot 10^{-22}:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;z \leq 1:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (- z))))
   (if (<= z -3.1e+171)
     (* y z)
     (if (<= z -5.2e+26)
       t_0
       (if (<= z -9.5e-22) (* y z) (if (<= z 1.0) x t_0))))))

C

double code(double x, double y, double z) {
	double t_0 = x * -z;
	double tmp;
	if (z <= -3.1e+171) {
		tmp = y * z;
	} else if (z <= -5.2e+26) {
		tmp = t_0;
	} else if (z <= -9.5e-22) {
		tmp = y * z;
	} else if (z <= 1.0) {
		tmp = x;
	} 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 <= (-3.1d+171)) then
        tmp = y * z
    else if (z <= (-5.2d+26)) then
        tmp = t_0
    else if (z <= (-9.5d-22)) then
        tmp = y * z
    else if (z <= 1.0d0) then
        tmp = x
    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 <= -3.1e+171) {
		tmp = y * z;
	} else if (z <= -5.2e+26) {
		tmp = t_0;
	} else if (z <= -9.5e-22) {
		tmp = y * z;
	} else if (z <= 1.0) {
		tmp = x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x * -z
	tmp = 0
	if z <= -3.1e+171:
		tmp = y * z
	elif z <= -5.2e+26:
		tmp = t_0
	elif z <= -9.5e-22:
		tmp = y * z
	elif z <= 1.0:
		tmp = x
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x * Float64(-z))
	tmp = 0.0
	if (z <= -3.1e+171)
		tmp = Float64(y * z);
	elseif (z <= -5.2e+26)
		tmp = t_0;
	elseif (z <= -9.5e-22)
		tmp = Float64(y * z);
	elseif (z <= 1.0)
		tmp = x;
	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 <= -3.1e+171)
		tmp = y * z;
	elseif (z <= -5.2e+26)
		tmp = t_0;
	elseif (z <= -9.5e-22)
		tmp = y * z;
	elseif (z <= 1.0)
		tmp = x;
	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, -3.1e+171], N[(y * z), $MachinePrecision], If[LessEqual[z, -5.2e+26], t$95$0, If[LessEqual[z, -9.5e-22], N[(y * z), $MachinePrecision], If[LessEqual[z, 1.0], x, t$95$0]]]]]

Derivation

1. Split input into 3 regimes

2. if z < -3.0999999999999999e171 or -5.20000000000000004e26 < z < -9.4999999999999994e-22

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 97.5%

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

      1. fma-def 100.0%

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

      2. mul-1-neg 100.0%

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

      3. *-commutative 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around 0 64.7%

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

   if -3.0999999999999999e171 < z < -5.20000000000000004e26 or 1 < z

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 65.3%

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

      1. mul-1-neg 65.3%

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

      2. unsub-neg 65.3%

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

   5. Simplified 65.3%

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

   6. Taylor expanded in z around inf 63.9%

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

      1. associate-*r* 63.9%

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

      2. mul-1-neg 63.9%

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

   8. Simplified 63.9%

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

   if -9.4999999999999994e-22 < z < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in z around 0 74.7%

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

4. Final simplification 69.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.1 \cdot 10^{+171}:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;z \leq -5.2 \cdot 10^{+26}:\\ \;\;\;\;x \cdot \left(-z\right)\\ \mathbf{elif}\;z \leq -9.5 \cdot 10^{-22}:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;z \leq 1:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(-z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 75.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.45 \cdot 10^{-133} \lor \neg \left(x \leq 9.2 \cdot 10^{-55}\right):\\ \;\;\;\;x \cdot \left(1 - z\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -2.45e-133) (not (<= x 9.2e-55))) (* x (- 1.0 z)) (* y z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -2.45e-133) || !(x <= 9.2e-55)) {
		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 <= (-2.45d-133)) .or. (.not. (x <= 9.2d-55))) 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 <= -2.45e-133) || !(x <= 9.2e-55)) {
		tmp = x * (1.0 - z);
	} else {
		tmp = y * z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -2.45e-133) or not (x <= 9.2e-55):
		tmp = x * (1.0 - z)
	else:
		tmp = y * z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -2.45e-133) || !(x <= 9.2e-55))
		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 <= -2.45e-133) || ~((x <= 9.2e-55)))
		tmp = x * (1.0 - z);
	else
		tmp = y * z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -2.45e-133], N[Not[LessEqual[x, 9.2e-55]], $MachinePrecision]], N[(x * N[(1.0 - z), $MachinePrecision]), $MachinePrecision], N[(y * z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.44999999999999998e-133 or 9.20000000000000046e-55 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 85.9%

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

      1. mul-1-neg 85.9%

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

      2. unsub-neg 85.9%

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

   5. Simplified 85.9%

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

   if -2.44999999999999998e-133 < x < 9.20000000000000046e-55

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 100.0%

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

      1. fma-def 100.0%

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

      2. mul-1-neg 100.0%

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

      3. *-commutative 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around 0 73.1%

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

4. Final simplification 81.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.45 \cdot 10^{-133} \lor \neg \left(x \leq 9.2 \cdot 10^{-55}\right):\\ \;\;\;\;x \cdot \left(1 - z\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 84.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.8 \cdot 10^{-20} \lor \neg \left(z \leq 3.2 \cdot 10^{-32}\right):\\ \;\;\;\;\left(y - x\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -1.8e-20) (not (<= z 3.2e-32))) (* (- y x) z) x))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1.8e-20) || !(z <= 3.2e-32)) {
		tmp = (y - 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 <= (-1.8d-20)) .or. (.not. (z <= 3.2d-32))) then
        tmp = (y - 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 <= -1.8e-20) || !(z <= 3.2e-32)) {
		tmp = (y - x) * z;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -1.8e-20) or not (z <= 3.2e-32):
		tmp = (y - x) * z
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -1.8e-20) || !(z <= 3.2e-32))
		tmp = Float64(Float64(y - x) * z);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -1.8e-20) || ~((z <= 3.2e-32)))
		tmp = (y - x) * z;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -1.8e-20], N[Not[LessEqual[z, 3.2e-32]], $MachinePrecision]], N[(N[(y - x), $MachinePrecision] * z), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if z < -1.79999999999999987e-20 or 3.2000000000000002e-32 < z

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 97.9%

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

      1. fma-def 99.3%

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

      2. mul-1-neg 99.3%

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

      3. *-commutative 99.3%

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

   5. Simplified 99.3%

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

   6. Taylor expanded in z around inf 97.1%

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

      1. mul-1-neg 97.1%

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

      2. unsub-neg 97.1%

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

   8. Simplified 97.1%

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

   if -1.79999999999999987e-20 < z < 3.2000000000000002e-32

   1. Initial program 100.0%

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

   3. Taylor expanded in z around 0 77.5%

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

4. Final simplification 88.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.8 \cdot 10^{-20} \lor \neg \left(z \leq 3.2 \cdot 10^{-32}\right):\\ \;\;\;\;\left(y - x\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1160000000 \lor \neg \left(z \leq 1\right):\\ \;\;\;\;\left(y - x\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -1160000000.0) (not (<= z 1.0))) (* (- y x) z) (+ x (* y z))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1160000000.0) || !(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 <= (-1160000000.0d0)) .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 <= -1160000000.0) || !(z <= 1.0)) {
		tmp = (y - x) * z;
	} else {
		tmp = x + (y * z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -1160000000.0) 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 <= -1160000000.0) || !(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 <= -1160000000.0) || ~((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, -1160000000.0], 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 < -1.16e9 or 1 < z

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 97.7%

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

      1. fma-def 99.2%

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

      2. mul-1-neg 99.2%

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

      3. *-commutative 99.2%

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

   5. Simplified 99.2%

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

   6. Taylor expanded in z around inf 99.0%

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

      1. mul-1-neg 99.0%

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

      2. unsub-neg 99.0%

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

   8. Simplified 99.0%

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

   if -1.16e9 < z < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf 98.7%

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

      1. *-commutative 98.7%

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

   5. Simplified 98.7%

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

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1160000000 \lor \neg \left(z \leq 1\right):\\ \;\;\;\;\left(y - x\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 60.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.75 \cdot 10^{-18} \lor \neg \left(z \leq 2.7 \cdot 10^{-32}\right):\\ \;\;\;\;y \cdot z\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -2.75e-18) (not (<= z 2.7e-32))) (* y z) x))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -2.75e-18) || !(z <= 2.7e-32)) {
		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 <= (-2.75d-18)) .or. (.not. (z <= 2.7d-32))) 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 <= -2.75e-18) || !(z <= 2.7e-32)) {
		tmp = y * z;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -2.75e-18) or not (z <= 2.7e-32):
		tmp = y * z
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -2.75e-18) || !(z <= 2.7e-32))
		tmp = Float64(y * z);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -2.75e-18) || ~((z <= 2.7e-32)))
		tmp = y * z;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -2.75e-18], N[Not[LessEqual[z, 2.7e-32]], $MachinePrecision]], N[(y * z), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if z < -2.75e-18 or 2.69999999999999981e-32 < z

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 97.9%

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

      1. fma-def 99.3%

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

      2. mul-1-neg 99.3%

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

      3. *-commutative 99.3%

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

   5. Simplified 99.3%

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

   6. Taylor expanded in x around 0 46.8%

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

   if -2.75e-18 < z < 2.69999999999999981e-32

   1. Initial program 100.0%

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

   3. Taylor expanded in z around 0 77.5%

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

4. Final simplification 60.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.75 \cdot 10^{-18} \lor \neg \left(z \leq 2.7 \cdot 10^{-32}\right):\\ \;\;\;\;y \cdot z\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
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 36.0%

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

4. Final simplification 36.0%

   \[\leadsto x \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024027 
(FPCore (x y z)
  :name "Diagrams.ThreeD.Shapes:frustum from diagrams-lib-1.3.0.3, B"
  :precision binary64
  (+ x (* (- y x) z)))